INDEPENDENCE-FRIENDLY EPISTEMIC LOGIC AND SOME NON ...

INDEPENDENCE-FRIENDLY EPISTEMIC LOGIC AND SOME NON-FIRSTORDERISABLE CONCEPTS AHTI PIETARINEN Abstract. Recent developments in epistemic logic and game-theoretic

semantics have led to the partial convergence of the two, as well as new powerful logical formalisms going beyond ordinary linear rst-order frameworks. It is argued that these new independencefriendly (IF) epistemic logics together with Hintikka{Sandu semantic games of imperfect information are needed in the formalisation of a class of concepts called mutual focussed attitudes of multiple agents, and that these concepts are dicult to adequately be represented using linear perfect information rst-order epistemic logics. In particular, the problem of intentional identity, a special case of focussed attitudes involving a common focus of two or more agents towards non-speci c individuals, admits of a resolution by means of epistemic logic with imperfect information semantic games. In general, the regulations on how information ows in games for epistemic logic is seen to give rise to various new IF epistemic languages. Apart from providing representational frameworks for several non- rstorderisable concepts, potential applications include information ow in communicating systems, knowledge representation in multi-agent architectures, and semantics of natural language in general.

1. Introduction While adding quanti ers to propositional epistemic logic adds a great deal of expressivity to the basic propositional framework for representing and reasoning about knowledge, there are surprisingly simple types of knowledge-related concepts and natural language expressions that cannot be represented even in the rst-order epistemic logic. For the adequate treatment of some of these concepts, one can use Hintikka's independence-friendly (IF) rst-order epistemic logic [3], [4], [5], [6] Date : April 1999.

The work on this paper has been supported by Osk. Huttunen Foundation, Jenny and Antti Wihuri Foundation and Osk. O und Foundation. I would like to thank Bill Keller, Gabriel Sandu, and anonymous referees for valuable comments. 1

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which extends the usual rst-order epistemic logic by relaxing the assumption that intensional epistemic operators, quanti ers and connectives are linearly ordered. IF epistemic logic allows for nonlinear ordering between logical components by using an outscoping method to scope some quanti ers, connectives and attitude operators out from the scopes of other attitude operators. For one thing, this would amount to partially ordered operator-quanti er structures, but in the course of this paper, it is seen that several other generalisations for possible patterns of dependence and independence are perfectly conceivable as well. Semantics for these languages are, following Hintikka{Sandu semantic games for partially ordered (Henkin) quanti ers, given by means of games of imperfect information, extending games also to cover possible worlds structures. These IF epistemic logics are studied here further. In particular, it is investigated what can be said in favour of them. It turns out that the classical problem of intentional identity can be captured in the multipleagent extension of IF epistemic logic. Also, the general notion of mutual focussed attitudes becomes expressible. It is argued that both notions resist any forthright formulation in ordinary linear intensional frameworks, and it is this what is meant here by \non- rstorderisability." However, the special case of cyclic operator-quanti er pre xes, which Hintikka has assumed to be the right logical counterparts for functional readings of the sentences such as \I know who everybody admires," cannot quite be expressed in this IF epistemic logic involving only a method of outscoping, nor are certain mutual nested focussed attitudes expressible. For these concepts, a new extension is proposed which involves a novel notion of inscoping. 2. Independence-friendly epistemic logics 2.1. Propositional subsystems. To start with, consider a propositional case of IF epistemic logic. Many observations in this section also carry over to the quanti ed case. The well-formed formulas (w) of an ordinary perfect information propositional epistemic logic LProp are formed as: ' ::= S j Ki ' j Li ' j ' ^ j ' _ j ': The formula Ki' is read as \an agent i knows that '," and Li', which is a dual Ki' is read as \it is possible, for all that an agent i knows that '." Let Mi 2 fKi; Lig, and let the set of all propositional symbols S be . A model for LProp is M = hW ; R; i, where is a valuation function assigning to each proposition letter S a subset (S ) of a set of possible worlds W = fw0 : : :wn g, and R = f1 : : :n g is a set of

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accessibility relations for each agent i = 1 : : : n, i W W . Let us write w1 2 [w0]i to denote that a possible world w1 is i-accessible from w0. Let Sub(') be an inductively de ned set of subformulas of '. The following de nes propositional epistemic logic with imperfect information. De nition 1. Let LProp be a propositional epistemic logic, and let Prop ' 2 L be in a negation normal form, where all the negation signs have been pushed inside the formulas to immediately precede atomic formulas. Let Mj be Sub('), and let A = fM1 : : :Mn g such that all Mi 2 A; i 2 f1 : : : ng exist in ' and Mj is in the scope of any Mi . Then IF propositional epistemic logic LProp O is obtained by the applications of the following: If B A, then (Mj =B ) 2 LProp O . An intuitive meaning of the slash operator \/" is that the information considering B is not usable at Mj , or not allowed to reach it. The slash is called an outscoping operator. Example 1. The followings are well-de ned formulas of LProp O : K1(K2 =K1) '; K1(' ^ (L1=K1 ) ): Since subformulas of LProp O may not receive an interpretation as such, a straightforward inductive truth-conditional semantics fails to provide interpretations for complex formulas. However, a fairly simple noncompositional method of giving semantics for LProp O is possible by means of games. De nition 2. A game is a tuple G = hP; M; Oi, where P = hV; F i is a tuple of two teams of players V (the veri er) and F (the falsi er). V and F themselves consist of nite sets Vi; Fj ; 1 i; j n of individual members of the teams. M is a set of well-de ned choices, and O is a set of well-de ned positions in a game. A game model is a game G('; w; ). The set M for LProp O consists of the following six rules, where the antecedents are the inputs I of each rule mi 2 M, and the consequents are the outputs of mi 2 M. m1 : If ' = , V and F change roles, and the next choice is in G( ; w; ). m2 : If ' = ('1 _ '2), V chooses i 2 f1; 2g, and the next choice is in G('i ; w; ).

