# Forgetting Formulas and Signature Elements in Epistemic States

Alexander Becker1 Gabriele Kern-Isberner1 Kai Sauerwald2 Christoph Beierle2
1TU Dortmund University, Germany
2FernUniversität in Hagen, Germany {alexander2.becker, gabriele.kern-
<EMAIL_ADDRESS>
{christoph.beierle<EMAIL_ADDRESS>
###### Abstract

Delgrande’s knowledge level account of forgetting provides a general approach
to forgetting syntax elements from sets of formulas with links to many other
forgetting operations, in particular, to Boole’s variable elimination. On the
other hand, marginalisation of epistemic states is a specific approach to
actively reduce signatures in more complex semantic frameworks, also aiming at
forgetting atoms that is very well known from probability theory. In this
paper, we bring these two perspectives of forgetting together by showing that
marginalisation can be considered as an extension of Delgrande’s approach to
the level of epistemic states. More precisely, we generalize Delgrande’s
axioms of forgetting to forgetting in epistemic states, and show that
marginalisation is the most specific and informative forgetting operator that
satisfies these axioms. Moreover, we elaborate suitable phrasings of
Delgrande’s concept of forgetting for formulas by transferring the basic ideas
of the axioms to forgetting formulas from epistemic states. However, here we
show that this results in trivial approaches to forgetting formulas. This
finding supports the claim that forgetting syntax elements is essentially
different from belief contraction, as e.g. axiomatized in the AGM belief
change framework.

##  1 Introduction

In the past decade, the popularity and presence of artificial intelligence
(AI) grew rapidly and thereby reached almost every part of our daily lives.
From product and media recommendations, voice assistants, and smart homes over
industrial optimizations, medical research, and traffic, to even criminal
prosecution. And most probably, the importance of AI will grow even further in
the near future, due to the ever-increasing amount of data that accumulates
day by day and the huge potential it carries. However, so far only little
attention was given to the concept of forgetting, even though it plays an
essential role in many areas of our daily lives as well. In 2018 the General
Data Protection Regulation (GDPR) became applicable, which gives every citizen
of the European Union the right to be forgotten (GDPR - Article 17). This
raises the question what it actually means to forget something, and whether it
is sufficient to only delete some data in order to forget certain information.
This is clearly not the case, since AI systems fitted on this data might still
be able to infer the information we like to forget. Thus, forgetting is far
more complex than just deleting data. From a cognitive point of view,
forgetting is an inextricable part of any learning process that helps handling
information overload, sort out irrelevant information, and resolve
contradictions. Moreover, it is also of importance when it comes to knowledge
management in organisational contexts (?), socio-digital systems (?), and
domains with highly dynamic information such as supply chain and network
management. These few examples illustrate the importance of forgetting in AI
systems to guarantee individual privacy and informational self-determination,
but also efficient reasoning by blinding out irrelevant information.

In the domain of logic and knowledge representation, several logic-specific
forgetting definitions exist, e.g. Boole’s variable elimination (?), fact
forgetting in first-order logic (?) and forgetting in modal logic (?).
However, none of these specific approaches argued about the general notions of
forgetting, but rather provided a way to compute its result. In (?), Delgrande
presented a general forgetting approach with the goal to unify many of the
hitherto existing logic-specific approaches. Moreover, he stated a set of
properties he refers to as _right_ and _desirable_ when it comes to the
notions of forgetting. In contrast to Delgrande’s approach, Beierle et al. (?)
presented a general framework for cognitively different kinds of forgetting,
which also consider the common-sense understanding of forgetting, and their
realisation by means of ordinal conditional functions. In the following, we
take this broad, common-sense motivated view of forgetting in contrast to the
viewpoint put forward by Delgrande, who explicitly states that e.g. the belief
change of contraction should not be considered as forgetting.

In this work, we show that Delgrande’s forgetting approach is included in and
even generalized by the cognitively different kinds of forgetting presented in
(?), concretely by means of the marginalisation. Moreover, we show that the
forgetting properties Delgrande refers to as _right_ and _desirable_ are not
suitable to axiomatise the general properties of all kinds of forgetting, but
only of those that aim to forget signature elements instead of formulas. Thus,
the here presented results form another step towards a general framework for
different kinds of forgetting, and provide a deeper understanding of their
properties and inherent differences.

Finally, we want to give an overview of how this work is structured. In
Section 2, we give all the preliminaries needed in the later sections
including model theoretical basics and ordinal conditional functions. Then we
will present both of the above-mentioned general forgetting approaches in
Section 3 and show that the marginalisation extends Delgrande’s forgetting to
epistemic states, since both approaches always result in the same posterior
beliefs. In Section 4, we will then generalize and extend the properties
stated by Delgrande to epistemic states, and show that the marginalisation
satisfies all of them. Moreover, we show that the marginalisation is the most
specific approach satisfying these properties. Finally, we extend the same
properties to forgetting formulas in epistemic states and show that they are
not suitable for axiomatizing general properties of forgetting, since they
imply trivial approaches of forgetting formulas. In Section 6, we present our
conclusions as well as some outlooks for future works.

##  2 Formal Basics

In the following, we introduce the formal basics as needed in this work. With
$\mathcal{L}_{\Sigma}$ we state a propositional language over the finite
signature $\Sigma$ with formulas $\varphi,\psi\in\mathcal{L}_{\Sigma}$. The
corresponding interpretations are denoted as $\Omega_{\Sigma}$. The
interpretations $\omega\in\Omega_{\Sigma}$ that satisfy a formula
$\varphi\in\mathcal{L}_{\Sigma}$, i.e. $\omega\models\varphi$, are called
models of $\varphi$ and are denoted as $\llbracket\varphi\rrbracket_{\Sigma}$.
If the signature of a model set is unambiguously given by the context, we also
write $\llbracket\varphi\rrbracket$ instead. The explicit declaration of the
corresponding signature is of particular importance when arguing about
different (sub-)signatures. Moreover, each model $\omega\in\Omega_{\Sigma}$
can also be considered as a conjunction of literals corresponding to the truth
values $\omega$ assigns to each signature element $\rho\in\Sigma$. Thus, we
can also write $\omega\models\omega^{\prime}$, where
$\omega,\omega^{\prime}\in\Omega_{\Sigma}$, but $\omega^{\prime}$ is
considered to be the conjunction of literals corresponding to the
interpretation. Note that we will make use of this notation several times in
this paper. When we specifically want to argue about some signature elements
in an interpretation $\omega\in\Omega_{\Sigma}$, we denote those signature
elements $\rho\in\Sigma$ as $\dot{\rho}$ for which the concrete truth
assignment is not needed, e.g. $p\dot{b}\dot{f}\in\Omega_{\Sigma}$ with
$\Sigma=\\{p,b,f\\}$. For two formulas $\varphi,\psi\in\mathcal{L}_{\Sigma}$,
we say that $\varphi$ infers $\psi$, denoted as $\varphi\models_{\Sigma}\psi$,
if and only if $\llbracket\varphi\rrbracket\subseteq\llbracket\psi\rrbracket$.
In case that both model sets are equal, $\varphi$ and $\psi$ are equivalent,
i.e. $\varphi\equiv\psi$, iff $\varphi\models\psi$ and $\psi\models\varphi$.
Furthermore, the deductively closed set of all formulas that can be inferred
from a formula $\varphi\in\mathcal{L}_{\Sigma}$ is given by
$Cn_{\Sigma}(\varphi)=\\{\psi\in\mathcal{L}_{\Sigma}\mid\varphi\models_{\Sigma}\psi\\}$.
Again, the signature in the index of the $Cn$ operator as well as $\models$
can be omitted when its clearly given by the context. Notice that a formula
$\varphi\in\mathcal{L}_{\Sigma}$ is always equivalent to its deductive
closure, since their models are equal. The deductive closure
$Cn_{\Sigma}(\varphi)$ of a formula $\varphi\in\mathcal{L}_{\Sigma}$ can also
be expressed by means of the theory
$Th(\llbracket\varphi\rrbracket)=\\{\psi\in\mathcal{L}_{\Sigma}\mid\llbracket\varphi\rrbracket\models\psi\\}$
of its models $\llbracket\varphi\rrbracket$. All of the above-mentioned formal
basics also hold for sets of formulas $\Gamma\subseteq\mathcal{L}_{\Sigma}$.

In order to argue about inferences and models in different (sub-)signatures,
further basic terms are needed. For two interpretations
$\omega,\omega^{\prime}\in\Omega_{\Sigma}$, we say that $\omega$ and
$\omega^{\prime}$ are elementary equivalent with the exception of the
signature elements $P$, denoted as $\omega\equiv_{P}\omega^{\prime}$, if and
only if they agree on the truth values they assign to all signature elements
in $\Sigma\setminus P$ (?). Furthermore, we define the reduction and expansion
of models in Def. 1, which allow us to argue about models in sub- or super-
signatures as well.

######  Definition 1.

(?) Let $\Sigma^{\prime}\subseteq\Sigma$ be signatures and
$\varphi\in\mathcal{L}_{\Sigma}$,
$\varphi^{\prime}\in\mathcal{L}_{\Sigma^{\prime}}$ formulas. The _reduction to
$\Sigma^{\prime}$_ of models $\llbracket\varphi\rrbracket_{\Sigma}$ is defined
as

$\displaystyle(\llbracket\varphi\rrbracket_{\Sigma})_{|\Sigma^{\prime}}=\\{\omega^{\prime}\in\Omega_{\Sigma^{\prime}}\mid\text{there
is }\omega\in\llbracket\varphi\rrbracket_{\Sigma}\text{ s.t.\
}\omega\models_{\Sigma}\omega^{\prime}\\}.$

The _expansion to $\Sigma$_ of models
$\llbracket\varphi^{\prime}\rrbracket_{\Sigma^{\prime}}$ is defined as

$(\llbracket\varphi^{\prime}\rrbracket_{\Sigma^{\prime}})_{\uparrow\Sigma}=\underset{\omega^{\prime}\in\llbracket\varphi^{\prime}\rrbracket_{\Sigma^{\prime}}}{\bigcup}\omega^{\prime}_{\uparrow\Sigma},$

where
$\omega^{\prime}_{\uparrow\Sigma}=\\{\omega\in\Omega_{\Sigma}\mid\omega\models_{\Sigma}\omega^{\prime}\\}$.
Thereby, $\omega\models_{\Sigma}\omega^{\prime}$ denotes that
$\omega\in\Omega_{\Sigma}$ is more specific than
$\omega^{\prime}\in\Omega_{\Sigma^{\prime}}$ w.r.t. $\Sigma$, which holds if
and only if $\omega_{|\Sigma^{\prime}}=\omega^{\prime}$.

