#
#  crnverifier/crn_bisimulation.py
#  Original source from the Nuskell compiler project
#
#  Authors:
#   - <NAME>
#   - <NAME>
#   - <NAME>
#
import logging
log = logging.getLogger(__name__)

import sys
import copy
import math
import random
from collections import Counter
from itertools import product, combinations, chain

from .utils import pretty_rxn
from .utils import interpret as interpretL

class SpeciesAssignmentError(Exception):
    pass

class EnumSpeciesAssignmentError(Exception):
    pass

class SearchDepthExceeded(Exception):
    pass

# Conversion between internal and external representation of implementation species.
def deformalize(k, fs):
    return f'i{{{k}}}' if k in fs else k

def formalize(intrp):
    intr = {}
    for sp in intrp:
        if sp[:2] == 'i{':
            isp = sp[2:-1]
            intr[isp] = intrp[sp]
        else:
            intr[sp] = intrp[sp]
    return intr

def subst(crn, intrp):
    """ Substitute implementation species with formal species in CRN.  """
    return [[interpretL(part, intrp) for part in rxn] for rxn in crn]

# Utils for list based multiset operations -> it's faster than counters ...
def subsetsL(x, reverse = True):
    """ Generate all (uniqe) multi-subsets of a list x. """
    if reverse:
        return chain(*[combinations(x, l) for l in range(len(x), -1, -1)])
    else:
        return chain(*[combinations(x, l) for l in range(0, len(x) + 1)])

def subtractL(a, b, strict = True): # Multiset a - b.
    """ Returns a new list: a - b. """
    a = a[:]
    for x in b:
        try:
            a.remove(x)
        except ValueError as err:
            if strict:
                raise err
    return a

def is_contained(a, b): # Multiset: a < b
    """ bool: True if list a is fully contained in list b. """
    b = b[:]
    try:
        [b.remove(s) for s in a]
    except ValueError as err:
        return False
    return True

def enumL(n, l, weights = None):
    """ Returns combinations to assign elements of list l to n variables. 

    The weights parameter determines how ofen a corresponding variable n
    appears. I.e if you have n = [x, y], weights = [2, 1], then you must assign
    a twice to the same value in l. In the above example, if l = [A, A, B] then
    the only assignments can be: 
        (x, y) = ((A,), (B,)), or 
        (x, y) = ((), (A, A, B)).

    Raises:
        EnumSpeciecsError('Cannot satisfy the assignment.')

    Yields: 
        tuple: all possible assignents.
    """
    if n == 0:
        yield []
        return

    if weights is None:
        weights = [1 for x in range(n)]
    w = weights[0]

    def ldiv(l, s):
        """ True, if all elements are multiples of s """
        if s == 1 or len(l) == 0:
            return l
        l = Counter(l)
        for k, v in l.items():
            if v % s:
                break
            l[k] = int(v/s)
        else:
            return tuple(l.elements())
        return None
        
    if n == 1:
        l = ldiv(l, w)
        if l is None:
            raise EnumSpeciesAssignmentError
        yield [tuple(l)]
        return

    for ss in set(subsetsL(l)):
        assert is_contained(ss, l)
        wss = ldiv(ss, w)
        if wss is None:
            continue
        try:
            for j in enumL(n-1, subtractL(l, ss), weights[1:]):
                yield [wss] + j
        except EnumSpeciesAssignmentError:
            continue
    return

#TODO(SB): When should we clear the cache?
#SRCACHE = dict()
#def same_reaction_cache(irxn, frxn, fs):
#    global SRCACHE
#    tirxn = tuple(map(tuple, irxn))
#    tfrxn = tuple(map(tuple, frxn))
#    if (tirxn, tfrxn) not in SRCACHE:
#        SRCACHE[(tirxn, tfrxn)] = same_reaction(irxn, frxn, fs)
#    return SRCACHE[(tirxn, tfrxn)]

def same_reaction(irxn, frxn, fs):
    """ Check if irxn *could* be interpreted as frxn.

    This assumes that irxn is already interpreted using the formal species fs.
    SB: Note, that I introduced a new expression expression here, see below.
    """
    ul = subtractL(irxn[0], frxn[0], strict = False) # exess implementation reactants 
    sl = subtractL(frxn[0], irxn[0], strict = False) # exess formal reactants 
    ur = subtractL(irxn[1], frxn[1], strict = False) # exess implementation products
    sr = subtractL(frxn[1], irxn[1], strict = False) # exess formal products

    ir = set(ul) # unassigned implementation reactants 
    fr = set(sl) # exess formal reactants 
    ip = set(ur) # unassingend implementation products
    fp = set(sr) # exess formal products
    if ir & fs or ip & fs:
        # There are excess formal reactants or excess formal products
        # in the implementation reaction, so this reaction cannot
        # implement the formal reaction.
        return False
    if len(fr) and len(ir) == 0:
        # There are excess formal reactants and 
        # no more implementation reactants.
        return False 
    if len(fp) and len(ip) == 0:
        # There are excess formal products and 
        # no more implementation products.
        return False
    if len(fp) and len(ir) and len(ip) and ir == ip:
        # NOTE: This line is new:
        # Example:    A -> B + C
        # cannot be:  A + y -> B + y
        return False

    try: # NOTE: this is new. The very same function we use later to assign species.
        # Left side check
        ul = Counter(ul)
        [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
        next(enumL(len(ul), sl, vl))

        # Right side check
        ur = Counter(ur)
        [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
        next(enumL(len(ur), sr, vr))
    except EnumSpeciesAssignmentError:
        return False
    return True

def trivial_reaction(irxn, fs):
    # Formal reactants and products.
    sl = subtractL(irxn[0], irxn[1], strict = False)
    sr = subtractL(irxn[1], irxn[0], strict = False)
    ir = set(sl) # "non-trivial" reactants
    ip = set(sr) # "non-trivial" products
    if (ir & fs and ip <= fs) or (ip & fs and ir <= fs):
        # If the implementation reactants contain a formal species and
        # all implementation products are (different) formal species, or
        # if the implementation products contain a formal species and
        # all implementation reactants are (different) formal species:
        return False
    else: # Could be trival
        try:
            ul = Counter([x for x in irxn[0] if x not in fs])
            sr = [x for x in sr if x in fs]
            [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
            next(enumL(len(ul), sr, vl))
            ur = Counter([x for x in irxn[1] if x not in fs])
            sl = [x for x in sl if x in fs]
            [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
            next(enumL(len(ur), sl, vr))
        except EnumSpeciesAssignmentError:
            return False
    return True
 
def makeT(fcrn, icrn, fs):
    """ Calculate a table with bool entries.

    Takes an implementation CRN where all implementation species have been
    replaced with the corresponding formal species. Thus, the implementation
    CRN contains a mix of interpreted species and/or uninterpreted species. 

    Returns:
        list[lists[bool]]: A table of bool entries with dimensions: 
            row number = len(icrn), col number = len(fcrn) + 1.
    """
    r = []
    for irxn in icrn:
        rr = [same_reaction(irxn, frxn, fs) for frxn in fcrn]
        rr.append(trivial_reaction(irxn, fs))
        r.append(rr)
    return r

def checkT(T):
    """ Check (partial) interpretation for the delimiting condition.

    Returns:
        bool: True if there is no row or (non-trivial) colum with all False values.
    """
    for row in T:
        if all(not b for b in row):
            return False
    for e, col in enumerate(zip(*T), 1):
        if e != len(T[0]) and all(not b for b in col):
            return False
    return True

def minimal_implementation_states(fstate, inter):
    """ Yields a set of minimal implementation states for a formal state. """
    def gen_supersets():
        if len(fstate) == 0:
            yield []
        else:
            fs = set(fstate).pop()
            fs1 = fstate[0]
            for k in inter:
                if fs1 in inter[k]:
                    for out in minimal_implementation_states(subtractL(fstate, 
                                                                       inter[k], 
                                                                       strict = False), inter):
                        yield [k] + out

    def filter_minimals(lst):
        seen = set()
        for x in lst:
            z = tuple(sorted(x))
            if z in seen:
                continue
            if len(x) == 0:
                yield x
                seen.add(z)
                continue
            for y in combinations(x, len(x)-1):
                if is_contained(fstate, interpretL(y, inter)):
                    break
            else:
                yield x
            seen.add(z)
        return
    return filter_minimals(gen_supersets())

def passes_modularity_condition(bisim, icrn, common_is, common_fs):
    """ Check if a bisimulation satisfies the modularity condition.

    The modularity condition is formulated with respect to subsets of 
    *common implementation* and *common formal* species, where each common
    implementation species must interpet to a common formal species. 
    "Common" should be understood as species wich are *shared* between two or
    more modules. 

    In order to guarantee that the permissive condition is satisfied when
    combining modules, this function asserts whether every implementation
    species can turn into "common" implementation species with the same
    interpretation via trivial reactions (that is the interpretation of the
    reaction must be trival). For details check JDW2019: Definition 3.3
    (Modularity condition).

    The implementation is basically the graphsearch algorithm to check the
    permissive condition, where all "minimal states" are each exactly one
    implementation species.

    Args:
        bisim (dict): An interpretation (= CRN bisimulation).
        icrn (list[lists]): An implementation CRN.
        common_is (set): A set of common implementation species.
        common_fs (set): A set of common formal species.

    Returns:
        bool: True if modularity condition is satisfied.
    """
    # All common implementation species are part of the interpretation.
    assert all(ci in bisim for ci in common_is)
    # All interpretations of common implementation species 
    # are part of common formal species.
    if not all(cf in common_fs for ci in common_is for cf in bisim[ci]):
        log.info('Unknown common formal species.')
        return False

    def Y(s):
        return s in common_is
    def Z(s):
        return len(set(bisim[s]) & common_fs) == 0

    trivial_irxns = [] # trivial reactions
    for rxn in icrn:
        iR = interpretL(rxn[0], bisim) 
        iP = interpretL(rxn[1], bisim) 
        if sorted(iR) == sorted(iP):
            trivial_irxns.append(rxn)

    done = {s for s in bisim if Y(s) or Z(s)}
    todo = {s: [set(), set()] for s in bisim if s not in done} 
    # todo[s] = [reach, produce]

    reset = len(todo) > 0
    while reset:
        reset = False
        for r in list(todo.keys()): # start at any of the non-common intermediates
            [r_reach, r_produce] = todo[r]
            for R, P in trivial_irxns:
                if r not in R:
                    continue
                nR = R[:]
                nR.remove(r)
                if set(nR) > r_produce:
                    # There is at least one more reactant which (as far as we
                    # have seen) cannot be produced by r.
                    continue

                nullsp, mystic = set(), set()
                for p in P:
                    if p in done or (not is_contained(bisim[r], bisim[p])): 
                        # If p is done, then any other p in P is either {} or
                        # it is also a reactant which must be checked
                        # separately. Alternatively, if the interpretation
                        # of products contains the interpretation
                        # of reactants and more, then we can call this one
                        # done and move on to check p.
                        done.add(r)
                        del todo[r]
                        break
                    elif len(bisim[p]) == 0:
                        nullsp.add(p)
                    else:
                        mystic.add(p)
                if r in done:
                    reset = True
                    break
                # Now investigate the mysterious "non-common implementation"
                # products which interpret to a formal species, but are not
                # known to be exit states.
                for p in mystic: 
                    if p not in r_reach:
                        r_reach.add(p)
                        reset = True
                    [p_reach, p_produce] = todo[p]
                    if not (p_reach <= r_reach):
                        # anything reachable by products is reachable by reactants.
                        r_reach |= p_reach
                        reset = True
                    if r in p_reach:
                        # we know r -> p -> r, so all r_nullsp and p_nullsp 
                        # can be produced at infinite amounts.
                        loopable = nullsp | p_produce
                        if not (loopable <= r_produce):
                            r_produce |= loopable
                            reset = True
    return len(todo) == 0

def check_permissive_bruteforce(fcrn, icrn, fs, inter, maxdepth = None):
    """ Check the permissive condition via a brute force path search.
    
    Args:
        fcrn: The formal CRN.
        icrn: The implementation CRN.
        fs: The formal species.
        inter: The interpretation.
        maxdepth: A maximum brute force recursion depth. Defaults to None.

    Raises:
        SearchDepthExceeded: If the hardcoded maxdepth is reached.

    Returns:
        bool: Whether the interpretation passes the permissive condition. 
        info: The search depth if successful, otherwise a tuple (R, S0).
    """
    sicrn = subst(icrn, inter)
    T = makeT(fcrn, sicrn, fs)
    assert checkT(T) # Assuming this has been checked before calling permissive.
    assert all(fs | set(v) == fs for k, v in inter.items())

    if maxdepth is None:
        maxdepth = sys.getrecursionlimit() - 10 # should it be one? should it be two?

    log.debug(f'The implementation CRN:\n' + '\n'.join(
        [f'{e} {t=} {pretty_rxn(sicrn[e])} ({pretty_rxn(icrn[e])})' \
                for e, t in enumerate(T)]))
    log.debug(f'{inter=}')

    def bruteforce(si, seen, trivials, depth = 0):
        if depth > maxdepth:
            raise SearchDepthExceeded('WARNING: brute-force permissive test exceeded recursion depth.')
        st = tuple(sorted(si))
        if st in seen:
            return False
        seen.add(st)
        for irxn in formal_irxns:
            if len(subtractL(irxn[0], si, False)) == 0:
                return True
        for irxn in trivials:
            if len(subtractL(irxn[0], si, False)) == 0:
                nS = subtractL(si, irxn[0]) + irxn[1]
                # Shuffle the crn to reduce the probability of getting stuck in
                # an infinite state space assuming that there is a way out ...
                nexttrivials = trivials[:]
                random.shuffle(nexttrivials)
                if bruteforce(nS, seen, nexttrivials, depth + 1):
                    return True
        return False

    # Find implementation reactions that interpret to trivial reactions.
    trivial_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][-1]]
    for i, frxn in enumerate(fcrn):
        # Find implementation reactions that interpret to this formal reaction.
        formal_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][i]]
        for S0 in minimal_implementation_states(frxn[0], inter):
            assert is_contained(frxn[0], interpretL(S0, inter))
            found = bruteforce(S0, set(), trivial_irxns)
            if not found: 
                return False, (frxn, S0)
    return True, maxdepth

def check_permissive_loopsearch(fcrn, icrn, fs, inter):
    """ The 'loopsearch' algorithm to check the permissive condition. 