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m3 : If ' = ('1 ^ '2), F chooses i 2 f1; 2g, and the next choice is in G('i ; w; ). m4 : If ' = Li , and the game has reached w, V chooses a possible world w1 2 [w]i , and the next choice is in G( ; w1; ). m5 : If ' = Ki , and the game has reached w, F chooses a possible world w1 2 [w]i , and the next choice is in G( ; w1; ). m6 : If ' is atomic, the game ends, and V wins if ' is true, and F wins if ' is false. These games can be represented in extensive form, which is a tree with labelled edges as choices and nodes as current positions in a game. The nonterminal nodes are partitioned into information sets Sij for each Pj , which represent the knowledge of the players such that for any two nodes at the same depth d in one information set, a respective player cannot distinguish between these nodes, and thus does not know everything that has happened earlier in a game. The non-singleton information sets thus give a semantic method of representing imperfect information, which syntactically is expressed in a language by means of the outscoping. De nition 3 (Non-repetitive extensive form of games). A game tree GT (together with the usual underlying components of a basic tree structure) for n-person game G on ' is a nite partially ordered set of well-de ned moves with the following constraints and de nitions: 1. the distinguished node v0 2 V is associated with the formula '; 2. each node v 2 V ? v0 is associated with Sub('); 3. all non-terminal nodes N of GT are partitioned into jP j + 1 sets S1 : : : SjP j, called the member sets ; 4. for each member Pj 2 P , Si is sub-partitioned into subsets Sij , called information sets, such that any two nodes in the same information set have the same number of immediate successors; 5. if the nodes = Sub(') and = Sub( ) lie on the same directed path, then 2 Sij and 2 Sik for all j 6= k. De nition 4. A play of a game G is a maximal set of choices in a path. The choices not in a play are alternatives to it. Immediate successors of a choice are alternative choices of the choice. The condition (5.) in De nition 3 ensures that GT s satisfy a nonrepetition hypothesis or non-absentmindedness: each node in the same information set may be visited at most once during one play. This guarantees that the information for individual members is persistent, that is, members of both teams of players have perfect memories in the sense that they do not forget information. However, at the level of


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players the information is non-persistent, and it is this way the notion of forgetting, or imperfect recall, which is an inevitable phenomenon of IF logics, is modelled. Example 2. Let K1L1(L2=K1 ) 2 LProp O . Since rst F picks up a possible world, say w1, then V picks up a possible world, say w2, and then V is to pick up a possible world without knowing which world F chose, she is not allowed to know her previous choice for L1 either, since this would indirectly tell her the choice for K1 as well. Therefore, in the third step, V splits in two, where V has the information about w1, and the new member V1 makes a choice for L2. Viewed as a whole, however, V is seen to forget some information during the play of the game. Consequently, one could speak of games of imperfect recall in addition to imperfect information. De nition 5. A strategy for Pi 2 P in a game G for ' is a function f : W \ r(N ) ! s(I ), where r : N ! fVi ; Fj g is a player function assigning to each non-terminal node v 2 N a member of a team Pk , and s : I ! W [f1; 2g[ r(N ) gives an output of an application of each rule mi 2 M, given an input I of each rule, which is a possible world, a value in f1; 2g, or an instruction to change roles at v 2 V . A winning strategy is a strategy f by which a player can make operational choices such that every play results a win for him or her, no matter how the opponent chooses. De nition 6. A formula ' is true in M, i.e. (M; w; ) j= ', i there exists a winning strategy f for the initial V , and false in M, i.e. (M; w; ) j= 6 ', i there exists a winning strategy f for the initial F . De nition 7. A game G is determined, i for every play on ', either V has a winning strategy f in G or F has a winning strategy f in G. Observation 1. Games G for LProp O are not determined.

Proof. De ne M such that w1 2 [w0]1 ; w2 2 [w0]1 ; w1 2 [w1]2 ; w2 2 [w2]2 ; w1 2 [w2]2 ; w2 2 [w1]2 , and ( ; w1) = true, ( ; w2) = false. Now, since S1V = fF [w1]; F [w2]g, she cannot always choose the world such that ( ; w1) = true. F does not have a winning strategy either, because ( ; w1) = true. 2 From this undeterminacy it follows that the law of the excluded middle fails in LProp O , as it usually does under the presence of imperfect information.