Notice that multiple subsequently performed reductions
$(\llbracket\varphi\rrbracket_{|\Sigma^{\prime}})_{|\Sigma^{\prime\prime}}$
can be reduced to a single reduction
$\llbracket\varphi\rrbracket_{|\Sigma^{\prime\prime}}$, if the signature
$\Sigma^{\prime\prime}$ is a subset of $\Sigma^{\prime}$.

In this work, we generally argue about epistemic states in the form of ordinal
conditional functions (OCFs) introduced in a more general form by Spohn (?).
An OCF $\kappa$ is a ranking function that assigns a rank $r\in\mathbb{N}_{0}$
to each interpretation $\omega\in\Omega_{\Sigma}$ with
$\kappa^{-1}(0)\neq\emptyset$. The rank of an interpretation can be understood
as a degree of plausibility, where $\kappa(\omega)=0$ means that $\omega$ is
most plausible. The most plausible interpretations according to an OCF
$\kappa$ are also called models of $\kappa$, and are therefore denoted by
$\llbracket\kappa\rrbracket_{\Sigma}$. The rank of formula
$\kappa(\varphi)=\min\\{\kappa(\omega)\mid\omega\in\llbracket\varphi\rrbracket\\}$
is given by the minimal rank of its models, where
$\kappa(\varphi\vee\psi)=\min\\{\kappa(\varphi),\kappa(\psi)\\}$. The beliefs
of an OCF
$Bel_{\Sigma}(\kappa)=\\{\varphi\in\mathcal{L}_{\Sigma}\mid\llbracket\kappa\rrbracket\models\varphi\\}$
is the deductively closed set of formulas $\varphi\in\mathcal{L}_{\Sigma}$
that are satisfied by the OCF’s models $\llbracket\kappa\rrbracket_{\Sigma}$.
Instead of $Bel_{\Sigma}(\kappa)\models\varphi$, we also write
$\kappa\models\varphi$.

##  3 Delgrande’s Forgetting and Marginalisation

In this section, we will first introduce Delgrande’s general forgetting
approach (?) as well as some of its most important properties. Afterwards, we
consider the OCF marginalisation as a kind of forgetting (?) and show that it
generalizes Delgrande’s definition to epistemic states.

###  3.1 Delgrande’s General Forgetting Approach

In (?), Delgrande defines a general forgetting approach with the goal to unify
many of the hitherto existing logic-specific forgetting definitions, e.g.
forgetting in propositional logic (?), first-order logic (?), or answer set
programming (?; ?). While most of these logic-specific approaches depend on
the syntactical structure of the knowledge, Delgrande defines forgetting on
the knowledge level itself, which means that it is independent of any
syntactical properties, and only argues about the beliefs that can be
inferred. Concretely, this is realized by arguing about the deductive closure
$Cn_{\Sigma}(\Gamma)$ of a set of formulas $\Gamma$ as seen in Def. 2

######  Definition 2.

(?) Let $\Sigma$ and $P$ be signatures, $\mathcal{L}_{\Sigma}$ a language with
corresponding consequence operator $Cn_{\Sigma}$, and
$\mathcal{L}_{\Sigma\setminus P}\subseteq\mathcal{L}_{\Sigma}$ a sub-language,
then _forgetting a signature_ $P$ in a set of formulas
$\Gamma\subseteq\mathcal{L}_{\Sigma}$ is defined as

$\mathcal{F}(\Gamma,P)=Cn_{\Sigma}(\Gamma)\cap\mathcal{L}_{\Sigma\setminus
P}.$

By intersecting the prior knowledge $Cn_{\Sigma}(\Gamma)$ with the sub-
language $\mathcal{L}_{\Sigma\setminus P}$ all formulas that mention any
signature element $\rho\in P$ will be removed. Therefore, forgetting according
to Def. 2 results in those consequences of $\Gamma$ that are included in the
reduced language $\mathcal{L}_{\Sigma\setminus P}$. However, since many of the
logic-specific forgetting approaches do not result in a sub-language,
Delgrande provides a second definition of forgetting that results in the
original language instead (Def. 3). This allows comparing the results of the
different forgetting approaches more easily.

######  Definition 3.

(?) Let $\Sigma$ and $P$ be signatures and $\mathcal{L}_{\Sigma}$ a language
with corresponding consequence operator $Cn_{\Sigma}$, then _forgetting a
signature $P$ in the original language $\mathcal{L}_{\Sigma}$_ in a set of
formulas $\Gamma\subseteq\mathcal{L}_{\Sigma}$ is defined as

$\mathcal{F}_{O}(\Gamma,P)=Cn_{\Sigma}(\mathcal{F}(\Gamma,P)).$

Thereby, forgetting in the original language $\mathcal{L}_{\Sigma}$ is defined
as the deductive closure of $\mathcal{F}(\Gamma,P)$ with respect to $\Sigma$.
Due to the syntax independent nature of Delgrande’s forgetting definition, it
is theoretically applicable to each logic with a well-defined consequence
operator. Note that even though the posterior knowledge still consists of
formulas mentioning the forgotten signature elements $P$, we know that they do
not provide any information about $P$, since forgetting in the original
signature results in knowledge equivalent the result of forgetting in the
reduced language, due to the deductive closure $Cn_{\Sigma}$. This also
follows from the model theoretical properties of both forgetting definitions
stated in Th. 1.

######  Theorem 1.

(?) Let $\Gamma\subseteq\mathcal{L}_{\Sigma}$ be a set of formulas and P a
signature, then the following equations hold:

  1. 1.

$\llbracket\mathcal{F}(\Gamma,P)\rrbracket_{\Sigma\setminus
P}=(\llbracket\Gamma\rrbracket_{\Sigma})_{|(\Sigma\setminus P)}$

  2. 2.

$\llbracket\mathcal{F}(\Gamma,P)\rrbracket_{\Sigma}=((\llbracket\Gamma\rrbracket_{\Sigma})_{|(\Sigma\setminus
P)})_{\uparrow\Sigma}$

From Th. 1, we can conclude that the models of forgetting in the original
language are equal to those of forgetting in the reduced language with respect
to $\Sigma$ (Cor. 1).

######  Corollary 1.

Let $\Gamma\subseteq\mathcal{L}_{\Sigma}$ be a set of formulas and $P$ a
signature, then the following holds:

$\llbracket\mathcal{F}_{O}(\Gamma,P)\rrbracket_{\Sigma}=(\llbracket\mathcal{F}(\Gamma,P)\rrbracket_{\Sigma\setminus
P})_{\uparrow\Sigma}=\llbracket\mathcal{F}(\Gamma,P)\rrbracket_{\Sigma}$

In Ex. 1 below, we illustrate the relations of both forgetting definitions
stated by Delgrande.

######  Example 1.

In this example, we illustrate both Delgrande’s forgetting in the reduced as
well as in the original language, and its effects on the model level. For
this, we consider the knowledge base $\Gamma=\\{p\rightarrow
b,f\rightarrow\overline{p},f\rightarrow
b,\overline{f}\rightarrow(p\vee\overline{b})\\}\subseteq\mathcal{L}_{\Sigma}$
with $\Sigma=\\{p,b,f\\}$, where the signature elements can be read as:

$\displaystyle p$ $\displaystyle-\text{the observed animal is a penguin},$
$\displaystyle b$ $\displaystyle-\text{the observed animal is a bird},$
$\displaystyle f$ $\displaystyle-\text{the observed animal can fly}.$

Thus, $\overline{f}\rightarrow(p\vee\overline{b})$ for example reads _if the
observed animal cannot fly, then it is a penguin or not a bird at all_. In the
following, we want to forget the subsignature $\\{p\\}\subseteq\Sigma$.
Forgetting $\\{p\\}$ in the reduced language
$\mathcal{L}_{\Sigma\setminus\\{p\\}}$ results in

$\mathcal{F}(\Gamma,\\{p\\})=Cn_{\Sigma}(\Gamma)\cap\mathcal{L}_{\Sigma\setminus\\{p\\}}=Th_{\Sigma}(\llbracket\Gamma\rrbracket_{\Sigma})\cap\mathcal{L}_{\Sigma\setminus\\{p\\}},$

where $\llbracket\Gamma\rrbracket_{\Sigma}=\\{\overline{p}\overline{b}\hskip
1.0pt\overline{f},pb\overline{f},\overline{p}bf\\}$. Concretely,
$\mathcal{F}(\Gamma,\\{p\\})$ consists of all conclusions that can be drawn
from $\Gamma$ and are part of the reduced language
$\mathcal{L}_{\Sigma\setminus\\{p\\}}$, i.e. those conclusions that do not
argue about penguins ($p$). According to Th. 1, we know that the models after
forgetting $\\{p\\}$ from $\Gamma$ correspond to the prior models
$\llbracket\Gamma\rrbracket_{\Sigma}$ reduced to $\Sigma\setminus\\{p\\}$:

$\displaystyle~{}\llbracket\mathcal{F}(\Gamma,\\{p\\})\rrbracket_{\Sigma\setminus\\{p\\}}=(\llbracket\Gamma\rrbracket_{\Sigma})_{|\Sigma\setminus\\{p\\}}$
$\displaystyle=$ $\displaystyle~{}\\{\overline{p}\overline{b}\hskip
1.0pt\overline{f},pb\overline{f},\overline{p}bf\\}_{|\Sigma\setminus\\{p\\}}=\\{\overline{b}\hskip
1.0pt\overline{f},b\overline{f},bf\\}.$

Thus, the posterior models after forgetting $\\{p\\}$ are obtain by mapping
each interpretation $\dot{p}\dot{b}\dot{f}$ to $\dot{b}\dot{f}$.