    TODO: Acutally this is a very different algorithm then described in the
    paper, so before using this in production, one has to go back and see that
    it is correct and what the time/space complexity of this algorithm is.

    """
    log.warning('This "loopsearch" algorithm is not implemented as described in JDW (2019).')
    sicrn = subst(icrn, inter)
    T = makeT(fcrn, sicrn, fs)
    assert checkT(T) # Assuming this has been checked before calling permissive.
    assert all(fs | set(v) == fs for k, v in inter.items())

    log.debug(f'The implementation CRN:\n' + '\n'.join(
        [f'{e} {t=} {pretty_rxn(sicrn[e])} ({pretty_rxn(icrn[e])})' \
                for e, t in enumerate(T)]))
    log.debug(f'{inter=}')

    # Potential null species.
    nulls = [k for k, v in inter.items() if len(v) == 0]
    # Find implementation reactions that interpret to trivial reactions.
    trivial_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][-1]]

    def reach_with_inf(start, goal = None, pickup = None, ignore = None, k = 0):
        """
        Search for a path from start to goal (states of non-null species) which
        produces at least pickup null species assuming it already has infinite
        copies of everything in ignore of length at most 2^k if goal is None,
        the goal is any reaction in formal_irxns.
        
        Args:
            start: an implementation state corresponding to a formal state.
            goal: a (different) implementation state corresponding to the same
                formal state, or None. If goal is None, then the goal is to find
                any implementation reaction corresponding to the formal reaction.
            pickup: xxx Something related to null species.
            ignore: Assuming there exist infinite copies of those species.
            k: Maximum search depth.
        """
        log.debug(f'CHECK: {start=} {goal=} {ignore=} {pickup=} {k=}')
        if ignore is None:
            ignore = set()
        if pickup is None:
            pickup = []

        #log.debug(f'{start=} {goal=} {ignore=}, {pickup=}, {k=}')
        if goal is None: 
            # Check if a implementation reaction is possible.
            for irxn in formal_irxns:
                # if irxn can happen: 
                if ignore >= set(subtractL(irxn[0], start, False)):
                    log.debug(f'True formal {irxn=}')
                    return True
        elif k == 0:
            # See if start can reach a pickup goal by trivial reactions.
            for irxn in trivial_irxns:
                # if irxn can happen: 
                if ignore >= set(subtractL(irxn[0], start, False)):
                    # and if goal is contained in the results of the trivial rxn.
                    if is_contained(goal+pickup, subtractL(start, irxn[0], False) + irxn[1]):
                        return True
        else:
            if reach_with_inf(start, goal, pickup, ignore, k-1):
                # If we can reach goal with trivial reactions, do it!
                log.debug(f'Shortcut {start=} {goal=} {pickup=} {ignore=} {k=}')
                return True

            # Divide the search into two by choosing a new state mid: 
            #   - find path from start to mid.
            #   - find path from mid to goal.
            for pickpart in map(list, subsetsL(pickup)):
              for mid in M:
                if mid == start or mid == goal: 
                    continue
                log.debug(f'Reach w inf: {start=} {mid=} {goal=} {pickpart=} {k=}')
                if reach_with_inf(start, mid, pickpart, ignore, k-1):
                    if reach_with_inf(mid, goal, subtractL(pickup, pickpart), ignore, k-1):
                        return True
        log.debug(f'FALSE: {start=} {goal=} {ignore=} {pickup=} {k=}')
        return False

    for i, frxn in enumerate(fcrn):
        # Find implementation reactions that interpret to this formal reaction.
        formal_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][i]]
        # From the paper M(r) is the set of minimal implementation states for a formal reaction r.
        M = list(map(sorted, minimal_implementation_states(frxn[0], inter)))

        k, roundequiv = 0, 1
        for nequiv in range(1, len(M)+1):
            if nequiv > roundequiv:
                k += 1
                roundequiv *= 2
        log.debug(f'LS: {k=} to find paths between {M} states: {frxn=}')

        for S0 in M:
            # The formal reaction must be able to happen.
            assert is_contained(frxn[0], interpretL(S0, inter))

            # Is the formal reaction possible w/o null species? 
            if reach_with_inf(S0, None):
                # This state can exit by itself.
                continue

            found = False
            reset = True
            ignore = set()
            # Can we reach a state from where the formal reaction is possible?
            while reset:
                log.debug(f'{reset=}, {found=}, {ignore=}')
                reset = False

                for pickup in map(list, nulls):
                    if set(pickup) & ignore:
                        continue
                    assert len(pickup) != 0
                    log.debug(f'{S0=}, {pickup=}, {ignore=}, {k=} ')

                    if reach_with_inf(S0, S0, pickup, ignore, k):
                        log.debug(f'Pickup! {S0=} {pickup=} {ignore=}.')
                        ignore |= set(pickup)
                        reset = True
                        if reach_with_inf(S0, None, ignore = ignore):
                            found = True
                            break

                for Si in M:
                    if Si == S0: 
                        continue
                    log.debug(f'Testing: {S0=} {Si=} {ignore=} {k=}.')
                    if reach_with_inf(S0, Si, ignore = ignore, k = k):
                        log.debug(f'True: {S0=} {Si=} {ignore=} {k=}.')
                        if reach_with_inf(Si, None, ignore = ignore):
                            log.debug(f'Found exit: {Si=}.')
                            found = True
                            break
                if found is True:
                    break

            if not found:
                return False, S0
    return True, None

def check_permissive_graphsearch(fcrn, icrn, fs, inter):
    """ The 'graphsearch' algorithm to check the permissive condition. 

    Args:
        fcrn: The formal CRN.
        icrn: The implementation CRN.
        fs: The formal species.
        inter: The interpretation.

    Returns:
        bool: Whether the interpretation passes the permissive condition. 
        info: The search depth if successful, otherwise a tuple (R, M).
    """

    sicrn = subst(icrn, inter)
    T = makeT(fcrn, sicrn, fs)
    assert checkT(T) # assuming this has been checked before calling permissive.
    assert all(fs | set(v) == fs for k, v in inter.items())

    log.debug(f'The implementation CRN:\n' + '\n'.join(
        [f'{e} {t=} {pretty_rxn(sicrn[e])} ({pretty_rxn(icrn[e])})' \
                for e, t in enumerate(T)]))
    log.debug(f'{inter=}')

    # Species that interpret to nothing.
    nullsp = set([k for k, v in inter.items() if len(v) == 0])

    # Find implementation reactions that interpret to trivial reactions.
    trivial_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][-1]]

    max_depth = 0
    for i, frxn in enumerate(fcrn):
        # Find implementation reactions that interpret to this formal reaction.
        formal_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][i]]
        # All implementation states which must permit the formal reaction.
        M = list(map(tuple, map(sorted, minimal_implementation_states(frxn[0], inter))))
        done = set() # set of species known to implement the current formal rxn.
        todo = {istate: [set(), set()] for istate in M}
        changed, depth = True, 0
        while changed:
            changed, depth = False, depth + 1
            for istate in M:
                if istate in done:
                    continue
                for irxn in formal_irxns:
                    # If the implementation state can implement the formal reaction.
                    if set(subtractL(irxn[0], istate, False)) <= todo[istate][0]: 
                        done.add(istate)
                        del todo[istate]
                        changed = True
                        break
                if istate in done:
                    continue
                for irxn in trivial_irxns:
                    # If the implementation state allows the implementation reaction.
                    if set(subtractL(irxn[0], istate, False)) <= todo[istate][0]: 
                        # apply the reaction to the current state.
                        nstate = subtractL(list(istate), irxn[0], False) + irxn[1]
                        for istate2 in M:
                            # if one of the other implementation states is reachable
                            if is_contained(istate2, nstate):
                                if istate2 in done:
                                    done.add(istate)
                                    del todo[istate]
                                    changed = True
                                    break
                                if istate in todo[istate2][1]: # I assume its a loop?
                                    s = todo[istate2][0] | (set(nstate) & nullsp)
                                    if not s <= todo[istate][0]:
                                        todo[istate][0] |= s
                                        changed = True
                                if istate2 not in todo[istate][1]:
                                    todo[istate][1].add(istate2)
                                    changed = True
                                if not (todo[istate2][1] <= todo[istate][1]):
                                    todo[istate][1] |= todo[istate2][1]
                                    changed = True
                    if istate in done:
                        break
        if todo: 
            return False, (frxn, M)
        if max_depth < depth: 
            max_depth = depth
    return True, max_depth
        
def passes_permissive_condition(fcrn, icrn, fs, inter, permcheck = 'default'):
    """ Tests an interpretation for the permissive condition.
 
    Args:
        fcrn: The formal CRN.
        icrn: The implementation CRN.
        fs: The formal species.
        inter: The interpretation.
        permcheck: Choice of permissive checker, select from ('default',
                   'grapsearch', 'bruteforce'). Defaults to 'default', which
                   tries the often faster bruteforce algorithm and uses
                   graphsearch whenever that algorithm exceeds the default
                   search depth.

    (TODO): The difficulty of checking the permissive condition scales with the
    number of minimal states (S0 in M) for any given formal reaction r = (R, P),
    which typically scales like (and scales no worse than) n**k where n = |S'|
    (the species of the implementation CRN) and k = |R|. (There may be up to
    order n^k minimal implementation states (M) for some formal reaction r and
    the trajectories by which formal r is implemented may have to pass through
    most or all of S0 in M.  When k is unbounded, checking an interpretation is
    PSPACE-complete.)

    We provide three algorithms: 
        * A bruteforce algorithm with unknown time bounds: a depth-first search
            for a path to implement each formal reaction. Space and time bounds
            are not known, but it appears faster than graphsearch in practice.
        * The graphsearch algorithm which takes poly(n^k) space and time:
            Construct a reachability graph for each formal reaction.
        * A preliminary implementation of the loopsearch algorithm, which
            should run in poly(n,k) space and poly(n^(kn)) time: search for
            "productive loops" using a space-efficient algorithm.
 
    Returns:
        bool: Whether the interpretation passes the permissive condition. 
        info: The search depth if successful, otherwise a tuple (r, M).
    """
    log.debug(f'Checking permissive condition using {permcheck=}.')
    if permcheck == 'graphsearch':
        passes, info = check_permissive_graphsearch(fcrn, icrn, fs, inter)
    elif permcheck == 'loopsearch':
        passes, info = check_permissive_loopsearch(fcrn, icrn, fs, inter)
    elif permcheck == 'bruteforce':
        passes, info = check_permissive_bruteforce(fcrn, icrn, fs, inter)
    elif permcheck == 'default':
        try:
            passes, info = check_permissive_bruteforce(fcrn, icrn, fs, inter)
        except SearchDepthExceeded as err:
            log.debug('Switching to graphsearch permissive checker.')
            passes, info = check_permissive_graphsearch(fcrn, icrn, fs, inter)
    else:
        raise SystemExit(f'Unknown algorithm for permissive condition: {permcheck}')
    return passes, info

def passes_delimiting_condition(fcrn, icrn, fs, inter):
    """ Tests an interpretation for the delimiting condition.

    Note that this function is actually never used, because typically you want
    to keep the table T. In any case, it returns the correct result.
    """
    sicrn = subst(icrn, inter)
    T = makeT(fcrn, sicrn, fs)
    return checkT(T)

def passes_atomic_condition(inter, fs):
    """ Tests an interpretation for the atomic condition.

    For every formal species there exists an implementation species which
    interprets to it.  
    """
    return all(any([f] == v for v in inter.values()) for f in fs)

def in_order(inter, order, unassigned):
    """ Check if an interpretation follows an order of assignments.

    Args:
        inter (dict): An interpretation dictionary.
        order (list): A list of implementation species. (They have to be
            assigned in the interpretation.)
        unassigend (set): A set of implementation species who's order has
            not yet been assigend.

    Returns:
        bool, info: True if interpretation is in order, updated order.
    """
    allthere = True
    for isp in order:
        if allthere:
            allthere = (isp in inter)
        elif isp in inter:
            return False, None
    if len(unassigned):
        for isp in inter:
            if isp in unassigned:
                order.append(isp)
                unassigned.remove(isp)
    return True, (order, unassigned)

def search_row(fcrn, icrn, fs, intrp, order = None, seen = None, depth = 0, permcheck = 'default'):
    """ Find full interpretations matching every irxn to one frxn or trxn.

    This "row search" finds all valid combinations of 
        implementation reaction -> formal reaction or 
        implementation reaction -> trivial reaction.
    Typically this function is called after search_column, which means we
    already have a partial interpretation that satisfies the delimiting
    condition. However, the delimiting condition may still break when an
    additional implementation reaction is introduced, so it is always necessary
    to check all three conditions.

    Yields:
        Iterator over all correct interpretations (CRN bisimulations).
    """
    log.debug(f'Searching row at {depth=}'.center(80-(2*depth), '~'))
    log.debug(f'Partial interpretation {intrp=}')
    log.debug(f'Ordered species {order=}')

    if order is None:
        itot = set().union(*[set().union(*rxn[:2]) for rxn in icrn])
        iordered = list(itot & set(intrp))  # must be an ordered list.
        iunassigned = itot - set(intrp)     # better a set.
    else:
        iordered, iunassigned = order
    valid, order = in_order(intrp, iordered, iunassigned)
    if not valid:
        raise SpeciesAssignmentError('Unordered input.')
    
