SOME SEMANTICS FOR A LOGICAL LANGUAGE FOR THE GAME OF DOMINOES

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1 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 frvq@uxmcc2.iimas.unam.mx Francisco Hernández-Quiroz Facultad de Ciencias Universidad Nacional Autónoma de México fhq@fciencias.unam.mx ABSTRACT Epistemic logic allows to reason not only about situations, but also about the knowledge that a set of agents have about situations. In later years, epistemic logic has been applied to study the role of knowledge in games and negotiations. The classical approach, the mathematical theory of games, does not facilitate explicit reasoning about the knowledge of the agents that interact in competitive situations. Dynamic epistemic logic has been used recently to provide formal methods to analyze how the knowledge of a set of agents changes as a consequence of actions. In this paper we develop a logical language for reasoning about the knowledge that flows during a match of dominoes; we call this language dynamic epistemic logical language for dominoes. Syntax of the language has been presented in a previous work. We define here a formal semantics for the epistemic part of our language. KEY WORDS Epistemic logic, Dynamic logic, Knowledge Acquisition, Knowledge Representation, Games, Dominoes 1 Introduction Epistemic logic [1] deals with reasoning about knowledge. This language is based on a set of atomic propositions (those that describe basic facts), a set of logical connectives (usually negation and conjunction) and a modal operator (denoted in this area by K). It allows to express not only situations, but also knowledge that an agent has about these situations. Epistemic logic is a modal logic in which the modal operator has an epistemic interpretation. Fagin et. al. began in [2] a tradition in computational logic that revived the modal approach for epistemic logic, developing generalized logical foundations and applications that had not occurred to philosophers. The idea is to introduce multiple agents, and to concentrate on reasoning about agent s knowledge about one another s knowledge. Epistemic logic became a multimodal language by using a modal operator K i for each agent i and the modal operator C for expressing common knowledge. [3] offers a comprehensive presentation of epistemic logic. Epistemic logic can be directly applied to the analysis of distributed multi-agent systems and, for that reason, its study belongs to a separate area of computer science; one that overlaps to some extent with AI. Epistemic logic has also played a role in studying knowledge in games and negotiations in economic theory as the classical approach (the mathematical theory of games [4]) does not allow explicit reasoning about the knowledge of the agents that interact in competitive situations. Dynamic epistemic logic is a relatively new offspring of epistemic logic. Its goal is to provide formal methods to analyze the way knowledge of a set of agents changes as a consequence of actions. This logic has two parts: the first deals with representation of knowledge while the second deals with the way knowledge changes. More recently there has been much work about information flow in games using logical tools. J. Gerbrandy and W. Groeneveld presented in [5] and [6] a different semantic model for epistemic logic, based on non-well-founded set theory. H. van Ditmarsch presented in [7] a dynamic epistemic logic for a specific kind of games called knowledge games: A knowledge game is defined by a deal of cards over players, a set of possible game actions (or moves), an order protocol to determine who is to move next, and a procedure to determine who wins. Although cards do not change hands during a match, they may be shown, so knowledge about cards does change. Cluedo is a concrete example of such a game. In [8], B. Kooi presented a probabilistic dynamic epistemic logic, which combines dynamic epistemic logic with probabilistic logic to analyze inferences about probabilistic information change. Our goal is to develop a logical language to reason about the knowledge flow during a match of dominoes. (A description of the rules of this popular game and its strategies can be found in [9]). The main point is to express how each player s knowledge is modified as a result of the actions that she and the other players carry out and what conclusions can be drawn as a result of this new knowledge. A summary of our language presenting the syntax and sketching its semantics has been presented in a previous paper [10]. We define here the formal semantics of the epistemic part of the language, leaving the formal definition of the dynamic part for a future paper. 2 Syntax of the Language Our language is based on the language of epistemic logic. As we said, epistemic logic comprises a set of atomic