If we forget $\\{p\\}$ in the original language $\mathcal{L}_{\Sigma}$
instead, we obtain

$\displaystyle~{}\mathcal{F}_{O}(\Gamma,\\{p\\})=Cn_{\Sigma}(\mathcal{F}(\Gamma,\\{p\\}))=Th(\llbracket\mathcal{F}(\Gamma,\\{p\\})\rrbracket_{\Sigma})$
$\displaystyle=$
$\displaystyle~{}Th((\llbracket\mathcal{F}(\Gamma,\\{p\\})\rrbracket_{\Sigma\setminus\\{p\\}})_{\uparrow\Sigma})=Th(((\llbracket\Gamma\rrbracket_{\Sigma})_{|\Sigma\setminus\\{p\\}})_{\uparrow\Sigma}).$

By means of the deductive closure of $\mathcal{F}(\Gamma,\\{p\\})$ with
respect to $\Sigma$, the result of forgetting in the reduced language is
extended by those formulas $\varphi\in\mathcal{L}_{\Sigma}$ in the original
language that can be inferred by it. However, due to the relations of the
prior models $\llbracket\Gamma\rrbracket_{\Sigma}$ and those after forgetting
$\\{p\\}$ in the reduced and the original language

$\displaystyle~{}\llbracket\mathcal{F}_{O}(\Gamma,\\{p\\})\rrbracket_{\Sigma}=((\llbracket\Gamma\rrbracket_{\Sigma})_{|\Sigma\setminus\\{p\\}})_{\uparrow\Sigma}$
$\displaystyle=$ $\displaystyle~{}(\\{\overline{p}\overline{b}\hskip
1.0pt\overline{f},pb\overline{f},\overline{p}bf\\}_{|\Sigma\setminus\\{p\\}})_{\uparrow\Sigma}=\\{\overline{b}\hskip
1.0pt\overline{f},b\overline{f},bf\\}_{\uparrow\Sigma}$ $\displaystyle=$
$\displaystyle~{}\\{\overline{p}\overline{b}\hskip
1.0pt\overline{f},p\overline{b}\hskip
1.0pt\overline{f},pb\overline{f},\overline{p}b\overline{f},\overline{p}bf,pbf\\},$

we see that $\mathcal{F}_{O}(\Gamma,\\{p\\})$ can only contain trivial
proposition about penguins ($p$), since we know that if
$p\dot{b}\dot{f}\in\llbracket\mathcal{F}_{O}(\Gamma,\\{p\\})\rrbracket$, then
$\overline{p}\dot{b}\dot{f}\in\llbracket\mathcal{F}_{O}(\Gamma,\\{p\\})\rrbracket$
must hold as well. This way non-trivial propositions about penguins are
prevented, which is why forgetting in the original language can still be
considered as forgetting ${p}$. We provide an overview of the different models
in Tab. 1.

$\llbracket\Gamma\rrbracket_{\Sigma}$ | $\llbracket\mathcal{F}(\Gamma,\\{p\\})\rrbracket_{\Sigma\setminus\\{p\\}}$ | $\llbracket\mathcal{F}_{O}(\Gamma,\\{p\\})\rrbracket_{\Sigma}$
---|---|---
$\overline{p}\overline{b}\hskip 1.0pt\overline{f}$, $pb\overline{f}$, $\overline{p}bf$ | $\overline{b}\hskip 1.0pt\overline{f}$, $b\overline{f}$, $bf$ |  $\overline{p}\overline{b}\hskip 1.0pt\overline{f}$, $p\overline{b}\hskip 1.0pt\overline{f}$, $\overline{p}b\overline{f}$,
$pb\overline{f}$, $\overline{p}bf$, $pbf$
Table 1:  Models of $\Gamma$, $\mathcal{F}(\Gamma,\\{p\\})$, and
$\mathcal{F}_{O}(\Gamma,\\{p\\})$ with respect to the corresponding signatures
of the languages, where $\Gamma=\\{p\rightarrow
b,f\rightarrow\overline{p},f\rightarrow
b,\overline{f}\rightarrow(p\vee\overline{b})\\}\subseteq\mathcal{L}_{\Sigma}$
and $\Sigma=\\{p,b,f\\}$.

Besides defining a general forgetting approach, Delgrande also states several
properties of his definition, which he refers to as _right_ and _desirable_
(?). In this work, we refer to these properties as (DFP-1)-(DFP-7) as stated
in Th. 2.

######  Theorem 2.

(?) Let $\mathcal{L}_{\Sigma}$ be a language over signature $\Sigma$ and
$Cn_{\Sigma}$ the corresponding consequence operator, then the following
relations hold for all sets of formulas
$\Gamma,\Gamma^{\prime}\subseteq\mathcal{L}_{\Sigma}$ and signatures
$P,P^{\prime}$.

(DFP-1)

$\Gamma\models\mathcal{F}(\Gamma,P)$

(DFP-2)

If $\Gamma\models\Gamma^{\prime}$, then
$\mathcal{F}(\Gamma,P)\models\mathcal{F}(\Gamma^{\prime},P)$

(DFP-3)

$\mathcal{F}(\Gamma,P)=Cn_{\Sigma\setminus P}(\mathcal{F}(\Gamma,P))$

(DFP-4)

If $P^{\prime}\subseteq P$, then
$\mathcal{F}(\Gamma,P)=\mathcal{F}(\mathcal{F}(\Gamma,P^{\prime}),P)$

(DFP-5)

$\mathcal{F}(\Gamma,P\cup
P^{\prime})=\mathcal{F}(\Gamma,P)\cap\mathcal{F}(\Gamma,P^{\prime})$

(DFP-6)

$\mathcal{F}(\Gamma,P\cup
P^{\prime})=\mathcal{F}(\mathcal{F}(\Gamma,P),P^{\prime})$

(DFP-7)

$\mathcal{F}(\Gamma,P)=\mathcal{F}_{O}(\Gamma,P)\cap\mathcal{L}_{\Sigma\setminus
P}$

(DFP-1) states the monotony of forgetting, which means that it is not possible
to obtain new knowledge by means of forgetting. (DFP-2) states that any
consequence relation $\Gamma\models\Gamma^{\prime}$ of prior knowledge sets is
preserved after forgetting a signature $P$ in both. (DFP-3) describes that
forgetting always results in a deductively closed knowledge set with respect
to the reduced signature. This also corresponds to Delgrande’s idea of
defining forgetting on the knowledge level – forgetting is applied to a
deductively closed set and results in such. In (DFP-4), Delgrande states that
forgetting two signatures $P^{\prime}$ and $P$ consecutively always equals the
forgetting of $P$, if $P^{\prime}$ is included in $P$. Thus, forgetting a
signature twice has no effect on the prior knowledge. (DFP-5) and (DFP-6)
argue about iterative and simultaneous forgetting. Finally, (DFP-7) describes
the relation between forgetting in the original and the reduced language by
stating that the result of forgetting in the reduced language can always be
obtained by intersecting the result of forgetting in the original language
with the reduced language. Note that we changed the notation of (DFP-7) in
order to make it more explicit. For more information on (DFP-1)-(DFP-7) we
refer to (?).

###  3.2 Marginalisation

A general framework of forgetting and its instantiation to an approach using
OCFs is developed in (?). For the purpose of this paper, we concentrate on the
marginalisation, which on a cognitive level corresponds to the notion of
focussing and can briefly be summarized as:

  1. 1.

Focussing on relevant aspects retains our beliefs about them.

  2. 2.

Focussing on relevant aspects (temporarily) changes our beliefs such that they
do not contain any information about irrelevant aspects anymore.

In practice, this notion of forgetting is useful when it comes to efficient
and focussed query answering by means of abstracting from irrelevant details,
e.g. marginalisation is crucially used in all inference techniques for
probabilistic networks. At this point, we consider the relevant aspects to be
given and focus on the marginalisation (Def. 4) as a kind of forgetting as
such.

######  Definition 4.

(?) Let $\kappa$ be an OCF over signature $\Sigma$ and
$\omega^{\prime}\in{\Omega_{\Sigma^{\prime}}}$ an interpretation with
$\Sigma^{\prime}\subseteq\Sigma$. $\kappa_{|\Sigma^{\prime}}$ is called a
_marginalisation_ of $\kappa$ to $\Sigma^{\prime}$ with

$\kappa_{|\Sigma^{\prime}}(\omega^{\prime})=\min\\{\kappa(\omega)\mid\omega\in\Omega_{\Sigma}\text{
with }\omega\models\omega^{\prime}\\}.$

By marginalising an OCF to a subsignature $\Sigma^{\prime}$, we consider
interpretations over $\Sigma^{\prime}$ as conjunctions and assign the
corresponding rank to them.

The first notion of focussing corresponds to Lem. 1, which states that a
formula over the reduced signature is believed after the marginalisation, if
and only if it is also believed by the prior OCF. Thus, the beliefs that only
argue about the relevant aspects $\Sigma^{\prime}$ are retained.

######  Lemma 1.

Let $\kappa$ be an OCF over $\Sigma$ and $\Sigma^{\prime}\subseteq\Sigma$,
then for each $\varphi\in\mathcal{L}_{\Sigma^{\prime}}$ the following holds:

$\kappa_{|\Sigma^{\prime}}\models\varphi\Leftrightarrow\kappa\models\varphi$

Similarly to Delgrande’s forgetting, marginalisation reduces beliefs to a
subsignature. Note that Lem. 1 directly follows from (?), where they already
stated that this relations generally holds for conditional beliefs.
Furthermore, Lem. 1 allows us to express the posterior beliefs analogously to
Delgrande’s forgetting definition (Prop. 1).

######  Proposition 1.

Let $\kappa$ be an OCF over signature $\Sigma$ and
$\Sigma^{\prime}\subseteq\Sigma$ a reduced signature.