    # We are in correct order, but there may still be duplicates...
    sicrn = subst(icrn, intrp)
    T = makeT(fcrn, sicrn, fs)
    if not checkT(T):
        log.debug(f'Delimiting condition not satisfied.')
        raise SpeciesAssignmentError('Delimiting condition not satisfied.')

    # The row indices where unassigned species can be found, but we 
    # remove duplicates of implementation reactions that solve for the
    # same set of implementation species!
    unknown = {tuple(sorted((set(sirxn[0]) | set(sirxn[1])) - fs)): i \
                for i, sirxn in enumerate(sicrn)} 
    unknown = [v for k, v in unknown.items() if len(k)]

    if unknown == []:
        if not passes_atomic_condition(intrp, fs):
            log.debug(f'Atomic condition not satisfied.')
            raise SpeciesAssignmentError('Atomic condition not satisfied.')
        correct, info = passes_permissive_condition(fcrn, icrn, fs, intrp, permcheck)
        if correct:
            log.debug(f'Permissive condition satisfied ({info=}).')
            yield intrp
            return
        log.debug(f'Permissive condition not satisfied ({info=}).')
        raise SpeciesAssignmentError('Permissive condition not satisfied.')

    unknown = sorted(unknown, key = lambda x: (T[x][-1], T[x][:-1].count(True)))
    # Start with the row with the minimal number of True's (excluding trivial reactions)
    log.debug(f'{unknown=} \n' + '\n'.join(
        [f'{e} {t=} {pretty_rxn(sicrn[e])} ({pretty_rxn(icrn[e])})' \
                for e, t in enumerate(T)]))

    later = dict() # less promising partial interpretations ...
    def search(later, n = 0, m = None):
        # use this to define a search hierarchy when sending results into the
        # next round. Currently, n and m are start and stop in a range of 
        # how many species interpet to a single species.
        while later:
            if m and n > m:
                break
            if n in later:
                for (inext, depth) in later[n]:
                    try:
                        for ir in search_row(fcrn, icrn, fs, inext, order,
                                                 depth = depth + 1, 
                                                 permcheck = permcheck):
                            yield ir
                    except SpeciesAssignmentError:
                        continue
                del later[n]
            n += 1

    alltriv = all(r is False for k in unknown for r in T[k][:-1]) 
    for k in unknown:
        irxn = sicrn[k] 
        if T[k][-1] is True: # Assign a trivial reaction.
            log.debug(f'Interpret: {pretty_rxn(irxn)} => trivial {fs=}')
            # Unassigned reactants and products.
            ul = Counter([x for x in irxn[0] if x not in fs])
            ur = Counter([x for x in irxn[1] if x not in fs])
            # If this reaction has no unassigned species, then we shouldn't be here!
            assert len(ul) or len(ur)
            # Formal reactants and products.
            sl = [x for x in subtractL(irxn[0], irxn[1], False) if x in fs]
            sr = [x for x in subtractL(irxn[1], irxn[0], False) if x in fs]
            log.debug(f'Interpret: {pretty_rxn(irxn)} => {sl=}, {sr=} | {ul=} {ur=}')
            [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
            [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
            tmpl = enumL(len(ul), sr, vl)
            tmpr = enumL(len(ur), sl, vr)
            for i, j in product(tmpl, tmpr):
                intrpleft = {k: tuple(sorted(v)) for k, v in zip(kl, i)}
                intrpright = {k: tuple(sorted(v)) for k, v in zip(kr, j)}
                for key in set(intrpleft) & set(intrpright):
                    if any([intrpleft[key][fsp] != \
                            intrpright[key][fsp] for fsp in fs]):
                        # Incompatible dictionaries!
                        break
                else:
                    inext = intrp.copy()
                    imove = tuple(sorted(chain(intrpleft.items(), intrpright.items())))
                    log.debug(f'Interpret: {imove}')
                    inext.update({k: list(v) for k, v in imove})
                    level = max(len(v) for k, v in imove)
                    later[level] = (later.get(level, [])) + [(inext, depth)]

            if len(ul) and len(ur):
                # An implementation reaction where species on both sides are
                # not interpreted yet, may also be interpreted as a trivial
                # reaction including an additional (single!!) formal species.
                for atom in fs:
                    log.debug(f'Interpret: {pretty_rxn(irxn)} => matching {atom=}')
                    [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
                    tmpl = enumL(len(ul), [atom]+sr, vl)
                    [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
                    tmpr = enumL(len(ur), [atom]+sl, vr)
                    try:
                        for i, j in product(tmpl, tmpr):
                            intrpleft = {k: tuple(sorted(v)) for k, v in zip(kl, i)}
                            intrpright = {k: tuple(sorted(v)) for k, v in zip(kr, j)}
                            for key in set(intrpleft) & set(intrpright):
                                if any([intrpleft[key][fsp] != \
                                        intrpright[key][fsp] for fsp in fs]):
                                    # Incompatible dictionaries!
                                    break
                            else:
                                inext = intrp.copy()
                                imove = tuple(sorted(chain(intrpleft.items(), intrpright.items())))
                                log.debug(f'Interpret: {imove}')
                                inext.update({k: list(v) for k, v in imove})
                                level = max(len(v) for k, v in imove)
                                later[level] = (later.get(level, [])) + [(inext, depth)]
                    except EnumSpeciesAssignmentError:
                        log.debug(f'Reaction {pretty_rxn(irxn)} => matching {atom=} ' + \
                                  f'was not compatible with trivial {fs=}!')

        for c, frxn in enumerate(fcrn): # Assign a formal reaction.
            if not T[k][c]:
                continue
            assert not alltriv
            log.debug(f'Interpret: {pretty_rxn(irxn)} => {pretty_rxn(frxn)}')
            # left 
            ul = Counter(subtractL(irxn[0], frxn[0], False))
            sl = subtractL(frxn[0], irxn[0], False)
            # right
            ur = Counter(subtractL(irxn[1], frxn[1], False))
            sr = subtractL(frxn[1], irxn[1], False)
            if len(ul) or len(ur):
                [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
                [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
                tmpl = enumL(len(ul), sl, vl)
                tmpr = enumL(len(ur), sr, vr)
                for i, j in product(tmpl, tmpr):
                    intrpleft = {k: tuple(sorted(v)) for k, v in zip(kl, i)}
                    intrpright = {k: tuple(sorted(v)) for k, v in zip(kr, j)}
                    for key in set(intrpleft) & set(intrpright):
                        if any([intrpleft[key][fsp] != \
                                intrpright[key][fsp] for fsp in fs]):
                            # Incompatible dictionaries!
                            break
                    else:
                        inext = intrp.copy()
                        inext.update({k: list(v) for k, v in intrpleft.items()})
                        inext.update({k: list(v) for k, v in intrpright.items()})
                        level = max(len(v) for k, v in chain(intrpleft.items(), 
                                                             intrpright.items()))
                        later[level] = (later.get(level, [])) + [(inext, depth)]
            for i in search(later, n = 0, m = 1):
                yield i
        for i in search(later, n = 0, m = 2):
            yield i
    for i in search(later, n = 2):
        yield i 
    assert len(later) == 0
    return

def search_column(fcrn, icrn, fs = None, intrp = None, unknown = None, depth = 0):
    """ Find all partial interpretations matching every frxn to one irxn.

    This "column search" finds all combinations where all formal reactions have
    one matching implementation reaction. For example, if there is one formal
    reaction and three compatible implementation reactions, then this function
    returns three interpretations that have exactly one reaction interpreted.
    Note that every result of the column search must pass the delimiting
    condition, but not necessarily atomic or permissive condition.

    Yields:
        Iterator over partial interpretations satisfying the delimiting condition.
    """
    if fs is None:
        fs = set().union(*[set().union(*rxn[:2]) for rxn in fcrn])
    if intrp is None:
        intrp = dict()
    log.debug(f'Searching column at {depth=}'.center(40-(2*depth), '*'))
    log.debug(f'Partial interpretation {intrp=}')

    sicrn = subst(icrn, intrp)
    T = makeT(fcrn, sicrn, fs)
    if not checkT(T):
        #log.debug(f'Delimiting condition not satisfied with {intrp=}')
        raise SpeciesAssignmentError('Delimiting condition not satisfied.')

    if unknown is None:
        unknown = [i for i in range(len(fcrn))]

    # Start with the unknown column with the minimal number of True's.
    # I.e. the implementation rxn that can be assigned to the least formal reactions.
    unknown = sorted(unknown, key = lambda x: sum(T[j][x] for j in range(len(sicrn))))

    if len(unknown) == 0: 
        # Note, we cannot test for the atomic condition at this point, because
        # some trivial reactions may be needed to satisfy the atomic condition.
        yield intrp
        return

    later = dict() # less promising partial interpretations ...
    def search(later, n = 0, m = None):
        # use this to define a search hierarchy when sending results into the
        # next round. Currently, n and m are start and stop in a range of 
        # how many species interpet to a single species.
        while later:
            if m and n > m:
                break
            if n in later:
                for (inext, unext, depth) in later[n]:
                    try:
                        for isuccess in search_column(fcrn, icrn, fs, 
                                                      inext, unext, depth + 1):
                            yield isuccess
                    except SpeciesAssignmentError:
                        continue
                del later[n]
            n += 1

    for c in unknown:
        frxn = fcrn[c]
        unext = [u for u in unknown if u != c]
        for k, irxn in enumerate(sicrn):
            if not T[k][c]:
                continue
            log.debug(f'Interpret: {pretty_rxn(irxn)} => {pretty_rxn(frxn)}')
            # left 
            ul = Counter(subtractL(irxn[0], frxn[0], False))
            sl = subtractL(frxn[0], irxn[0], False)
            # right
            ur = Counter(subtractL(irxn[1], frxn[1], False))
            sr = subtractL(frxn[1], irxn[1], False)

            if len(ul) or len(ur):
                [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
                [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
                tmpl = enumL(len(ul), sl, vl)
                tmpr = enumL(len(ur), sr, vr)
                for i, j in product(tmpl, tmpr):
                    intrpleft = {k: tuple(sorted(v)) for k, v in zip(kl, i)}
                    intrpright = {k: tuple(sorted(v)) for k, v in zip(kr, j)}
                    for key in set(intrpleft) & set(intrpright):
                        if any([sorted(intrpleft[key][fsp]) != \
                                sorted(intrpright[key][fsp]) for fsp in fs]):
                            # Incompatible dictionaries!
                            break
                    else:
                        inext = intrp.copy()
                        inext.update({k: list(v) for k, v in intrpleft.items()})
                        inext.update({k: list(v) for k, v in intrpright.items()})
                        level = max(len(v) for k, v in chain(intrpleft.items(), 
                                                             intrpright.items()))
                        later[level] = (later.get(level, [])) + [(inext, unext, depth)]
            else: # Just in case the full interpretation is provided.
                later[0] = (later.get(0, [])) + [(intrp.copy(), unext, depth)]
            for isuccess in search(later, n = 0, m = 1):
                yield isuccess
        for isuccess in search(later, n = 1, m = 2):
            yield isuccess
    for isuccess in search(later, n = 2):
        yield isuccess
    assert len(later) == 0
    return

def crn_bisimulations(fcrn, icrn, 
                      interpretation = None,
                      formals = None, 
                      permissive = 'default'):
    """ Iterate over all crn bisimulations.

    for e, bisim in enumerate(crn_bisimulations(fcrn, icrn), 1):
        print(f'Bisimulation {e} = {bisim}')

    Note that it is possible to provide more formal species than actually
    appear in the formal CRN. This is because you can still return trivial
    reactions involving this formal species. Hence, the algorithm does try to
    match trivial reactions between formal species, but only those that involve
    one (additional) formal species at a time.

    Limitations: The CRN bisimulation algorithm finds *all* CRN bisimulations
        compatible with a fully enforced partial interpretation. That means if
        you want to constrain the CRN bisimulations to some subset where e.g
        m(x) = A or m(x) = B, but m(x) != C, then you must leave m(x)
        unconstrained and filter the results afterwards.

    Args:
        fcrn (list): A formal CRN.
        icrn (list): An implementation CRN.
        interpretation (dict, optional): A (partial) interpretation with the
            format interpretation[is] = list[fs,..]. Defaults to None. 
        formals (set, optional): The set of formal species. 
            Defaults to None: all species in the formal CRN.
        permissive (string, optional): Choice of permissive checker, select
            from ('default', 'grapsearch', 'bruteforce'). Defaults to
            'default', which tries the often faster bruteforce algorithm and
            uses graphsearch whenever that algorithm exceeds the default search
            depth.

