Knowledge Constructions for Artificial Intelligence

Transcription

1 Knowledge Constructions for Artificial Intelligence Ahti Pietarinen Department of Philosophy, University of Helsinki P.O. Box 9, FIN University of Helsinki Abstract. Some new types of knowledge constructions in epistemic logic are defined, and semantics given by combining game-theoretic notions with modal models. One such notion introduced is focussed knowledge, which arises from imperfect information in quantified epistemic logics. This notion is useful in knowledge representation schemes in artificial intelligence and multi-agent systems with uncertainty. In general, in all the logics considered here, the imperfect information is seen to give rise to partiality, including partial common and partial distributed knowledge. A game-theoretic method of creating non-monotonicity will then be suggested, based on the partialised notion of only knowing and inaccessible possible worlds. The overall purpose is to show the extent in which games combine with a given variety of knowledge constructions. 1 Introduction The underlying motivation for this work can perhaps be illustrated by noting that classical logic is a logic of perfect information transmission. This fact is of course true of propositional and predicate logics, but it is also true of intensional modal logics and logics of epistemic notions (knowledge and belief). By perfect information transmission, it is meant that in transmitting semantic information from one logical component to another, that information is never lost. The aim is to show that once we adopt semantics that is suitable not only for the received logics with perfect information, we are able to produce new logics with new, expressive resources that capture an interesting variety of constructions of knowledge. Some of such constructions are needed in representing knowledge of multi-agent systems. The distinction between perfect and imperfect information transmission can be made precise within the framework of gametheoretic semantics (gts, see e.g. [3,7,8]), which operationalises a semantic game between two players, the team of Verifiers (V, loise) and the team of Falsifiers (F, belard). These semantic games provide an evaluation method that can be defined for a variety of logics. Research in epistemic logic and reasoning about knowledge has played an important role in AI, and uncertainty has been a major topic in reasoning even longer. The purpose of this paper is to combine the two. It is argued that logics of knowledge can represent multi-agent uncertainty, and it is suggested how M.-S. Hacid et al. (Eds.): ISMIS 2002, LNAI 2366, pp , c Springer-VerlagBerlin Heidelberg2002

2 304 A. Pietarinen a unifying semantics based on the notion of games can be defined to a wide variety of knowledge constructions. These notions include focussed, common and distributed knowledge, and the nonmonotonic notion of only knowing. Epistemic logics with imperfect information are partial, that is, there are sentences that are neither true nor false. Partiality itself has game-theoretic roots: if the associated semantic games are non-determined, all attempts of trying to verify or falsify a formula can be defeated. In contrast to received partial modal logics [5], games give rise to partiality even if the underlying models are complete. One outcome is that some new multi-agent logics for notions of knowledge that need games for their interpretation will be available. Far from being a purely technical enterprise, these logics are motivated by those knowledge representation schemes in multi-agent systems that virtually necessitate the introduction of new knowledge constructions. For example, focussed knowledge involves inherent uncertainty, and is argued to play an important role in representing knowledge in multi-agent systems where agents, such as communicating processors, do not always know the content of a message that has been sent to them. 2 From Perfect to Imperfect Information: Knowledge and Multi-agent Systems The inauguration of epistemic logic [2] has led to a proliferation of knowledge in logic, philosophy, computer science and artificial intelligence. Nowadays we find notions like common, shared and distributed knowledge. Furthermore, the distinction between de dicto and de re knowledge is widely spread. The well-formed formulas of ordinary propositional epistemic logic L are constructed by φ ::= p ϕ ψ ϕ K i ϕ. K i ϕ is read an agent i knows ϕ. Let ϕ, ψ be formulas of classical propositional epistemic logic L. A model is M = W,R,g, where g is a total valuation function g : W (Φ {True, False}), assigning to each proposition letter a subset of a set of possible worlds W = {w 0...w n } for which {w i g(w)(p) = True,w W}. R = {ρ 1...ρ n } is a set of accessibility relations for each i =1...n, ρ i W W. Let w 1 [w 0 ] ρi denote that a possible world w 1 is i-accessible from w 0. M,w = p iff {w i g(w)(p) =True,w W},p Φ. M,w = ϕ iff M,w = ϕ. M,w = ϕ ψ iff M,w = ϕ or M,w = ψ. M,w = K i ϕ iff M,w = ϕ, for all w [w] ρi. Let K j ψ be an L-formula, and let A = {K 1...K n },K i A, i {1...n} such that K j is in the syntactic scope of K i.nowifb A, then (K j /B) ψ L,K j / B. For example, K 1 (K 2 /K 1 ) ϕ and K 1 (ϕ (K 2 /K 1 ) ψ) are wffs of L. Every L -formula ϕ defines a game G(ϕ, w, g) onamodelm between two teams of players, the team of falsifiers F = {F 1...F n } and the team of verifiers V = {V 1...V k }, where w is a world and g is an assignment to the propositional letters. The game G(ϕ, w, g) is defined by the following rules.