2 propositions, a set of logical connectives and a set of modal operators. The aim is to express the knowledge that a group of agents can have about different situations, and even more, the knowledge they can have about the knowledge of others agents. We offer here a brief summary of the syntax of our language; see [10] for a full presentation. The set of atomic propositions for dominoes, those that express the basic facts about the game, is defined as follows. Definition 1 (Atomic propositions for dominoes) We define the set of players in a match as J = { a, b, c, d } and the set of tiles as F = { 0 0, 0 1,..., 5 6, 6 6 } (note that x y y is the same tile as x ). The set of atomic propositions for dominoes (Φ D ) defined as Φ D = { x y i, x y t, tiles i n, tiles t n, pts i n, ptst n, fewerpoints i,j, turn i, (u, v), x y i (u,v), x y i } (u,v) where i, j J, x y F and u, v are possible free-ends on the table. Each proposition has the next intuitive meaning: x y i Player i has tile x y in her hand. x y t Tile x y is on the table. tiles i n Player i has n tiles in her hand. tiles t n There are n tiles on the table. pts i The tiles player i has in her hand sum n n points. pts t The tiles on the table sum n points. n fewerpoints i,j The tiles of player i and player j sum fewer points than those of the other players. turn i It is player i s turn. (u, v) The free-ends on the table are u and v. x y i (u,v) The free-ends on the table were u and v before player i threw tile x y. x y i (u,v) The free-ends on the table were u and v after player i threw tile x y. Based on the set Φ D, we define the language of epistemic logic for dominoes EL D. Definition 2 (Epistemic logical language for dominoes) The formulae ϕ of the epistemic logical language for dominoes (EL D ) are given by the rule: ϕ ::= p ϕ (ϕ ψ) K i ϕ C B ϕ where p Φ D, i J, B J and ϕ, ψ EL D. The logical connectives and have the usual meaning ( is the proposition that is always true). The formula K i ϕ means player i knows that ϕ while the formula C B ϕ means it is common knowledge between players in B that ϕ (everybody in B knows ϕ, everybody in B knows that everybody in B knows ϕ and so on). For expressing the way actions modify the state of a match and the knowledge of the players, we need to express these actions in our language. The basic actions of the game are defined in the following way. Definition 3 (Basic actions for dominoes) The set of basic actions for dominoes (A D ) is defined as A D = {α i x y, α i ɛ} where α i x y has the intended meaning player i draws tile x y on side x of the table and α i ɛ has the intended meaning player i passes without draw any tile. Note that α i x y and α i y x are different actions. These actions can be combined to get new actions by using sequential composition and non-deterministic choice. We define the language of actions for dominoes as follows. Definition 4 (Action language for dominoes) The actions α of the action language for dominoes (AL D ) are given by the rule α ::= ε ω (α; β) (α β) where ω A D and α, β AL D. The action ε is the do nothing action, (α; β) represents the sequential composition of the actions α and β, and (α β) represents the non-deterministic choice between α and β. Based on the atomic propositions on Φ D and the language of actions AL D, we define the dynamic epistemic logical language for dominoes (DEL D ) for expressing the game s situation at certain stage, the knowledge of agents and how this situation and this knowledge change as a consequence of the actions carried out. Definition 5 (Language DEL D ) The formulae ϕ of the dynamic epistemic logical language for dominoes (DEL D ) are given by the following rule: ϕ ::= p ϕ (ϕ ψ) K i ϕ C B ϕ [α]ϕ where p Φ D, i J, B J, ϕ, ψ DEL D and α AL D. The formula [α]ϕ means after the action α is executed, ϕ holds. The connectives, and are defined as usual. The proposition and the modal operators P, E and α are defined in the following way. Let i J, B J, α AL D and ϕ EL D : E B ϕ i B (K iϕ) P i ϕ K i ϕ α ϕ [α] ϕ 3 Semantics of EL D As a modal (multimodal) logic, epistemic logic s meaning is found in structures called Kripke models or possible world models [1], [11]. The intuition behind is that there are a number of other possible situations or worlds besides the real situation. Given her current information, an agent

