Knowledge Constructions for Artificial Intelligence
|
|
- Kristina Miles
- 6 years ago
- Views:
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
A Game Semantics for a Non-Classical Logic
Can BAŞKENT INRIA, Nancy can@canbaskent.net www.canbaskent.net October 16, 2013 Outlook of the Talk Classical (but Extended) Game Theoretical Semantics for Negation Classical Game Theoretical Semantics
More informationModal Dependence Logic
Modal Dependence Logic Jouko Väänänen Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands J.A.Vaananen@uva.nl Abstract We
More informationSOME SEMANTICS FOR A LOGICAL LANGUAGE FOR THE GAME OF DOMINOES
SOME SEMANTICS FOR A LOGICAL LANGUAGE FOR THE GAME OF DOMINOES Fernando R. Velázquez-Quesada Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas Universidad Nacional Autónoma de México
More informationAdding Modal Operators to the Action Language A
Adding Modal Operators to the Action Language A Aaron Hunter Simon Fraser University Burnaby, B.C. Canada V5A 1S6 amhunter@cs.sfu.ca Abstract The action language A is a simple high-level language for describing
More informationValentin Goranko Stockholm University. ESSLLI 2018 August 6-10, of 29
ESSLLI 2018 course Logics for Epistemic and Strategic Reasoning in Multi-Agent Systems Lecture 5: Logics for temporal strategic reasoning with incomplete and imperfect information Valentin Goranko Stockholm
More informationAn Inquisitive Formalization of Interrogative Inquiry
An Inquisitive Formalization of Interrogative Inquiry Yacin Hamami 1 Introduction and motivation The notion of interrogative inquiry refers to the process of knowledge-seeking by questioning [5, 6]. As
More informationRough Sets for Uncertainty Reasoning
Rough Sets for Uncertainty Reasoning S.K.M. Wong 1 and C.J. Butz 2 1 Department of Computer Science, University of Regina, Regina, Canada, S4S 0A2, wong@cs.uregina.ca 2 School of Information Technology
More informationSurvey on IF Modal Logic
Survey on IF Modal Logic Tero Tulenheimo Laboratoire Savoirs, Textes, Langage CNRS Université Lille 3 France Seminario de Lógica y Lenguaje Universidad de Sevilla 19.10.2009 Outline 1 Syntactic vs. semantic
More informationLogic: Propositional Logic (Part I)
Logic: Propositional Logic (Part I) Alessandro Artale Free University of Bozen-Bolzano Faculty of Computer Science http://www.inf.unibz.it/ artale Descrete Mathematics and Logic BSc course Thanks to Prof.
More informationAmbiguous Language and Differences in Beliefs
Proceedings of the Thirteenth International Conference on Principles of Knowledge Representation and Reasoning Ambiguous Language and Differences in Beliefs Joseph Y. Halpern Computer Science Dept. Cornell
More informationBisimulation for conditional modalities
Bisimulation for conditional modalities Alexandru Baltag and Giovanni Ciná Institute for Logic, Language and Computation, University of Amsterdam March 21, 2016 Abstract We give a general definition of
More informationMaximal Introspection of Agents
Electronic Notes in Theoretical Computer Science 70 No. 5 (2002) URL: http://www.elsevier.nl/locate/entcs/volume70.html 16 pages Maximal Introspection of Agents Thomas 1 Informatics and Mathematical Modelling
More informationTowards Tractable Inference for Resource-Bounded Agents
Towards Tractable Inference for Resource-Bounded Agents Toryn Q. Klassen Sheila A. McIlraith Hector J. Levesque Department of Computer Science University of Toronto Toronto, Ontario, Canada {toryn,sheila,hector}@cs.toronto.edu
More informationNested Epistemic Logic Programs
Nested Epistemic Logic Programs Kewen Wang 1 and Yan Zhang 2 1 Griffith University, Australia k.wang@griffith.edu.au 2 University of Western Sydney yan@cit.uws.edu.au Abstract. Nested logic programs and
More informationCorrelated Information: A Logic for Multi-Partite Quantum Systems
Electronic Notes in Theoretical Computer Science 270 (2) (2011) 3 14 www.elsevier.com/locate/entcs Correlated Information: A Logic for Multi-Partite Quantum Systems Alexandru Baltag 1,2 Oxford University
