Properties of Knowledge Forgetting

Transcription

1 Properties of Knowledge Forgetting Yan Zhang and Yi Zhou Intelligent Systems Laboratory University of Western Sydney, Australia Abstract In this paper we propose a formal theory of knowledge forgetting based on the single agent S5 modal logic. We first present a model theoretic definition of knowledge forgetting and study its essential semantic properties. We show that knowledge forgetting is a generalization of variable forgetting in propositional logic and can be precisely characterized by four forgetting postulates. We then investigate the computational properties of knowledge forgetting. We observe that each propositional S5 formula can be transformed into a kind of disjunctive normal form - this result leads us to develop an algorithm for computing the syntactic representation of knowledge forgetting. By studying the major decision problems, we prove that the complexity of model checking problem for knowledge forgetting is NP complete, and the complexity of its related inference problems varies from co- NP complete to Π P 2 complete. Key words: belief revision and update, knowledge update, computational aspects of knowledge representation Introduction Epistemic reasoning concerns the problem of how to reason about agents epistemic states (knowledge) in a dynamic environment. In the last decade, it has been demonstrated that epistemic reasoning has many important applications in computer science and AI fields (Meyer & van der Hoek 1995). Amongst various theories and approaches, one major assumption in the study of epistemic reasoning is that agents always remember their previous knowledge (i.e. agents have perfect recall). However, as pointed by Fagin et al.: There are often scenarios of interest where we want to model the fact that certain information is discarded. In practice, for example, an agent may simply not have enough memory capacity to remember everything he has learned. (page 129 in (Fagin et al. 1995)). Hence knowledge forgetting is an important behaviour for an agent under certain circumstances. An earlier formal study on the concept of knowledge forgetting was due to Baral and Zhang s work on knowledge update ((Baral & Zhang 2005)), where they treated knowledge forgetting as a special form of update with the effect Kφ K φ: after knowledge forgetting φ (φ is a propositional formula), the agent would neither know φ nor φ. Copyright c 2008, Association for the Advancement of Artificial Intelligence ( All rights reserved. Since Baral and Zhang s knowledge forgetting is specified from a knowledge update viewpoint, it does not seem to obey the general intuition of forgetting in classical propositional theories (Lin & Reiter 1994), and their semantics of forgetting update does not always represent an intuitive meaning. As a logical notion, forgetting was first formally defined in propostional and first order logics by Lin and Reiter (Lin & Reiter 1994). Over the years, researchers have demonstrated many useful applications of the propositional forgetting theory in abductive reasoning, belief revision/update, and reasoning about knowledge (Lang, Liberator, & Marquis 2003; Lin 2001; Shu, Lv, & Zhang 2004). In recent years, various theories of forgetting have also been proposed under the answer set programming semantics and used in solving logic program conflicts (Eiter & Wang 2008 to appear; Zhang & Foo 2006). From these works, we can see that forgetting is an important and useful concept in knowledge representation and reasoning. Nevertheless, it is not difficult to see that existing forgetting definitions in propositional logic and answer set programming are not directly applicable in modal logics. For instance, in propositional forgetting theory, forgetting atom q from T (p q) ((q r) s) is equivalent to a formula T [q/ ] T [q/ ], where T [q/ ] is a formula obtained from T by replacing each q with and T [q/ ] is obtained from T by replacing each q with, which is (r s) p. However, this method cannot be extended to a S5 modal logic formula. Consider an S5 formula T Kp Kq K q. If we want to forget atom q from T by using the above method, we would have T [q/ ] T [q/ ]. This is obviously not correct because after forgetting q, the agent s knowledge set should not become inconsistent! From the above discussions, in order to have a formal theory of of forgetting in propositional single agent S5 modal logic, some new approach must be developed. In this paper, we present a semantic definition of knowledge forgetting and propose four forgetting postulates that we argue that the underlying knowledge forgetting should obey. We prove a representaiton theorem showing that our knowledge forgetting is precisely characterized by these four postulates. We also study computational properties of knowledge forgetting in details. We first develop an algorithm to compute the syntactic representation of knowledge forgetting from a

