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##  Appendix A Complexity Theory

This section provides an overview of complexity classes. A promise problem is
a decision problem where the input is to be selected from all possible input
strings. A problem with an explicitly defined structure or characteristics of
input is defined in advance, making the problem environment-entrusted with
certain types of promised input with no instance of input sent in unexpectedly
or out of definition. All promise problems are assigned to a complexity class
suitable to their nature and requirements. Formally, complexity classes are a
group of computational problems that have similar resource-based complexity
[165]. In classical computing, there exist the following classes:

  * •

Polynomial Class(P): It contains all decision problems that can be solved by a
deterministic Turing machine using a polynomial amount of computation time, or
polynomial time.

  * •

Non-deterministic Polynomial (NP): It is the collection of decision problems
that can be solved by a non-deterministic machine in polynomial time but these
problems of NP can be verified by a Turing machine in polynomial time.

  * •

NP-hard: An NP-hard problem is at least as hard as the hardest problem in NP
and it is a class of problems such that every problem in NP reduces to NP-
hard. Since it takes a long time to verify their solution, all NP-hard
problems are not in NP.

  * •

NP-complete: A problem is NP-complete if it is both NP and NP-hard. NP-
complete problems are the hardest problems in the NP space. If one could solve
an NP-complete problem in polynomial time, then one could also solve any NP
problem in polynomial time.