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[Kluge et al.2019], socio-digital systems [Ellwart et al.2019], 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 [Boole1854], fact forgetting in first-order logic [Lin and Reiter1994] and forgetting in modal logic [Baral and Zhang2005]. However, none of these specific approaches argued about the general notions of forgetting, but rather provided a way to compute its result. In [Delgrande2017], 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. beierle2019towards 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 [Beierle et al.2019], 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 we state a propositional language over the finite signature with formulas . The corresponding interpretations are denoted as . The interpretations that satisfy a formula , i.e. , are called models of and are denoted as . If the signature of a model set is unambiguously given by the context, we also write instead. The explicit declaration of the corresponding signature is of particular importance when arguing about different (sub-)signatures. Moreover, each model can also be considered as a conjunction of literals corresponding to the truth values assigns to each signature element . Thus, we can also write , where , but 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 , we denote those signature elements as for which the concrete truth assignment is not needed, e.g. with . For two formulas , we say that infers , denoted as , if and only if . In case that both model sets are equal, and are equivalent, i.e. , iff and . Furthermore, the deductively closed set of all formulas that can be inferred from a formula is given by . Again, the signature in the index of the operator as well as can be omitted when its clearly given by the context. Notice that a formula is always equivalent to its deductive closure, since their models are equal. The deductive closure of a formula can also be expressed by means of the theory of its models . All of the above-mentioned formal basics also hold for sets of formulas .
In order to argue about inferences and models in different (sub-)signatures, further basic terms are needed. For two interpretations , we say that and are elementary equivalent with the exception of the signature elements , denoted as , if and only if they agree on the truth values they assign to all signature elements in [Delgrande2017]. 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.
[Delgrande2017] Let be signatures and , formulas. The reduction to of models is defined as
The expansion to of models is defined as
where . Thereby, denotes that is more specific than w.r.t. , which holds if and only if .
Notice that multiple subsequently performed reductions can be reduced to a single reduction , if the signature is a subset of .
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 spohn1988ordinal. An OCF is a ranking function that assigns a rank to each interpretation with . The rank of an interpretation can be understood as a degree of plausibility, where means that is most plausible. The most plausible interpretations according to an OCF are also called models of , and are therefore denoted by . The rank of formula is given by the minimal rank of its models, where . The beliefs of an OCF is the deductively closed set of formulas that are satisfied by the OCF’s models . Instead of , we also write .
3 Delgrande’s Forgetting and Marginalisation
In this section, we will first introduce Delgrande’s general forgetting approach [Delgrande2017] as well as some of its most important properties. Afterwards, we consider the OCF marginalisation as a kind of forgetting [Beierle et al.2019] and show that it generalizes Delgrande’s definition to epistemic states.
3.1 Delgrande’s General Forgetting Approach
In [Delgrande2017], 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 [Boole1854], first-order logic [Lin and Reiter1994], or answer set programming [Wong2009, Zhang and Foo2006]. 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 of a set of formulas as seen in Def. 2
[Delgrande2017] Let and be signatures, a language with corresponding consequence operator , and a sub-language, then forgetting a signature in a set of formulas is defined as
By intersecting the prior knowledge with the sub-language all formulas that mention any signature element will be removed. Therefore, forgetting according to Def. 2 results in those consequences of that are included in the reduced language . 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.
[Delgrande2017] Let and be signatures and a language with corresponding consequence operator , then forgetting a signature in the original language in a set of formulas is defined as
Thereby, forgetting in the original language is defined as the deductive closure of with respect to . 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 , we know that they do not provide any information about , since forgetting in the original signature results in knowledge equivalent the result of forgetting in the reduced language, due to the deductive closure . This also follows from the model theoretical properties of both forgetting definitions stated in Th. 1.
[Delgrande2017] Let be a set of formulas and P a signature, then the following equations hold:
Let be a set of formulas and a signature, then the following holds:
In Ex. 1 below, we illustrate the relations of both forgetting definitions stated by Delgrande.
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 with , where the signature elements can be read as:
Thus, 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 . Forgetting in the reduced language results in
where . Concretely, consists of all conclusions that can be drawn from and are part of the reduced language , i.e. those conclusions that do not argue about penguins (). According to Th. 1, we know that the models after forgetting from correspond to the prior models reduced to :
Thus, the posterior models after forgetting are obtain by mapping each interpretation to .
If we forget in the original language instead, we obtain
By means of the deductive closure of with respect to , the result of forgetting in the reduced language is extended by those formulas in the original language that can be inferred by it. However, due to the relations of the prior models and those after forgetting in the reduced and the original language
we see that can only contain trivial proposition about penguins (), since we know that if , then 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 . We provide an overview of the different models in Tab. 1.
|, ,||, ,||, , ,|
Besides defining a general forgetting approach, Delgrande also states several properties of his definition, which he refers to as right and desirable [Delgrande2017]. In this work, we refer to these properties as (DFP-1)-(DFP-7) as stated in Th. 2.