More informationPrinciples of Knowledge Representation and Reasoning
Principles of Knowledge Representation and Reasoning Modal Logics Bernhard Nebel, Malte Helmert and Stefan Wölfl Albert-Ludwigs-Universität Freiburg May 2 & 6, 2008 Nebel, Helmert, Wölfl (Uni Freiburg)
More informationIndependence-Friendly Cylindric Set Algebras
Independence-Friendly Cylindric Set Algebras by Allen Lawrence Mann B.A., Albertson College of Idaho, 2000 M.A., University of Colorado at Boulder, 2003 A thesis submitted to the Faculty of the Graduate
More informationFinite information logic
Finite information logic Rohit Parikh and Jouko Väänänen April 5, 2002 Work in progress. Please do not circulate! Partial information logic is a generalization of both rst order logic and Hintikka-Sandu
More informationTR : Public and Private Communication Are Different: Results on Relative Expressivity
City University of New York CUNY) CUNY Academic Works Computer Science Technical Reports The Graduate Center 2008 TR-2008001: Public and Private Communication Are Different: Results on Relative Expressivity
More informationFinite information logic
Finite information logic Rohit Parikh and Jouko Väänänen January 1, 2003 Abstract: we introduce a generalization of Independence Friendly (IF) logic in which Eloise is restricted to a nite amount of information
More informationChanging Types. Dominik Klein Eric Pacuit. April 24, 2011
Changing Types Dominik Klein Eric Pacuit April 24, 2011 The central thesis of the epistemic program in game theory (Brandenburger, 2007) is that the basic mathematical models of a game situation should
More informationGame Semantical Rules for Vague Proportional Semi-Fuzzy Quantifiers
Game Semantical Rules for Vague Proportional Semi-Fuzzy Quantifiers Matthias F.J. Hofer Vienna University of Technology Vague quantifier expressions like about half or almost all, and their semantics,
More informationFirst-Degree Entailment
March 5, 2013 Relevance Logics Relevance logics are non-classical logics that try to avoid the paradoxes of material and strict implication: p (q p) p (p q) (p q) (q r) (p p) q p (q q) p (q q) Counterintuitive?
More informationcis32-ai lecture # 18 mon-3-apr-2006
cis32-ai lecture # 18 mon-3-apr-2006 today s topics: propositional logic cis32-spring2006-sklar-lec18 1 Introduction Weak (search-based) problem-solving does not scale to real problems. To succeed, problem
More informationMonodic fragments of first-order temporal logics
Outline of talk Most propositional temporal logics are decidable. But the decision problem in predicate (first-order) temporal logics has seemed near-hopeless. Monodic fragments of first-order temporal
More informationPropositional logic. First order logic. Alexander Clark. Autumn 2014
Propositional logic First order logic Alexander Clark Autumn 2014 Formal Logic Logical arguments are valid because of their form. Formal languages are devised to express exactly that relevant form and
More informationTowards A Multi-Agent Subset Space Logic
Towards A Multi-Agent Subset Space Logic A Constructive Approach with Applications Department of Computer Science The Graduate Center of the City University of New York cbaskent@gc.cuny.edu www.canbaskent.net
More informationMULTI-AGENT ONLY-KNOWING
MULTI-AGENT ONLY-KNOWING Gerhard Lakemeyer Computer Science, RWTH Aachen University Germany AI, Logic, and Epistemic Planning, Copenhagen October 3, 2013 Joint work with Vaishak Belle Contents of this
More informationLöwenheim-Skolem Theorems, Countable Approximations, and L ω. David W. Kueker (Lecture Notes, Fall 2007)
Löwenheim-Skolem Theorems, Countable Approximations, and L ω 0. Introduction David W. Kueker (Lecture Notes, Fall 2007) In its simplest form the Löwenheim-Skolem Theorem for L ω1 ω states that if σ L ω1
More informationEncoding formulas with partially constrained weights in a possibilistic-like many-sorted propositional logic