3 Knowledge Constructions for Artificial Intelligence 305 (G. ) If ϕ = ψ, V and F change roles, and the next choice is in G(ψ, w, g). (G. ) If ϕ =(ψ θ), V chooses Left or Right, and the next choice is in G(ψ, w, g) if Left and G(θ, w, g) if Right. (G.K i ) If ϕ = K i ψ, and the game has reached w, F j F chooses w 1 [w] ρi, and the next choice is in G(ψ, w 1,g). (G.(K i /B)) If ϕ =(K i /B) ψ, K i B, and the game has reached w, then F l F chooses w 1 W independently of the choices made for the elements in B. (G.at) If ϕ is atomic, the game ends, and V wins if ϕ true, and F wins if ϕ false. The formulas (K i /B) ψ signal imperfect information: player choosing for K i on the left-hand side of the slash is not informed of the choices made for the elements in B earlier in the game. Nothing is said about the accessibility relation, since we want to leave the interpretation of these modalities open. The purpose of V is to show that ϕ is true in M ( M,w = + ϕ), and the purpose of F is to show that ϕ is false in M ( M,w = ϕ). If M,w = + ϕ, V wins, and if M,w = ϕ, F wins. A strategy for a player in G(ϕ, w, g) is a function assigning to each non-atomic subformula a member of the team, outputting a possible world, a value in {Left, Right} (the connective information), or an instruction to change roles (negation). A winning strategy is a strategy by which a player can make operational choices such that every play results in a win for him or her, no matter how the opponent chooses. Let ϕ be an L -formula. For any M,w W, M,w = + ϕ iff a strategy exists which is winning for V in G(ϕ, w, g), and M,w = ϕ iff a strategy exists which is winning for F in G(ϕ, w, g). A game is determined, iff for every play on ϕ, either V has a winning strategy in G or F has a winning strategy in G. It is easy to see that games for L are not determined. From non-determinacy it follows that the law of excluded middle ϕ ϕ fails in L. This is a common thing to happen in logics with imperfect information. Non-determinacy is related to partiality. A partial model is a triple M = W,R,g, where g is a partial valuation function g : W (Φ {True, False}), assigning to each proposition letter in Φ a subset g(φ) of a set of possible worlds W = {w 0...w n }. Partiality means that M,w = + K i ϕ iff M,w = + ϕ for all w W,w [w] ρi. M,w = K i ϕ iff M,w = ϕ for some w W,w [w] ρi. An alternative way to approach partiality is by gts of imperfect information, where partiality arises at the level of complex formulas, dispensing with partial models. One consequence is that semantic games generalise received partial modal logics [5] to complete models. 3Focussed Knowledge and Multi-agent Systems There are important non-technical motivations as to why one should be interested in combining games with various modalities. For one thing, in quantified extensions the combination gives rise to focussed knowledge.

4 306 A. Pietarinen 3.1 Language and Semantics Let the syntax for first-order epistemic logic L ωω consist of a signature, a logical vocabulary, and rules for building up formulas: φ ::= P K i ϕ xϕ xϕ ϕ ψ ϕ x y. Let Qψ, Q { x j, y j,k i } be an L ωω -formula in the syntactic scope of the elements in A = {K 1...K n, x k, y k }. Then L ωω consists of wffs of L ωω together with: if B A, then (Q/B) ψ is an L ωω-formula, Q B. For example, K 1 y(k 2 /K 1 y) Sxy L ωω. This hides the information about the choice for K 1 and y at K 2. Models and valuations have to take the world-boundedness of individuals into account. A non-empty world-relative domain consisting of aspects of individuals is D wj. We skip the formal definitions here. The semantics needs to be enriched by a finite number of identifying functions (world lines), which extend the valuation g to a (partial) mapping from worlds to individuals, that is, to g : X D W w i, such that if w W and g is an identifying function, then g(w) D w. These functions imply that individuals have local manifestations in any possible world. The interpretation of the equality sign (identifying functional) is: M,w 0,g = x y iff for some w i,w j W, f h such that f(w i)=h(w j). That is, two individuals are identical iff there are world lines f and h that pick the same individuals in w i and in w j. World lines can meet at some world but then pick different individuals in other worlds: the two-place identifying functional operation spells out when they meet. Individuals within a domain of a possible world are local and need to be cross-identified in order to be global and specific. The informal game rules for L ωω are: (G. x...k i ) If K i is in the syntactic scope of x and the game has reached w, the individual picked for x by a verifying player V has to be defined and exist in all worlds accessible from the current one. This rule is motivated by the fact that the course of the play reached at a certain point is unbeknownst to F choosing for K i. This approach leads to the notion of specific focus. (G.K i... x) If x is in the scope of K i, the individual picked for x has to be defined and exist in the world chosen for K i. This leads to the notion of non-specific focus. Finally, the rule for the hidden information says that (G.Q/B) If ϕ =(Q/B) ψ, Q { x, K i }, and the game has reached w, then if Q = x, F 1 F chooses an individual from D w1, where w 1 is the world from which the first world in B has departed. The next choice is in G(ψ, w, g). If Q = K 1 then F 1 F chooses a world w 1 independently of the choices made for the elements in B, and the next choice is in G(ψ, w 1,g). Likewise for V 1 V.