[Delgrande2017] Let be a language over signature and the corresponding consequence operator, then the following relations hold for all sets of formulas and signatures .
If , then
If , then
(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 of prior knowledge sets is preserved after forgetting a signature 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 and consecutively always equals the forgetting of , if is included in . 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 [Delgrande2017].
A general framework of forgetting and its instantiation to an approach using OCFs is developed in [Beierle et al.2019]. 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:
Focussing on relevant aspects retains our beliefs about them.
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.
[Beierle et al.2019] Let be an OCF over signature and an interpretation with . is called a marginalisation of to with
By marginalising an OCF to a subsignature , we consider interpretations over 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 are retained.
Let be an OCF over and , then for each the following holds:
Similarly to Delgrande’s forgetting, marginalisation reduces beliefs to a subsignature. Note that Lem. 1 directly follows from [Beierle et al.2019], 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).
Let be an OCF over signature and a reduced signature.
Thereby, Prop. 1 also corresponds to the second notion of focussing, due to the intersection with reduced language . 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 (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.
Let be an OCF over signature and a subsignature. Then holds.
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.
Let be an OCF over signature . A lifting of to , denoted by , is uniquely defined by for all .
By means of lifting an OCF over signature to a signature with , we (re-)introduce new signature elements to in a way that acts invariantly towards them. This is guaranteed by the fact that all interpretations that only differ in the truth value they assign to the new signature elements 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.
Let be an OCF over signature . Then the models of the lifted are the expanded models of , i.e., .
Proof of Prop. 3.
due to (Def. 1). Since we know that if there is an interpretation that is equivalent to , then is included in as well, and vice-versa, this last set is the same as
again by definition. ∎
Therefore, we also know that the beliefs after lifting are equivalent to the prior with respect to , which can also be denoted as the deductive closure of the prior beliefs with respect to (Prop. 4).
Let be an OCF over signature and be a lifting of to , then the beliefs of are given by .
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).
Let be a set of formulas and an OCF over signature with , then
holds for each signature .
The equivalence of the prior knowledge for both approaches can be stated as , which means that the set of formulas Delgrande’s forgetting is applied to must be equivalent to the prior beliefs . 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 over the signature , 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).
In this example, we illustrate a marginalisation and a consecutively performed lifting of the OCF over (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 .
First of all, we want to note that the beliefs of are equivalent to the knowledge base (Ex. 1), since their corresponding models are the same:
Marginalising to results in as given in Tab. 2. There it can be seen that the posterior most plausible interpretation correspond to those of when omitting , i.e. each interpretation is mapped to . This exactly corresponds to the way Delgrande’s forgetting in the reduced language affects the models of the given knowledge base :
In conclusion, we that know the posterior beliefs of the marginalisation and the result of Delgrande’s forgetting must be equal:
When we lift the marginalised OCF back to the original signature , the posterior most plausible interpretations can be obtained by mapping each interpretation to (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 :
Therefore, the result of Delgrande’s forgetting in the original language is equal to the beliefs after marginalising and lifting :
|0||, ,||, ,||, , ,|
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 (Th. 4).
[Boole1854] Let be a formula and be an atom. Forgetting in is then defined as
where denotes the substitution of by , and the substitution by .
[Delgrande2017] Let be the language in propositional logic with signature and let be an atom.
Let be an OCF over signature and a formula with , then
holds for each atom .
4 Postulates for Forgetting Signatures in Epistemic States
In [Delgrande2017], 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)-(DFPes-6), and show that the marginalisation satisfies all of them. For this, let be epistemic states, signatures, and an arbitrary operator that maps an epistemic state together with a signature to a new epistemic state:
If , then
If , then
For a detailed explanation of the above-stated postulates (DFPes-1)-(DFPes-6), 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)-(DFPes-6). Furthermore, we expressed the forgetting in the original signature in (DFP-7) as the beliefs after forgetting and lifting the posterior epistemic state back to the original signature. The models of are equal to the models of forgetting in in the reduced signature lifted back to the original signature, i.e. (Cor. 1). When we consider the models of , i.e. , we see that this also describes the models after forgetting lifted back to the original signature. Therefore, (DFPes-6) exactly matches the property originally stated by (DFP-7). In the following, we refer to those operators satisfying (DFPes-1)-(DFPes-6) as signature forgetting operators.
Next, we show in Th. 5 that the marginalisation satisfies (DFPes-1)-(DFPes-6), 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).
Let be an OCF over signature and a signature. The marginalisation to a subsignature satisfies (DFPes-1)-(DFPes-6).
Proof of Th. 5.
In the following, we assume the epistemic states and to be OCFs, since the marginalisation is specifically defined over OCFs, denoted as and , and further denote the marginalisation as .
Regarding (DFPes-3), we have the following equalities due to Prop. 2, and because of :