3 (player, in this case) may not be able to distinguish between this real world and the other possible (but not real) worlds. She only knows what is true in all the worlds she considers possible. In the game of dominoes, these possible worlds have a concrete and direct interpretation: all the possible distribution of the tiles among the four players. Usually, given a set of agents A and a set of atomic propositions Φ, a possible world model M is defined as a tuple M = (W, R i, V ) where W is the set of possible worlds. R i (W W ) is the accessibility relation for each agent i A. V : Φ 2 W is a function that assigns to every atomic proposition a subset of W. The pair (W, R i ) is called a possible world frame, and a pair (M, w) where w W is called a pointed possible world model. To define the semantics for the epistemic language for dominoes EL D, we need to make some modifications to the possible world model. These are the main reasons: 1. In a possible world given, each atomic proposition has a truth value independently of the truth value of the others atomic propositions. In dominoes, the truth values of some atomic propositions are related (like 0 2 a and 0 2 c, it is not possible that two players have the same tile). We need to define truth values for an atomic proposition in a way that the truth values of its related propositions are related. 2. Even if we know the distribution of the tiles at a certain stage, we can not give truth value to some atomic propositions (turn i, (u, v), x y i (u,v) and x y (u,v)). i To give truth value to these propositions we need to keep a record of the actions that have been executed during the match. Note that, except for the distribution of tiles, all the actions in a dominoes match are public actions, and therefore become public knowledge among the players. As we said, each possible world corresponds to one of the possible distribution of tiles. We can characterize each distribution with a function that takes a tile as argument and tells us which player has it. This function must assign seven tiles to each player. During a match, tiles are played by putting them on the table, so it is also important to know if a tile has been played or stills in a player s hand. Definition 6 (Function of distribution of tiles) Let δ be a function from the set of tiles to the cartesian product of the set of players and the set {0, 1}: δ : F J {0, 1} Let Fδ i be the set of tiles assigned to player i according to the function δ: Fδ i = { x y F δ( x y ) = (i, ) } The function δ is said to be a function of distribution of tiles if it assigns exactly seven tiles to each player, i.e., if F i δ = 7 for i J A function of distribution of tiles tells us which player has (or had) each tile, and if the tile is on the table (1) or not (0). Each possible world w W is then associated with a function δ w that characterize the distribution of the tiles in that world. With this function, propositions like 0 2 a and 0 2 c cannot be true in the same world (that would imply that δ w ( 4 5 ) = (c, 0) and δ w ( 4 5 ) = (d, 0)). The truth value for atomic propositions that are related to the distribution of tiles ( x y i, x y t, tiles i n, tiles t n, pts i n, ptst n and fewerpoints i,j ) is defined in terms of δ w. A possible world frame (W, R i ) in which each world w W has associated a function of distribution of tiles δ w is called a possible world model for dominoes and it is denoted by M D. For the remaining atomic propositions (turn i, (u, v), x y i (u,v) and x y (u,v)) i we need to keep a record of the actions that have been realized during the match. We define the history of a match in the following way. Definition 7 (History of a match) The history of a match H is defined as a (possibly empty) sequence in which for each actions that takes place during the match we add two entries. The first entry, of the form α i f, indicates the action that has been executed (i J and f F {ɛ}), while the second entry, of the form (u, v), indicates the free-ends on the table after the action. Action α i x y will indicate that player i placed tile x y on the table s free-end x whereas α i ɛ will indicate that player i passed without playing any tile. Some restrictions apply for H to be an history of a valid match. 1. If (u, v)α i x y appears in H, then x = u or x = v. 2. If α i x y (u, v) appears in H, then y = u or y = v. 3. If (u 1, v 1 )α i ɛ(u 2, v 2 ) appears in H, then u 1 = u 2 and v 1 = v If x y appears in H, then nor x y neither its permutation ( x ) appears again in y H. 5. Actions in H respect the order protocol. If H satisfies these restrictions, then it is called a history of a valid match. Let α i x y (u, v) be a subsequence of H. Then the subsequences α i y x (u, v), α i x y (v, u) and α i y x (v, u) are called its permutations. For a subsequence of the form (u, v)α i x y, its permutations are defined in an analogous way. We define the structure where formulae of EL D have meaning.