    Yields:
        Iterator over all correct interpretations (CRN bisimulations).
    """
    if interpretation is None:
        interpretation = dict()
    if formals is None:
        formals = set().union(*[set().union(*rxn[:2]) for rxn in fcrn])

    # Now this is a funny one. An interpretation contains a "formal" species
    # that does not appear in the formal CRN and is also not present in
    # extra "formals". May happen if you are not careful combining modules!
    assert all(set(v) & formals == set(v) for v in interpretation.values())

    log.debug(f'Finding CRN bisimulations using {permissive} algorithm:')
    log.debug('Original formal CRN:')
    [log.debug('  {}'.format(pretty_rxn(r))) for r in fcrn]
    log.debug('Original implementation CRN:')
    [log.debug('  {}'.format(pretty_rxn(r))) for r in icrn]
    log.debug(f'Formal species: {formals}')

    if icrn == [] and fcrn != []:
        return

    if permissive not in ['default', 'graphsearch', 'loopsearch', 'bruteforce']:
        raise ValueError('Unknown option: {}'.format(
            'the permissive test should be {}'.format(
            '"graphsearch", "loopsearch", or "bruteforce".')))
    new = []
    for [r, p] in icrn:
        nr = [deformalize(k, formals) for k in r]
        np = [deformalize(k, formals) for k in p]
        assert sorted(nr) != sorted(np)
        new.append([nr, np])
    icrn = new
    log.debug('Internal implementation CRN:')
    [log.debug('  {}'.format(pretty_rxn(r))) for r in icrn]
    inter = {deformalize(k, formals): v for k, v in interpretation.items()
                                                } if interpretation else {}
    log.debug('Internal interpretation:')
    [log.debug('  {}: {}'.format(k,v)) for (k, v) in inter.items()]

    log.info(f'Searching for bisimulation.')
    try: # Needed if you start with a wrong partial interpretation.
        seen = set()
        for parti in search_column(fcrn, icrn, formals, inter):
            hpart = tuple(sorted(tuple([k,tuple(sorted(v))]) for k, v in parti.items()))
            if hpart in seen:
                continue
            seen.add(hpart)
            try:
                for bisim in search_row(fcrn, icrn, formals, parti, 
                                        permcheck = permissive):
                    yield formalize(bisim)
            except SpeciesAssignmentError:
                continue
        log.debug(f'Done after checking {len(seen)} partial interpretations.')
    except SpeciesAssignmentError:
        log.info(f'Unable to satisfy delimiting condition with initial partial interpretation.')
    return

def modular_crn_bisimulation_test(fcrns, icrns, 
                                  formals = None, 
                                  interpretation = None, 
                                  permissive = 'default'):
    """ Check if a modulular CRN bisimulation exists. 

    Note: There are a few modifications to the original source:
        - the arguments to check CRN bisimulation for each module changed:
            - only the formal species present in the module are passed on
        - the modularity condition input changed
            - isc (former ispCommon) are now all implementation species that
              are both in the current module and in at least one other module.
            - fsc are all formal species that are both in the current module
              and in at least one other module.
        - it is still possible to provide one extra implementation module. In
          that case, all implementation reactions have to interpret to trivial
          reactions of the form {} -> {}, UNLESS, there are interpreted
          implementation species in the last module. Those may be trivial
          reactions involving only that species.
    
    Raises:
        NotImplementedError: If there are shared implementation species between
        modules for which no partial interpretation is provided, then there is 
        no guarantee that two modules can be combined. 

    Args:
        see crn_bisimulations()

    Returns:
        [bool, dict/None]: Whether a modular CRN bisimulation exists, the bisimulation or None.
    """
    # Let's make a copy here to avoid modification of the partial interpretation.
    inter = {k : v for k, v in interpretation.items()} if interpretation else {}

    # Identify common implementation species.
    ispc = dict() # Store for every implementation species in which module it appears: 
    for e, module in enumerate(icrns, 1):
        mspecies = set().union(*[set().union(*rxn[:2]) for rxn in module])
        for isp in mspecies:
            ispc[isp] = ispc.get(isp, []) + [e]
    log.debug(f'ispc = {ispc}')

    # Identify common formal species.
    fspc = dict() # Store for every formal species in which module it appears: 
    for e, module in enumerate(fcrns, 1):
        mspecies = set().union(*[set().union(*rxn[:2]) for rxn in module])
        for fsp in mspecies:
            fspc[fsp] = fspc.get(fsp, []) + [e]
    log.debug(f'fspc = {fspc}')

    if formals is None:
        formals = list(fspc.keys())

    if len(fcrns) == len(icrns)-1:
        # If one extra icrn is provided, that means that all implementation
        # rxns have to interpret trivial rxns of the form "{} -> {}". 
        # UNLESS, we have dedicated implementation species in the last module
        # which are assigned to formal species in the partial interpretation!
        # Those may be trivial reactions involving only that species.
        fcrns.append([])
        imspecies = set().union(*[set().union(*rxn[:2]) for rxn in icrns[-1]])
        fmspecies = set().union(*[set().union(inter.get(isp, [])) for isp in imspecies])
        for fsp in fmspecies:
            fspc[fsp] = fspc.get(fsp, []) + [len(fcrns)]

    # Now a quick check if there are common implementation species without interpretation:
    for e, (fcrn, icrn) in enumerate(zip(fcrns, icrns), 1):
        mfs = {k for k in formals if e in fspc[k]}
        minter = {k: v for k, v in inter.items() if e in ispc[k]}
        fsc = {f for f, m in fspc.items() if e in m and len(m) > 1} # S0
        isc = {i for i, m in ispc.items() if e in m and len(m) > 1} # S'0
        if not all(i in minter for i in isc):
            raise NotImplementedError('Modular CRN bisimulation: ' + \
                f'please provide an interpretation for all shared implementation species: {isc}')

    for e, (fcrn, icrn) in enumerate(zip(fcrns, icrns), 1):
        # Prepare inputs for crn bisimulation of this module
        mfs = {k for k in formals if e in fspc[k]}
        minter = {k: v for k, v in inter.items() if e in ispc[k]}
        for bisim in crn_bisimulations(fcrn, icrn, 
                                       interpretation = minter, 
                                       formals = mfs, 
                                       permissive = permissive):
            # Get all formal and implementation species that are in
            # common with at least one other module.
            fsc = {f for f, m in fspc.items() if e in m and len(m) > 1} # S0
            isc = {i for i, m in ispc.items() if e in m and len(m) > 1} # S'0
            if passes_modularity_condition(bisim, icrn, isc, fsc):
                inter.update(bisim)
                break
            log.debug(f'Skipping non-modular bisimulation: {bisim}')
        else:
            return False, None
    return True, inter

def crn_bisimulation_test(fcrn, icrn, 
                          formals = None, 
                          interpretation = None,
                          permissive = 'default'):
    """ Backward compatible CRN bisimulation interface.

    Args:
        see crn_bisimulations()

    Returns:
        [bool, dict/None]: Whether a modular CRN bisimulation exists, the bisimulation or None.
    """
    iterator = crn_bisimulations(fcrn, icrn, 
                                 interpretation = interpretation,
                                 formals = formals, 
                                 permissive = permissive)

    try:
        bisim = next(iterator)
        return True, bisim
    except StopIteration:
        return False, None


#
#  crnverifier/crn_bisimulation.py
#  Original source from the Nuskell compiler project
#
#  Authors:
#   - <NAME>
#   - <NAME>
#   - <NAME>
#
import logging
log = logging.getLogger(__name__)

import sys
import copy
import math
import random
from collections import Counter
from itertools import product, combinations, chain

from .utils import pretty_rxn
from .utils import interpret as interpretL

class SpeciesAssignmentError(Exception):
    pass

class EnumSpeciesAssignmentError(Exception):
    pass

class SearchDepthExceeded(Exception):
    pass

# Conversion between internal and external representation of implementation species.
def deformalize(k, fs):
    return f'i{{{k}}}' if k in fs else k

def formalize(intrp):
    intr = {}
    for sp in intrp:
        if sp[:2] == 'i{':
            isp = sp[2:-1]
            intr[isp] = intrp[sp]
        else:
            intr[sp] = intrp[sp]
    return intr

def subst(crn, intrp):
    """ Substitute implementation species with formal species in CRN.  """
    return [[interpretL(part, intrp) for part in rxn] for rxn in crn]

# Utils for list based multiset operations -> it's faster than counters ...
def subsetsL(x, reverse = True):
    """ Generate all (uniqe) multi-subsets of a list x. """
    if reverse:
        return chain(*[combinations(x, l) for l in range(len(x), -1, -1)])
    else:
        return chain(*[combinations(x, l) for l in range(0, len(x) + 1)])

def subtractL(a, b, strict = True): # Multiset a - b.
    """ Returns a new list: a - b. """
    a = a[:]
    for x in b:
        try:
            a.remove(x)
        except ValueError as err:
            if strict:
                raise err
    return a

def is_contained(a, b): # Multiset: a < b
    """ bool: True if list a is fully contained in list b. """
    b = b[:]
    try:
        [b.remove(s) for s in a]
    except ValueError as err:
        return False
    return True

def enumL(n, l, weights = None):
    """ Returns combinations to assign elements of list l to n variables. 

    The weights parameter determines how ofen a corresponding variable n
    appears. I.e if you have n = [x, y], weights = [2, 1], then you must assign
    a twice to the same value in l. In the above example, if l = [A, A, B] then
    the only assignments can be: 
        (x, y) = ((A,), (B,)), or 
        (x, y) = ((), (A, A, B)).

    Raises:
        EnumSpeciecsError('Cannot satisfy the assignment.')

    Yields: 
        tuple: all possible assignents.
    """
    if n == 0:
        yield []
        return

    if weights is None:
        weights = [1 for x in range(n)]
    w = weights[0]

    def ldiv(l, s):
        """ True, if all elements are multiples of s """
        if s == 1 or len(l) == 0:
            return l
        l = Counter(l)
        for k, v in l.items():
            if v % s:
                break
            l[k] = int(v/s)
        else:
            return tuple(l.elements())
        return None
        
    if n == 1:
        l = ldiv(l, w)
        if l is None:
            raise EnumSpeciesAssignmentError
        yield [tuple(l)]
        return

    for ss in set(subsetsL(l)):
        assert is_contained(ss, l)
        wss = ldiv(ss, w)
        if wss is None:
            continue
        try:
            for j in enumL(n-1, subtractL(l, ss), weights[1:]):
                yield [wss] + j
        except EnumSpeciesAssignmentError:
            continue
    return

#TODO(SB): When should we clear the cache?
#SRCACHE = dict()
#def same_reaction_cache(irxn, frxn, fs):
#    global SRCACHE
#    tirxn = tuple(map(tuple, irxn))
#    tfrxn = tuple(map(tuple, frxn))
#    if (tirxn, tfrxn) not in SRCACHE:
#        SRCACHE[(tirxn, tfrxn)] = same_reaction(irxn, frxn, fs)
#    return SRCACHE[(tirxn, tfrxn)]

def same_reaction(irxn, frxn, fs):
    """ Check if irxn *could* be interpreted as frxn.

    This assumes that irxn is already interpreted using the formal species fs.
    SB: Note, that I introduced a new expression expression here, see below.
    """
    ul = subtractL(irxn[0], frxn[0], strict = False) # exess implementation reactants 
    sl = subtractL(frxn[0], irxn[0], strict = False) # exess formal reactants 
    ur = subtractL(irxn[1], frxn[1], strict = False) # exess implementation products
    sr = subtractL(frxn[1], irxn[1], strict = False) # exess formal products

    ir = set(ul) # unassigned implementation reactants 
    fr = set(sl) # exess formal reactants 
    ip = set(ur) # unassingend implementation products
    fp = set(sr) # exess formal products
    if ir & fs or ip & fs:
        # There are excess formal reactants or excess formal products
        # in the implementation reaction, so this reaction cannot
        # implement the formal reaction.
        return False
    if len(fr) and len(ir) == 0:
        # There are excess formal reactants and 
        # no more implementation reactants.
        return False 
    if len(fp) and len(ip) == 0:
        # There are excess formal products and 
        # no more implementation products.
        return False
    if len(fp) and len(ir) and len(ip) and ir == ip:
        # NOTE: This line is new:
        # Example:    A -> B + C
        # cannot be:  A + y -> B + y
        return False

    try: # NOTE: this is new. The very same function we use later to assign species.
        # Left side check
        ul = Counter(ul)
        [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
        next(enumL(len(ul), sl, vl))

        # Right side check
        ur = Counter(ur)
        [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
        next(enumL(len(ur), sr, vr))
    except EnumSpeciesAssignmentError:
        return False
    return True

def trivial_reaction(irxn, fs):
    # Formal reactants and products.
    sl = subtractL(irxn[0], irxn[1], strict = False)
    sr = subtractL(irxn[1], irxn[0], strict = False)
    ir = set(sl) # "non-trivial" reactants
    ip = set(sr) # "non-trivial" products
    if (ir & fs and ip <= fs) or (ip & fs and ir <= fs):
        # If the implementation reactants contain a formal species and
        # all implementation products are (different) formal species, or
        # if the implementation products contain a formal species and
        # all implementation reactants are (different) formal species:
        return False
    else: # Could be trival
        try:
            ul = Counter([x for x in irxn[0] if x not in fs])
            sr = [x for x in sr if x in fs]
            [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
            next(enumL(len(ul), sr, vl))
            ur = Counter([x for x in irxn[1] if x not in fs])
            sl = [x for x in sl if x in fs]
            [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
            next(enumL(len(ur), sl, vr))
        except EnumSpeciesAssignmentError:
            return False
    return True
 
def makeT(fcrn, icrn, fs):
    """ Calculate a table with bool entries.

    Takes an implementation CRN where all implementation species have been
    replaced with the corresponding formal species. Thus, the implementation
    CRN contains a mix of interpreted species and/or uninterpreted species. 

    Returns:
        list[lists[bool]]: A table of bool entries with dimensions: 
            row number = len(icrn), col number = len(fcrn) + 1.
    """
    r = []
    for irxn in icrn:
        rr = [same_reaction(irxn, frxn, fs) for frxn in fcrn]
        rr.append(trivial_reaction(irxn, fs))
        r.append(rr)
    return r

def checkT(T):
    """ Check (partial) interpretation for the delimiting condition.