2 given knowledge set, and then investigate the complexity of decision problems in relation to knowledge forgetting. In particular, we show that the complexity of model checking problem for knowledge forgetting is NP complete, and the complexity of its related inference problem varies from co- NP complete to Π P 2 complete. The rest of the paper is organized as follows. Section 2 presents a model based semantic definition of knowledge forgetting, while section 3 proves an important representation theorem for knowledge forgetting and studies other related semantic properties. Section 4 then addresses major computational properties of knowledge forgetting. Finally section 5 concludes this paper with some remarks. Defining Knowledge Forgetting Our knowledge forgetting will be defined on a basis of the finite proportional modal logic S5. Let Atom be a finite set of atoms (also called variables). The finite language L of propositional S5 modal logic is defined recursively by Atom, classical connectives,, and a modal operator K as follows: φ ::= p φ φ ψ Kφ, where p Atom., φ ψ and φ ψ, are defined as the standard way. Elements in L are called formulas. Formulas without modal operators are called objective formulas. A knowledge set is a finite set of formulas. Literals are atoms and their negations. Let φ be a formula and Γ a knowledge set, we write V ar(φ) and V ar(γ) to denote the set of atoms occurred in φ and Γ respectively. For convenience, we usually use a, b, c,, p, q, to denote atoms; φ, ψ, υ, to denote formulas; and Γ, T, to denote knowledge sets. Sometimes, we also use Γ = φ 1 φ n to represent a finite set of formulas {φ 1,, φ n }. A Kripke structure is a triple S = W, R, L, where W is a set of possible worlds, R an equivalence relation on W, and L a set of interpretations for each world in W. As illustrated in (Meyer & van der Hoek 1995), an S5 Kripke interpretation for single agent may be simplified as M = W, w, where W is the set of all possible worlds, each world is identified as a set of atoms, and w W is called the actual world. In this case, we call M = W, w a k-interpretation. The satisfaction relation = between k-interpretations and formulas in L is defined recursively as follows 1 : (1) W, w = ; (2) W, w = p iff p w, where p Atom; (3) W, w = φ iff W, w = φ; (4) W, w = φ ψ iff W, w = φ or W, w = ψ; (5) W, w = Kφ iff w W, W, w = φ. We say that M is a k-model of φ iff M = φ. We write Mod(φ) (or Mod(Γ) if Γ is a finite set of formulas) to denote the set of all k-models of φ (or Γ resp.). We say that two S5 formulas (knowledge sets) φ and ψ are equivalent, denoted by φ ψ, iff M od(φ) = M od(ψ). Let Γ and Γ be two knowledge sets. We write Γ = Γ iff Mod(Γ) Mod(Γ ). 1 We write W, w = F if it is not the case that W, w = F. To present a formal definition of knowledge forgetting, we first introduce a useful notion. Let w and w be two worlds identified as two sets of atoms respectively, and V Atom a set of atoms. We say that w and w are identical with exception on V, denoted by w V w, if for all p Atom \ V, p w iff p w. Let M = W, w and M = W, w be two k-interpretations, and V Atom a set of atoms, we say that M and M are bisimilar with exception on V, denoted by M V M, iff there exists a binary relation σ W W such that: 1. σ(w, w ); 2. w W, w W such that σ(w, w ) (the forth condition); and 3. w W, w W such that σ(w, w ) (the back condition). 4. if σ(w, w ) then w V w. Proposition 1 The relation V is an equivalence relation. Proof: It is easy to see that V satisfies reflexivity and symmetry. We now show that it satisfies transitivity as well. Suppose that M 0, M 1, M 2 are three Kripke interpretations such that M 0 V M 1 via the binary relation σ 1 and M 1 V M 2 via the binary relation σ 2. Construct a binary relation σ 3 W 0 W 2 : for any two interpretations w 0 W 0, w 2 W 2, σ 3 (w 0, w 2 ) iff there exists w 1 W 1 such that σ 1 (w 0, w 1 ) and σ 2 (w 1, w 2 ). It is easy to check that M 0 M 2 via σ 3. Now we define knowledge forgetting as follows. Definition 1 (Knowledge forgetting) Let Γ be a knowledge set and p Atom an atom. A knowledge set, denoted as KForget(Γ, p), is the result of knowledge forgetting p from Γ, if the following condition holds: M od(kforget(γ, p)) = M. M Mod(Γ),M {p} M Similarly to model based update formulations (Baral & Zhang 2005; Winslett 1988), KForget(Γ, p) denotes the result of knowledge forgetting atom p from Γ, where the set of all k-models of KForget(Γ, p) is specified as in the definition. Since we restrict to a finite propositional S5 modal logic, such knowledge set KForget(Γ, p) always exists. In section 5, we also provide an algorithm to compute a concrete S5 formula φ such that φ KForget(Γ, p). Example 1 Consider knowledge sets K(p q), Kp Kq and K(p q). From Definition 1, it is easy to check that KForget(K(p q), p), KForget(Kp Kq, p), and KForget(K(p q), p) Kq. We may extend Definition 1 to the case of knowledge forgetting an arbitrary set V of atoms from Γ. Definition 2 Let Γ be a knowledge set and V Atom a set of atoms. A knowledge set, denoted as KForget(Γ, V ), is the result of knowledge forgetting V from Γ, if the following condition holds: Mod(KForget(Γ, V )) = M. M Mod(Γ),M V M