$Bel(\kappa_{|\Sigma^{\prime}})=Bel(\kappa)\cap\mathcal{L}_{\Sigma^{\prime}}$

###### Proof of Prop. 1.

Due to Lemma 1, we have
$Bel(\kappa)\cap\mathcal{L}_{\Sigma^{\prime}}=Bel(\kappa_{|\Sigma^{\prime}})\cap\mathcal{L}_{\Sigma^{\prime}}=Bel(\kappa_{|\Sigma^{\prime}})$
because
$(Bel(\kappa_{|\Sigma^{\prime}})\subseteq\mathcal{L}_{\Sigma^{\prime}})$. ∎

Thereby, Prop. 1 also corresponds to the second notion of focussing, due to
the intersection with reduced language $\mathcal{L}_{\Sigma^{\prime}}$. The
above-stated relations of the prior and posterior beliefs further imply that
the models of the posterior beliefs are equal to the those of the prior when
reducing them to $\Sigma^{\prime}$ (Prop. 2). This rather technical property
allows us to freely switch between the models of the marginalised and the
prior OCF, which will be useful in later proofs.

######  Proposition 2.

Let $\kappa$ be an OCF over signature $\Sigma$ and
$\Sigma^{\prime}\subseteq\Sigma$ a subsignature. Then
$\llbracket\kappa_{|\Sigma^{\prime}}\rrbracket=\llbracket\kappa\rrbracket_{|\Sigma^{\prime}}$
holds.

###### Proof of Prop. 2.

By definition,

$\llbracket\kappa_{|\Sigma^{\prime}}\rrbracket=\\{\omega^{\prime}\in\Omega_{\Sigma^{\prime}}\mid\kappa_{|\Sigma^{\prime}}(\omega^{\prime})=0\\},$

so applying Def. 4 yields

$\displaystyle\llbracket\kappa_{|\Sigma^{\prime}}\rrbracket$ $\displaystyle=$
$\displaystyle\\{\omega^{\prime}\in\Omega_{\Sigma^{\prime}}\mid\min\\{\kappa(\omega)\mid\omega\in\Omega_{\Sigma}\text{
with }\omega\models\omega^{\prime}\\}=0\\},$

which is the same as

$\displaystyle\\{\omega^{\prime}\in\Omega_{\Sigma^{\prime}}\mid\exists\omega\in\Omega_{\Sigma}\text{
with }\omega\models\omega^{\prime}\text{ and }\kappa(\omega)=0\\}$
$\displaystyle=$
$\displaystyle\\{\omega^{\prime}\in\Omega_{\Sigma^{\prime}}\mid\exists\omega\in\Omega_{\Sigma}\text{
with }\omega\models\omega^{\prime}\text{ and
}\omega\in\llbracket\kappa\rrbracket\\}$ $\displaystyle=$
$\displaystyle\\{\omega^{\prime}\in\Omega_{\Sigma^{\prime}}\mid\exists\omega\in\llbracket\kappa\rrbracket\text{
with
}\omega\models\omega^{\prime}\\}=\llbracket\kappa\rrbracket_{|\Sigma^{\prime}}.$

∎

Similar to Delgrande’s idea of forgetting in the original language, we might
be interested in arguing about the original signature after focussing, e.g.
for reasons of comparability. Thus, we define the concept of lifting an OCF in
Def. 5 below.

######  Definition 5.

Let $\kappa^{\prime}$ be an OCF over signature
$\Sigma^{\prime}\subseteq\Sigma$. A _lifting_ of $\kappa^{\prime}$ to
$\Sigma$, denoted by $\kappa^{\prime}_{\uparrow\Sigma}$, is uniquely defined
by
$\kappa^{\prime}_{\uparrow\Sigma}(\omega)=\kappa^{\prime}(\omega_{|\Sigma^{\prime}})$
for all $\omega\in\Omega_{\Sigma}$.

By means of lifting an OCF $\kappa^{\prime}$ over signature $\Sigma^{\prime}$
to a signature $\Sigma$ with $\Sigma^{\prime}\subseteq\Sigma$, we
(re-)introduce new signature elements to $\kappa^{\prime}$ in a way that
$\kappa^{\prime}_{\uparrow\Sigma}$ acts invariantly towards them. This is
guaranteed by the fact that all interpretations $\omega\in\Omega_{\Sigma}$
that only differ in the truth value they assign to the new signature elements
$\Sigma\setminus\Sigma^{\prime}$ are assigned to the same rank. Analogously to
Prop. 2, we show in Prop. 3 that the models of a lifted OCF are equal to the
prior models when expanded to the super-signature.

######  Proposition 3.

Let $\kappa^{\prime}$ be an OCF over signature
$\Sigma^{\prime}\subseteq\Sigma$. Then the models of the lifted
$\kappa^{\prime}$ are the expanded models of $\kappa^{\prime}$, i.e.,
$\llbracket\kappa^{\prime}_{\uparrow\Sigma}\rrbracket=\llbracket\kappa^{\prime}\rrbracket_{\uparrow\Sigma}$.

###### Proof of Prop. 3.

By definition,

$\llbracket\kappa^{\prime}\rrbracket_{\uparrow\Sigma}=\underset{\omega^{\prime}\in\llbracket\kappa^{\prime}\rrbracket}{\bigcup}\\{\omega\in\Omega_{\Sigma}\mid\omega\models\omega^{\prime}\\},$

and hence

$\displaystyle\llbracket\kappa^{\prime}\rrbracket_{\uparrow\Sigma}$
$\displaystyle=\\{\omega\in\Omega_{\Sigma}\mid\exists\omega^{\prime}\in\llbracket\kappa^{\prime}\rrbracket_{\Sigma^{\prime}}\text{
with }\omega\models\omega^{\prime}\\}$
$\displaystyle=\\{\omega\in\Omega_{\Sigma}\mid\exists\omega^{\prime}\in\llbracket\kappa^{\prime}\rrbracket_{\Sigma^{\prime}}\text{
with }\omega_{|\Sigma^{\prime}}\equiv\omega^{\prime}\\},$

due to
$\omega\models\omega^{\prime}\Leftrightarrow\omega_{|\Sigma^{\prime}}=\omega^{\prime}$
(Def. 1). Since we know that if there is an interpretation
$\omega^{\prime}\in\llbracket\kappa^{\prime}\rrbracket_{\Sigma^{\prime}}$ that
is equivalent to $\omega_{|\Sigma^{\prime}}$, then $\omega_{|\Sigma^{\prime}}$
is included in $\llbracket\kappa^{\prime}\rrbracket_{\Sigma^{\prime}}$ as
well, and vice-versa, this last set is the same as

$\displaystyle\\{\omega\in\Omega_{\Sigma}\mid\omega_{|\Sigma^{\prime}}\in\llbracket\kappa^{\prime}\rrbracket_{\Sigma^{\prime}}\\}=\\{\omega\in\Omega_{\Sigma}\mid\kappa^{\prime}(\omega_{|\Sigma^{\prime}})=0\\}$
$\displaystyle=$
$\displaystyle\\{\omega\in\Omega_{\Sigma}\mid\kappa^{\prime}_{\uparrow\Sigma}(\omega)=0\\}=\llbracket\kappa^{\prime}_{\uparrow\Sigma}\rrbracket,$

again by definition. ∎

Therefore, we also know that the beliefs after lifting are equivalent to the
prior with respect to $\Sigma$, which can also be denoted as the deductive
closure of the prior beliefs with respect to $\Sigma$ (Prop. 4).

######  Proposition 4.

Let $\kappa^{\prime}$ be an OCF over signature
$\Sigma^{\prime}\subseteq\Sigma$ and $\kappa^{\prime}_{\uparrow\Sigma}$ be a
lifting of $\kappa^{\prime}$ to $\Sigma$, then the beliefs of
$\kappa^{\prime}_{\uparrow\Sigma}$ are given by
$Bel(\kappa^{\prime}_{\uparrow\Sigma})=Cn_{\Sigma}(Bel(\kappa^{\prime}))$.

###### Proof of Prop. 4.

In a straightforward way, we obtain from Prop. 3

$\displaystyle~{}Bel(\kappa^{\prime}_{\uparrow\Sigma})=Th(\llbracket\kappa^{\prime}_{\uparrow\Sigma}\rrbracket)$
$\displaystyle=$
$\displaystyle~{}Cn_{\Sigma}(\underset{\omega\in\llbracket\kappa^{\prime}_{\uparrow\Sigma}\rrbracket}{\bigvee}\omega)=Cn_{\Sigma}(\underset{\omega\in\llbracket\kappa^{\prime}\rrbracket_{\uparrow\Sigma}}{\bigvee}\omega)$
$\displaystyle=$
$\displaystyle~{}Cn_{\Sigma}(\underset{\omega\in\underset{\omega^{\prime}\in\llbracket\kappa^{\prime}\rrbracket}{\bigcup}\omega^{\prime}_{\uparrow\Sigma}}{\bigvee}\omega)=Cn_{\Sigma}(\underset{\omega^{\prime}\in\llbracket\kappa^{\prime}\rrbracket}{\bigvee}(\underset{\omega\in\omega^{\prime}_{\uparrow\Sigma}}{\bigvee}\omega))$
$\displaystyle=$
$\displaystyle~{}Cn_{\Sigma}(\underset{\omega^{\prime}\in\llbracket\kappa^{\prime}\rrbracket}{\bigvee}\omega^{\prime})=Cn_{\Sigma}(Cn_{\Sigma^{\prime}}(\underset{\omega^{\prime}\in\llbracket\kappa^{\prime}\rrbracket}{\bigvee}\omega^{\prime}))$
$\displaystyle=$
$\displaystyle~{}Cn_{\Sigma}(Th(\llbracket\kappa^{\prime}\rrbracket))=Cn_{\Sigma}(Bel(\kappa^{\prime})).$

∎

Prop. 4 clearly shows that the beliefs of a marginalised OCF relate to those
after lifting it to the original signature again in the same way Delgrande’s
forgetting in the original language relates to forgetting in the reduced
language (see Def. 3).

Finally, we can show that the marginalisation generalizes Delgrande’s
forgetting definition to epistemic states, since both forgetting approaches
result in equivalent posterior beliefs when applied to the same prior
knowledge (Th. 3).

######  Theorem 3.

Let $\Gamma\subseteq\mathcal{L}_{\Sigma}$ be a set of formulas and $\kappa$ an
OCF over signature $\Sigma$ with $Bel(\kappa)\equiv\Gamma$, then

$\mathcal{F}(\Gamma,P)=Bel(\kappa_{|(\Sigma\setminus P)})$

holds for each signature $P$.