4 Definition 8 (Structure for a dominoes match) A tuple (H, M D ) where H is the history of a valid match and M D = (W, R i ) is a possible world model for dominoes is called a structure for a dominoes match. The tuple (H, M D, w) where w W is called a pointed structure for a dominoes match. A structure for a dominoes match is said to be a valid structure for a dominoes match if for every w W 1. δ w ( x y ) = (i, 1) if and only if either α i x y or α i y appears on H. 2. δ w ( x y ) = (i, 0) if and only if neither α i x y nor α i y appears on H. Given a pointed structure for a dominoes match (H, M D, w), we can define truth values for atomic propositions in the following way: x y i is true if and only if x y F i w, where F i w is the set of tiles that player i has in her hand at possible world w: Fw i = { x y F δ w ( x y ) = (i, 0) } x y t is true if and only if x y Fw, t where Fw t is the set of tiles that are on the table at possible world w: Fw t = { x y F δ w ( x y ) = (, 1) } tiles i n is true if and only if F i w = n. tiles t n is true if and only if F t w = n. pts i n is true if and only if PTS(F i w) = n, where the function PTS : 2 F N returns the points that sum a set of tiles. pts t n is true if and only if PTS(F t w) = n. fewerpoints i,j is true if and only if the sum of PTS(Fw) i and PTS(Fw) j is less than the sum of the points of the other players. turn i is true if and only if the number of actions that have been executed plus one modulo 4 result i. (u, v) is true if and only if H = (u, v) or H = (v, u). x y i is true if and only if (u, v)α i x y or one of its permutations appears in H. (u,v) x y i (u,v) is true if and only if α i x y (u, v) or one of its permutations appears in H. Now we can give truth value to every formulae in the epistemic language EL D. Let (H, M D, w) be a pointed structure for a dominoes match where M D = (W, R i ): 1. The formula is true in every tuple (H, M D, w). x x 2. The truth value for atomic propositions in a tuple (H, M D, w) is defined as before. 3. A formula ϕ is true in (H, M D, w) if and only if ϕ is not true in (H, M D, w). 4. A formula ϕ ψ is true in (H, M D, w) if and only if ϕ is true in (H, M D, w) or ψ is true in (H, M D, w). 5. A formula K i ϕ is true in (H, M D, w) if and only if ϕ is true in all (H, M D, u) such as (w, u) R i. 6. A formula C B ϕ is true in (H, M D, w) if and only if ϕ is true in all (H, M D, u) such as (w, u) R + B. When ϕ is true in (H, M D, w), we will write (H, M D, w) = ϕ. Otherwise we will write (H, M D, w) = ϕ and say that ϕ is false in (H, M D, w). The dynamic epistemic language DEL D is syntactically defined as the epistemic language plus a new rule that allows us to build formulae to express the way actions modifies the situation and the knowledge of the players: if ϕ is a formula in DEL D and α is an action in AL D, then [α]ϕ is also a formula in DEL D. To give a truth value to formulae like these, we need to specify how each action modify the structure for a dominoes match. Following ideas from [12], we consider each action as a map that takes the structure for a dominoes match that describes the situation of the game before the action, and returns the structure for a dominoes match that describes the situation of the game after the action. A formal definition of these actions as maps will appear in a future paper. 4 Example of a Match We give an example of how the language DEL D can express the situation of the game and the knowledge of the players in a match of dominoes. Given that the semantics of the language has not been fully defined, our example is only syntactic. We will use the next abbreviations: J i = {j J j i} x y J x y a x y b x y c x y d x i x 0 i x 1 i x 2 i x 3 i x 5 i x 6 i closed (x, x ) x J tiles J i n j J (tiles j n) i x 4 i Remember that the game is played by teams. Player a and player c are one team and players b and d are the other team. We define a formulae to indicate that one of the teams have won. won a,c (tiles a 0 tiles J a 0 ) (tiles c 0 tiles J c 0 ) (closed fewerpoints a,c ) won b,d (tiles b 0 tiles J b 0 ) (tiles d 0 tiles J d 0 ) (closed fewerpoints b,d )