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2.1.1. From games to Kripke models. What are the Kripke models for epistemic logic with imperfect information? In other words, how does the hiding of possible worlds in above imperfect information games aect Kripke structures? These questions are best answered if one rst has some way of translating an extensive form of semantic games into Kripke models. Let us propose one way of doing so. De nition 8. Let GT be an extensive form game tree for 2 LProp O , and let d be a depth of a node v 2 V in GT . Then the translation M(GT ()) of a game tree GT of into a Kripke model M is done by applications of (1){(5). 1. (a) For = ('1 ^ '2) at v 2 V , remove edges i 2 f1; 2g, and attach f'1; '2g to v. (b) For = ('1 _ '2) at v 2 V , remove edges i 2 f1; 2g, and attach f'1; :'2g or f:'1; '2g to v. 2. Merge all v1 : : : vn in GT for which d(v1) = = d(vn ) together with their predecessor edges which have the same label wi. 3. (a) Name the root with w0. (b) Change edges to directed edges, following the direction of game choices. (c) Move all the labels from edges to the nodes at the end of edges, and label edges with accessibility relations j corresponding to Pj making a choice in that edge. 4. Change every directed edge with wi occurring at both ends of an edge to a re exive relation wi 2 [wi]j . 5. For those worlds wi which correspond to nodes vi in the same information set Skj , create an equivalence relation j between every i. The translation M(GT ()) will be called a game-Kripke translation, and its primary purpose is to simplify games and view them as Kripke structures. Proposition 1. 9f , given ' 2 LProp O which is winning for r(v0 ) = Vi in G, i (M; w; ) j= ', and 9f , given ' 2 LProp O which is winning for r(v0) = Fi in G, i (M; w; ) j= 6 '. Proof. Note rst that the merging of nodes in (2) does not aect truth-values, since the subtrees in all the nodes v1 : : :vn to be merged are the same. The proof is by induction on : = ' is true, i (M; w0; ) j= ', i (M; w0; ) j= 6 '; = ('1 _ '2) is true, i 9f which is winning for r(v0) = Vi in G, i (M; w0; ) j= '1 or (M; w0; ) j= '2, i (ind.ass.) (M; w0; ) j= ('1 _ '2); = Li' is true, i 9f which is winning for r(v0) = Vi in G, i (ind.ass.) 9w0; w0 2 [w0]i such that


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(M; w0; ) j= '. The cases for r(v0) = Fi are dual. Finally, the step (5) adds an equivalence relations to Kripke models, but since subtrees for nodes within the same information set Sij are the same, all the possibilities are available in the same manner for Pi 's as they are w.r.t. corresponding i's in M. 2 What do the IF formulas of LProp O generally mean? Does this language extend the expressive power of propositional epistemic logic, and if so, how much? Intuitively, the formula K1(K2=K1 )' says that 1 knows that ', and that 2 knows that ', but that there is no nesting, that is, 1 does not know that 2 knows '. Therefore, it seems that this formula is only a shorthand and a more compact version of the formula K1 ' ^ K2' (or K1' ^ K2'^ K1K2 '), and other formulas could be rewritten in a similar manner. But where do the extra conjunctions come from, for the former represents only one proposition which is the focus of two knowers? I suggest that the rst formula is best read as \1 and 2 know that '," but it does not mean the same as the distributive version \1 knows that ' and 2 knows that '." They both imply that no nesting occurs, but in the former, there is no conjunction involved, as there is no conjunction present in the former formula. These readings really are dierent, for the former answers to the question \Who knows that '?" whereas the latter is not as good an answer to the same question. For example, I can ask \Who knows that it will be rainy tomorrow?" and you can answer \Sam and Peter know that it will be rainy tomorrow," but if you answer \Sam knows that it will be rainy tomorrow and Peter knows that it will be rainy tomorrow" you give a wrong impression after asserting the rst conjunct, namely that Peter knows something else than \it will be rainy tomorrow." Consider next a formula K1(' ^ (K2=K1 ) ). One can likewise try to rewrite this as K1(' ^ ) ^ K2 . So what do we gain by asserting the former version? Again, the rst formula would perhaps be read as \Sam knows that it will be rainy tomorrow and Sam and Peter know that it will be cold tomorrow," and the latter as \Sam knows that it will be rainy and cold tomorrow and Peter knows that it will be cold tomorrow." Now, if I am to ask \Who knows that it will be rainy and cold tomorrow?" the latter seems to be a better answer for this question than the former, concordant with the earlier example. 2.1.2. Bisimulations. The answer to the question of whether the two Prop really are equivalent is given by considering languages LProp O and L their models. An ecient way to do so is by way of bisimulations, which gives an important relation between models in a modal setting.

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De nition 9. Let M = hW ; R; i and M0 = hW 0 ; R0; 0i be two modProp els of a language L . A non-empty binary relation Z W W 0 0

is a bisimulation between M and M , if the following three conditions hold. 1. If w0 2 [w]Z , then w and w0 satisfy the same proposition letters p 2 . 2. If w0 2 [w]Z and v 2 [w]i , then there exists some v0 2 W 0 such that v0 2 [v]Z and v0 2 [w0]i . 3. If w0 2 [w]Z and w0 2 [v0]i , then there exists some v 2 W such that v 2 [v0]Z and v 2 [w]i . The bisimulation allows mutual tracing in two models of LProp. When Z is a bisimulation between two worlds w and w0, the worlds are bisimilar. Bisimulation is notated as M M0, and bisimilarity as w w0. The de nition is easily extended to the general case with many accessibility relations i and with all the possible worlds in a model. Let us assume that bisimulation is de ned in the same manner for LProp O . Now, we can see that the linguistic considerations are somewhat instructive, and the alleged rewriting of the formulas in LProp O into their perfect information counterparts fails, as predicted per the failure of determinacy. Proposition 2. The class of models for formulas in LProp O and their Prop rewrite form in L are not bisimilar. Proof. By a counterexample. First, consider = K 1(K 1=K 2)' and its game-Kripke translation M(GT ()). In the resulting model N, w1 2 [w0]1 ; w1 2 [w1]2 ; w1 2 [w2]2 and w2 2 [w0]1 ; w2 2 [w2]2 ; w2 2 [w1]2 . Also, ('; w1) = true, ('; w2) = false. Now, consider = K1' ^ K2 ', and a model M, for which w10 2 [w00 ]1 ; w20 2 [w00 ]1 , and ('; w10 ) = true, 6 N, since for w10 2 [w1]Z and w1 2 [w1]1 , ('; w20 ) = false. Now, M