5 Knowledge Constructions for Artificial Intelligence 307 The notion of choosing independently is explained below. Other game rules are ordinary. Independent modalities mean that the player choosing for K i is not informed of the choices made for K j, and hence K i s are exempted from the syntactic scope of K j. This can be captured by taking hints from the theory of games. We will apply a partitional information structure (I i ) i N in the corresponding extensiveform games, which partitions sequences of actions (histories) h H into equivalence classes (information sets) {Sj i Si j (I i) i N,h i h Sj i,h,h H}. The purpose of equivalence classes is to denote which histories are indistinguishable to players. Payoff functions u i (h) associate a pair of truth-values in {1, 1} to terminal h H. Strategies are functions f i : P 1 ({i}) A from histories where players move to sequences of actions in A. Ifi is planning his decisions within the equivalence class Sj i annotated for him, his strategies are further required to be uniform on indistinguishable histories h, h Sj i, that is, f i(h) =f i (h ),i N. This leads to an informal observation: tracing uniform strategies along the game histories reveals in which worlds the specific focus is located. To see this, it suffices to correlate information sets of an extensive game with world lines. The clause choosing independently that appeared in the above game rules now means that players strategies have to be uniform on indistinguishable histories, that is, on worlds that players cannot distinguish. The notion of uniformity puts some constraints on allowable models. At any modal depth (defined in a standard way) there has to be the same number of departing worlds. If we assume that players can observe the set of available choices, the uniformity of strategies also requires that the departing worlds have to coincide for all indistinguishable worlds. Independent K i s can either refer to simultaneous worlds accessible from the current one, or to detached submodels of M. In the latter case we evaluate formulas in M,(w0,w w0 n ),g. The models would break into concurrent submodels, whence the designated worlds in each submodel become independent. 3.2 A Case for Imperfect Information in Multi-agent Systems Understanding knowledge of communicating multi-agent system benefits from the kind of concurrency outlined above. Suppose a process U 2 sends a message x to U 1. We ought to report this by saying that U 2 knows what x is, and U 1 knows that it has been sent (U 1 might knows this because a communication channel is open). This is already a rich situation involving all kinds of knowledge. However, the knowledge involved in this two-agent system cannot be captured in ordinary (first-order) epistemic logic. In this system, what is involved is U 2 knows what has been sent, as well as U 1 knows that something has been sent. However, what is not involved is U 1 knows that U 2 knows, nor U 2 knows that U 1 knows. How do we combine these clauses? It is easy to see that the three formulas xk U2 Mess(x) K U1 ymess(y), K U1 x(mess(x) K U2 Mess(x)), and x(k U2 Mess(x) K U1 ymess(y) x