3 The following proposition ensures that Definitions 1 and 2 are semantically coherent. Proposition 2 Let Γ be a knowledge set and V a set of atoms and p an atom such that p V. Then KForget(Γ, {p} V ) KForget(KForget(Γ, p), V ). Proof: Let M 1 = W 1, s 1 be a k-model of KForget(Γ, {p} V ). By the definition, there exists a k- model M = W, s of Γ, such that M {p} V M 1 via a binary relation σ. We construct a k-interpretation M 2 = W 2, s 2 as follows: (1) for all pairs w W and w 1 W 1 such that σ(w, w 1 ), let w 2 W 2 and (a) p w 2 iff p w 1, (b) for all atoms q V, q w 2 iff q w, (c) for all other atoms q, q w 2 iff q w 1 iff q w; (2) delete duplicated k-interpretations in W 2 ; (3) let s 2 be the world such that (a) p s 2 iff p s 1, (b) for all atoms q V, q s 2 iff q s, (c) for all other atoms q, q s 2 iff q s 1 iff q s. Then we have the following results: M {p} M 2. Let σ 1 W W 2 be a binary relation such that σ(w, w 2 ) iff w 2 is constructed based on w (see the construction of M 2 above). It is easy to see that M {p} M 2 via σ 1. M 2 V M 1. Let σ 2 W 2 W 1 be a binary relation such that σ(w 2, w 1 ) iff w 2 is specified based on w 1 (see the construction of M 2 above). It is observed that M 2 V M 1 via σ 2. Thus, M 2 is a k-model of KForget(Γ, p). This follows that M 1 is a k-model of KForget(KForget(Γ, p), V ). On the other hand, suppose that M 1 is a k-model of KForget(KForget(Γ, p), V ), then there exists M 2 such that M 2 is a k-model of KForget(Γ, p) and M 2 V M 1, and there exists M such that M is a k-model of Γ and M {p} M 2. Therefore, M {p} V M 1, and consequently, M 1 is also a k-model of KForget(Γ, {p} V ). Semantic characterizations In this section we study essential semantic properties of knowledge forgetting. We will first propose a set of postulates and show that these postulates precisely characterize the semantics of knowledge forgetting. We then discuss other desired properties that our knowledge forgetting satisfies. A representation theorem Consider a formula φ. Intuitively, if a propositional atom (variable) a does not occur in V ar(φ), we may consider that φ is irrelevant to variable a. It is not surprising that the notion of irrelevance plays an important role to characterize the semantics of knowledge forgetting. We first give the following formal definition. Definition 3 (Irrelevance) Let Γ be a knowledge set and V a set of atoms. We say that Γ is irrelevant to V, denoted by IR(Γ, V ), if there exists a knowledge set Γ such that Γ Γ and V ar(γ ) V =. Let Γ and Γ be two knowledge sets, V a set of atoms. Now we propose the following postulates: (W) Weakening: Γ = Γ. (PP) Positive Persistence: if IR(φ, V ) and Γ = φ, then Γ = φ. (NP) Negative Persistence: if IR(φ, V ) and Γ = φ, then Γ = φ. (IR) Irrelevance: IR(Γ, V ). By specifying Γ KForget(Γ, V ), we call (W), (PP), (NP) and (IR) are four postulates for knowledge forgetting. Let us take a closer look at these postulates. (W) seems an essential requirement for knowledge forgetting: after forgetting some knowledge from a knowledge set, the resulting knowledge set then becomes weaker. Indeed, as demonstrated in propositional variable forgetting (Lin & Reiter 1994; Lin 2001), forgetting weakens the original formula. The postulates of positive persistence (PP) and negative persistence (NP) simply state that knowledge forgetting a set of atoms should not affect those positive or negative information respectively that is irrelevant to this set of atoms. Also note that that (PP) and (NP) can be combined as: if IR(φ, V ), then Γ = φ iff Γ = φ. Finally, irrelevance (IR) means that after knowledge forgetting, the resulting knowledge set should be irrelevant to those atoms which we have (knowledge) forgotten. We argue that these postulates precisely capture the basic properties that knowledge forgetting should satisfy. Lemma 1 Let V be a set of atoms, φ a formula such that V ar(φ) V = and M a model of φ. Then for any M such that M V M, M is also a model of φ. Proof: We prove this assertion by induction on the structure of φ. If φ is, this assertion holds obviously. If φ is an atom p, then M = W, w = p iff p w iff p w, where M = W, w and M V M via the binary relation σ iff M = p. If φ is ψ, then M = φ iff M = ψ iff M = ψ iff M = φ. If φ is φ 1 φ 2, then M = φ iff M = φ 1 or M = φ 2 iff M = φ 1 or M = φ 2 iff M = φ. If φ is Kψ, then M = φ iff for all w W, W, w = ψ. Suppose that M V M via the binary relation σ, given w W, there exists w W such that σ(w, w ) and w V w. Thus, W, w V W, w. By induction hypothesis, W, w = ψ. Thus, M = φ. This completes the induction proof. Lemma 2 Let V be a set of atoms, φ a formula such that IR(φ, V ) and M a k-model of φ. Then for any M such that M V M, M is also a k-model of φ. Proof: This assertion follows directly from Lemma 1 Now we have the following representation theorem which states that our forgetting postulates precisely characterize our knowledge forgetting semantics.