2 there is no world w 2 [w10 ]1 for Z . For example, K (L=K ) and LK are not bisimilar in models where W = fw1; w2g; W = fw10 ; w20 g, since w2 and w20 do not satisfy the same proposition letters. This is how things should be: the formulas are not strongly equivalent (have the same truth-value in the class of all models) under semantic games, since in the latter, F has a broader set of winning strategies available. The next question concerns the expressivity of LProp O . 2.1.3. Modal translation. One way of measuring the expressive power of propositional epistemic logic is by translating it to other languages, and looking at the resulting relation, which would tell which properties of 0

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models can be de ned using epistemic language. An obvious candidate for this translation is rst-order logic, since for ordinary propositional modal logic, the standard translation to the special fragment of rstorder logic says that a class of models is de nable by modal formulas only if the class of models is de nable by a rst-order class of models. In particular, all propositional modal formulas are equivalent to rstorder formulas with one free variable, n +1-ary relations corresponding to each modal operators, and unary predicates corresponding to each propositional symbols. The most reasonable way of proceeding with the translation of LProp O is to carry the imperfect information over to translated rst-order formulas. First, we have to de ne such a rst-order language. Let us adopt an outscoping device again, which, as one thing, allows us to express Henkin quanti ers in a linearised format. De nition 10. Let ' 2 L0 be a well-formed rst-order formula in a negation normal form, and let Qx ; Q 2 f8; 9g and ; 2 f^; _g be Sub(') which are in the scope of Q1x1 : : : Qnxn. Let A = fx1 : : : xng. Then the w`s of L are formed by the applications of the following. If B A, then f(Qx=B ) ; (=B )g 2 L. The set fx1 : : : xng is written as x1 : : :xn . Let us call L IF ( rst-order) logic (for a comprehensive treatment of similar logic, see [4], and an alternative \trump semantics" to game semantics presented in [7], [8]). Example 3. The followings are well-formed formulas of IF logic: 8x1 : : : 8xn(9y=x1 : : :xn ) Sx1 : : : xny; 8x9y (S1x(_=x)S2y): The Henkin quanti er H is de ned in IF logic as follows: 8 x 9 y 8x9y(8z=x; y)(9u=x; y) Sxyzu 8z 9u Sxyzu: De nition 11. Let x; y; yi be new variables, and let Var(')Prop be inductively de ned set of variables (free and bound) of ' 2 LO . The translation Tx from ' to IF logic with unary predicates and n + 1-ary relation symbols is de ned as follows. Tx(p) = Sx Tx( ) = Tx( ) Tx( ^ ) = Tx() ^ Tx( ) Tx(Li ) = 9y(Rxy ^ Ty ( )) Tx(Ki ) = 8y(Rxy ^ Ty ( )) Tx((Mi=Mj ) ) = (Qyi=zj )(Rxyi ^ Ty ( )):

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In the last clause, Q = 8 if Mi = Ki , and Q = 9, if Mi = Li. Since Tx proceeds from outside in, and (Mi=Mj ) is Sub(') where zj 2 Var(') at the last clause of the translation, it is known at that point which variable the right-hand side of the slash should be in the translated clause. Example 4. Consider rst the propositional formula K1L2 . Its translation goes on as follows: Tx(K1L2 ) = 8y(Rxy ^ Ty (L2 )) = 8y(Rxy ^ 9y1(Ryy1 ^ Ty1 ( ))) = 8y(Rxy ^ 9y1(Ryy1 ^ Sy1)): Example 5. Consider next the IF formula K1(L2=K1) . Its translation goes on as follows: Tx(K1(L2=K1 ) ) = 8y(Rxy ^ Ty ((L2=K1 ) )) = 8y(Rxy ^ (9y1=y)(Ryy1 ^ Ty1 ( ))) = 8y(Rxy ^ (9y1=y)(Ryy1 ^ Sy1)): Since in the output of the translation the existential quanti er 9y1 is independent from the universal 8y only, it is true in the same models as 9y18y(Rxy ^ Ryy1 ^ Sy1), which is in L0. Example 6. As an example of a formula whose translation cannot be rewritten in a linear rst-order notation, consider K1 L1K2 (L2=K1) . Its translation (omitting intermediate steps) is as follows. Tx(K1L1K2(L2=K1 ) ) = 8y(R1xy ^ 9y1(R1yy1 ^ 8y2(R2y1y2^ (9y3=y)(R2y2y3 ^ Sy3)))): This IF formula is equivalent to the Henkin quanti er formula (1): 8y 9y1 (R1xy ^ R1yy1 ^ R2y1y2 ^ R2y2y3 ^ Sy3): (1) 8y2 9y3 It follows that the language LProp O is indeed strong, as it even goes beyond ordinary rst-order logic. However, the resulting Henkin quanti er language is only a fragment of an unrestricted language of Henkin quanti ers LH , for it requires only a nite number of variables, one free variable, unary predicate symbols, and n-ary relation symbols. Proposition 3. If LProp O does not contain more than n distinct epistemic operators Mj ; 1 j n, every ' 2 LProp O is equivalent to a Henkin quanti er formula H with at most n + 1 variables. Proof. Let W V = fM1 : : : Mng, where ' contains all Mj 2 V , and let Q 2 f8; 9g. The cases for boolean operator are obvious. Consider a sequence of translations Tx1 ((M1=W ) ) = Qx2(R1x1x2 : : : ),