###### Proof of Th. 3.

Due to Prop. 1, we have $Bel(\kappa_{|(\Sigma\setminus
P)})=Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus P}$. Since
$Bel(\kappa)\equiv\Gamma$, this is the same as
$Cn_{\Sigma}(\Gamma)\cap\mathcal{L}_{\Sigma\setminus
P}=\mathcal{F}(\Gamma,P)$, by definition. ∎

The equivalence of the prior knowledge for both approaches can be stated as
$Bel(\kappa)\equiv\Gamma$, which means that the set of formulas Delgrande’s
forgetting is applied to must be equivalent to the prior beliefs
$Bel(\kappa)$. Furthermore, note that Delgrande’s forgetting definition argues
about the elements that should be forgotten, while the marginalisation argues
about the remaining subsignature.

In Ex. 2 below, we illustrate the marginalisation as well as a subsequently
performed lifting of an OCF $\kappa$ over the signature $\Sigma=\\{p,b,f\\}$,
and show how marginalisation and lifting corresponds to Delgrande’s forgetting
definitions. For this we refer to the example on Delgrande’s forgetting (Ex.
1).

######  Example 2.

In this example, we illustrate a marginalisation and a consecutively performed
lifting of the OCF $\kappa$ over $\Sigma=\\{p,b,f\\}$ (see Ex. 1) given in
Tab. 2, as well as the relations to Delgrande’s forgetting definitions. In the
following, we want to forget the subsignature $\\{p\\}\subseteq\Sigma$.

First of all, we want to note that the beliefs of $\kappa$ are equivalent to
the knowledge base $\Gamma$ (Ex. 1), since their corresponding models are the
same:

$\displaystyle~{}Bel_{\Sigma}(\kappa)=Th(\llbracket\kappa\rrbracket_{\Sigma})=Th(\\{\overline{p}\overline{b}\hskip
1.0pt\overline{f},pb\overline{f},\overline{p}bf\\})$ $\displaystyle=$
$\displaystyle~{}Th(\llbracket\Gamma\rrbracket_{\Sigma})=Cn_{\Sigma}(\Gamma)\equiv\Gamma$

Marginalising $\kappa$ to $\Sigma\setminus P$ results in
$\kappa_{|(\Sigma\setminus P)}$ as given in Tab. 2. There it can be seen that
the posterior most plausible interpretation correspond to those of $\kappa$
when omitting $p$, i.e. each interpretation
$\dot{p}\dot{b}\dot{f}\in\llbracket\kappa\rrbracket$ is mapped to
$\dot{b}\dot{f}\in\llbracket\kappa_{|(\Sigma\setminus\\{p\\})}\rrbracket$.
This exactly corresponds to the way Delgrande’s forgetting in the reduced
language affects the models of the given knowledge base $\Gamma$:

$\displaystyle~{}\llbracket\kappa_{|(\Sigma\setminus\\{p\\})}\rrbracket_{\Sigma\setminus
P}=\llbracket\kappa\rrbracket_{|(\Sigma\setminus\\{p\\})}=\\{\overline{p}\overline{b}\hskip
1.0pt\overline{f},pb\overline{f},\overline{p}bf\\}_{|(\Sigma\setminus\\{p\\})}$
$\displaystyle=$ $\displaystyle~{}\\{\overline{b}\hskip
1.0pt\overline{f},b\overline{f},bf\\}=\llbracket\mathcal{F}(\Gamma,\\{p\\})\rrbracket_{\Sigma\setminus\\{p\\}}$

In conclusion, we that know the posterior beliefs of the marginalisation and
the result of Delgrande’s forgetting must be equal:

$\displaystyle~{}Bel(\kappa_{|(\Sigma\setminus\\{p\\})})=Th(\llbracket\kappa_{|(\Sigma\setminus\\{p\\})}\rrbracket_{\Sigma\setminus\\{p\\}})$
$\displaystyle=$ $\displaystyle~{}Th(\\{\overline{b}\hskip
1.0pt\overline{f},b\overline{f},bf\\})=Th(\llbracket\mathcal{F}(\Gamma,\\{p\\})\rrbracket_{\Sigma\setminus
P})=\mathcal{F}(\Gamma,\\{p\\})$

When we lift the marginalised OCF $\kappa_{|(\Sigma\setminus\\{p\\})}$ back to
the original signature $\Sigma$, the posterior most plausible interpretations
can be obtained by mapping each interpretation
$\dot{b}\dot{f}\in\llbracket\kappa_{|(\Sigma\setminus\\{p\\})}\rrbracket_{\Sigma\setminus\\{p\\}}$
to
$\\{p\dot{b}\dot{f},\overline{p}\dot{b}\dot{f}\\}\subseteq\llbracket(\kappa_{|(\Sigma\setminus\\{p\\})})_{\uparrow\Sigma}\rrbracket_{\Sigma}$
(see Tab. 2). Just as for the marginalisation, this exactly corresponds to the
way Delgrande’s forgetting in the original language affects the prior models
of the knowledge base $\Gamma$:

$\displaystyle~{}\llbracket(\kappa_{|(\Sigma\setminus\\{p\\})})_{\uparrow\Sigma}\rrbracket_{\Sigma}=\\{\overline{b}\hskip
1.0pt\overline{f},b\overline{f},bf\\}_{\uparrow\Sigma}$ $\displaystyle=$
$\displaystyle~{}\\{\overline{p}\overline{b}\hskip
1.0pt\overline{f},p\overline{b}\hskip
1.0pt\overline{f},pb\overline{f},\overline{p}b\overline{f},\overline{p}bf,pbf\\}=\llbracket\mathcal{F}_{O}(\Gamma,\\{p\\})\rrbracket_{\Sigma}$

Therefore, the result of Delgrande’s forgetting in the original language is
equal to the beliefs after marginalising and lifting $\kappa$:

$\displaystyle~{}Bel_{\Sigma}((\kappa_{|(\Sigma\setminus\\{p\\})})_{\uparrow\Sigma})=Th(\llbracket(\kappa_{|(\Sigma\setminus\\{p\\})})_{\uparrow\Sigma}\rrbracket_{\Sigma})$
$\displaystyle=$ $\displaystyle~{}Th(\\{\overline{p}\overline{b}\hskip
1.0pt\overline{f},p\overline{b}\hskip
1.0pt\overline{f},pb\overline{f},\overline{p}b\overline{f},\overline{p}bf,pbf\\})$
$\displaystyle=$
$\displaystyle~{}Th(\llbracket\mathcal{F}_{O}(\Gamma,\\{p\\})\rrbracket_{\Sigma})=\mathcal{F}_{O}(\Gamma,\\{p\\})$

| $\kappa$ | $\kappa_{|(\Sigma\setminus\\{p\\})}$ | $(\kappa_{|(\Sigma\setminus\\{p\\})})_{\uparrow\Sigma}$
---|---|---|---
2 |  $pbf$, $\overline{p}b\overline{f}$ | - | -
1 |  $p\overline{b}\hskip 1.0pt\overline{f}$, $\overline{p}\overline{b}f$, $p\overline{b}f$ | $\overline{b}f$ |  $p\overline{b}f$, $\overline{p}\overline{b}f$
0 | $\overline{p}\overline{b}\hskip 1.0pt\overline{f}$, $pb\overline{f}$, $\overline{p}bf$ | $\overline{b}\hskip 1.0pt\overline{f}$, $b\overline{f}$, $bf$ |  $\overline{p}\overline{b}\hskip 1.0pt\overline{f}$, $p\overline{b}\hskip 1.0pt\overline{f}$, $pb\overline{f}$,
$\overline{p}b\overline{f}$, $\overline{p}bf$, $pbf$
Table 2: OCFs $\kappa$ over signature $\Sigma=\\{p,b,f\\}$, as well as its
marginalisation $\kappa_{|(\Sigma\setminus\\{p\\})}$ and the corresponding
lifting $(\kappa_{|(\Sigma\setminus\\{p\\})})_{\uparrow\Sigma}$.

From the equivalence stated in Th. 3, we know that all relations of the logic-
specific forgetting approaches and Delgrande’s general approach that can be
traced back to the equivalence of the results must hold for the
marginalisation as well. In the following, we exemplarily state this for
Boole’s atom forgetting in propositional (Def. 6), of which we know that it
can also be described by means of $\mathcal{F}$ (Th. 4).

######  Definition 6.

(?) Let $\varphi\in\mathcal{L}_{\Sigma}$ be a formula and
$\rho\in\mathcal{L}_{\Sigma}$ be an atom. _Forgetting_ $\rho$ in $\varphi$ is
then defined as

$forget(\varphi,\rho)=\varphi[\rho/\top]\vee\varphi[\rho/\bot],$

where $\varphi[\rho/\top]$ denotes the substitution of $\rho$ by $\top$, and
$\varphi[\rho/\bot]$ the substitution by $\bot$.

######  Theorem 4.

(?) Let $\mathcal{L}_{\Sigma}$ be the language in propositional logic with
signature $\Sigma$ and let $\rho\in\Sigma$ be an atom.

$forget(\varphi,\rho)\equiv\mathcal{F}(\varphi,\\{\rho\\})$

From Th. 3 and Th. 4, we can directly conclude that Boole’s forgetting
definition can also be realized by means of a marginalisation (Cor. 2).

######  Corollary 2.

Let $\kappa$ be an OCF over signature $\Sigma$ and
$\varphi\in\mathcal{L}_{\Sigma}$ a formula with $Bel(\kappa)\equiv\varphi$,
then

$forget(\varphi,\rho)\equiv Bel(\kappa_{|\Sigma\setminus\\{\rho\\}})$

holds for each atom $\rho\in\Sigma$.