Encoding formulas with partially constrained weights in a possibilistic-like many-sorted propositional logic Salem Benferhat CRIL-CNRS, Université d Artois rue Jean Souvraz 62307 Lens Cedex France benferhat@criluniv-artoisfr
More informationModal logics: an introduction
Modal logics: an introduction Valentin Goranko DTU Informatics October 2010 Outline Non-classical logics in AI. Variety of modal logics. Brief historical remarks. Basic generic modal logic: syntax and
More informationDeductive Algorithmic Knowledge
Deductive Algorithmic Knowledge Riccardo Pucella Department of Computer Science Cornell University Ithaca, NY 14853 riccardo@cs.cornell.edu Abstract The framework of algorithmic knowledge assumes that
More informationHINTIKKA'S "THE PRINCIPLES OF MATHEMATICS REVISITED" Harrie de SWART, Tom VERHOEFF and Renske BRANDS
Logique & Analyse 159 (1997), 281-289 HINTIKKA'S "THE PRINCIPLES OF MATHEMATICS REVISITED" Harrie de SWART, Tom VERHOEFF and Renske BRANDS Abstract In this book, published by Cambridge University Press,
More informationLogic and Artificial Intelligence Lecture 6
Logic and Artificial Intelligence Lecture 6 Eric Pacuit Currently Visiting the Center for Formal Epistemology, CMU Center for Logic and Philosophy of Science Tilburg University ai.stanford.edu/ epacuit
More informationCommon Knowledge in Update Logics
Common Knowledge in Update Logics Johan van Benthem, Jan van Eijck and Barteld Kooi Abstract Current dynamic epistemic logics often become cumbersome and opaque when common knowledge is added for groups
More informationAutomata, Logic and Games: Theory and Application
Automata, Logic and Games: Theory and Application 1. Büchi Automata and S1S Luke Ong University of Oxford TACL Summer School University of Salerno, 14-19 June 2015 Luke Ong Büchi Automata & S1S 14-19 June
More informationMathematics 114L Spring 2018 D.A. Martin. Mathematical Logic
Mathematics 114L Spring 2018 D.A. Martin Mathematical Logic 1 First-Order Languages. Symbols. All first-order languages we consider will have the following symbols: (i) variables v 1, v 2, v 3,... ; (ii)
More informationPreliminaries. Introduction to EF-games. Inexpressivity results for first-order logic. Normal forms for first-order logic
Introduction to EF-games Inexpressivity results for first-order logic Normal forms for first-order logic Algorithms and complexity for specific classes of structures General complexity bounds Preliminaries
More informationAn Extended Interpreted System Model for Epistemic Logics
Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008) An Extended Interpreted System Model for Epistemic Logics Kaile Su 1,2 and Abdul Sattar 2 1 Key laboratory of High Confidence
More informationApproximations of Modal Logic K
WoLLIC 2005 Preliminary Version Approximations of Modal Logic K Guilherme de Souza Rabello 2 Department of Mathematics Institute of Mathematics and Statistics University of Sao Paulo, Brazil Marcelo Finger
More informationKRIPKE S THEORY OF TRUTH 1. INTRODUCTION
KRIPKE S THEORY OF TRUTH RICHARD G HECK, JR 1. INTRODUCTION The purpose of this note is to give a simple, easily accessible proof of the existence of the minimal fixed point, and of various maximal fixed
More informationTwo New Definitions of Stable Models of Logic Programs with Generalized Quantifiers
Two New Definitions of Stable Models of Logic Programs with Generalized Quantifiers Joohyung Lee and Yunsong Meng School of Computing, Informatics and Decision Systems Engineering Arizona State University,
More informationLecture 3: MSO to Regular Languages
Lecture 3: MSO to Regular Languages To describe the translation from MSO formulas to regular languages one has to be a bit more formal! All the examples we used in the previous class were sentences i.e.,
More informationEQUIVALENCE OF THE INFORMATION STRUCTURE WITH UNAWARENESS TO THE LOGIC OF AWARENESS. 1. Introduction