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Tx2 ((M2=W ) ) = Qx3(R2x2x3 : : : ), Tx3 ((M3=W ) ) = Qx4(R3x3x4 : : : ) : : : Txn ((Mn=W ) ) = Qxn+1(Rn xnxn+1 : : : ). Thus, after n operators Mn the translation has used n + 1 variables. 2 Just how much LProp O extends the rst-order logic is an open question, as the properties of such a restricted fragment of Henkin quanti ers are generally unknown. It is known that in a logic with Henkin quanti ers with unary predicates only (without equality), Henkin quanti ers can be eliminated (see [10]), resulting in an equivalent logic with an ordinary rst-order one. On the other hand, the theory of in nitely many unary relations in a logic with Henkin quanti ers is decidable, but the theory of in nitely many binary relations is undecidable (see [9]). In fact, almost all theories are undecidable in LH . If LProp O is undecidable, the formulas in it are not preserved under bisimulation. Remark. Rather than considering the properties of a fragment of LH , that is, some properties of a fragment of second-order 11-languages, one should also pay attention to possible restricted subfragments of fragments of LH , much in the same manner as one considers guarded fragments of rst-order logic. In such guarded fragments of Henkin quanti ers, it makes a dierence whether a guarded formula 9y(Rxy ^ xy) (x; y are nite sequences of variables) exists in the scope of another guarded formula, such as 8z(Ruz ^ uz), since variables in the guard Rxy may become free in the matrix xy for some x1 2 x z, if imperfect information is allowed. However, if variables in y1 in (9y=y1)(Rxy ^ xy) do not occur in the matrix the presence of imperfect information seems admissible. 2.1.4. Translation with connectives. Consider next an extension of Prop LO to imperfect information with connectives. There, one can have formulas such as K1('(_=K1 ) ). Let us call the language LProp C . Now, K1('(_=K1 ) ) obviously is logically equivalent with K1'_K1 . These can both be read as \John knows that it will be rainy tomorrow or John knows that it will be cold tomorrow." However, this means that \John knows whether it will be rainy or cold tomorrow." Consequently, the mechanism of imperfect information seems to provide a formulation for knows whether locutions. However, more complex formulas such as K1K2 ('(_=K1) ) do not generally reduce to propositional epistemic logic. One cannot rewrite K1K2('(_=K1 ) ) such that 2 knows ' _ , and 1 knows ' or 1 knows , and that the proper nesting K1K2 is preserved. This formula would actually mean something like \John knows whether it will be rainy or cold tomorrow, and that Sam knows that it will be rainy or cold."

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There are no clear-cut examples of a type of knowledge with regard to conjunction, although the same distributiveness indeed holds, i.e. K1' ^ K1 K1('(^=K1 ) ). Without the slash, the latter sentence would mean, for example, that \Mary knows that one should not drink and drive," but it does not mean the same as \Mary knows that one should not drink and Mary knows that one should not drive," for she may fail to conjoin the two propositions which together would mean something more. Again, linguistic examples suggest ways of reasonable interpretation. Proposition 4.Prop The class of models of formulas in LProp C and their rewrite forms L are not bisimilar. 2 Since the extension LProp C to connectives does not reduce to propositional epistemic logic, a question again arises of the strength of it. The translation needs to be calibrated for the slashed IF connectives, but I shall refrain from doing so here. Remark. Restricting attention to partially ordered structures is problematic, for in general formulas also involve the \attitudinal" binding scope (subsumption), which indicates which predicates fall under which operators. It may happen that even though some partial ordering is possible for considering the epistemic operators only, the occurrences of predicates in a matrix prevent bringing the operators together into the same pre x as in Henkin quanti ers. Further, modal operators themselves block the quanti ers o moving to prenex normal form positions. These are yet another examples of such situations where the perspective of game-theoretic semantics is useful, for it allows for an almost inde nite generalisation of the logically possible orderings of components. 2.2. First-order case. We assume the basics of rst-order logic. A quanti ed intensional language LO of imperfect information is formed as follows. Let L be a rst-order intensional language with identity for each signature as: ' ::= S j Ki ' j Li ' j 8x' j 9x' j ' _ j ' ^ j ': De nition 12. Let ' be a formula of L in a negation normal form, and let Qx ; Q 2 f8; 9g be Sub(') in the scope of A = fM1 : : :Mn g. Then LO consist of a well-formed formulas of L together with If B A, then f(Qx=B ) ; (Mi=B ) g 2 LO . A model for LO is an ordered tuple hA; =i, where A is a -structure hW ; Di of a signature of a nonempty set of possible worlds W and a