6 308 A. Pietarinen y) all fail. So does an attempt that distinguishes between a message whose content is known ( Cont(x) ), and a message that has been sent ( Sent(y) ): x y((k U1 Cont(x) =y) K U2 Sent(x)). For now U 2 comes to know what has been sent, which is too strong. Hence, what we need is information hiding: xk U2 (K U1 /K U2 x)( y/k U2 x)(mess(x) x y). (1) In concurrent processing for quantified multi-modal epistemic logic, the notion of focussed knowledge is thus needed. 4 Further Knowledge Constructions and Semantic Games Here we move back to propositional logics and observe how games can be applied to various other constructions of knowledge. What we get is a range of partial logics for different modalities. The purpose is to show that (i) gts is useful for epistemic logics in artificial intelligence as it unifies the semantic outlook to different notions of knowledge; (ii) if games are non-determined, one gets partialised versions of these logics; (iii) if the possible-worlds semantics is augmented with inaccessible worlds, non-monotonic epistemic logic can be built upon game-theoretic principles. 4.1 Partial Only Knowing Let us begin with the logic of only knowing, partialise it, and then define semantics game rules for it. Only knowing (O i ϕ, or exactly knowing ) means that we do not have worlds in a model where ϕ could be true other than the accessible ones [4,6]. Informally, such a description picks a model where the set of possible worlds is as large as possible. This is because the larger set of possible worlds, the less knowledge an agent has. To partialise the logic of only knowing, we define M,w = + O i ϕ iff M,w = + ϕ w [w] ρi, for all w W. M,w = O i ϕ iff M,w = ϕ w [w] ρi, for some w W. Let a logic based on these be L O. The operator O i can also be understood in terms of another operator N i : O i ϕ ::= K i ϕ N i ϕ. M,w = + N i ϕ iff for all w W, M,w = + ϕ, M,w = N i ϕ iff for some w W, M,w = ϕ, where W is the set of inaccessible worlds, W = W W, W is the set of accessible worlds. A game G(ϕ, w, g) for L O -formulas ϕ, with a world w Wand an assignment g to the propositional letters is defined as a set of classical rules plus: (G.O i ) If ϕ = O i ψ, and the game has reached w W, F chooses between K i ψ and N i ϕ, and the game continues with respect to that choice.

7 Knowledge Constructions for Artificial Intelligence 309 (G.N i ) If ϕ = N i ψ, and the game has reached w W, F chooses w W, and the next choice is in G( ψ, w,g). (G.at) If ϕ is atomic, the game ends. F wins, if M,w = + ϕ, w W, or if not: M,w = + ϕ, w W. V wins, if M,w = ϕ, w W, orif M,w = + ϕ, w W. Strategies will operate on all worlds, including inaccessible ones. In general, we dispense with the accessibility relation and assume that also inaccessible worlds can be chosen. This is natural, because player knowledge and agent knowledge mean different things. (Further restrictions can be that any such inaccessible world can be chosen only once within a play of the game, etc.) Letting ϕ be an L O -formula, then for any model M, a valuation g, and w W, ϕ is true iff there exists a strategy which is winning for V in G(ϕ, w, g), and false iff there exists a strategy which is winning for F in G(ϕ, w, g). By imposing the uniformity condition on strategies, the logic of only knowing becomes partial and the underlying games non-determined, even if the propositions were completely interpreted. Traditionally, the logic of only knowing was developed in order to create semantic non-monotonicity by using stable sets [6]. An alternative game-theoretic method of creating non-monotonicity can thus be obtained by assuming that players can choose inaccessible worlds in addition to the accessible ones. 4.2 Partial Common Knowledge The modal operator E I ϕ means that everyone in the group of agents I Ag (the set of all agents) knows ϕ, and C I ϕ means that it is common knowledge among the group of agents I that ϕ. The partialised version of the logic augmented with these operators has M,w = + E I ϕ iff M,w = + K i ϕ for all i I. M,w = E I ϕ iff M,w = K i ϕ for some i I. Let EI ϕ be a reflexive and transitive closure on E0 I ϕ E1 I ϕ Ek+1 I ϕ, where EI 0 ϕ = ϕ, E1 I = E I ϕ, and E k+1 I ϕ = E I EI k ϕ. Hence: M,w = + C I ϕ iff M,w = + EI ϕ. M,w = C I ϕ iff M,w = K i ϕ for some i I. A game-theoretisation of common knowledge is this. A world w is I-reachable from w if there exists a sequence of worlds w 1...w k,w 1 = w, w k = w and for all j, 0 j k 1, there is i I Ag for which w j+1 [w j ] ρi for some k 0 [1]. A game G(ϕ, w, g) for formulas of L C, with a world w and an assignment g to the propositional letters has two additional rules. (The latter is to have duality. Since there can be infinite plays in the game tree, winning strategies are to be slightly modified in order to account for this.) (G.C I ) If ϕ = C I ψ, and the game has reached w, F chooses some w that is I-reachable from w, and the game continues as G(ψ, w,g). (G. C I ) If ϕ = C I ψ, and the game has reached w, V chooses some w that is I-reachable from w, and the game continues as G(ψ, w,g).