5 Let us suppose that player a won the last match so she starts the new one. Before the distribution of the tiles, the next formula is true: K a turn a C J turn a After the distribution, a has the tiles 1 6, 5 5, 3 4, 0 6, 0 1 and 2 3. Then we have K a ( 1 6 a 5 5 a a ) K a ( 6 6 a 5 6 a a 0 0 a ) 2 6, Player a knows that b (also c and d) does not know which tiles she has K a ( K b Also it is true that 1 6 a K b 5 5 a... K b 2 3 a ) tiles a 7 [α a x y ]tiles a 6 Player a plays 2 6, b plays 6 6 and then c plays 0 2. At this point the next formulae are true: t 0 2 t C J ( t 0 2 t ) C J ( 2 6 a 2 6 b 2 6 c 2 6 d ) (0, 6) turn d ((0, 6) turn d ) [α d ɛ ]C J ( 0 d 6 d ) C J (tiles a 6 tiles b 6 tiles c 6 tiles d 7) Player d passes. Then C J ( 0 d 6 d ) The next round is played as follows: a throws 0 6, b throws 3 6, c 3 5 and d 4 5. Player a knows which tiles she has and which ones have been played K a ( 0 6 t 2 6 t 3 6 t 6 6 t ) K a 1 6 a K a 6 d K a ( 4 6 a 5 6 a ) K a ( c ) K a ( 5 6 K d ( c ) b Player a plays 5 6 c ) 3 4. Then the next formulae are true (3, 6) turn b ((3, 6) turn b ) [α b ɛ]c J 6 6 d [α b ɛ]k a ( 4 6 c 5 6 c ) Player b does not pass, so a cannot conclude that 4 6 c 5 6 c. She (b) plays 5 6, c plays 0 5 and d plays 3 3. It is a s turn, and she throws 2 3. At this point it is true that 2 3 t 3 3 t 3 4 t 3 5 t 3 6 t C J ( 2 3 t 3 3 t 3 4 t 3 5 t 3 6 t ) K a ( 0 3 a 1 3 a ) P a 0 3 b P a 0 3 c P a 1 3 b P a 1 3 c P a 1 3 d d Then b plays 0 0, c plays 0 4 and d plays 2 4. Now it is true (after four rounds) that 0 0 t 0 2 t 0 4 t 0 5 t K a ( 0 1 a 0 3 a ) C J (tiles a 3 tiles b 3 tiles c 3 tiles d 4) 0 6 t Also it is true that (turn a (4, 4) 4 a ). Player a passes and b passes too, but c does not and throws 4 6. Then d throws 4 4. Now the next formulae hold: C J ( 4 a 4 b ) C J ( 4 0 t 4 2 t 4 3 t 4 4 t 4 5 t C J ( 4 1 c 4 1 d ) Now that (turn a (4, 6) and throws 1 6. Given that (turn a (4, 6)) [α a 6 u ](u, 4) t ) a ) is true, a can play holds, then after our turn (1, 4) is true. We know that 4 b and therefore [α b 4 ] is true. Then b plays 1 3 and c pass when (3, 4) is true. This makes true that K a ( 3 and given that c 1 4 t 4 4 c ) a 4 b is true, then it is true that K a 1 4 d. At this point it is also true that 0 0 t 0 2 t 0 4 t 0 5 t 0 6 t K a 0 1 a K a 0 3 a K a 3 c Ka ( 3 c 3 0 c ) K a 0 d Ka ( 0 d 3 0 d ) K a 0 3 b After d plays 4 c 0 3 t 1 4, the following formulae hold turn a (3, 1) tiles a a 5 5 a [α a 1 0 ](3, 0) K a 0 3 ( b K a [α a 1 0 ; (α b 3 0 α b 0 3 )](closed pts a 10 )) Player a does the only thing she can do. She throw 0 1 and b throws 0 3. Then closed and C J (tiles a 1 tiles b 1 tiles c 2 tiles d 2) are true. Every player shows her tiles and then: C J ( 5 5 a 1 2 c 1 5 c ) C J ( 2 2 b 1 1 d 2 5 d ) (pts a 10 ptsb 4 ptsc 9 ptsd 9 ) (pts a 10 ptsb 4 ptsc 9 ptsd 9 ) fewerpointsb,d won b,d