EQUIVALENCE OF THE INFORMATION STRUCTURE WITH UNAWARENESS TO THE LOGIC OF AWARENESS SANDER HEINSALU Abstract. Here it is shown that the unawareness structure in Li (29) is equivalent to a single-agent
More informationDescription Logics. Foundations of Propositional Logic. franconi. Enrico Franconi
(1/27) Description Logics Foundations of Propositional Logic Enrico Franconi franconi@cs.man.ac.uk http://www.cs.man.ac.uk/ franconi Department of Computer Science, University of Manchester (2/27) Knowledge
More informationA Note on Graded Modal Logic
A Note on Graded Modal Logic Maarten de Rijke Studia Logica, vol. 64 (2000), pp. 271 283 Abstract We introduce a notion of bisimulation for graded modal logic. Using these bisimulations the model theory
More informationarxiv: v1 [math.lo] 15 Feb 2008
Perfect IFG-formulas arxiv:0802.2128v1 [math.lo] 15 Feb 2008 ALLEN L. MANN Department of Mathematics Colgate University 13 Oak Drive Hamilton, NY 13346 USA E-mail: amann@mail.colgate.edu October 26, 2018
More informationPhilosophy 244: Modal Logic Preliminaries
Philosophy 244: Modal Logic Preliminaries By CI Lewis in his 1910 Harvard dis- sertation The Place of Intuition in Knowledge (advised by Josiah Royce). Lewis credits earlier work of Hugh MacColl s. What
More informationThe Mother of All Paradoxes
The Mother of All Paradoxes Volker Halbach Truth and Intensionality Amsterdam 3rd December 2016 A theory of expressions The symbols of L are: 1. infinitely many variable symbols v 0, v 1, v 2, v 3,...
More informationFirst-order resolution for CTL
First-order resolution for Lan Zhang, Ullrich Hustadt and Clare Dixon Department of Computer Science, University of Liverpool Liverpool, L69 3BX, UK {Lan.Zhang, U.Hustadt, CLDixon}@liverpool.ac.uk Abstract
More informationLogics of Rational Agency Lecture 3
Logics of Rational Agency Lecture 3 Eric Pacuit Tilburg Institute for Logic and Philosophy of Science Tilburg Univeristy ai.stanford.edu/~epacuit July 29, 2009 Eric Pacuit: LORI, Lecture 3 1 Plan for the
More informationC. Modal Propositional Logic (MPL)
C. Modal Propositional Logic (MPL) Let s return to a bivalent setting. In this section, we ll take it for granted that PL gets the semantics and logic of and Ñ correct, and consider an extension of PL.
More informationUsing Counterfactuals in Knowledge-Based Programming
Using Counterfactuals in Knowledge-Based Programming Joseph Y. Halpern Cornell University Dept. of Computer Science Ithaca, NY 14853 halpern@cs.cornell.edu http://www.cs.cornell.edu/home/halpern Yoram
More informationStabilizing Boolean Games by Sharing Information
John Grant Sarit Kraus Michael Wooldridge Inon Zuckerman Stabilizing Boolean Games by Sharing Information Abstract. We address the issue of manipulating games through communication. In the specific setting
More informationSome Remarks on Alternating Temporal Epistemic Logic
Some Remarks on Alternating Temporal Epistemic Logic Corrected version: July 2003 Wojciech Jamroga Parlevink Group, University of Twente, Netherlands Institute of Mathematics, University of Gdansk, Poland
More informationProduct Update and Looking Backward
Product Update and Looking Backward Audrey Yap May 21, 2006 Abstract The motivation behind this paper is to look at temporal information in models of BMS product update. That is, it may be useful to look
More informationIntegrating State Constraints and Obligations in Situation Calculus
Integrating State Constraints and Obligations in Situation Calculus Robert Demolombe ONERA-Toulouse 2, Avenue Edouard Belin BP 4025, 31055 Toulouse Cedex 4, France. Robert.Demolombe@cert.fr Pilar Pozos
More informationToday. Next week. Today (cont d) Motivation - Why Modal Logic? Introduction. Ariel Jarovsky and Eyal Altshuler 8/11/07, 15/11/07
Today Introduction Motivation- Why Modal logic? Modal logic- Syntax riel Jarovsky and Eyal ltshuler 8/11/07, 15/11/07 Modal logic- Semantics (Possible Worlds Semantics): Theory Examples Today (cont d)