    Returns:
        bool: True if there is no row or (non-trivial) colum with all False values.
    """
    for row in T:
        if all(not b for b in row):
            return False
    for e, col in enumerate(zip(*T), 1):
        if e != len(T[0]) and all(not b for b in col):
            return False
    return True

def minimal_implementation_states(fstate, inter):
    """ Yields a set of minimal implementation states for a formal state. """
    def gen_supersets():
        if len(fstate) == 0:
            yield []
        else:
            fs = set(fstate).pop()
            fs1 = fstate[0]
            for k in inter:
                if fs1 in inter[k]:
                    for out in minimal_implementation_states(subtractL(fstate, 
                                                                       inter[k], 
                                                                       strict = False), inter):
                        yield [k] + out

    def filter_minimals(lst):
        seen = set()
        for x in lst:
            z = tuple(sorted(x))
            if z in seen:
                continue
            if len(x) == 0:
                yield x
                seen.add(z)
                continue
            for y in combinations(x, len(x)-1):
                if is_contained(fstate, interpretL(y, inter)):
                    break
            else:
                yield x
            seen.add(z)
        return
    return filter_minimals(gen_supersets())

def passes_modularity_condition(bisim, icrn, common_is, common_fs):
    """ Check if a bisimulation satisfies the modularity condition.

    The modularity condition is formulated with respect to subsets of 
    *common implementation* and *common formal* species, where each common
    implementation species must interpet to a common formal species. 
    "Common" should be understood as species wich are *shared* between two or
    more modules. 

    In order to guarantee that the permissive condition is satisfied when
    combining modules, this function asserts whether every implementation
    species can turn into "common" implementation species with the same
    interpretation via trivial reactions (that is the interpretation of the
    reaction must be trival). For details check JDW2019: Definition 3.3
    (Modularity condition).

    The implementation is basically the graphsearch algorithm to check the
    permissive condition, where all "minimal states" are each exactly one
    implementation species.

    Args:
        bisim (dict): An interpretation (= CRN bisimulation).
        icrn (list[lists]): An implementation CRN.
        common_is (set): A set of common implementation species.
        common_fs (set): A set of common formal species.

    Returns:
        bool: True if modularity condition is satisfied.
    """
    # All common implementation species are part of the interpretation.
    assert all(ci in bisim for ci in common_is)
    # All interpretations of common implementation species 
    # are part of common formal species.
    if not all(cf in common_fs for ci in common_is for cf in bisim[ci]):
        log.info('Unknown common formal species.')
        return False

    def Y(s):
        return s in common_is
    def Z(s):
        return len(set(bisim[s]) & common_fs) == 0

    trivial_irxns = [] # trivial reactions
    for rxn in icrn:
        iR = interpretL(rxn[0], bisim) 
        iP = interpretL(rxn[1], bisim) 
        if sorted(iR) == sorted(iP):
            trivial_irxns.append(rxn)

    done = {s for s in bisim if Y(s) or Z(s)}
    todo = {s: [set(), set()] for s in bisim if s not in done} 
    # todo[s] = [reach, produce]

    reset = len(todo) > 0
    while reset:
        reset = False
        for r in list(todo.keys()): # start at any of the non-common intermediates
            [r_reach, r_produce] = todo[r]
            for R, P in trivial_irxns:
                if r not in R:
                    continue
                nR = R[:]
                nR.remove(r)
                if set(nR) > r_produce:
                    # There is at least one more reactant which (as far as we
                    # have seen) cannot be produced by r.
                    continue

                nullsp, mystic = set(), set()
                for p in P:
                    if p in done or (not is_contained(bisim[r], bisim[p])): 
                        # If p is done, then any other p in P is either {} or
                        # it is also a reactant which must be checked
                        # separately. Alternatively, if the interpretation
                        # of products contains the interpretation
                        # of reactants and more, then we can call this one
                        # done and move on to check p.
                        done.add(r)
                        del todo[r]
                        break
                    elif len(bisim[p]) == 0:
                        nullsp.add(p)
                    else:
                        mystic.add(p)
                if r in done:
                    reset = True
                    break
                # Now investigate the mysterious "non-common implementation"
                # products which interpret to a formal species, but are not
                # known to be exit states.
                for p in mystic: 
                    if p not in r_reach:
                        r_reach.add(p)
                        reset = True
                    [p_reach, p_produce] = todo[p]
                    if not (p_reach <= r_reach):
                        # anything reachable by products is reachable by reactants.
                        r_reach |= p_reach
                        reset = True
                    if r in p_reach:
                        # we know r -> p -> r, so all r_nullsp and p_nullsp 
                        # can be produced at infinite amounts.
                        loopable = nullsp | p_produce
                        if not (loopable <= r_produce):
                            r_produce |= loopable
                            reset = True
    return len(todo) == 0

def check_permissive_bruteforce(fcrn, icrn, fs, inter, maxdepth = None):
    """ Check the permissive condition via a brute force path search.
    
    Args:
        fcrn: The formal CRN.
        icrn: The implementation CRN.
        fs: The formal species.
        inter: The interpretation.
        maxdepth: A maximum brute force recursion depth. Defaults to None.

    Raises:
        SearchDepthExceeded: If the hardcoded maxdepth is reached.

    Returns:
        bool: Whether the interpretation passes the permissive condition. 
        info: The search depth if successful, otherwise a tuple (R, S0).
    """
    sicrn = subst(icrn, inter)
    T = makeT(fcrn, sicrn, fs)
    assert checkT(T) # Assuming this has been checked before calling permissive.
    assert all(fs | set(v) == fs for k, v in inter.items())

    if maxdepth is None:
        maxdepth = sys.getrecursionlimit() - 10 # should it be one? should it be two?

    log.debug(f'The implementation CRN:\n' + '\n'.join(
        [f'{e} {t=} {pretty_rxn(sicrn[e])} ({pretty_rxn(icrn[e])})' \
                for e, t in enumerate(T)]))
    log.debug(f'{inter=}')

    def bruteforce(si, seen, trivials, depth = 0):
        if depth > maxdepth:
            raise SearchDepthExceeded('WARNING: brute-force permissive test exceeded recursion depth.')
        st = tuple(sorted(si))
        if st in seen:
            return False
        seen.add(st)
        for irxn in formal_irxns:
            if len(subtractL(irxn[0], si, False)) == 0:
                return True
        for irxn in trivials:
            if len(subtractL(irxn[0], si, False)) == 0:
                nS = subtractL(si, irxn[0]) + irxn[1]
                # Shuffle the crn to reduce the probability of getting stuck in
                # an infinite state space assuming that there is a way out ...
                nexttrivials = trivials[:]
                random.shuffle(nexttrivials)
                if bruteforce(nS, seen, nexttrivials, depth + 1):
                    return True
        return False

    # Find implementation reactions that interpret to trivial reactions.
    trivial_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][-1]]
    for i, frxn in enumerate(fcrn):
        # Find implementation reactions that interpret to this formal reaction.
        formal_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][i]]
        for S0 in minimal_implementation_states(frxn[0], inter):
            assert is_contained(frxn[0], interpretL(S0, inter))
            found = bruteforce(S0, set(), trivial_irxns)
            if not found: 
                return False, (frxn, S0)
    return True, maxdepth

def check_permissive_loopsearch(fcrn, icrn, fs, inter):
    """ The 'loopsearch' algorithm to check the permissive condition. 

    TODO: Acutally this is a very different algorithm then described in the
    paper, so before using this in production, one has to go back and see that
    it is correct and what the time/space complexity of this algorithm is.

    """
    log.warning('This "loopsearch" algorithm is not implemented as described in JDW (2019).')
    sicrn = subst(icrn, inter)
    T = makeT(fcrn, sicrn, fs)
    assert checkT(T) # Assuming this has been checked before calling permissive.
    assert all(fs | set(v) == fs for k, v in inter.items())

    log.debug(f'The implementation CRN:\n' + '\n'.join(
        [f'{e} {t=} {pretty_rxn(sicrn[e])} ({pretty_rxn(icrn[e])})' \
                for e, t in enumerate(T)]))
    log.debug(f'{inter=}')

    # Potential null species.
    nulls = [k for k, v in inter.items() if len(v) == 0]
    # Find implementation reactions that interpret to trivial reactions.
    trivial_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][-1]]

    def reach_with_inf(start, goal = None, pickup = None, ignore = None, k = 0):
        """
        Search for a path from start to goal (states of non-null species) which
        produces at least pickup null species assuming it already has infinite
        copies of everything in ignore of length at most 2^k if goal is None,
        the goal is any reaction in formal_irxns.
        
        Args:
            start: an implementation state corresponding to a formal state.
            goal: a (different) implementation state corresponding to the same
                formal state, or None. If goal is None, then the goal is to find
                any implementation reaction corresponding to the formal reaction.
            pickup: xxx Something related to null species.
            ignore: Assuming there exist infinite copies of those species.
            k: Maximum search depth.
        """
        log.debug(f'CHECK: {start=} {goal=} {ignore=} {pickup=} {k=}')
        if ignore is None:
            ignore = set()
        if pickup is None:
            pickup = []

        #log.debug(f'{start=} {goal=} {ignore=}, {pickup=}, {k=}')
        if goal is None: 
            # Check if a implementation reaction is possible.
            for irxn in formal_irxns:
                # if irxn can happen: 
                if ignore >= set(subtractL(irxn[0], start, False)):
                    log.debug(f'True formal {irxn=}')
                    return True
        elif k == 0:
            # See if start can reach a pickup goal by trivial reactions.
            for irxn in trivial_irxns:
                # if irxn can happen: 
                if ignore >= set(subtractL(irxn[0], start, False)):
                    # and if goal is contained in the results of the trivial rxn.
                    if is_contained(goal+pickup, subtractL(start, irxn[0], False) + irxn[1]):
                        return True
        else:
            if reach_with_inf(start, goal, pickup, ignore, k-1):
                # If we can reach goal with trivial reactions, do it!
                log.debug(f'Shortcut {start=} {goal=} {pickup=} {ignore=} {k=}')
                return True

            # Divide the search into two by choosing a new state mid: 
            #   - find path from start to mid.
            #   - find path from mid to goal.
            for pickpart in map(list, subsetsL(pickup)):
              for mid in M:
                if mid == start or mid == goal: 
                    continue
                log.debug(f'Reach w inf: {start=} {mid=} {goal=} {pickpart=} {k=}')
                if reach_with_inf(start, mid, pickpart, ignore, k-1):
                    if reach_with_inf(mid, goal, subtractL(pickup, pickpart), ignore, k-1):
                        return True
        log.debug(f'FALSE: {start=} {goal=} {ignore=} {pickup=} {k=}')
        return False

    for i, frxn in enumerate(fcrn):
        # Find implementation reactions that interpret to this formal reaction.
        formal_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][i]]
        # From the paper M(r) is the set of minimal implementation states for a formal reaction r.
        M = list(map(sorted, minimal_implementation_states(frxn[0], inter)))

        k, roundequiv = 0, 1
        for nequiv in range(1, len(M)+1):
            if nequiv > roundequiv:
                k += 1
                roundequiv *= 2
        log.debug(f'LS: {k=} to find paths between {M} states: {frxn=}')

        for S0 in M:
            # The formal reaction must be able to happen.
            assert is_contained(frxn[0], interpretL(S0, inter))

            # Is the formal reaction possible w/o null species? 
            if reach_with_inf(S0, None):
                # This state can exit by itself.
                continue

            found = False
            reset = True
            ignore = set()
            # Can we reach a state from where the formal reaction is possible?
            while reset:
                log.debug(f'{reset=}, {found=}, {ignore=}')
                reset = False

                for pickup in map(list, nulls):
                    if set(pickup) & ignore:
                        continue
                    assert len(pickup) != 0
                    log.debug(f'{S0=}, {pickup=}, {ignore=}, {k=} ')

                    if reach_with_inf(S0, S0, pickup, ignore, k):
                        log.debug(f'Pickup! {S0=} {pickup=} {ignore=}.')
                        ignore |= set(pickup)
                        reset = True
                        if reach_with_inf(S0, None, ignore = ignore):
                            found = True
                            break

                for Si in M:
                    if Si == S0: 
                        continue
                    log.debug(f'Testing: {S0=} {Si=} {ignore=} {k=}.')
                    if reach_with_inf(S0, Si, ignore = ignore, k = k):
                        log.debug(f'True: {S0=} {Si=} {ignore=} {k=}.')
                        if reach_with_inf(Si, None, ignore = ignore):
                            log.debug(f'Found exit: {Si=}.')
                            found = True
                            break
                if found is True:
                    break

            if not found:
                return False, S0
    return True, None

def check_permissive_graphsearch(fcrn, icrn, fs, inter):
    """ The 'graphsearch' algorithm to check the permissive condition. 

    Args:
        fcrn: The formal CRN.
        icrn: The implementation CRN.
        fs: The formal species.
        inter: The interpretation.