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nonempty domain D. = is an interpretation h; Ri, where attaches to each possible world a -structure together with an assignment : T ! D from terms to the domain, and R = f1 : : : n g is a set of accessibility relations. Truth is again de ned by games. In addition to propositional case, there are two additional rules in M: m7 : If ' = 9x x, V chooses an individual of the domain of the structure A, gives it a name, e.g., a, and the next choice is in G( [x=a]; w; ; A). m8 : If ' = 8x x, F chooses an individual of the domain of the structure A, gives it a name, e.g., a, and the next choice is in G( [x=a]; w; ; A). De nition 13. Strategies for elements of P are now f : (W X ) \ r(N ) ! s(I ), where r : N ! fVi ; Fj g is as before, X is a set of variables in , and s : I ! D [f1; 2g[W[ r(N ) gives an output of an application of each move. Winning strategies are as before. Proposition 5. Let a formula ' 2 LO be in a negation normal form. Then ' is true, i there exists a winning strategy f for V , and false, i there exists a winning strategy f for F . Proof. By induction on '. If ' is a literal, then the established interpretation = determines its truth value; ' = ( _ ): Now (M; w; ) j= ', i (M; w; ) j= or (M; w; ) j= , i (ind.ass.) 9f which is winning for V in G( ; w; ; A) or 9f which is winning for V in G(; w; ; A), i 9f which is winning for V in G('; w; ; A), i true, since V can win G('; w; ; A) from both possible positions; ' = (9x) [x]: Now (M; w; ) j= ', i (M; w; [x=c]) j= [x] for some individual constant c 2 C , i (ind.ass.) 9f which is winning for V in G('; w; ; A), i (M; w; ) j= [c], where c is a Skolem constant, i (M; w; ) j= [x=c], i true; ' = Mi : Now (M; wi; ) j= ', i there exists wj 2 [wi]i such that (M; wj ; ) j= ', i V can win G('; wj ; ; A), i true, since V makes a move and there can always exist a win-making choice for her. The cases for F are dual. 2 Again, the only dierence between perfect information games and imperfect information games is that in the latter, winning strategies are de ned on a shorter sequence than in perfect information games, for some elements in fW X g may be hidden from the subsequent players. Example 7. A strategic (skolemised) form of a perfect information formula K18x9y(S1xy _ K29zS2xyz) is 9fghk K18x((S1xf (w1; x) ^

14

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g(w1; x) = 1) _ K2(S2xf (w1; x)h(w1; w2; x) ^ g(w1; x) = 0)). A strategic form of an imperfect information formula K18x(9y=K1) Sxy is 9fK18x Sxf (x). Without the slash, however, it is 9fK18x Sxf (w1; x). The dependencies of Ki 's from Li's, or Mi's from Mj (when Mi = Mj = Km or Mi = Mj = Lm ), do not make dierence in a strategic form, since despite the imperfect information these formulas are equivalent up to truth. The dependencies of L's from K 's do make a dierence, for it was seen that the models in general are not bisimilar, but that this kind of strategic dierence nonetheless does not have to be formalised in this account, for the fairly ne-grained notions of \game-logical equivalences" going beyond usual logical equivalences are not essential here. Several consequences ensue. First, the semantics only seems to be operational under a common domain assumption. This is seen by considering the formula Ki(9x=Ki) Sx. First, F chooses a possible world, and then V chooses an individual, without knowing which possible world F has been chosen. But V cannot make a move if she is ignorant of the relevant domain she should operate on, unless knowledge of the domain is forced to be the same for every possible world. Second, the formula Ki (9x=Ki) Sx is only weakly equivalent with 9xKi Sx, since they do not have to be false in the same models, because in the latter case the choice for the individual is included in the information set of F , expanding his available set of winning strategies. In fact, Ki(9x=Ki ) Sx is not strongly equivalent with any linear ordering of components. There is yet a further interesting interplay between a possible worlds model and semantic games. Since the players are able to demonstrate the truth of some formulas against any choice of the possible world by their opponent (as in Ki(9x=Ki ) Sx), they can also be said to have some kind of knowledge at hand. Indeed, it turns out that the game model can be recast in a possible worlds fashion also in the manner where alternatives are all the situations that, for all that a player Pi knows, Pj might have been in. Furthermore, for any two related worlds w0 2 [w]i , Pi follows the same strategies, i.e. f (w0) = f (w). This approach would lead to the closer investigation of the connections between the eventbased Aumann structures and Kripke structures, for instance. Since rst-order modal logic is not preserved under bisimulations, it is not reasonable to expect that our rst-order epistemic logic of imperfect information, which is a more expressive formalism, would be preserved under bisimulations, either. Indeed, the translation Tx, adapted to the rst-order case for a formula K19z1K2 (9z2=K1) Sz1z2