More information09 Modal Logic II. CS 3234: Logic and Formal Systems. October 14, Martin Henz and Aquinas Hobor
Martin Henz and Aquinas Hobor October 14, 2010 Generated on Thursday 14 th October, 2010, 11:40 1 Review of Modal Logic 2 3 4 Motivation Syntax and Semantics Valid Formulas wrt Modalities Correspondence
More informationReasoning About Knowledge of Unawareness
Reasoning About Knowledge of Unawareness Joseph Y. Halpern Computer Science Department, Cornell University, Ithaca, NY, 14853, U.S.A. e-mail: halpern@cs.cornell.edu Leandro C. Rêgo Statistics Department,
More informationCHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS
CHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS 1 Language There are several propositional languages that are routinely called classical propositional logic languages. It is due to the functional dependency
More informationNeighborhood Semantics for Modal Logic Lecture 5
Neighborhood Semantics for Modal Logic Lecture 5 Eric Pacuit ILLC, Universiteit van Amsterdam staff.science.uva.nl/ epacuit August 17, 2007 Eric Pacuit: Neighborhood Semantics, Lecture 5 1 Plan for the
More informationIntroduction to Temporal Logic. The purpose of temporal logics is to specify properties of dynamic systems. These can be either
Introduction to Temporal Logic The purpose of temporal logics is to specify properties of dynamic systems. These can be either Desired properites. Often liveness properties like In every infinite run action
More informationClassical First-Order Logic
Classical First-Order Logic Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) First-Order Logic (Classical) MFES 2008/09
More informationKnowledge Based Obligations RUC-ILLC Workshop on Deontic Logic
Knowledge Based Obligations RUC-ILLC Workshop on Deontic Logic Eric Pacuit Stanford University November 9, 2007 Eric Pacuit: Knowledge Based Obligations, RUC-ILLC Workshop on Deontic Logic 1 The Kitty
More informationBelief revision: A vade-mecum
Belief revision: A vade-mecum Peter Gärdenfors Lund University Cognitive Science, Kungshuset, Lundagård, S 223 50 LUND, Sweden Abstract. This paper contains a brief survey of the area of belief revision
More informationNormal Forms for Priority Graphs
Johan van Benthem and Davide rossi Normal Forms for Priority raphs Normal Forms for Priority raphs Johan van Benthem and Davide rossi Institute for Logic, Language and Computation d.grossi@uva.nl Abstract
More informationA logical formalism for the subjective approach in a multi-agent setting
logical formalism for the subjective approach in a multi-agent setting Guillaume ucher Université Paul Sabatier, Toulouse (F) University of Otago, Dunedin (NZ) aucher@irit.fr bstract. Representing an epistemic
More informationProving Completeness for Nested Sequent Calculi 1
Proving Completeness for Nested Sequent Calculi 1 Melvin Fitting abstract. Proving the completeness of classical propositional logic by using maximal consistent sets is perhaps the most common method there
More informationThe Lambek-Grishin calculus for unary connectives
The Lambek-Grishin calculus for unary connectives Anna Chernilovskaya Utrecht Institute of Linguistics OTS, Utrecht University, the Netherlands anna.chernilovskaya@let.uu.nl Introduction In traditional
More informationA Logic for Cooperation, Actions and Preferences
A Logic for Cooperation, Actions and Preferences Lena Kurzen Universiteit van Amsterdam L.M.Kurzen@uva.nl Abstract In this paper, a logic for reasoning about cooperation, actions and preferences of agents
More informationInfinite-game Semantics for Logic Programming Languages
Infinite-game Semantics for Logic Programming Languages Chrysida Galanaki National and Kapodistrian University of Athens Department of Informatics and Telecommunications chrysida@di.uoa.gr Abstract. This
More informationModal Logics. Most applications of modal logic require a refined version of basic modal logic.