4 Theorem 1 (Representation theorem) Let Γ and Γ be two knowledge sets and V Atom a set of atoms. Then the following statements are equivalent: 1. Γ KForget(Γ, V ); 2. Γ {φ Γ = φ, IR(φ, V )}; 3. Postulates (W), (PP), (NP) and (IR) hold. Theorem 1 is significant in the sense that it provides an if and only if characterization on knowledge forgetting. That is, given a knowledge set Γ and a set of atoms V, an S5 formula Γ represents a result of knowledge forgetting V from Γ if Γ satisfies postulates (W), (PP), (NP) and (IR), and vice versa. We have observed that no such theorem has been proved in the previous work regarding propositional forgetting (Lang, Liberator, & Marquis 2003; Lin & Reiter 1994) and logic programming forgetting (Eiter & Wang 2008 to appear; Zhang & Foo 2006). In order to prove Theorem 1, we need to first define the notion of characteristic formula that plays a central role in the proof of our representation theorem for knowledge forgetting. Intuitively, given a propositional interpretation π, π s characteristic formula C(π, V ) on a set V of atoms is a propositional formula which represents π by a restricted sources of atoms V. Formally, let π be an interpretation and V a set of atoms, the characteristic formula of π on V, denoted by C(π, V ), is defined as: a b. a π,a V b π,b V It is clear that π = C(π, V ). Now we consider a k- interpretation M = W, w and a set V of atoms. Then the characteristic formula of M on V, denoted by C(M, V ), is defined as: C(w, V ) w W K C(w, V ) { w 2 Atom w W, w W s.t.w Atom\V w } K C(w, V ). It is not difficult to check that M = C(M, V ). Moreover, C(M, V ) is irrelevant to Atom\V, i.e. IR(C(M, V ), Atom \ V ). Proposition 3 Let M and M be two k-interpretations and V a set of atoms. M = C(M, V ) iff M Atom\V M. Proof: Firstly, given two propositional interpretations π and π and a set V of atoms, it is clear that π = C(π, V ) iff π Atom\V π. Let M = W, s and M = W, s. Now suppose that M = C(M, Atom\V ). Then we have: M = C(s, V ), and hence s = C(s, V ). So s Atom\V s. For all w W, M = K C(w, V ). That is, there exists some w W such that w = C(w, V ). This follows w Atom\V w. For all w W such that there does not exist w 1 W and w Atom\V w 1, M = K C(w, V ). Thus, there does not exist w W such that w = C(w, V ). That means that there does not exist w such that w Atom\V w. Now we construct a binary relation σ W W such that σ(w, w ) iff w Atom\V w. Then we can see that M Atom\V M via the binary relation σ because: 1. σ(s, s ). 2. For all w W, there exists w W such that w Atom\V w. 3. For all w W, there exists w W such that w Atom\V w. Otherwise, C(M, V ) = K C(w, V ). Thus, M = M(C(w, V )), a contradiction. 4. Finally, by the definition, for all pairs w, w such that σ(w, w ), w Atom\V w. On the other hand, we suppose that M Atom\V M via the given binary relation σ above, then: We have that s Atom\V s. Thus s = C(s, V ). Thus M = C(s, V ). For each w W, there exists w W such that σ(w, w ) and w Atom\V w. Thus, w = C(w, V ). Thus M = K C(w, V ). For each w W such that there does not exist w 1 W and w Atom\V w 1. We have that M = K C(w, V ). Otherwise, M = K C(w, V ). There exists w W such that w = C(w, V ). Since M Atom\V M, there exits w 1 W such that σ(w 1, w ), which means that w 1 Atom\V w. Thus w 1 = C(w, V ), w 1 Atom\V w, a contradiction. This shows that M = C(M, V ). Corollary 1 Let M be a k-interpretation and V a set of atoms. M = C(M, V ). Proof: This assertion follows directly from Proposition 3 since M Atom\V M. Proposition 4 Let Γ be a knowledge base. Then Γ C(M, Atom). M Mod(Γ) Proof: Let M be a k-model of Γ. By Corollary 1, M = C(M, Atom). Thus, M is also a k-model of M Mod(Γ) C(M, Atom). On the other hand, suppose that M is a k-model of M Mod(Γ) C(M, Atom). Then there is a M Mod(Γ) such that M = C(M, Atom). By Proposition 3, M M. Therefore, M is also a k-model of Γ. Proof of Theorem 1 To prove 1 2, We will show that Mod(KForget(Γ, V )) = Mod({φ Γ = φ, IR(φ, V )}) = Mod( M =Γ C(M, Atom\V )). Firstly, suppose that M is also a model of KForget(Γ, V ). By Definition 2, there exists a k-interpretation M such that M is a k-mode of Γ and M V M. By Lemma 2, for all formula φ such that Γ = φ and IR(φ, V ), M = φ. Thus, M is a k-model of {φ Γ = φ, IR(φ, V )}. Secondly, suppose that M is a k-model of {φ Γ = φ, IR(φ, V )}. Thus,