##  4 Postulates for Forgetting Signatures in Epistemic States

In (?), Delgrande argues that the properties (DFP-1)-(DFP-7) (Th. 2) of his
forgetting definition are _right_ and _desirable_ for describing the general
notions of forgetting. Since we already proved that his definition can be
generalised to epistemic states by means of the marginalisation, we also
present an extended and generalised form of (DFP-1)-(DFP-7), namely
(DFPes-1)Σ-(DFPes-6)Σ, and show that the marginalisation satisfies all of
them. For this, let $\Psi,\Phi$ be epistemic states, $P,P’,P_{1},P_{2}$
signatures, and $\circ_{f}^{\Sigma}$ an arbitrary operator that maps an
epistemic state together with a signature to a new epistemic state:

(DFPes-1)Σ

$Bel(\Psi)\models Bel(\Psi\circ_{f}^{\Sigma}P)$

(DFPes-2)Σ

If $Bel(\Psi)\models Bel(\Phi)$, then $Bel(\Psi\circ_{f}^{\Sigma}P)\models
Bel(\Phi\circ_{f}^{\Sigma}P)$

(DFPes-3)Σ

If $P^{\prime}\subseteq P$, then
$Bel((\Psi\circ_{f}^{\Sigma}P^{\prime})\circ_{f}^{\Sigma}P)\equiv
Bel(\Psi\circ_{f}^{\Sigma}P)$

(DFPes-4)Σ

$Bel(\Psi\circ_{f}^{\Sigma}(P_{1}\cup P_{2}))\equiv
Bel(\Psi\circ_{f}^{\Sigma}P_{1})\cap Bel(\Psi\circ_{f}^{\Sigma}P_{2})$

(DFPes-5)Σ

$Bel(\Psi\circ_{f}^{\Sigma}(P_{1}\cup P_{2}))\equiv
Bel((\Psi\circ_{f}^{\Sigma}P_{1})\circ_{f}^{\Sigma}P_{2})$

(DFPes-6)Σ

$Bel(\Psi\circ_{f}^{\Sigma}P)\equiv
Bel((\Psi\circ_{f}^{\Sigma}P)_{\uparrow\Sigma})\cap\mathcal{L}_{\Sigma\setminus
P}$

For a detailed explanation of the above-stated postulates
(DFPes-1)Σ-(DFPes-6)Σ, we refer to the explanations of the postulates
(DFP-1)-(DFP-7) as originally stated by Delgrande. However, there are a few
points we want to emphasise in particular. First, since the beliefs of an
epistemic state are deductively closed by definition, it is not necessary to
maintain (DFP-3). Notice that due to omitting (DFP-3) the postulates
(DFP-4)-(DFP-7) correspond to (DFPes-3)Σ-(DFPes-6)Σ. Furthermore, we expressed
the forgetting in the original signature $\mathcal{F}_{O}(\Gamma,P)$ in
(DFP-7) as the beliefs after forgetting $P$ and lifting the posterior
epistemic state back to the original signature. The models of
$\mathcal{F}_{O}(\Gamma,P)$ are equal to the models of forgetting $P$ in
$\Gamma$ in the reduced signature lifted back to the original signature, i.e.
$\llbracket\mathcal{F}(\Gamma,P)\rrbracket_{\uparrow\Sigma}$ (Cor. 1). When we
consider the models of $Bel((\Psi\circ_{f}^{\Sigma}P)_{\uparrow\Sigma})$, i.e.
$\llbracket\Psi\circ_{f}^{\Sigma}P\rrbracket_{\uparrow\Sigma}$, we see that
this also describes the models after forgetting $P$ lifted back to the
original signature. Therefore, (DFPes-6)Σ exactly matches the property
originally stated by (DFP-7). In the following, we refer to those operators
satisfying (DFPes-1)Σ-(DFPes-6)Σ as signature forgetting operators.

Next, we show in Th. 5 that the marginalisation satisfies
(DFPes-1)Σ-(DFPes-6)Σ, and therefore not only yields results equivalent to
those of Delgrande’s forgetting definition, but also corresponds to the
notions of forgetting stated by Delgrande by means of (DFP-1)-(DFP-7).

Note that there exist forgetting approaches that yield results semantically
equivalent to those of Delgrande’s approach, but do not satisfy
(DFP-1)-(DFP-7). An example is Boole’s atom forgetting (Def. 6), which
violates (DFP-3).

######  Theorem 5.

Let $\kappa$ be an OCF over signature $\Sigma$ and $P$ a signature. The
marginalisation $\kappa_{|(\Sigma\setminus P)}$ to a subsignature
$(\Sigma\setminus P)\subseteq\Sigma$ satisfies (DFPes-1)Σ-(DFPes-6)Σ.

###### Proof of Th. 5.

In the following, we assume the epistemic states $\Psi$ and $\Phi$ to be OCFs,
since the marginalisation is specifically defined over OCFs, denoted as
$\kappa$ and $\kappa^{\prime}$, and further denote the marginalisation
$\kappa_{|\Sigma\setminus P}$ as $\kappa\circ_{f}^{\Sigma,m}P$.

For (DFPes-1)Σ, we need to show $Bel(\kappa)\models
Bel(\kappa\circ_{f}^{\Sigma,m}P)$, which means $Bel(\kappa)\models
Bel(\kappa_{|(\Sigma\setminus P)})$. This holds due to Lem. 1. For (DFPes-2)Σ,
we presuppose $Bel(\kappa)\models Bel(\kappa^{\prime})$. Then also
$Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus P}\models
Bel(\kappa^{\prime})\cap\mathcal{L}_{\Sigma\setminus P}$ which is equivalent
to $Bel(\kappa_{|(\Sigma\setminus P)})\models
Bel(\kappa^{\prime}_{|(\Sigma\setminus P)})$ because of Prop. 1, and hence by
definition, $Bel(\kappa\circ_{f}^{\Sigma,m}P)\models
Bel(\kappa^{\prime}\circ_{f}^{\Sigma,m}P)$.

Regarding (DFPes-3)Σ, we have the following equalities due to Prop. 2, and
because of $P^{\prime}\subseteq P$:

$\displaystyle~{}Bel((\kappa\circ_{f}^{\Sigma,m}P^{\prime})\circ_{f}^{\Sigma,m}P)$
$\displaystyle=$ $\displaystyle~{}Bel(\kappa_{\Sigma\setminus
P^{\prime}}\circ_{f}^{\Sigma,m}P)=Bel((\kappa_{|\Sigma\setminus
P^{\prime}})_{|(\Sigma\setminus P^{\prime})\setminus P})$ $\displaystyle=$
$\displaystyle~{}Bel((\kappa_{|\Sigma\setminus
P^{\prime}})_{|(\Sigma\setminus(P^{\prime}\cup
P))})=Th(\llbracket(\kappa_{|\Sigma\setminus
P^{\prime}})_{|(\Sigma\setminus(P^{\prime}\cup P))}\rrbracket)$
$\displaystyle=$
$\displaystyle~{}\\{\varphi\in\mathcal{L}_{\Sigma\setminus(P^{\prime}\cup
P)}\mid\llbracket(\kappa_{|\Sigma\setminus
P^{\prime}})_{|(\Sigma\setminus(P^{\prime}\cup
P))}\rrbracket\models\varphi\\}$ $\displaystyle=$
$\displaystyle~{}\\{\varphi\in\mathcal{L}_{\Sigma\setminus(P^{\prime}\cup
P)}\mid\llbracket\kappa_{|\Sigma\setminus
P^{\prime}}\rrbracket_{|\Sigma\setminus(P^{\prime}\cup P)}\models\varphi\\}$
$\displaystyle=$
$\displaystyle~{}\\{\varphi\in\mathcal{L}_{\Sigma\setminus(P^{\prime}\cup
P)}\mid(\llbracket\kappa\rrbracket_{|\Sigma\setminus
P^{\prime}})_{|\Sigma\setminus(P^{\prime}\cup P)}\models\varphi\\}$
$\displaystyle=$
$\displaystyle~{}\\{\varphi\in\mathcal{L}_{\Sigma\setminus(P^{\prime}\cup
P)}\mid\llbracket\kappa\rrbracket_{|\Sigma\setminus(P^{\prime}\cup
P)}\models\varphi\\}$ $\displaystyle=$
$\displaystyle~{}\\{\varphi\in\mathcal{L}_{\Sigma\setminus
P}\mid\llbracket\kappa\rrbracket_{|\Sigma\setminus P}\models\varphi\\}$
$\displaystyle=$ $\displaystyle~{}\\{\varphi\in\mathcal{L}_{\Sigma\setminus
P}\mid\llbracket\kappa_{|\Sigma\setminus P}\rrbracket\models\varphi\\}$
$\displaystyle=$ $\displaystyle~{}Th(\llbracket\kappa_{|\Sigma\setminus
P}\rrbracket)=Bel(\kappa_{|\Sigma\setminus
P})=Bel(\kappa\circ_{f}^{\Sigma,m}P).$

The proof of (DFPes-4)Σ is mainly based on Prop. 1, here we compute

$\displaystyle~{}Bel(\kappa\circ_{f}^{\Sigma,m}(P_{1}\cup
P_{2}))=Bel(\kappa_{|\Sigma\setminus(P_{1}\cup P_{2})})$ $\displaystyle=$
$\displaystyle~{}Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus(P_{1}\cup
P_{2})}=Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus
P_{1}}\cap\mathcal{L}_{\Sigma\setminus P_{2}}$ $\displaystyle=$
$\displaystyle~{}Bel(\kappa)\cap Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus
P_{1}}\cap\mathcal{L}_{\Sigma\setminus P_{2}}$ $\displaystyle=$
$\displaystyle~{}(Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus
P_{1}})\cap(Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus P_{2}})$
$\displaystyle=$ $\displaystyle~{}Bel(\kappa_{|\Sigma\setminus P_{1}})\cap
Bel(\kappa_{|\Sigma\setminus P_{2}})$ $\displaystyle=$
$\displaystyle~{}Bel(\kappa\circ_{f}^{\Sigma,m}P_{1})\cap
Bel(\kappa\circ_{f}^{\Sigma,m}P_{2}).$