8 310 A. Pietarinen By letting ϕ be an L C -formula, then for any model M, a valuation g, and w W, ϕ is true iff there exists a strategy which is winning for V in G(ϕ, w, g), and false iff there exists a strategy which is winning for F in G(ϕ, w, g). This is how we bring gts to bear on common knowledge, certainly an important notion both in AI and Game Theory. It is known that common knowledge cannot be attained if communication is not taken to be reliable. It should be noted that any reference to communication pertains to agents knowledge, not to the information players have, and hence it is safe for us to consider partialised common knowledge. 4.3 Partial Distributed Knowledge Let the operator D I ϕ mean it is distributed knowledge among the group of agents I that ϕ. Partial distributed knowledge based on this notion can be defined as follows: M,w = + D I ϕ iff M,w = + ϕ for each w i I [w] ρ i. M,w = D I ϕ iff M,w = ϕ for some w i I [w] ρ i. A game G(ϕ, w, g) for L C+D, with a world w and an assignment g to the propositional letters has two additional rules: (G.D I ) If ϕ = D I ψ, and the game has reached w, F chooses some w i I [w] ρ i, and the game continues as G(ψ, w,g). (G. D I ) If ϕ = D I ψ, and the game has reached w, V chooses some w i I [w] ρ i, and the game continues as G(ψ, w,g). This definition amounts to partiality, as games for L C+D are non-determined. 4.4 Some Further Variations Finally, some further knowledge constructions can be envisaged, by a combination of previous systems. For instance, we can have a logic with only common knowledge (CI O ϕ). To see this, let us start with the notion of everybody only knows (EI O ϕ): M,w = + EI O ϕ iff M,w =+ O i ϕ for all i I. M,w = EI O ϕ iff M,w = O i ϕ for some i I. Only common knowledge is now closure on everybody only knows, along the lines already described. For falsification, it suffices that only knowing fails. Thus: M,w = + CI O ϕ iff M,w =+ EI ϕ, for all i I. M,w = CI O ϕ iff M,w = O i ϕ, for some i I. One can then devise games for these in a straightforward way. Also other combinations are possible. In general, we can have non-standard partiality: M,w = + Ki ϕ iff M,w = + ϕ for all w W,w [w] ρi. M,w = Ki ϕ iff not M,w = + ϕ for some w W,w [w] ρi.

9 Knowledge Constructions for Artificial Intelligence 311 M,w = + K # i ϕ iff not M,w = ϕ for all w W,w [w] ρi. M,w = K # i ϕ iff M,w = ϕ for some w W,w [w] ρi. The formula Ki ϕ captures the idea that the sentence is true when ϕ is true in all accessible worlds, and false when ϕ is not true in some accessible world. K # i ϕ, on the other hand, says that the sentence is true when ϕ is not false in every accessible worlds, and false when ϕ is false in some accessible world. Clearly the standard interpretation subsumes the truth-conditions for Ki ϕ and the falsity-conditions for K # i ϕ. The duals L i and L# i are defined accordingly. Games for non-standard clauses change the rules for winning conditions to weaker ones. For -modalities and #-modalities we have, respectively: (G.at ) If ϕ atomic, game ends. V wins if ϕ true, and F wins if ϕ not true. (G.at # ) If ϕ atomic, game ends. V wins if ϕ not false, and F wins if ϕ false. 5 Concluding Remarks Our approach is useful for sentences using non-compositional information hiding. The general perspective is that games unify the semantics for modalities, and that their imperfect information versions partialise logics extended with new operators. Further uses and computational aspects of the above knowledge constructions have to be left for future occasions. In general, areas of computer science and AI where these constructions and gts may turn out to be useful include uncertainty in AI and in distributed systems, intensional dimensions of knowledge representation arising in inter-operation, unification of verification languages for multi-agent systems [9], reasoning about secure information flow, strategic meanings of programs, modularity, and other information dependencies. References 1. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. Cambridge: MIT Press (1995) 2. Hintikka, J.: Knowledge and Belief. Cornell University Press, Ithaca (1962) 3. Hintikka, J., Sandu, G.: Game-theoretical semantics. In: van Benthem, J., ter Meulen, A. (eds): Handbook of Logic and Language. Elsevier, Amsterdam (1997) Humberstone, I.L.: Inaccessible worlds. Notre Dame Journal of Formal Logic 24. (1981) Jaspars, J., Thijsse, E.: Fundamentals of partial modal logic. In: Doherty, P. (ed.): Partiality, Modality, and Nonmonotonicity. Stanford, CSLI (1996) Levesque, H.J.: All I know: A study in autoepistemic logic. Artificial Intelligence 42. (1990) Pietarinen, A.: Intentional identity revisited. Nordic Journal of Philosophical Logic 6. (2001) Sandu, G., Pietarinen, A.: Partiality and games: Propositional logic. Logic Journal of the IGPL 9. (2001) Wooldridge, M.: Semantic issues in the verification of agent communication languages. Journal of Autonomous Agents and Multi-Agent Systems 3. (2000) 9 31