    Returns:
        bool: Whether the interpretation passes the permissive condition. 
        info: The search depth if successful, otherwise a tuple (R, M).
    """

    sicrn = subst(icrn, inter)
    T = makeT(fcrn, sicrn, fs)
    assert checkT(T) # assuming this has been checked before calling permissive.
    assert all(fs | set(v) == fs for k, v in inter.items())

    log.debug(f'The implementation CRN:\n' + '\n'.join(
        [f'{e} {t=} {pretty_rxn(sicrn[e])} ({pretty_rxn(icrn[e])})' \
                for e, t in enumerate(T)]))
    log.debug(f'{inter=}')

    # Species that interpret to nothing.
    nullsp = set([k for k, v in inter.items() if len(v) == 0])

    # Find implementation reactions that interpret to trivial reactions.
    trivial_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][-1]]

    max_depth = 0
    for i, frxn in enumerate(fcrn):
        # Find implementation reactions that interpret to this formal reaction.
        formal_irxns = [irxn for e, irxn in enumerate(icrn) if T[e][i]]
        # All implementation states which must permit the formal reaction.
        M = list(map(tuple, map(sorted, minimal_implementation_states(frxn[0], inter))))
        done = set() # set of species known to implement the current formal rxn.
        todo = {istate: [set(), set()] for istate in M}
        changed, depth = True, 0
        while changed:
            changed, depth = False, depth + 1
            for istate in M:
                if istate in done:
                    continue
                for irxn in formal_irxns:
                    # If the implementation state can implement the formal reaction.
                    if set(subtractL(irxn[0], istate, False)) <= todo[istate][0]: 
                        done.add(istate)
                        del todo[istate]
                        changed = True
                        break
                if istate in done:
                    continue
                for irxn in trivial_irxns:
                    # If the implementation state allows the implementation reaction.
                    if set(subtractL(irxn[0], istate, False)) <= todo[istate][0]: 
                        # apply the reaction to the current state.
                        nstate = subtractL(list(istate), irxn[0], False) + irxn[1]
                        for istate2 in M:
                            # if one of the other implementation states is reachable
                            if is_contained(istate2, nstate):
                                if istate2 in done:
                                    done.add(istate)
                                    del todo[istate]
                                    changed = True
                                    break
                                if istate in todo[istate2][1]: # I assume its a loop?
                                    s = todo[istate2][0] | (set(nstate) & nullsp)
                                    if not s <= todo[istate][0]:
                                        todo[istate][0] |= s
                                        changed = True
                                if istate2 not in todo[istate][1]:
                                    todo[istate][1].add(istate2)
                                    changed = True
                                if not (todo[istate2][1] <= todo[istate][1]):
                                    todo[istate][1] |= todo[istate2][1]
                                    changed = True
                    if istate in done:
                        break
        if todo: 
            return False, (frxn, M)
        if max_depth < depth: 
            max_depth = depth
    return True, max_depth
        
def passes_permissive_condition(fcrn, icrn, fs, inter, permcheck = 'default'):
    """ Tests an interpretation for the permissive condition.
 
    Args:
        fcrn: The formal CRN.
        icrn: The implementation CRN.
        fs: The formal species.
        inter: The interpretation.
        permcheck: Choice of permissive checker, select from ('default',
                   'grapsearch', 'bruteforce'). Defaults to 'default', which
                   tries the often faster bruteforce algorithm and uses
                   graphsearch whenever that algorithm exceeds the default
                   search depth.

    (TODO): The difficulty of checking the permissive condition scales with the
    number of minimal states (S0 in M) for any given formal reaction r = (R, P),
    which typically scales like (and scales no worse than) n**k where n = |S'|
    (the species of the implementation CRN) and k = |R|. (There may be up to
    order n^k minimal implementation states (M) for some formal reaction r and
    the trajectories by which formal r is implemented may have to pass through
    most or all of S0 in M.  When k is unbounded, checking an interpretation is
    PSPACE-complete.)

    We provide three algorithms: 
        * A bruteforce algorithm with unknown time bounds: a depth-first search
            for a path to implement each formal reaction. Space and time bounds
            are not known, but it appears faster than graphsearch in practice.
        * The graphsearch algorithm which takes poly(n^k) space and time:
            Construct a reachability graph for each formal reaction.
        * A preliminary implementation of the loopsearch algorithm, which
            should run in poly(n,k) space and poly(n^(kn)) time: search for
            "productive loops" using a space-efficient algorithm.
 
    Returns:
        bool: Whether the interpretation passes the permissive condition. 
        info: The search depth if successful, otherwise a tuple (r, M).
    """
    log.debug(f'Checking permissive condition using {permcheck=}.')
    if permcheck == 'graphsearch':
        passes, info = check_permissive_graphsearch(fcrn, icrn, fs, inter)
    elif permcheck == 'loopsearch':
        passes, info = check_permissive_loopsearch(fcrn, icrn, fs, inter)
    elif permcheck == 'bruteforce':
        passes, info = check_permissive_bruteforce(fcrn, icrn, fs, inter)
    elif permcheck == 'default':
        try:
            passes, info = check_permissive_bruteforce(fcrn, icrn, fs, inter)
        except SearchDepthExceeded as err:
            log.debug('Switching to graphsearch permissive checker.')
            passes, info = check_permissive_graphsearch(fcrn, icrn, fs, inter)
    else:
        raise SystemExit(f'Unknown algorithm for permissive condition: {permcheck}')
    return passes, info

def passes_delimiting_condition(fcrn, icrn, fs, inter):
    """ Tests an interpretation for the delimiting condition.

    Note that this function is actually never used, because typically you want
    to keep the table T. In any case, it returns the correct result.
    """
    sicrn = subst(icrn, inter)
    T = makeT(fcrn, sicrn, fs)
    return checkT(T)

def passes_atomic_condition(inter, fs):
    """ Tests an interpretation for the atomic condition.

    For every formal species there exists an implementation species which
    interprets to it.  
    """
    return all(any([f] == v for v in inter.values()) for f in fs)

def in_order(inter, order, unassigned):
    """ Check if an interpretation follows an order of assignments.

    Args:
        inter (dict): An interpretation dictionary.
        order (list): A list of implementation species. (They have to be
            assigned in the interpretation.)
        unassigend (set): A set of implementation species who's order has
            not yet been assigend.

    Returns:
        bool, info: True if interpretation is in order, updated order.
    """
    allthere = True
    for isp in order:
        if allthere:
            allthere = (isp in inter)
        elif isp in inter:
            return False, None
    if len(unassigned):
        for isp in inter:
            if isp in unassigned:
                order.append(isp)
                unassigned.remove(isp)
    return True, (order, unassigned)

def search_row(fcrn, icrn, fs, intrp, order = None, seen = None, depth = 0, permcheck = 'default'):
    """ Find full interpretations matching every irxn to one frxn or trxn.

    This "row search" finds all valid combinations of 
        implementation reaction -> formal reaction or 
        implementation reaction -> trivial reaction.
    Typically this function is called after search_column, which means we
    already have a partial interpretation that satisfies the delimiting
    condition. However, the delimiting condition may still break when an
    additional implementation reaction is introduced, so it is always necessary
    to check all three conditions.

    Yields:
        Iterator over all correct interpretations (CRN bisimulations).
    """
    log.debug(f'Searching row at {depth=}'.center(80-(2*depth), '~'))
    log.debug(f'Partial interpretation {intrp=}')
    log.debug(f'Ordered species {order=}')

    if order is None:
        itot = set().union(*[set().union(*rxn[:2]) for rxn in icrn])
        iordered = list(itot & set(intrp))  # must be an ordered list.
        iunassigned = itot - set(intrp)     # better a set.
    else:
        iordered, iunassigned = order
    valid, order = in_order(intrp, iordered, iunassigned)
    if not valid:
        raise SpeciesAssignmentError('Unordered input.')
    
    # We are in correct order, but there may still be duplicates...
    sicrn = subst(icrn, intrp)
    T = makeT(fcrn, sicrn, fs)
    if not checkT(T):
        log.debug(f'Delimiting condition not satisfied.')
        raise SpeciesAssignmentError('Delimiting condition not satisfied.')

    # The row indices where unassigned species can be found, but we 
    # remove duplicates of implementation reactions that solve for the
    # same set of implementation species!
    unknown = {tuple(sorted((set(sirxn[0]) | set(sirxn[1])) - fs)): i \
                for i, sirxn in enumerate(sicrn)} 
    unknown = [v for k, v in unknown.items() if len(k)]

    if unknown == []:
        if not passes_atomic_condition(intrp, fs):
            log.debug(f'Atomic condition not satisfied.')
            raise SpeciesAssignmentError('Atomic condition not satisfied.')
        correct, info = passes_permissive_condition(fcrn, icrn, fs, intrp, permcheck)
        if correct:
            log.debug(f'Permissive condition satisfied ({info=}).')
            yield intrp
            return
        log.debug(f'Permissive condition not satisfied ({info=}).')
        raise SpeciesAssignmentError('Permissive condition not satisfied.')

    unknown = sorted(unknown, key = lambda x: (T[x][-1], T[x][:-1].count(True)))
    # Start with the row with the minimal number of True's (excluding trivial reactions)
    log.debug(f'{unknown=} \n' + '\n'.join(
        [f'{e} {t=} {pretty_rxn(sicrn[e])} ({pretty_rxn(icrn[e])})' \
                for e, t in enumerate(T)]))

    later = dict() # less promising partial interpretations ...
    def search(later, n = 0, m = None):
        # use this to define a search hierarchy when sending results into the
        # next round. Currently, n and m are start and stop in a range of 
        # how many species interpet to a single species.
        while later:
            if m and n > m:
                break
            if n in later:
                for (inext, depth) in later[n]:
                    try:
                        for ir in search_row(fcrn, icrn, fs, inext, order,
                                                 depth = depth + 1, 
                                                 permcheck = permcheck):
                            yield ir
                    except SpeciesAssignmentError:
                        continue
                del later[n]
            n += 1

    alltriv = all(r is False for k in unknown for r in T[k][:-1]) 
    for k in unknown:
        irxn = sicrn[k] 
        if T[k][-1] is True: # Assign a trivial reaction.
            log.debug(f'Interpret: {pretty_rxn(irxn)} => trivial {fs=}')
            # Unassigned reactants and products.
            ul = Counter([x for x in irxn[0] if x not in fs])
            ur = Counter([x for x in irxn[1] if x not in fs])
            # If this reaction has no unassigned species, then we shouldn't be here!
            assert len(ul) or len(ur)
            # Formal reactants and products.
            sl = [x for x in subtractL(irxn[0], irxn[1], False) if x in fs]
            sr = [x for x in subtractL(irxn[1], irxn[0], False) if x in fs]
            log.debug(f'Interpret: {pretty_rxn(irxn)} => {sl=}, {sr=} | {ul=} {ur=}')
            [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
            [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
            tmpl = enumL(len(ul), sr, vl)
            tmpr = enumL(len(ur), sl, vr)
            for i, j in product(tmpl, tmpr):
                intrpleft = {k: tuple(sorted(v)) for k, v in zip(kl, i)}
                intrpright = {k: tuple(sorted(v)) for k, v in zip(kr, j)}
                for key in set(intrpleft) & set(intrpright):
                    if any([intrpleft[key][fsp] != \
                            intrpright[key][fsp] for fsp in fs]):
                        # Incompatible dictionaries!
                        break
                else:
                    inext = intrp.copy()
                    imove = tuple(sorted(chain(intrpleft.items(), intrpright.items())))
                    log.debug(f'Interpret: {imove}')
                    inext.update({k: list(v) for k, v in imove})
                    level = max(len(v) for k, v in imove)
                    later[level] = (later.get(level, [])) + [(inext, depth)]

            if len(ul) and len(ur):
                # An implementation reaction where species on both sides are
                # not interpreted yet, may also be interpreted as a trivial
                # reaction including an additional (single!!) formal species.
                for atom in fs:
                    log.debug(f'Interpret: {pretty_rxn(irxn)} => matching {atom=}')
                    [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
                    tmpl = enumL(len(ul), [atom]+sr, vl)
                    [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
                    tmpr = enumL(len(ur), [atom]+sl, vr)
                    try:
                        for i, j in product(tmpl, tmpr):
                            intrpleft = {k: tuple(sorted(v)) for k, v in zip(kl, i)}
                            intrpright = {k: tuple(sorted(v)) for k, v in zip(kr, j)}
                            for key in set(intrpleft) & set(intrpright):
                                if any([intrpleft[key][fsp] != \
                                        intrpright[key][fsp] for fsp in fs]):
                                    # Incompatible dictionaries!
                                    break
                            else:
                                inext = intrp.copy()
                                imove = tuple(sorted(chain(intrpleft.items(), intrpright.items())))
                                log.debug(f'Interpret: {imove}')
                                inext.update({k: list(v) for k, v in imove})
                                level = max(len(v) for k, v in imove)
                                later[level] = (later.get(level, [])) + [(inext, depth)]
                    except EnumSpeciesAssignmentError:
                        log.debug(f'Reaction {pretty_rxn(irxn)} => matching {atom=} ' + \
                                  f'was not compatible with trivial {fs=}!')

        for c, frxn in enumerate(fcrn): # Assign a formal reaction.
            if not T[k][c]:
                continue
            assert not alltriv
            log.debug(f'Interpret: {pretty_rxn(irxn)} => {pretty_rxn(frxn)}')
            # left 
            ul = Counter(subtractL(irxn[0], frxn[0], False))
            sl = subtractL(frxn[0], irxn[0], False)
            # right
            ur = Counter(subtractL(irxn[1], frxn[1], False))
            sr = subtractL(frxn[1], irxn[1], False)
            if len(ul) or len(ur):
                [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
                [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
                tmpl = enumL(len(ul), sl, vl)
                tmpr = enumL(len(ur), sr, vr)
                for i, j in product(tmpl, tmpr):
                    intrpleft = {k: tuple(sorted(v)) for k, v in zip(kl, i)}
                    intrpright = {k: tuple(sorted(v)) for k, v in zip(kr, j)}
                    for key in set(intrpleft) & set(intrpright):
                        if any([intrpleft[key][fsp] != \
                                intrpright[key][fsp] for fsp in fs]):
                            # Incompatible dictionaries!
                            break
                    else:
                        inext = intrp.copy()
                        inext.update({k: list(v) for k, v in intrpleft.items()})
                        inext.update({k: list(v) for k, v in intrpright.items()})
                        level = max(len(v) for k, v in chain(intrpleft.items(), 
                                                             intrpright.items()))
                        later[level] = (later.get(level, [])) + [(inext, depth)]
            for i in search(later, n = 0, m = 1):
                yield i
        for i in search(later, n = 0, m = 2):
            yield i
    for i in search(later, n = 2):
        yield i 
    assert len(later) == 0
    return

def search_column(fcrn, icrn, fs = None, intrp = None, unknown = None, depth = 0):
    """ Find all partial interpretations matching every frxn to one irxn.