Similarly for (DFPes-5)Σ, we have

$\displaystyle~{}Bel(\kappa\circ_{f}^{\Sigma,m}P_{1}\cup
P_{2})=Bel(\kappa_{|\Sigma\setminus(P_{1}\cup P_{2})})$ $\displaystyle=$
$\displaystyle~{}Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus(P_{1}\cup
P_{2})}=Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus(P_{1}\cup(P_{1}\cup
P_{2}))}$ $\displaystyle=$
$\displaystyle~{}Bel(\kappa)\cap(\mathcal{L}_{\Sigma\setminus
P_{1}}\cap\mathcal{L}_{\Sigma\setminus(P_{1}\cup P_{2})})$ $\displaystyle=$
$\displaystyle~{}(Bel(\kappa)\cap\mathcal{L}_{\Sigma\setminus
P_{1}})\cap\mathcal{L}_{\Sigma\setminus(P_{1}\cup P_{2})}$ $\displaystyle=$
$\displaystyle~{}Bel(\kappa_{|\Sigma\setminus
P_{1}})\cap\mathcal{L}_{\Sigma\setminus(P_{1}\cup
P_{2})}=Bel(\kappa_{|\Sigma\setminus P_{1}})\cap\mathcal{L}_{(\Sigma\setminus
P_{1})\setminus P_{2}}$ $\displaystyle=$
$\displaystyle~{}Bel((\kappa_{|\Sigma\setminus P_{1}})_{|(\Sigma\setminus
P_{1})\setminus
P_{2}})=Bel((\kappa\circ_{f}^{\Sigma,m}P_{1})_{|(\Sigma\setminus
P_{1})\setminus P_{2}})$ $\displaystyle=$
$\displaystyle~{}Bel((\kappa\circ_{f}^{\Sigma,m}P_{1})\circ_{f}^{\Sigma,m}P_{2}).$

Finally, Prop. 4 is used for proving (DFPes-6)Σ:

$\displaystyle~{}Bel((\kappa\circ_{f}^{\Sigma,m}P)_{\uparrow\Sigma})\cap\mathcal{L}_{\Sigma\setminus
P}=Bel((\kappa_{|\Sigma\setminus
P})_{\uparrow\Sigma})\cap\mathcal{L}_{\Sigma\setminus P}$ $\displaystyle=$
$\displaystyle~{}Cn_{\Sigma}(Bel(\kappa_{|\Sigma\setminus
P}))\cap\mathcal{L}_{\Sigma\setminus P}=Bel(\kappa_{|\Sigma\setminus
P})\cap\mathcal{L}_{\Sigma\setminus P}$ $\displaystyle=$
$\displaystyle~{}Bel(\kappa_{|\Sigma\setminus
P})=Bel(\kappa\circ_{f}^{\Sigma,m}P).$

∎

From Th. 5 above, we can also conclude that the marginalisation forms the
signature forgetting operator that only induces minimal changes to the prior
beliefs.

######  Proposition 5.

Let $\kappa$ be an OCF over signature $\Sigma$, $P\subseteq\Sigma$ a
subsignature, and $\circ_{f}^{\Sigma}$ an operator satisfying
(DFPes-1)Σ-(DFPes-6)Σ, where $\kappa\circ_{f}^{\Sigma}P$ is an OCF over the
reduced signature $\Sigma\setminus P$, then the following relation holds:

$Bel(\kappa_{|\Sigma\setminus P})\models Bel(\kappa\circ_{f}^{\Sigma}P)$

###### Proof of Prop. 5.

Because of (DFPes-1)Σ and (DFPes-6)Σ, we have $Bel(\kappa)\models
Bel(\kappa\circ_{f}^{\Sigma}P)=Bel((\kappa\circ_{f}^{\Sigma}P)_{\uparrow\Sigma})\cap\mathcal{L}_{\Sigma\setminus
P}$, hence $\kappa\models\varphi$ for all $\varphi\in
Bel((\kappa\circ_{f}^{\Sigma}P)_{\uparrow\Sigma})\cap\mathcal{L}_{\Sigma\setminus
P}$. But then also, due to Lem. 1, $\kappa_{|\Sigma\setminus P}\models\varphi$
for all $\varphi\in
Bel((\kappa\circ_{f}^{\Sigma}P)_{\uparrow\Sigma})\cap\mathcal{L}_{\Sigma\setminus
P}$, which means $Bel(\kappa_{|\Sigma\setminus P})\models
Bel((\kappa\circ_{f}^{\Sigma}P)_{\uparrow\Sigma})\cap\mathcal{L}_{\Sigma\setminus
P}$, and therefore again due to (DFPes-6)Σ, $Bel(\kappa_{|\Sigma\setminus
P})\models Bel(\kappa\circ_{f}^{\Sigma}P)$, which was to be shown. ∎

Thus, we know that any signature forgetting operator other than the
marginalisation must induce further belief changes for some epistemic states
and signatures. Such signature forgetting operators could for example depend
on some model prioritisation in addition to the epistemic state and the
signature itself.

##  5 Forgetting Signatures vs. Forgetting Formulas – A Triviality Result

In the following, we want to discuss (DFP-1)-(DFP-7) displaying the right
properties to describe the general notions forgetting. In our opinion, these
properties might display the _right_ properties when assuming forgetting as a
reduction of the language, or as forgetting signature elements, respectively.
Delgrande also comments on this, and argues that other (belief change)
operators that could be considered as some kind of forgetting, e.g.
contraction, should simply not be considered as forgetting. We think that this
view on the concept of forgetting as such is debatable, since there exist
multiple intuitively and cognitively different kinds of forgetting (?) from
which Delgrande’s approach, which corresponds to the notion of focussing (Th.
3), only forms one. Therefore, it is still to be investigated whether
(DFP-1)-(DFP-7) also states the right properties for other kinds of
forgetting.

Following the overview of cognitively different kinds of forgetting presented
in (?), it can be seen that the concept of focussing, i.e. the
marginalisation, forms the only kind of forgetting that describes forgetting
with respect to signatures. Thus, in order to investigate Delgrande’s
forgetting properties for those kinds of forgetting that argue about formulas
instead, we have to generalise and extend (DFP-1)-(DFP-7) such that they not
only argue about arbitrary epistemic states and operators, but also about
formulas. We refer to them as (DFPes-1)L-(DFPes-6)L. For this, let $\Psi,\Phi$
be epistemic states, $\varphi,\psi\in\mathcal{L}$ formulas, and
$\circ_{f}^{\mathcal{L}}$ an arbitrary belief change operator:

(DFPes-1)L

$Bel(\Psi)\models Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)$

(DFPes-2)L

If $Bel(\Psi)\models Bel(\Phi)$, then
$Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\models
Bel(\Phi\circ_{f}^{\mathcal{L}}\varphi)$

(DFPes-3)L

If $\varphi\models\psi$, then $Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv
Bel((\Psi\circ_{f}^{\mathcal{L}}\psi)\circ_{f}^{\mathcal{L}}\varphi)$

(DFPes-4)L

$Bel(\Psi\circ_{f}^{\mathcal{L}}(\varphi\vee\psi))\equiv
Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\cap
Bel(\Psi\circ_{f}^{\mathcal{L}}\psi)$

(DFPes-5)L

$Bel(\Psi\circ_{f}^{\mathcal{L}}(\varphi\vee\psi))\equiv
Bel((\Psi\circ_{f}^{\mathcal{L}}\varphi)\circ_{f}^{\mathcal{L}}\psi)$

(DFPes-6)L

If $\varphi\not\equiv\top$, then
$Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\hskip 2.00749pt\mid\neq\hskip
1.00374pt\varphi$

While the extension to (DFPes-1)L-(DFPes-6)L works almost analogously to the
extension to (DFPes-1)Σ-(DFPes-6)Σ, there exist some crucial differences,
which we will address in the following. In (DFP-4), Delgrande argues about
forgetting signature $P,P’$ for which we assume that $P’$ is fully included in
$P$. In order to extend and generalise this property, we have to examine how
this notion can be described with respect to formulas. We found it most
accurate to generalise this relation of the information we would like to
forget by means of the specificity of formulas, i.e. $\varphi\models\psi$.
Thereby, we say that the knowledge described by $\psi$ is fully included in
that of $\varphi$, if and only if $\psi$ can be inferred from $\varphi$. More
formally, this can be stated by means of the deductive closures of $\varphi$
and $\psi$, i.e. $\varphi\models\psi\Leftrightarrow Cn(\psi)\subseteq
Cn(\varphi)$.

In (DFP-5) and (DFP-6), Delgrande argues about forgetting two signatures
$P,P’$ at once, which is described as forgetting $P\cup P’$. On a more
intuitive level this can be viewed as only forgetting a single piece of
information that consist of both the information we actual like to forget.
When arguing about formulas instead of signatures, this can be expressed by
means of a disjunction $\varphi\vee\psi$, where $\varphi$ and $\psi$ are the
two formulas we actually want to forget. Even though it might seem more
appropriate to describe this idea by means of a conjunction
$\varphi\wedge\psi$, it is not sufficient to forget the conjunction in order
to forget both $\varphi$ and $\psi$, since it is generally sufficient to
forget one of the formulas in order to forget the conjunction as well. Thus,
describing the unification of two pieces of information by means of a
disjunction guarantees that both formulas can no longer be inferred after
forgetting.

Just as for the postulates for forgetting signatures, we omit (DFP-3), since a
belief set is already deductively closed by definition. Furthermore, we omit
(DFP-7) since it argues about the relation of forgetting in the reduced and in
the original language, which is not applicable in case of forgetting formulas.
Instead, we introduce an additional postulate (DFPes-6)L that explicitly
states the success of the forgetting operator, i.e. after forgetting a non-
tautologous formula $\varphi$, we are no longer able to infer $\varphi$.

When extending (DFP-1)-(DFP-7) to forgetting formulas, Delgrande’s idea that
forgetting should be performed on the knowledge level, and therefore should be
independent of the syntactic structure of the given knowledge also extends to
the knowledge we want to forget. Thus, we show in Th. 6 the syntax
independence implied by (DFPes-1)L-(DFPes-6)L.