5 M = M Mod(Γ) C(M, Atom\V ), since M Mod(Γ) C(M, Atom\V ) is irrelevant to V. Finally, suppose that M is a k-model of M Mod(Γ) C(M, Atom\V ). Then there exists M Mod(Γ) such that M = C(M, Atom\V ). By Proposition 3, M V M. Thus M is also a k-model of KForget(Γ, V ). Now we show 3 2. Suppose that all postulates hold. By Positive Persistence, Γ = {φ Γ = φ, IR(φ, V )}. Now we show that {φ Γ = φ, IR(φ, V )} = Γ. Otherwise, there exists formula ψ such that Γ = ψ but {φ Γ = φ, IR(φ, V )} = ψ. There are three cases: 1. ψ is relevant to V. Thus, Γ is also relevant to V, a contradiction to Irrelevance. 2. ψ is irrelevant to V and Γ = ψ. This contradicts to our assumption. 3. ψ is irrelevant to V and Γ = ψ. By Negative Persistence, Γ = ψ, a contradiction. Thus, Γ is equivalent to {φ Γ = φ, IR(φ, V )}. Finally 2 3 is quite obvious. Corollary 2 Let Γ be a knowledge base, φ a formula and V a set of atoms. KForget(Γ, V ) = φ iff IR(φ, V ) and Γ = φ. Other semantic properties As we have mentioned in Introduction, the notion of forgetting has been defined and used in a variety of contexts under propositional logic (Lang, Liberator, & Marquis 2003; Lin & Reiter 1994). It is important to know the relationship between variable forgetting in propositional logic and knowledge forgetting in S5 propositional modal logic. Let φ be an objective formula. We use φ[p/ ] and φ[p/ ] to denote the formulas obtained from φ by replacing atom p with and respectively. Then formula Forget(φ, p) is obtained from φ by forgetting p from φ, if Forget(φ, p) φ[p/ ] φ[p/ ]. By forgetting a set of atoms V in φ, we recursively define Forget(φ, V {p}) = Forget(Forget(φ, p), V ), where Forget(φ, ) =. We first have the following result. Theorem 2 Let φ be an objective formula and V a set of atoms. Then we have KForget(φ, V ) Forget(φ, V ) and KForget(Kφ, V ) K(Forget(φ, V )). Proof: We prove Result 1 as follows. Suppose that M = W, w is a model of KF orget(φ, V ). Then there exists M = W, w Mod(φ) such that M V M. Thus, w V w. Since w = φ, w = F orget(φ, V ). Hence, M is also a model of F orget(φ, V ). On the other hand, suppose that M = W, w is a model of F orget(φ, V ). Then there exists w = φ and w V w. Construct a k-interpretation M = W, w such that W = W {w }\{w}. It s clear that M is a model of φ and M V M. Hence, M is also a model of KF orget(φ, V ). Now consider Result 2. Suppose that M = W, w is a model of KF orget(kφ, V ). Then there exists M = W, w Mod(Kφ) such that M V M. We have that for all w 1 W, w 1 = φ. Since M V M, for all w 1 W, there exists w 1 W such that w 1 V w 1. Therefore w 1 = F orget(φ, V ). This shows that M = K(F orget(φ, V )). On the other hand, suppose that M = W, w is a model of K(F orget(φ, V )). Then for all w 1 W, w 1 = F orget(φ, V ), and there exists w 1 = φ and w 1 V w. Construct a k-interpretation M = W, w such that W is the set of all w 1 mentioned above and w V w. It s clear that M is a model of K(φ) and M V M. Hence, M is also a model of KF orget(φ, V ). Theorem 2 simply reveals that propositional variable forgetting is a special case of knowledge forgetting, and also knowledge forgetting of formulas with the form Kφ (where φ is objective) can be achieved through the corresponding propositional variable forgetting. However, the syntactic definition of variable forgetting Forget(φ, p) φ[p/ ] φ[p/ ] cannot be extended to knowledge forgetting. Consider that we wan to knowledge forget atom p from formula K(p q). According to Definition 1, we have KForget(K(p q), p), while K(p q)[p/ ] K(p q)[p/ ] Kq K q. The following results further illustrate other essential semantic properties of knowledge forgetting. Theorem 3 Let Γ, Γ 1 and Γ 2 be three knowledge sets, φ 1 and φ 2 two formulas, and V a set of atoms. Then the following results hold: 1. KForget(Γ, V ) is satisfiable iff Γ is satisfiable; 2. If Γ 1 Γ 2, then KForget(Γ 1, V ) KForget(Γ 2, V ); 3. if Γ 1 = Γ 2, then KForget(Γ 1, V ) = KForget(Γ 2, V ); 4. KForget(φ 1 φ 2, V ) KForget(φ 1, V ) KForget(φ 2, V ); 5. KForget(φ 1 φ 2, V ) = KForget(φ 1, V ) KForget(φ 2, V ). Proof: To prove Result 1, suppose that M is a k-model of Γ. Then M is also a model of KForget(Γ, V ). This shows that KForget(Γ, V ) is satisfiable. On the other hand, suppose that Γ is unsatisfiable. Then, Mod(Γ) =. It follows that Mod(KForget(Γ, V )) =. Result 2 directly follows from Definition 1 and the fact Mod(Γ 1 ) = Mod(Γ 2 ). Now we prove Result 3. Suppose that M is a k-model of KForget(Γ 1, V ), then there exists a k-model M of Γ 1, V such that M V M. Since Γ 1 = Γ 2, M is also a k-model of Γ 2. Hence, M is a k-model of KForget(Γ 2, V ) as well. To prove Result 4, we need to show Mod(KForget(φ 1 φ 2, V )) = Mod(KForget (φ 1, V ) KForget(φ 2, V )). Suppose that M is a k-model of KForget(φ ψ, V ), then there exists a k-model M 0 of φ ψ such that M 0 V M. Since M 0 is a k-model of φ ψ, M 0 is a k-model of φ or ψ. Without loss of generality, suppose that M 0 is a k-model of φ. We have that M is a k-model of KForget(φ, V ). Thus, M is a k-model of KForget(φ, V ) KForget(ψ, V ). On the other hand, suppose that M is a k-model of KForget(φ, V ) KForget(ψ, V ), then M is a k-model of