    This "column search" finds all combinations where all formal reactions have
    one matching implementation reaction. For example, if there is one formal
    reaction and three compatible implementation reactions, then this function
    returns three interpretations that have exactly one reaction interpreted.
    Note that every result of the column search must pass the delimiting
    condition, but not necessarily atomic or permissive condition.

    Yields:
        Iterator over partial interpretations satisfying the delimiting condition.
    """
    if fs is None:
        fs = set().union(*[set().union(*rxn[:2]) for rxn in fcrn])
    if intrp is None:
        intrp = dict()
    log.debug(f'Searching column at {depth=}'.center(40-(2*depth), '*'))
    log.debug(f'Partial interpretation {intrp=}')

    sicrn = subst(icrn, intrp)
    T = makeT(fcrn, sicrn, fs)
    if not checkT(T):
        #log.debug(f'Delimiting condition not satisfied with {intrp=}')
        raise SpeciesAssignmentError('Delimiting condition not satisfied.')

    if unknown is None:
        unknown = [i for i in range(len(fcrn))]

    # Start with the unknown column with the minimal number of True's.
    # I.e. the implementation rxn that can be assigned to the least formal reactions.
    unknown = sorted(unknown, key = lambda x: sum(T[j][x] for j in range(len(sicrn))))

    if len(unknown) == 0: 
        # Note, we cannot test for the atomic condition at this point, because
        # some trivial reactions may be needed to satisfy the atomic condition.
        yield intrp
        return

    later = dict() # less promising partial interpretations ...
    def search(later, n = 0, m = None):
        # use this to define a search hierarchy when sending results into the
        # next round. Currently, n and m are start and stop in a range of 
        # how many species interpet to a single species.
        while later:
            if m and n > m:
                break
            if n in later:
                for (inext, unext, depth) in later[n]:
                    try:
                        for isuccess in search_column(fcrn, icrn, fs, 
                                                      inext, unext, depth + 1):
                            yield isuccess
                    except SpeciesAssignmentError:
                        continue
                del later[n]
            n += 1

    for c in unknown:
        frxn = fcrn[c]
        unext = [u for u in unknown if u != c]
        for k, irxn in enumerate(sicrn):
            if not T[k][c]:
                continue
            log.debug(f'Interpret: {pretty_rxn(irxn)} => {pretty_rxn(frxn)}')
            # left 
            ul = Counter(subtractL(irxn[0], frxn[0], False))
            sl = subtractL(frxn[0], irxn[0], False)
            # right
            ur = Counter(subtractL(irxn[1], frxn[1], False))
            sr = subtractL(frxn[1], irxn[1], False)

            if len(ul) or len(ur):
                [kl, vl] = zip(*ul.items()) if len(ul) else [[], []]
                [kr, vr] = zip(*ur.items()) if len(ur) else [[], []]
                tmpl = enumL(len(ul), sl, vl)
                tmpr = enumL(len(ur), sr, vr)
                for i, j in product(tmpl, tmpr):
                    intrpleft = {k: tuple(sorted(v)) for k, v in zip(kl, i)}
                    intrpright = {k: tuple(sorted(v)) for k, v in zip(kr, j)}
                    for key in set(intrpleft) & set(intrpright):
                        if any([sorted(intrpleft[key][fsp]) != \
                                sorted(intrpright[key][fsp]) for fsp in fs]):
                            # Incompatible dictionaries!
                            break
                    else:
                        inext = intrp.copy()
                        inext.update({k: list(v) for k, v in intrpleft.items()})
                        inext.update({k: list(v) for k, v in intrpright.items()})
                        level = max(len(v) for k, v in chain(intrpleft.items(), 
                                                             intrpright.items()))
                        later[level] = (later.get(level, [])) + [(inext, unext, depth)]
            else: # Just in case the full interpretation is provided.
                later[0] = (later.get(0, [])) + [(intrp.copy(), unext, depth)]
            for isuccess in search(later, n = 0, m = 1):
                yield isuccess
        for isuccess in search(later, n = 1, m = 2):
            yield isuccess
    for isuccess in search(later, n = 2):
        yield isuccess
    assert len(later) == 0
    return

def crn_bisimulations(fcrn, icrn, 
                      interpretation = None,
                      formals = None, 
                      permissive = 'default'):
    """ Iterate over all crn bisimulations.

    for e, bisim in enumerate(crn_bisimulations(fcrn, icrn), 1):
        print(f'Bisimulation {e} = {bisim}')

    Note that it is possible to provide more formal species than actually
    appear in the formal CRN. This is because you can still return trivial
    reactions involving this formal species. Hence, the algorithm does try to
    match trivial reactions between formal species, but only those that involve
    one (additional) formal species at a time.

    Limitations: The CRN bisimulation algorithm finds *all* CRN bisimulations
        compatible with a fully enforced partial interpretation. That means if
        you want to constrain the CRN bisimulations to some subset where e.g
        m(x) = A or m(x) = B, but m(x) != C, then you must leave m(x)
        unconstrained and filter the results afterwards.

    Args:
        fcrn (list): A formal CRN.
        icrn (list): An implementation CRN.
        interpretation (dict, optional): A (partial) interpretation with the
            format interpretation[is] = list[fs,..]. Defaults to None. 
        formals (set, optional): The set of formal species. 
            Defaults to None: all species in the formal CRN.
        permissive (string, optional): Choice of permissive checker, select
            from ('default', 'grapsearch', 'bruteforce'). Defaults to
            'default', which tries the often faster bruteforce algorithm and
            uses graphsearch whenever that algorithm exceeds the default search
            depth.

    Yields:
        Iterator over all correct interpretations (CRN bisimulations).
    """
    if interpretation is None:
        interpretation = dict()
    if formals is None:
        formals = set().union(*[set().union(*rxn[:2]) for rxn in fcrn])

    # Now this is a funny one. An interpretation contains a "formal" species
    # that does not appear in the formal CRN and is also not present in
    # extra "formals". May happen if you are not careful combining modules!
    assert all(set(v) & formals == set(v) for v in interpretation.values())

    log.debug(f'Finding CRN bisimulations using {permissive} algorithm:')
    log.debug('Original formal CRN:')
    [log.debug('  {}'.format(pretty_rxn(r))) for r in fcrn]
    log.debug('Original implementation CRN:')
    [log.debug('  {}'.format(pretty_rxn(r))) for r in icrn]
    log.debug(f'Formal species: {formals}')

    if icrn == [] and fcrn != []:
        return

    if permissive not in ['default', 'graphsearch', 'loopsearch', 'bruteforce']:
        raise ValueError('Unknown option: {}'.format(
            'the permissive test should be {}'.format(
            '"graphsearch", "loopsearch", or "bruteforce".')))
    new = []
    for [r, p] in icrn:
        nr = [deformalize(k, formals) for k in r]
        np = [deformalize(k, formals) for k in p]
        assert sorted(nr) != sorted(np)
        new.append([nr, np])
    icrn = new
    log.debug('Internal implementation CRN:')
    [log.debug('  {}'.format(pretty_rxn(r))) for r in icrn]
    inter = {deformalize(k, formals): v for k, v in interpretation.items()
                                                } if interpretation else {}
    log.debug('Internal interpretation:')
    [log.debug('  {}: {}'.format(k,v)) for (k, v) in inter.items()]

    log.info(f'Searching for bisimulation.')
    try: # Needed if you start with a wrong partial interpretation.
        seen = set()
        for parti in search_column(fcrn, icrn, formals, inter):
            hpart = tuple(sorted(tuple([k,tuple(sorted(v))]) for k, v in parti.items()))
            if hpart in seen:
                continue
            seen.add(hpart)
            try:
                for bisim in search_row(fcrn, icrn, formals, parti, 
                                        permcheck = permissive):
                    yield formalize(bisim)
            except SpeciesAssignmentError:
                continue
        log.debug(f'Done after checking {len(seen)} partial interpretations.')
    except SpeciesAssignmentError:
        log.info(f'Unable to satisfy delimiting condition with initial partial interpretation.')
    return

def modular_crn_bisimulation_test(fcrns, icrns, 
                                  formals = None, 
                                  interpretation = None, 
                                  permissive = 'default'):
    """ Check if a modulular CRN bisimulation exists. 

    Note: There are a few modifications to the original source:
        - the arguments to check CRN bisimulation for each module changed:
            - only the formal species present in the module are passed on
        - the modularity condition input changed
            - isc (former ispCommon) are now all implementation species that
              are both in the current module and in at least one other module.
            - fsc are all formal species that are both in the current module
              and in at least one other module.
        - it is still possible to provide one extra implementation module. In
          that case, all implementation reactions have to interpret to trivial
          reactions of the form {} -> {}, UNLESS, there are interpreted
          implementation species in the last module. Those may be trivial
          reactions involving only that species.
    
    Raises:
        NotImplementedError: If there are shared implementation species between
        modules for which no partial interpretation is provided, then there is 
        no guarantee that two modules can be combined. 

    Args:
        see crn_bisimulations()

    Returns:
        [bool, dict/None]: Whether a modular CRN bisimulation exists, the bisimulation or None.
    """
    # Let's make a copy here to avoid modification of the partial interpretation.
    inter = {k : v for k, v in interpretation.items()} if interpretation else {}

    # Identify common implementation species.
    ispc = dict() # Store for every implementation species in which module it appears: 
    for e, module in enumerate(icrns, 1):
        mspecies = set().union(*[set().union(*rxn[:2]) for rxn in module])
        for isp in mspecies:
            ispc[isp] = ispc.get(isp, []) + [e]
    log.debug(f'ispc = {ispc}')

    # Identify common formal species.
    fspc = dict() # Store for every formal species in which module it appears: 
    for e, module in enumerate(fcrns, 1):
        mspecies = set().union(*[set().union(*rxn[:2]) for rxn in module])
        for fsp in mspecies:
            fspc[fsp] = fspc.get(fsp, []) + [e]
    log.debug(f'fspc = {fspc}')

    if formals is None:
        formals = list(fspc.keys())

    if len(fcrns) == len(icrns)-1:
        # If one extra icrn is provided, that means that all implementation
        # rxns have to interpret trivial rxns of the form "{} -> {}". 
        # UNLESS, we have dedicated implementation species in the last module
        # which are assigned to formal species in the partial interpretation!
        # Those may be trivial reactions involving only that species.
        fcrns.append([])
        imspecies = set().union(*[set().union(*rxn[:2]) for rxn in icrns[-1]])
        fmspecies = set().union(*[set().union(inter.get(isp, [])) for isp in imspecies])
        for fsp in fmspecies:
            fspc[fsp] = fspc.get(fsp, []) + [len(fcrns)]

    # Now a quick check if there are common implementation species without interpretation:
    for e, (fcrn, icrn) in enumerate(zip(fcrns, icrns), 1):
        mfs = {k for k in formals if e in fspc[k]}
        minter = {k: v for k, v in inter.items() if e in ispc[k]}
        fsc = {f for f, m in fspc.items() if e in m and len(m) > 1} # S0
        isc = {i for i, m in ispc.items() if e in m and len(m) > 1} # S'0
        if not all(i in minter for i in isc):
            raise NotImplementedError('Modular CRN bisimulation: ' + \
                f'please provide an interpretation for all shared implementation species: {isc}')

    for e, (fcrn, icrn) in enumerate(zip(fcrns, icrns), 1):
        # Prepare inputs for crn bisimulation of this module
        mfs = {k for k in formals if e in fspc[k]}
        minter = {k: v for k, v in inter.items() if e in ispc[k]}
        for bisim in crn_bisimulations(fcrn, icrn, 
                                       interpretation = minter, 
                                       formals = mfs, 
                                       permissive = permissive):
            # Get all formal and implementation species that are in
            # common with at least one other module.
            fsc = {f for f, m in fspc.items() if e in m and len(m) > 1} # S0
            isc = {i for i, m in ispc.items() if e in m and len(m) > 1} # S'0
            if passes_modularity_condition(bisim, icrn, isc, fsc):
                inter.update(bisim)
                break
            log.debug(f'Skipping non-modular bisimulation: {bisim}')
        else:
            return False, None
    return True, inter

def crn_bisimulation_test(fcrn, icrn, 
                          formals = None, 
                          interpretation = None,
                          permissive = 'default'):
    """ Backward compatible CRN bisimulation interface.