Modal Logics Most applications of modal logic require a refined version of basic modal logic. Definition. A set L of formulas of basic modal logic is called a (normal) modal logic if the following closure
More informationUnderstanding the Brandenburger-Keisler Belief Paradox
Understanding the Brandenburger-Keisler Belief Paradox Eric Pacuit Institute of Logic, Language and Information University of Amsterdam epacuit@staff.science.uva.nl staff.science.uva.nl/ epacuit March
More informationAn Infinite-Game Semantics for Well-Founded Negation in Logic Programming
An Infinite-Game Semantics for Well-Founded Negation in Logic Programming Chrysida Galanaki a, Panos Rondogiannis a and William W. Wadge b a Department of Informatics & Telecommunications, University of
More informationFormal Epistemology: Lecture Notes. Horacio Arló-Costa Carnegie Mellon University
Formal Epistemology: Lecture Notes Horacio Arló-Costa Carnegie Mellon University hcosta@andrew.cmu.edu Logical preliminaries Let L 0 be a language containing a complete set of Boolean connectives, including
More informationReasoning about Fuzzy Belief and Common Belief: With Emphasis on Incomparable Beliefs
Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence Reasoning about Fuzzy Belief and Common Belief: With Emphasis on Incomparable Beliefs Yoshihiro Maruyama Department
More informationRestricted truth predicates in first-order logic
Restricted truth predicates in first-order logic Thomas Bolander 1 Introduction It is well-known that there exist consistent first-order theories that become inconsistent when we add Tarski s schema T.
More informationFrom Bi-facial Truth to Bi-facial Proofs
S. Wintein R. A. Muskens From Bi-facial Truth to Bi-facial Proofs Abstract. In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [7] distinguish between an ontological
More informationINTENSIONS MARCUS KRACHT
INTENSIONS MARCUS KRACHT 1. The Way Things Are This note accompanies the introduction of Chapter 4 of the lecture notes. I shall provide some formal background and technology. Let a language L be given
More informationAwareness, Negation and Logical Omniscience
Awareness, Negation and Logical Omniscience Zhisheng Huang and Karen Kwast Department of Mathematics and Computer Science University of Amsterdam Plantage Muidergracht 24 1018TV Amsterdam, The Netherlands
More informationDecomposing Modal Logic
Decomposing Modal Logic Gabriel G. Infante-Lopez Carlos Areces Maarten de Rijke Language & Inference Technology Group, ILLC, U. of Amsterdam Nieuwe Achtergracht 166, 1018 WV Amsterdam Email: {infante,carlos,mdr}@science.uva.nl
More informationFuzzy and Rough Sets Part I
Fuzzy and Rough Sets Part I Decision Systems Group Brigham and Women s Hospital, Harvard Medical School Harvard-MIT Division of Health Sciences and Technology Aim Present aspects of fuzzy and rough sets.
More informationTemporal logics and explicit-state model checking. Pierre Wolper Université de Liège
Temporal logics and explicit-state model checking Pierre Wolper Université de Liège 1 Topics to be covered Introducing explicit-state model checking Finite automata on infinite words Temporal Logics and
More informationAgency and Interaction in Formal Epistemology
Agency and Interaction in Formal Epistemology Vincent F. Hendricks Department of Philosophy / MEF University of Copenhagen Denmark Department of Philosophy Columbia University New York / USA CPH / August
More informationSeminar in Semantics: Gradation & Modality Winter 2014
1 Subject matter Seminar in Semantics: Gradation & Modality Winter 2014 Dan Lassiter 1/8/14 Handout: Basic Modal Logic and Kratzer (1977) [M]odality is the linguistic phenomenon whereby grammar allows
More informationCOMP219: Artificial Intelligence. Lecture 19: Logic for KR
COMP219: Artificial Intelligence Lecture 19: Logic for KR 1 Overview Last time Expert Systems and Ontologies Today Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof
More informationHow Restrictive Are Information Partitions?