######  Theorem 6 (Syntax Independence).

Let $\Psi$ be an epistemic state and $\circ_{f}^{\mathcal{L}}$ a belief change
operator satisfying (DFPes-1)L-(DFPes-6)L. Further, let
$\varphi,\psi\in\mathcal{L}$ be formulas, then the following holds:

$\text{If }\varphi\equiv\psi,\text{ then
}Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv
Bel(\Psi\circ_{f}^{\mathcal{L}}\psi).$

###### Proof of Th. 6.

From $\varphi\equiv\psi$, we obtain with (DFPes-3)L and (DFPes-5)L:

$\displaystyle~{}Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)=Bel((\Psi\circ_{f}^{\mathcal{L}}\psi)\circ_{f}^{\mathcal{L}}\varphi)=Bel(\Psi\circ_{f}^{\mathcal{L}}\psi\vee\varphi)$

and

$\displaystyle~{}Bel(\Psi\circ_{f}^{\mathcal{L}}\psi)=Bel((\Psi\circ_{f}^{\mathcal{L}}\varphi)\circ_{f}^{\mathcal{L}}\psi)=Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi\vee\psi).$

Therefore,

$\displaystyle Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)$
$\displaystyle=Bel(\Psi\circ_{f}^{\mathcal{L}}\psi)\cap
Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)$
$\displaystyle=Bel(\Psi\circ_{f}^{\mathcal{L}}\psi),$

due to (DFPes-4)L. ∎

Next, we show that there cannot exist any non-trivial belief change operator
satisfying (DFPes-1)L-(DFPes-6)L. For this, we first show that (DFPes-3)L
together with (DFPes-5)L imply that the forgetting of any conjunction
$\varphi\wedge\psi$ must result in beliefs equivalent to just forgetting
$\varphi$ or $\psi$ (Prop. 6).

######  Proposition 6.

Let $\Psi$ be an epistemic state and $\circ_{f}^{\mathcal{L}}$ a belief change
operator satisfying (DFPes-1)L-(DFPes-6)L, then

$Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv
Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi\wedge\psi)\equiv
Bel(\Psi\circ_{f}^{\mathcal{L}}\psi)$

holds for all formulas $\varphi,\psi\in\mathcal{L}$.

###### Proof of (Prop. 6).

Using (DFPes-3)L, (DFPes-5)L, and Th. 6, we compute

$\displaystyle~{}Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi\wedge\psi)=Bel((\Psi\circ_{f}^{\mathcal{L}}\varphi)\circ_{f}^{\mathcal{L}}\varphi\wedge\psi)$
$\displaystyle=$
$\displaystyle~{}Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi\vee(\varphi\wedge\psi))=Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi).$

This holds for $\psi$ analogously. Thus, we can conclude
$Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv
Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi\wedge\psi)\equiv
Bel(\Psi\circ_{f}^{\mathcal{L}}\psi)$. ∎

From Prop. 6 we can especially conclude that forgetting according to
(DFPes-1)L-(DFPes-6)L must be independent of the formula we actually like to
forget (Cor. 3).

######  Corollary 3.

Let $\Psi$ be an epistemic state and $\circ_{f}^{\mathcal{L}}$ a belief change
operator satisfying (DFPes-1)L-(DFPes-6)L, then

$Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv
Bel(\Psi\circ_{f}^{\mathcal{L}}\psi)$

holds for all formulas $\varphi,\psi\in\mathcal{L}$.

Therefore, we know that a forgetting operator satisfying (DFPes-1)L-(DFPes-6)L
must always forget all prior beliefs except for tautologies (Th. 7).

######  Theorem 7 (Triviality Result).

Let $\Psi$ be an epistemic state. A belief change operator
$\circ_{f}^{\mathcal{L}}$ satisfies (DFPes-1)L-(DFPes-6)L, if and only if
$Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv\top$ holds for each
$\varphi\in\mathcal{L}$.

###### Proof of Th. 7.

We prove Th. 7 in two steps. First, we show that if a belief change operator
satisfies (DFPes-1)L-(DFPes-6)L, then it must always result in posterior
beliefs $Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)$ equivalent to $\top$.
Second, we show that each belief change operator $\circ_{f}^{\mathcal{L}}$
with $Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv\top$ for each
$\varphi\in\mathcal{L}$ satisfies (DFPes-1)L-(DFPes-6)L. We refer to these two
steps as $(\Rightarrow)$ and $(\Leftarrow)$. Note that we assume all formulas
$\varphi,\psi\in\mathcal{L}$ to be non-tautologous.

Case $(\Rightarrow)$: From Cor. 3, we know that applying
$\circ_{f}^{\mathcal{L}}$ to an epistemic state $\Psi$ must result in
equivalent beliefs for all formulas $\varphi,\psi\in\mathcal{L}$. From
(DFPes-6)L we know that after forgetting a formula $\varphi$, we are no longer
able to infer $\varphi$. Since the posterior beliefs are equivalent for all
formulas, we can conclude that after applying $\circ_{f}^{\mathcal{L}}$ to
$\Psi$, we are not able to infer any formula, but tautologies.

$\displaystyle\begin{aligned} &Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv
Bel(\Psi\circ_{f}^{\mathcal{L}}\psi),\\\ &\quad\text{ for all
}\varphi,\psi\in\mathcal{L}\end{aligned}$ (Cor. 3) $\displaystyle\Rightarrow$
$\displaystyle Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\hskip
2.00749pt\mid\neq\hskip 1.00374pt\varphi,\psi,\text{ for all
}\varphi,\psi\in\mathcal{L}$ (DFPes-1)L $\displaystyle\Leftrightarrow$
$\displaystyle~{}Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv\top,\text{ for
all }\varphi\in\mathcal{L}$

Case $(\Leftarrow)$: Let $\Psi$ and $\Phi$ be epistemic states and
$\varphi,\psi\in\mathcal{L}$ be non-tautologous formulas, and
$\circ_{f}^{\mathcal{L}}$ a belief change operator with
$Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv\top$ for all epistemic states
$\Psi$ and formulas $\varphi$. The fact that $\circ_{f}^{\mathcal{L}}$
satisfies (DFPes-1)L-(DFPes-6)L directly concludes from the assumption
$Bel(\Psi\circ_{f}^{\mathcal{L}}\varphi)\equiv\top$, for all
$\varphi\in\mathcal{L}$.

We showed that both cases $(\Rightarrow)$ and $(\Leftarrow)$ hold, and
therefore proved the triviality result stated in Th. 7. ∎

##  6 Conclusion

We discussed two of the existing approaches towards a general forgetting
framework. The first approach was that of Delgrande (?) in which he gives a
general forgetting definition that argues about forgetting on the knowledge
level, and is capable of representing several of the hitherto existing logic-
specific forgetting approaches, such as Boole’s atom forgetting in
propositional logic (?). The second approach was that of Beierle et al. (?).
In contrast to Delgrande’s approach, Beierle et al. define several cognitively
different kinds of forgetting in a general OCF framework, which is generally
more expressive than just arguing about knowledge sets. Thereby, we concretely
focussed on the marginalisation or the concept of focussing as one kind of
forgetting, respectively, which is of importance when it comes to efficient
and focussed query answering. We showed that the marginalisation generalizes
Delgrande’s forgetting definition to epistemic states by resulting in
equivalent posterior beliefs, as well as holding the same properties, which
Delgrande referred to as _right_ and _desirable_. Furthermore, this implies
that the relations Delgrande elaborated between his and the logic-specific
approaches also hold for the marginalisation. We exemplarily showed this by
means of Boole’s atom forgetting in propositional logic. We think that
(DFP-1)-(DFP-7), or (DFPes-1)Σ-(DFPes-6)Σ respectively, describe properties
that are _right_ and _desirable_ as long as we consider the forgetting of
signature elements. However, we showed that these properties are not suitable
for postulating properties for any kind of forgetting formulas, since
generalizing these properties to formulas (DFPes-1)L-(DFPes-6)L implies the
triviality result stated in Th. 7.

In principle, we agree with Delgrande insofar that belief change operators
like contraction are essentially different from the notion of forgetting as it
is implemented by Delgrande’s approach. However, we argue that Delgrande’s
approach and in general, approaches based on variable elimination, are too
narrow to cover cognitive forgetting in its full generality. As our triviality
result shows, Delgrande’s postulates seem to be unsuitable for describing the
forgetting of formulas. Nevertheless, as the works of Beierle et al. (?) show,
very different kinds of forgetting are realizable in a common framework,
distinguishable by different properties. So, as part of our future work, we
pursue the research question which of Delgrande’s postulates (which all seem
very rational at first sight) need to be modified or omitted to make the idea
of forgetting by variable elimination reconcilable to other forms of
forgetting and how Delgrande’s forgetting definition itself could be amended
to satisfy the adapted postulates.

## References

  * 2005 Baral, C., and Zhang, Y.  2005\.  Knowledge updates: Semantics and complexity issues.  Artificial Intelligence 164(1-2):209–243.
  * 2019 Beierle, C.; Kern-Isberner, G.; Sauerwald, K.; Bock, T.; and Ragni, M.  2019\.  Towards a general framework for kinds of forgetting in common-sense belief management.  KI-Künstliche Intelligenz 33(1):57–68.
  * 1854 Boole, G.  1854\.  An investigation of the laws of thought: on which are founded the mathematical theories of logic and probabilities.  Dover Publications.
  * 2017 Delgrande, J. P.  2017\.  A knowledge level account of forgetting.  Journal of Artificial Intelligence Research 60:1165–1213.
  * 2019 Ellwart, T.; Ulfert, A.-S.; Antoni, C. H.; Becker, J.; Frings, C.; Göbel, K.; Hertel, G.; Kluge, A.; Meeßen, S. M.; Niessen, C.; et al.  2019\.  Intentional forgetting in socio-digital work systems: system characteristics and user-related psychological consequences on emotion, cognition, and behavior.  AIS Transactions on Enterprise Systems 4(1).
  * 2019 Kluge, A.; Schüffler, A. S.; Thim, C.; Haase, J.; and Gronau, N.  2019\.  Investigating unlearning and forgetting in organizations.  The Learning Organization.
  * 1994 Lin, F., and Reiter, R.  1994\.  Forget it.  In Working Notes of AAAI Fall Symposium on Relevance, 154–159.
  * 1988 Spohn, W.  1988\.  Ordinal conditional functions: A dynamic theory of epistemic states.  In Causation in decision, belief change, and statistics. Springer.  105–134.
  * 2009 Wong, K.-S.  2009\.  Forgetting in logic programs.  Ph.D. Dissertation, Ph. D. thesis, The University of New South Wales.
  * 2006 Zhang, Y., and Foo, N. Y.  2006\.  Solving logic program conflict through strong and weak forgettings.  Artificial Intelligence 170(8-9):739–778.