6 KForget(φ, V ) or a k-model of KForget(ψ, V ). Without loss of generality, suppose that M is a k-model of KForget(φ, V ), then there exists a model M 0 of φ such that M 0 V M. M 0 is also a k-model of φ ψ. Thus, M is a k-model of KForget(φ ψ, V ). Finally we prove Result 5. Suppose that M is a k-model of KForget(φ ψ, V ), then there exists a k-model M 0 of φ ψ such that M 0 V M. Therefore, M 0 is a k-model of φ. Thus, M is also a k-model of KForget(φ, V ). Similarly, M is a k-model of KForget(ψ, V ) as well. In Theorem 3, we should note that the converse of Result 5 in Theorem 3 does not hold generally. For instance, let φ be q p; φ be q r. Then, KForget(φ ψ, p) is equivalent to q r, while KForget(φ, V ) KForget(ψ, V ). Computational properties In this section, we will develop an algorithm to compute the syntactic representation of knowledge forgetting. We then prove the complexity results of major decision problems in relation to knowledge forgetting, and also characterize some tractable subclasses of knowledge forgetting problems. An algorithm for computing knowledge forgetting From a technical consideration, we first introduce modal operator B in our language. B is viewed as a dual modal operator of K, where Bφ can be read as the agent believes φ. Its semantics is defined as follows: W, w = Bφ iff w W, W, w = φ. It is easy to see that Kφ B φ. Then we extend the defintion of S5 formulas which may also contain modal operator B. Obviously, introducing modal operator B in the language does not affect the original semantics of finite propositional S5 modal logic. Proposition 5 Every S5 formula can be equivalently transformed into a formula without nested modal operators in polynomial time. Proof: By applying the following rules, we can transform each S5 formula into a form that only consists of propositional literals, connectives and and modal operators K and B, and such transformation can be done in polynomial time. That is, we may remove negations from the original formula. These rules are: 1. (φ ψ) φ ψ. 2. (φ ψ) φ ψ. 3. Kφ B φ. 4. Bφ K φ. Then we remove nested nested modal operators in the following way: for every formula obtained in the above step, it can be further transformed into a equivalent formula without nested modal operators according to the following transformation rules. Again, such transformation can be done in polynomial time. These rules are as follows: 1. K(φ ψ) Kφ Kψ. 2. B(φ ψ) Bφ Bψ. 3. K(φ Kψ) Kφ Kψ. 4. K(φ Bψ) Kφ Bψ. 5. B(φ Kψ) Bφ Kψ. 6. B(φ Bψ) Bφ Bψ. Proposition 6 Every S5 formula can be equivalently transformed into a disjunction of clauses of the following form: φ 0 Kφ 1 Bφ 2... Bφ n, (1) where all φ i (0 i n) are propositional formulas, and any of φ i may be absent. Proof: By Proposition 5, every formula can be transformed into an S5 formula without nested modal operators. By replacing each modal sub-formula with a new atom, we can view this formula as a propositional formula. Then this formula can be transformed into a disjunctive normal form (DNF). Notice that every clause in the DNF is actually an S5 formula, which can be transformed into the form 1 using rules illustrated in the proof of Proposition 5. Now we propose the following algorithm to compute a syntactic representation of knowledge forgetting. Algorithm 1 Input: an S5 formula φ; an atom p. Output: the S5 formula obtained from φ by knowledge forgetting p. 1. transform φ into a set of clauses of form (1); 2. for each formula of the form: ψ 0 Kψ 1 Bψ 2... Bψ n, obtained from Step 1, replace it with the following formula: Forget(ψ 0, p) K(Forget(ψ 1, p)) B(Forget(ψ 1 ψ 2, p))... B(Forget(ψ 1 ψ n ), p). 3. return the disjunction of all formulas obtained from Step 2. From Algorithm 1, we can see that knowledge forgetting can be precisely computed through propositional variable forgetting. Example 2 Let Γ Ka (Kb Kc) be a knowledge set. We consider to knowledge forget c from Γ by using Algorithm 1. First, Γ is transformed into the form of (1): K(a b) Ka B c. Then for each clause we do the transformation as showed in Step 2 in the algorithm. That is, clause K(a b) is replaced by KForget(a b, c), and clause Ka B c is replaced by KForget(a, c) BForget(a c, c), where Forget(a b, c) K(a b), and KForget(a, c) BForget(a c, c) Ka Ba Ka. So by using Algorithm 1, we have KForget(Γ, c) K(a b) Ka Ka, which is also equivalent to the result obtained using Definition 1. Theorem 4 Algorithm 1 is sound.