    Args:
        see crn_bisimulations()

    Returns:
        [bool, dict/None]: Whether a modular CRN bisimulation exists, the bisimulation or None.
    """
    iterator = crn_bisimulations(fcrn, icrn, 
                                 interpretation = interpretation,
                                 formals = formals, 
                                 permissive = permissive)

    try:
        bisim = next(iterator)
        return True, bisim
    except StopIteration:
        return False, None


# #  crnverifier/crn_bisimulation.py #  Original source from the Nuskell compiler project # #  Authors: #   - <NAME> #   - <NAME> #   - <NAME> # # Conversion between internal and external representation of implementation species. Substitute implementation species with formal species in CRN. # Utils for list based multiset operations -> it's faster than counters ... Generate all (uniqe) multi-subsets of a list x. # Multiset a - b. Returns a new list: a - b. # Multiset: a < b bool: True if list a is fully contained in list b. Returns combinations to assign elements of list l to n variables.       The weights parameter determines how ofen a corresponding variable n     appears. I.e if you have n = [x, y], weights = [2, 1], then you must assign     a twice to the same value in l. In the above example, if l = [A, A, B] then     the only assignments can be:          (x, y) = ((A,), (B,)), or          (x, y) = ((), (A, A, B)).      Raises:         EnumSpeciecsError('Cannot satisfy the assignment.')      Yields:          tuple: all possible assignents. True, if all elements are multiples of s #TODO(SB): When should we clear the cache? #SRCACHE = dict() #def same_reaction_cache(irxn, frxn, fs): #    global SRCACHE #    tirxn = tuple(map(tuple, irxn)) #    tfrxn = tuple(map(tuple, frxn)) #    if (tirxn, tfrxn) not in SRCACHE: #        SRCACHE[(tirxn, tfrxn)] = same_reaction(irxn, frxn, fs) #    return SRCACHE[(tirxn, tfrxn)] Check if irxn *could* be interpreted as frxn.      This assumes that irxn is already interpreted using the formal species fs.     SB: Note, that I introduced a new expression expression here, see below. # exess implementation reactants # exess formal reactants # exess implementation products # exess formal products # unassigned implementation reactants # exess formal reactants # unassingend implementation products # exess formal products # There are excess formal reactants or excess formal products # in the implementation reaction, so this reaction cannot # implement the formal reaction. # There are excess formal reactants and # no more implementation reactants. # There are excess formal products and # no more implementation products. # NOTE: This line is new: # Example:    A -> B + C # cannot be:  A + y -> B + y # NOTE: this is new. The very same function we use later to assign species. # Left side check # Right side check # Formal reactants and products. # "non-trivial" reactants # "non-trivial" products # If the implementation reactants contain a formal species and # all implementation products are (different) formal species, or # if the implementation products contain a formal species and # all implementation reactants are (different) formal species: # Could be trival Calculate a table with bool entries.      Takes an implementation CRN where all implementation species have been     replaced with the corresponding formal species. Thus, the implementation     CRN contains a mix of interpreted species and/or uninterpreted species.       Returns:         list[lists[bool]]: A table of bool entries with dimensions:              row number = len(icrn), col number = len(fcrn) + 1. Check (partial) interpretation for the delimiting condition.      Returns:         bool: True if there is no row or (non-trivial) colum with all False values. Yields a set of minimal implementation states for a formal state. Check if a bisimulation satisfies the modularity condition.      The modularity condition is formulated with respect to subsets of      *common implementation* and *common formal* species, where each common     implementation species must interpet to a common formal species.      "Common" should be understood as species wich are *shared* between two or     more modules.       In order to guarantee that the permissive condition is satisfied when     combining modules, this function asserts whether every implementation     species can turn into "common" implementation species with the same     interpretation via trivial reactions (that is the interpretation of the     reaction must be trival). For details check JDW2019: Definition 3.3     (Modularity condition).      The implementation is basically the graphsearch algorithm to check the     permissive condition, where all "minimal states" are each exactly one     implementation species.      Args:         bisim (dict): An interpretation (= CRN bisimulation).         icrn (list[lists]): An implementation CRN.         common_is (set): A set of common implementation species.         common_fs (set): A set of common formal species.      Returns:         bool: True if modularity condition is satisfied. # All common implementation species are part of the interpretation. # All interpretations of common implementation species # are part of common formal species. # trivial reactions # todo[s] = [reach, produce] # start at any of the non-common intermediates # There is at least one more reactant which (as far as we # have seen) cannot be produced by r. # If p is done, then any other p in P is either {} or # it is also a reactant which must be checked # separately. Alternatively, if the interpretation # of products contains the interpretation # of reactants and more, then we can call this one # done and move on to check p. # Now investigate the mysterious "non-common implementation" # products which interpret to a formal species, but are not # known to be exit states. # anything reachable by products is reachable by reactants. # we know r -> p -> r, so all r_nullsp and p_nullsp # can be produced at infinite amounts. Check the permissive condition via a brute force path search.          Args:         fcrn: The formal CRN.         icrn: The implementation CRN.         fs: The formal species.         inter: The interpretation.         maxdepth: A maximum brute force recursion depth. Defaults to None.      Raises:         SearchDepthExceeded: If the hardcoded maxdepth is reached.      Returns:         bool: Whether the interpretation passes the permissive condition.          info: The search depth if successful, otherwise a tuple (R, S0). # Assuming this has been checked before calling permissive. # should it be one? should it be two? # Shuffle the crn to reduce the probability of getting stuck in # an infinite state space assuming that there is a way out ... # Find implementation reactions that interpret to trivial reactions. # Find implementation reactions that interpret to this formal reaction. The 'loopsearch' algorithm to check the permissive condition.       TODO: Acutally this is a very different algorithm then described in the     paper, so before using this in production, one has to go back and see that     it is correct and what the time/space complexity of this algorithm is. # Assuming this has been checked before calling permissive. # Potential null species. # Find implementation reactions that interpret to trivial reactions. Search for a path from start to goal (states of non-null species) which         produces at least pickup null species assuming it already has infinite         copies of everything in ignore of length at most 2^k if goal is None,         the goal is any reaction in formal_irxns.                  Args:             start: an implementation state corresponding to a formal state.             goal: a (different) implementation state corresponding to the same                 formal state, or None. If goal is None, then the goal is to find                 any implementation reaction corresponding to the formal reaction.             pickup: xxx Something related to null species.             ignore: Assuming there exist infinite copies of those species.             k: Maximum search depth. #log.debug(f'{start=} {goal=} {ignore=}, {pickup=}, {k=}') # Check if a implementation reaction is possible. # if irxn can happen: # See if start can reach a pickup goal by trivial reactions. # if irxn can happen: # and if goal is contained in the results of the trivial rxn. # If we can reach goal with trivial reactions, do it! # Divide the search into two by choosing a new state mid: #   - find path from start to mid. #   - find path from mid to goal. # Find implementation reactions that interpret to this formal reaction. # From the paper M(r) is the set of minimal implementation states for a formal reaction r. # The formal reaction must be able to happen. # Is the formal reaction possible w/o null species? # This state can exit by itself. # Can we reach a state from where the formal reaction is possible? The 'graphsearch' algorithm to check the permissive condition.       Args:         fcrn: The formal CRN.         icrn: The implementation CRN.         fs: The formal species.         inter: The interpretation.      Returns:         bool: Whether the interpretation passes the permissive condition.          info: The search depth if successful, otherwise a tuple (R, M). # assuming this has been checked before calling permissive. # Species that interpret to nothing. # Find implementation reactions that interpret to trivial reactions. # Find implementation reactions that interpret to this formal reaction. # All implementation states which must permit the formal reaction. # set of species known to implement the current formal rxn. # If the implementation state can implement the formal reaction. # If the implementation state allows the implementation reaction. # apply the reaction to the current state. # if one of the other implementation states is reachable # I assume its a loop? Tests an interpretation for the permissive condition.       Args:         fcrn: The formal CRN.         icrn: The implementation CRN.         fs: The formal species.         inter: The interpretation.         permcheck: Choice of permissive checker, select from ('default',                    'grapsearch', 'bruteforce'). Defaults to 'default', which                    tries the often faster bruteforce algorithm and uses                    graphsearch whenever that algorithm exceeds the default                    search depth.      (TODO): The difficulty of checking the permissive condition scales with the     number of minimal states (S0 in M) for any given formal reaction r = (R, P),     which typically scales like (and scales no worse than) n**k where n = |S'|     (the species of the implementation CRN) and k = |R|. (There may be up to     order n^k minimal implementation states (M) for some formal reaction r and     the trajectories by which formal r is implemented may have to pass through     most or all of S0 in M.  When k is unbounded, checking an interpretation is     PSPACE-complete.)      We provide three algorithms:          * A bruteforce algorithm with unknown time bounds: a depth-first search             for a path to implement each formal reaction. Space and time bounds             are not known, but it appears faster than graphsearch in practice.         * The graphsearch algorithm which takes poly(n^k) space and time:             Construct a reachability graph for each formal reaction.         * A preliminary implementation of the loopsearch algorithm, which             should run in poly(n,k) space and poly(n^(kn)) time: search for             "productive loops" using a space-efficient algorithm.       Returns:         bool: Whether the interpretation passes the permissive condition.          info: The search depth if successful, otherwise a tuple (r, M). Tests an interpretation for the delimiting condition.      Note that this function is actually never used, because typically you want     to keep the table T. In any case, it returns the correct result. Tests an interpretation for the atomic condition.      For every formal species there exists an implementation species which     interprets to it. Check if an interpretation follows an order of assignments.      Args:         inter (dict): An interpretation dictionary.         order (list): A list of implementation species. (They have to be             assigned in the interpretation.)         unassigend (set): A set of implementation species who's order has             not yet been assigend.      Returns:         bool, info: True if interpretation is in order, updated order. Find full interpretations matching every irxn to one frxn or trxn.      This "row search" finds all valid combinations of          implementation reaction -> formal reaction or          implementation reaction -> trivial reaction.     Typically this function is called after search_column, which means we     already have a partial interpretation that satisfies the delimiting     condition. However, the delimiting condition may still break when an     additional implementation reaction is introduced, so it is always necessary     to check all three conditions.      Yields:         Iterator over all correct interpretations (CRN bisimulations). # must be an ordered list. # better a set. # We are in correct order, but there may still be duplicates... # The row indices where unassigned species can be found, but we # remove duplicates of implementation reactions that solve for the # same set of implementation species! # Start with the row with the minimal number of True's (excluding trivial reactions) # less promising partial interpretations ... # use this to define a search hierarchy when sending results into the # next round. Currently, n and m are start and stop in a range of # how many species interpet to a single species. # Assign a trivial reaction. # Unassigned reactants and products. # If this reaction has no unassigned species, then we shouldn't be here! # Formal reactants and products. # Incompatible dictionaries! # An implementation reaction where species on both sides are # not interpreted yet, may also be interpreted as a trivial # reaction including an additional (single!!) formal species. # Incompatible dictionaries! # Assign a formal reaction. # left # right # Incompatible dictionaries! Find all partial interpretations matching every frxn to one irxn.      This "column search" finds all combinations where all formal reactions have     one matching implementation reaction. For example, if there is one formal     reaction and three compatible implementation reactions, then this function     returns three interpretations that have exactly one reaction interpreted.     Note that every result of the column search must pass the delimiting     condition, but not necessarily atomic or permissive condition.      Yields:         Iterator over partial interpretations satisfying the delimiting condition. #log.debug(f'Delimiting condition not satisfied with {intrp=}') # Start with the unknown column with the minimal number of True's. # I.e. the implementation rxn that can be assigned to the least formal reactions. # Note, we cannot test for the atomic condition at this point, because # some trivial reactions may be needed to satisfy the atomic condition. # less promising partial interpretations ... # use this to define a search hierarchy when sending results into the # next round. Currently, n and m are start and stop in a range of # how many species interpet to a single species. # left # right # Incompatible dictionaries! # Just in case the full interpretation is provided. Iterate over all crn bisimulations.      for e, bisim in enumerate(crn_bisimulations(fcrn, icrn), 1):         print(f'Bisimulation {e} = {bisim}')      Note that it is possible to provide more formal species than actually     appear in the formal CRN. This is because you can still return trivial     reactions involving this formal species. Hence, the algorithm does try to     match trivial reactions between formal species, but only those that involve     one (additional) formal species at a time.      Limitations: The CRN bisimulation algorithm finds *all* CRN bisimulations         compatible with a fully enforced partial interpretation. That means if         you want to constrain the CRN bisimulations to some subset where e.g         m(x) = A or m(x) = B, but m(x) != C, then you must leave m(x)         unconstrained and filter the results afterwards.      Args:         fcrn (list): A formal CRN.         icrn (list): An implementation CRN.         interpretation (dict, optional): A (partial) interpretation with the             format interpretation[is] = list[fs,..]. Defaults to None.          formals (set, optional): The set of formal species.              Defaults to None: all species in the formal CRN.         permissive (string, optional): Choice of permissive checker, select             from ('default', 'grapsearch', 'bruteforce'). Defaults to             'default', which tries the often faster bruteforce algorithm and             uses graphsearch whenever that algorithm exceeds the default search             depth.      Yields:         Iterator over all correct interpretations (CRN bisimulations). # Now this is a funny one. An interpretation contains a "formal" species # that does not appear in the formal CRN and is also not present in # extra "formals". May happen if you are not careful combining modules! # Needed if you start with a wrong partial interpretation. Check if a modulular CRN bisimulation exists.       Note: There are a few modifications to the original source:         - the arguments to check CRN bisimulation for each module changed:             - only the formal species present in the module are passed on         - the modularity condition input changed             - isc (former ispCommon) are now all implementation species that               are both in the current module and in at least one other module.             - fsc are all formal species that are both in the current module               and in at least one other module.         - it is still possible to provide one extra implementation module. In           that case, all implementation reactions have to interpret to trivial           reactions of the form {} -> {}, UNLESS, there are interpreted           implementation species in the last module. Those may be trivial           reactions involving only that species.          Raises:         NotImplementedError: If there are shared implementation species between         modules for which no partial interpretation is provided, then there is          no guarantee that two modules can be combined.       Args:         see crn_bisimulations()      Returns:         [bool, dict/None]: Whether a modular CRN bisimulation exists, the bisimulation or None. # Let's make a copy here to avoid modification of the partial interpretation. # Identify common implementation species. # Store for every implementation species in which module it appears: # Identify common formal species. # Store for every formal species in which module it appears: # If one extra icrn is provided, that means that all implementation # rxns have to interpret trivial rxns of the form "{} -> {}". # UNLESS, we have dedicated implementation species in the last module # which are assigned to formal species in the partial interpretation! # Those may be trivial reactions involving only that species. # Now a quick check if there are common implementation species without interpretation: # S0 # S'0 # Prepare inputs for crn bisimulation of this module # Get all formal and implementation species that are in # common with at least one other module. # S0 # S'0 Backward compatible CRN bisimulation interface.      Args:         see crn_bisimulations()      Returns:         [bool, dict/None]: Whether a modular CRN bisimulation exists, the bisimulation or None.