How Restrictive Are Information Partitions? Peter J. Hammond Department of Economics, Stanford University, CA 94305 6072, U.S.A. e-mail: peter.hammond@stanford.edu Preliminary Version: July 1996; revisions:
More informationLecture 3: Semantics of Propositional Logic
Lecture 3: Semantics of Propositional Logic 1 Semantics of Propositional Logic Every language has two aspects: syntax and semantics. While syntax deals with the form or structure of the language, it is
More informationSchematic Validity in Dynamic Epistemic Logic: Decidability
This paper has been superseded by W. H. Holliday, T. Hoshi, and T. F. Icard, III, Information dynamics and uniform substitution, Synthese, Vol. 190, 2013, 31-55. Schematic Validity in Dynamic Epistemic
More informationAction, Failure and Free Will Choice in Epistemic stit Logic
Action, Failure and Free Will Choice in Epistemic stit Logic Jan Broersen Department of Information and Computing Sciences Utrecht University, The Netherlands Symposium "What is really possible?" Utrecht,
More informationNonmonotonic Reasoning in Description Logic by Tableaux Algorithm with Blocking
Nonmonotonic Reasoning in Description Logic by Tableaux Algorithm with Blocking Jaromír Malenko and Petr Štěpánek Charles University, Malostranske namesti 25, 11800 Prague, Czech Republic, Jaromir.Malenko@mff.cuni.cz,
More informationReflections on Agent Beliefs
Reflections on Agent Beliefs JW Lloyd 1 and KS Ng 2 1 Computer Sciences Laboratory Research School of Information Sciences and Engineering The Australian National University jwl@mailrsiseanueduau 2 Symbolic
More informationPrefixed Tableaus and Nested Sequents
Prefixed Tableaus and Nested Sequents Melvin Fitting Dept. Mathematics and Computer Science Lehman College (CUNY), 250 Bedford Park Boulevard West Bronx, NY 10468-1589 e-mail: melvin.fitting@lehman.cuny.edu
More informationTeam Semantics and Recursive Enumerability
Team Semantics and Recursive Enumerability Antti Kuusisto University of Wroc law, Poland, Technical University of Denmark Stockholm University, Sweden antti.j.kuusisto@uta.fi Abstract. It is well known
More informationModel Checking for Modal Intuitionistic Dependence Logic
1/71 Model Checking for Modal Intuitionistic Dependence Logic Fan Yang Department of Mathematics and Statistics University of Helsinki Logical Approaches to Barriers in Complexity II Cambridge, 26-30 March,
More informationPreference and its Dynamics
Department of Philosophy,Tsinghua University 28 August, 2012, EASLLC Table of contents 1 Introduction 2 Betterness model and dynamics 3 Priorities and dynamics 4 Relating betterness and priority dynamics
More informationAhti Pietarinen and Gabriel Sandu GAMES IN PHILOSOPHICAL LOGIC
Ahti Pietarinen and Gabriel Sandu GAMES IN PHILOSOPHICAL LOGIC Semantic games are an important evaluation method for a wide range of logical languages, and are frequently resorted to when traditional methods
More informationAn Intuitionistic Epistemic Logic for Sequential Consistency on Shared Memory
An Intuitionistic Epistemic Logic for Sequential Consistency on Shared Memory Yoichi Hirai 2010-04-27, Dakar Motivation Ÿ Ÿ Treating asynchronous communication using an epistemic logic. Shared memory consistency
More informationDon t Plan for the Unexpected: Planning Based on Plausibility Models
Don t Plan for the Unexpected: Planning Based on Plausibility Models Thomas Bolander, DTU Informatics, Technical University of Denmark Joint work with Mikkel Birkegaard Andersen and Martin Holm Jensen
More information