7 Proof: By Result (5) in Theorem 3, we only need to prove that KForget(ψ 0 Kψ 1 Bψ 2... Bψ n, p) Forget(ψ 0, p) K(Forget(ψ 1, p)) B(Forget(ψ 1 ψ 2, p))... B(Forget(ψ 1 ψ n ), p). First suppose that M = W, s is a k-model of KForget(ψ 0 Kψ 1 Mψ 2... Mψ n, p), then there exists a k-model M 1 = W 1, s 1 of ψ 0 K(ψ 1 ) B(ψ 2 )... B(ψ n ) such that M 1 {p} M via a binary relation σ. We will show that M is also a k-model of Forget(ψ 0, p), K(Forget(ψ 1, p)), and B(Forget(ψ 1 ψ i, p)) (2 i n) respectively. We consider the following cases. Case 1. From Result (5) in Theorem 3, we know that M = KForget(ψ 0, p). Since ψ 0 is a propositional formula, from Theorem 2, we have M = Forget(ψ 0, p). Case 2. M 1 = K(ψ 1 ). That is, for every w 1 W 1, w 1 = ψ 1. On the other hand, for each w W, there exists w 1 W 1 such that σ(w 1, w), i.e. w 1 {p} w. So we have w = Forget(ψ 1, p), which follows M = K(Forget(ψ 1, p)). Case 3. For any i (2 i n), M 1 = B(ψ i ). Thus, there exists some w 1 W 1 such that w 1 = ψ i. Moreover, w 1 = ψ 1 ψ i. On the other hand, there also exists some w W such that σ(w 1, w). That is, w 1 {p} w. So we have w = Forget(ψ 1 ψ i, p), which follows M = B(Forget(ψ 1 ψ i, p)). Hence M is a k-model of Forget(ψ 0, p) K(Forget(ψ 1, p)) B(Forget(ψ 1 ψ 2, p))... B(Forget(ψ 1 ψ n ), p). Now we consider a k-model M = W, s of formula Forget(ψ 0, p) K(Forget(ψ 1, p)) B(Forget(ψ 1 ψ 2, p))... B(Forget(ψ 1 ψ n ), p). For this purpose, we construct a k-interpretation M 1 = W 1, s 1 and a binary relation σ simultaneously as follows: (a) Since M = Forget(ψ 0, p), s = Forget(ψ 0, p). That is, s = ψ 0 [p/ ] ψ 0 [p/ ]. Without loss of generality, suppose that s = ψ 0 [p/ ]. Let s 1 be the interpretation obtained from s by assigning p to. We have that s 1 = ψ 0 and s 1 {p} s. Let σ(s 1, s). (b) For each i (2 i n, M = B(Forget(ψ 1 ψ i, p)). Thus, there exists some w W such that w = Forget(ψ 1 ψ i, p). Similarly, we can construct a w i such that w i = ψ 1 ψ i and w i {p} w. Let w i W 1 and σ(w i, w). (c) For each winw such that for all i (2 ileqn) w does not satisfy Forget(ψ 1 ψ i, ψ i, p), we have that w = Forget(ψ 1, p). Similarly, we can construct a w j such that w j = ψ 1 and w j {p} w. Let w j W 1 and σ(w j, w). Then it is not difficult to verify that M 1 = ψ 0 Kψ 1 Bψ 2... Bψ n and M 1 {p} M. This shows that M is a k-model of KForget(ψ 0 Kψ 1 Bψ 2... Bψ n, p). This completes our proof. From Proposition 2 in section 2, we know that forgetting a set V of atoms from a given knowledge set Γ can be sequentially computed by forgetting a single atom of V from Γ one by one using Algorithm 1. This means that Algorithm 1 can be used for a general knowledge forgetting computation. It is not difficult to observe that the time complexity of Algorithm 1 is generally exponential in the size of input knowledge set Γ because transforming Γ into the form (1) (Step 1 in the algorithm) could be expensive. However, if the input knowledge set Γ is already in the form (1), computing knowledge forgetting of atom p from Γ can be done in polynomial time. Main complexity results In this subsection we will mainly study the complexity of three major decision problems in relation to knowledge forgetting: model checking, inference and irrelevance. First we introduce some useful notions of complexity theory. Two basic complexity classes are P and NP. The class of P includes all decision problems solvable by a polynomialtime deterministic Turing machine, while the class of NP includes all decision problems solvable by a polynomial-time nondeterministic Turing machine. Suppose C is a class of decision problem. The class P C consists of the problems solvable by a polynomial-time deterministic Turing machine with an oracle for a problem from C, and the class NP C includes the problems solvable by a nondeterministic Turing machine with an oracle for a problem in C. By co-c we mean the class that consists of the complements of the problems in C. The classes Σ P k and ΠP k of the polynomial hierarchy are defined as follows: Σ P 0 = ΠP 0 = P, and Σ P k = NPΣP k 1, Π P k =co-σ P k for all k > 1. Note that NP=Σ P 1 and co-np=π P 1. Theorem 5 (Model checking complexity) Let Γ be a knowledge set, V a set of atoms and M a k-interpretation. Deciding whether M is a k-model of KForget(Γ, V ) is NPcomplete. Proof: The problem can be determined by first guessing a k-interpretation M polynomial in the size of M and then checking if M = Γ and M V M. It is easy to see that the checking part can be done in polynomial time. Hence, the problem is in NP. For the hardness, we show that model checking for propositional variable forgetting is NP hard (considering that propositional variable forgetting is a special case of knowledge forgetting). Now we show that an objective formula φ is satisfiable iff π = Forget(φ, V ar(φ)), where π is a propositional interpretation in which all atoms occurring in φ are assigned true. Firstly, if φ is satisfiable, then Forget(φ, V ar(φ)) becomes, which certainly should be satisfied in π. Second, if π = Forget(φ, V ar(φ)), then from Result (1) in Theorem 3 we know that ψ must be satisfiable. This follows that the model checking for propositional variable forgetting is NP-hard. Consequently, the model checking for knowledge forgetting is also NP-hard.