6 5 Conclusions and Related Work We have defined a logical language DEL D to describe situations during a game of dominoes, including the epistemic part of its semantics. In a further paper a full semantics will be offered. The language presents an objective point of view of the game, i.e,, that of an external observer who knows what each player knows at each stage of the match. In order to give a truth value to formulae in epistemic logic, we need to know both the possible world model and the real world. A subjective point of view is needed to understand better the evolution of a single player s knowledge not only about the situation of the game but also about the knowledge of the other players. Extending ideas from S. Druiven ([13]), we can distinguish four kinds of knowledge in a game environment: game knowledge (knowledge players have about the rules of the game they are playing), definite knowledge (knowledge about the situation of the game, developed as a consequence of the actions in the game), strategic knowledge (knowledge about strategies over the game) and historical knowledge (knowledge about previous matches). For the time being, DEL D can only express definite knowledge. We intend to extend the language and its semantic model to express strategic and historical knowledge too. We are particularly interested in how strategic and historical knowledge influences how a single player s definite knowledge evolves. Some kind of abductive and inductive reasoning may play a role in this respect. A complete assessment of an automatic deduction system based on our logic is pending. For the time being, it can be said that such a system would face a high amount of calculations (very likely more than polynomial) in the early stages of a match, but this amount will reduce dramatically in later stages. There exists previous work relating dominoes and logic. [14] presents a partial formalization of a game of dominoes using first order predicate logic. From the opposite viewpoint, [15] shows how dominoes games can be used to describe proofs in a natural deduction system. This proposal includes an interactive theorem prover. References [5] J. Gerbrandy and W. Groeneveld, Reasoning about information change, Journal of Logic, Language, and Information, 6, 1997, [6] J. Gerbrandy, Bisimulations on Planet Kripke (PhD thesis), (ILLC, University of Amsterdam, 1999). [7] H. P. van Ditmarsch, Knowledge games (PhD thesis) (ILLC, University of Amsterdam, 2000). [8] B. P. Kooi, Knowledge, chance, and change (PhD thesis) (ILLC, University of Amsterdam, 2003). [9] J. L. González Sanz, El arte del dominó (Editorial Paidotribo, 2000). [10] F. R. Velázquez-Quesada and F. Hernández-Quiroz, A Logical Language for Dominoes, Short paper at 12th LPAR, Montego Bay, Jamaica, 2005, geoff/conferences/lpar- 12/ShortPapers.pdf [11] S. A. Kripke, Semantical analysis of modal logic I. Normal modal propositional calculi, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 9, 1963, [12] A. Baltag, L. S. Moss, S. Solecki, The Logic of Public Announcements, Common Knowledge and Private Suspicious (Technical Report SEN-R9922, CWI, Amsterdam, 1999). [13] S. Druiven, Knowledge development in games of imperfect information (MsC thesis) (Institute for Knowledge and Agent Technology (University Maastricht) & Artificial Intelligence (University of Groningen), 2002). [14] E. Schwarz, An instance of a complete communication cycle within co-operative games: the case of Domino, Draft, 2001 [15] M. S. Benkmann, Visualization of Natural Deduction as a Game of Dominoes, article.html [1] J. Hintikka, Knowledge and Belief (Ithaca, N.Y.: Cornell University Press, 1962). [2] R. Fagin, J. Halpern, M. Vardi, A model theoretic analysis of knowledge, Proc. 25th Annual Symp. on the Foundations of Computer Science, [3] R. Fagin, J. Y. Halpern, Y. Moses, M. Y. Vardi, Reasoning about knowledge (The MIT Press, 1995). [4] J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior (Princenton University Press, 1944).