8 Now we consider two types of decision problems related to the inference of knowledge forgetting. Given a knowledge set Γ, an S5 formula φ and a set of atoms V, we like to decide: (1) whether KForget(Γ, V ) = φ; and (2) whether φ = KForget(Γ, V ). Quite interestingly, our following result shows that these two types of inference have different complexity. Theorem 6 (Inference complexity) Let Γ be a knowledge set, φ a formula, and V a set of atoms. Then we have the results: (1) deciding whether KForget(Γ, V ) = φ is co-np complete; and (2) deciding whether φ = KForget(Γ, V ) is Π P 2 -complete. Proof: For Result (1), the hardness is easy to see by setting KForget(Γ, V ar(γ)). For membership, from Corollary 2, we have KForget(Γ, V ) = φ iff Γ = φ and IR(φ, V ). Clearly, in single agent S5 modal logic, deciding Γ = φ is in co-np. We show that deciding whether IR(φ, V ) is also in co-np. Without loss of generality, we assume that φ is satisfiable. Then φ has a k-model in the polynomial size of φ. We consider the complement of the problem: deciding whether φ is not irrelevant to V. It is easy to see that φ is not irrelevant to V iff there exist a k-model M of φ and an k-interpretation M in the polynomial size of φ such that M V M and M = φ. So checking whether φ is not irrelevant to V can be achieved in the following steps: (1) guess two k-interpretations M and M in the polynomial size of φ, (2) check if M = φ and M = φ, and (3) check M = φ. Obviously (1) can be done in polynomial time with a non-deterministic Turing machine while (2) and (3) can be done in polynomial time. Now we consider Result (2). Membership. First it is well known that an S5 formula ψ is not valid iff there is a polynomial size k-interpretation M such that M = ψ (Fagin et al. 1995). We consider the complement of the problem. We may guess a polynomial size model M and check whether M = φ and M = KForget(Γ, V ). From Theorem 5, we know that this is in Σ P 2. So the original problem is in Π P 2. Hardness. Let φ. Then the problem is reduced to decide KForget(Γ, V ) s validity. Since a propositional variable forgetting is a special case knowledge forgetting, the hardness is directly followed from the proof of Proposition 24 in (Lang, Liberator, & Marquis 2003). As we showed in Section 3, irrelevance plays an important role in characterizing knowledge forgetting. Recall that a formula φ is irrelevant to a set of atoms V if there is a φ such that φ φ and V ar(φ ) V =. In the following, we consider the complexity of deciding whether an S5 formula is irrelevant to a set of atoms V. Theorem 7 (Irrelevance complexity) Let φ be an S5 formula and V a set of atoms. Then deciding whether φ is irrelevant to V is co-np complete. Proof: From the proof of Theorem 6, we know that deciding whether IR(φ, V ) is in co-np. For the hardness, we show that for any propositional formula φ, φ is valid iff q (p φ) is irrelevant to p, where p, q are two different atoms not occurring in V ar(φ). First, if φ is valid, then q (p φ) q which is irrelevant to p. If q (p φ) is irrelevant to p, again from Corollary 2, we know that KForget(q (p φ), p) = q (p φ). Then from Theorem 2, we can derive q q (p φ). This implies that φ must be valid. Conclusion In this paper, we have developed a formal theory of knowledge forgetting under the propositional S5 single agent modal logic. We showed that knowledge forgetting is not only an important action that an agent may need in certain situations, but also a useful notion in representing various knowledge changes. Although the basic concept and intuition of knowledge forgetting were discussed in previous research. e.g. (Baral & Zhang 2005; Fagin et al. 1995), its formal semantics and computational properties have not been thoroughly studied. Our work presented in this paper provided a first formalization on knowledge forgetting and addressed its computational properties. Some related issues remain for our further study. In this paper we only considered the problem of knowledge forgetting in a single agent S5 modal logic. In a multi-agent system, it is more common that an agent not only needs to forget his own knowledge due to a memory limit, but also has to forget other agents knowledge for various reasons. So generalizing our knowledge forgetting to the mutli-agent S5 modal logic (and other multi-agent modal logics) will be a challenge. One particular concern we should take into account in this development is common knowledge which does not occur in single agent modal logic. References Baral, C., and Zhang, Y Knowledge updates: Semantic and complexity issues. Artificial Intelligence 164: Eiter, T., and Wang, K (to appear). Semantic forgetting in answer set programming. Artificial Intelligence. 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