15

in LO yields to (2): 8y0(R1xy0 ^ 9z18y1(R2y0y1 ^ (9z2=y0) Sz1z2)); (2) which is de nable in LH , although again in a restricted fragment of it. From this translation into IF rst-order logic it is seen that: Proposition 6. LO is not preserved under bisimulations. 2 3. Some non-firstorderisable concepts Hintikka argues that an epistemic IF expression (9x=Ki ) is the right logical counterpart for knowing wh- locutions (such as who, what, where, when), and (_=Ki ) for knowing which or knowing whether phrases (cf. [3], [4]). These expressions sometimes reduce to rst-order formats, but here I give examples of knowledge statements in multiple agents contexts which are not expressible at a linear rst-order level at all. 3.1. Focussed attitudes. As an example, consider the following sentence: John knows that someone has cheated and Mary knows (3) who she is. This sentence can be analysed as an instance of a situation involving both de dicto and de re knowledge. These terms are notoriously vague without further quali cations, so it is taken that they both manifest dierent aspects of knowledge of individuals (nonspeci c vs. crossidenti ed) rather than the distinction between nonspeci c vs. singular propositions. Consequently, both kinds of knowledge can be directed towards the same individual and thus have the same focus. The sentence (3) can now be represented in IF notation as (4): K19x(K1=K2) Cx: (4) The sentences like (3) whose representation calls for IF notation are termed concepts with mutual focussed attitudes. 3.2. Intentional identities. An adjacent concept for mutual focussed attitudes is a well-known notion of intentional identity. The preferred reading of the statements of intentional identity represents situations where two or more attitudes are directed towards the same but possibly nonspeci c individual, regardless of whether there is such an individual. The concept was rst introduced in [2], and the classical example is (5): Hob knows that a witch has blighted Bob's mare, and Nob (5) knows that she has killed Cob's sow.

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Because of the scoping problems together with the anaphoric pronoun in the last conjunct, (5) cannot be captured in rst-order logic. One could nonetheless try to analyse (5) as involving imperfect information, as in (6):

TH 9x(Wx ^ Bx ^ (TN =TH ) Kx):

(6)

Here, the second though-operator TN is exempted from the scope of the rst operator, to prevent the nesting. However, since the choices prompted by TH and TN are made by the same team of players, this formula is weakly equivalent to the perfect information formula where the slash is omitted. Thus, the latter thought is towards a speci c individual, which is against the preferred reading. Therefore, a better formulation would resort to identity clause and two existentially quanti ed variables, together with imperfect information. This approach amounts to (7):

TH 9x(Wx ^ Bx ^ (TN =TH )(9y=TH )(Wy ^ Ky ^ x ' y)):

(7)

In (7), all the requirements for the preferred reading are met. Only nonspeci c individuals are concerned with regard to Hob's and Nob's thoughts. The anaphoric binding is taken care of by using an identity statement. No free variables occur, and no excess thoughts are created. A semantic explanation by means of games remains to be done. Let be (7). The game G(; w0; ; A), given a supply of possible worlds fw0 : : :wng 2 W , is as follows. 1. F chooses a possible world w1 2 [w0]H , and the game continues as G(9x(Wx ^ Bx ^ (TN =TH )(9y=TH )(Wy ^ Ky ^ x ' y)); w1; ; A). 2. V chooses an individual, say a, and the game continues as G(Wa^ Ba ^ (TN =TH )(9y=TH )(Wy ^ Ky ^ a ' y); w1; ; A). 3. F chooses a conjunct, say the last one, and the game continues as G((TN =TH )(9y=TH )(Wy ^ Ky ^ a ' y); w1; ; A). 4. F introduces a new member F1, who chooses a possible world w2 2 [w1]N , without knowing F 's rst move, i.e. S1F1 = fF [w1] : : :F [wn]g (so he knows which conjunct he is working on), and the game continues as G((9y=TH )(Wy ^Ky ^a ' y); w2; ; A). 5. V introduces V1 (9x and 9y are assumed independent), who chooses an individual, say b, having S1F1 = fF [w1] : : :F [wn]g (this individual b has to exist in w2), and the game continues as G(Wb ^ Kb ^ a ' b; w2; ; A). 6. F chooses one of the conjuncts, say Kb. Given an interpretation =, if (M; w2; ) j= Kb, V wins, if (M; w2; ) j=6 Kb, F wins.


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The Hob-Nob sentence (7) is true, i there exists a winning strategy for V , i.e. i 9fg TH (Wf (w1)^Bf (w1)^(TN =TH )(Wg(w2)^Kg(w2)^f (w1) ' g(w2))): 4. Further extensions and concepts In this section, I propose another extension of IF epistemic logic which allows even more freedom for the possible patterns of dependence and independence between attitudes and quanti ers. Indeed, the outscoping device yields partial orders, but in general there is no hindrance in considering (possibly non-transitive) multi-graphs, too. This does not make sense in rst-order logic, since all the possible orders reduce to partial orders (ignoring the subtle dependencies between the components associated with the same team of players), but in intensional rst-order logic we have more ingredients, which complicates the matter. For example, one could think of orders such as (8): 9y ; K1 ; 8x ; K2 ; 9y: ( ; denotes the information ow) (8) Here the associated dependency graph is cyclic. In general, one can have situations where an expression is contextually dependent on something which resides inside the formula, and not only further out in a formula, as in the previous examples of IF logics with an outscoping mechanism. For the sake of simplicity, connectives are ignored. Our new IF epistemic logic has the following ws. De nition 14. Let '; 2 LO be in a negation normal form and let (Qx=Mi) be Sub('). Let A = fx1 : : : xng; B = fy1 : : :ymg be possibly non-disjoint sets such that for all xi 2 A; yi 2 B; xi 2 Var(') and yi 2 Var('). Let C = fM1 : : : Mk ; Mk+1 : : : Mmg such that each Mi ; i = 1 : : : k occurs in '. Then LOI consists of well-de ned formulas of LO together with: For any xi 2 W A for which (Qxi=Mk+1 ), (Mk+1 nW )' 2 LOI . For any yi 2 B; Mj 2 C occurring outside the priority scopes of M1 : : :Mk , if U fB; C g, then (Mi nU ) 2 LOI; i = 1 : : : k. Example 8. The following are ws of LOI : (K1nx)(K2=K1)9x Sx; 9xK1(K2nK1; x) '; 8x(L1nx; y) Sxy ^ K19y Sy: The following is not a w of LOI: (K1nx)(K2nK1)8x Sx:

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The last example gives an example of a sentence which is not wellformed. There are reasons why it cannot be, for it does not make sense to say that some one individual is simultaneously both de dicto and de re for the same agent, although the usual conventions QxKi ' ) KiQx ' and Ki Qx ' ) 6 QxKi ' might hold. Whereas the forward slash \=" is called an outscoping operator, the backslash \n" shall be termed an inscoping operator. The meaning of outscoping is as before. The intuitive meaning of inscoping is that it explicitly indicates which information is allowed to ow into the context of intensional operators. In particular, it allows quanti cation into the binding scopes of those knowledge operators whose scopes the variables are outscoped. Consequently, using inscoping it is possible, for some variables, to be quanti ed into the scopes of modal operators outside the scope of the quanti ers that bind those variables, so one has a

avour of dynamics here. In another words, intensional operators can entrap those variables that are outscoped from it but that may occur in a scope of some other intensional operators or quanti ers that are part of the subformulas of those inscopable operators. Semantic games for inscoping need fairly radical adjustments, since games are no longer just from outside in, as there might be some goto jumps inside the formula to nd an eective starting point for the evaluation. Further, if the associated dependency graph is cyclic, there are no eective starting points at all. In this case one needs to play n concurrent games for n logical components in a cyclic pre x. By playing concurrent games, we are able to prevent repetition, that is, the reoccurrences of choice points within the same information set. We thus also avoid in nite plays of the game, whose characteristics are not well studied. The details of game-theoretic semantics for LOI are presented elsewhere.

Proposition 7. The set of valid formulas of LOI is not recursively enumerable.

Proof. K1(L1nK1)K2(L2nK2) Sxyzu i 9fg K1 K2 Sw1f (w1)w2g(w2), which de nes a Henkin quanti er H = fX W j 9fg on W such that for all w1; w2 2 W , hw1; f (w1); w2; g(w2)i 2 X g: 2

Corollary 1. The formulas of LOI are not preserved under bisimula-

tion.

2

4.1. Nested mutual focussed attitudes. The logic LOI allows representations of concepts that resist, I submit, any formulation in LO


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(or likewise in L). A case in point is (9): John knows who the thief is and that Mary knows that she (9) exists. The representation for this in LOI is (10): (K1n9x)(K2nK1)(9x=K1) Tx: (10) The information ow in (10) is cyclic, and therefore it needs concurrent semantic games for its adequate interpretation. 4.2. Functional readings. Functional readings of sentences represent one class of concepts that also resist linear representations, and require inscoping. Functional readings take place if the natural continuation of a sentence such as (11) is a function, such as his best friend, but not an individual like a proper name, or a pair-list continuation such as \Bill invites Suzy, Tom invites John ..." I know who everybody invited. (11) However, a functional reading of (11) involves mutually dependent components 9xK (\knows who"), K 8x (\everybody, not knowing who"), and 8x9y (\whom everybody"). To put all these in one formula requires not only outscoping, but also inscoping to account for cyclic, symmetric dependencies. Thus, the representation of (11) would be a formula of LOI : (K ny)8x(9y=K1) Sxy: (12) There are now two choices to represent (12) in a strategic form. One can represent the existence of a winning strategy either as (13) or (14): 9fK 8x Sxf (x): (13) K 9f 8x Sxf (x): (14) I opt for the latter representation, since it implies that the strategy function itself is known. This is what the functional reading suggests. The information concerning V 's choice for y is passed on to F who makes a choice of a possible world, which indicates that F knows the former choice | at the level of agents, (14) really is the only way of representing this kind of knowledge in strategic form. Further, one could perhaps make a use of a second-order (15) axiom in analysing dierent kinds of knowledge: Ki9f ' ) 9fKi ' (15) This axiom (15) would say that \if i knows the strategy for succeeding in ', then succeeding in ' makes i to know '." However, the converse is likely to fail.

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Finally, in a multi-agent setting, one can consider (16) and its representation in IF epistemic logic with inscoping (17): I know who everybody invited and you know that they (16) (she) exist(s). (K ny)8xK2(9y=K1) Sxy: (17) In conclusion, those concepts that capture the idea of functional readings, such as (11) and (16), as well as those with nested mutual focussed attitudes as (9) present us with one class of concepts that appear to require this kind of novel framework for their adequate modelling. 5. Conclusion Several of the logics presented here are useful in understanding information ow in multi-agent environments, and representing knowledge possessed by individual processes in communicative systems, for instance. This paper aims at providing theoretical frameworks with pointers to potential applications. For one thing, it often happens that one process knows that something has been sent to it by another process, and only the latter process knows the value of the message. This surprisingly simple schema is, however, an instance of IF epistemic logic, not a traditional perfect information one. Consequently, many other non-linear situations are predicted to arise in a logical study of knowledge in mobile processes, some of which are pointed out in this paper. Apart from Hintikka's work, references for nonlinear modal operators are few and far between (but see [11]). In [1, p. 67] (the only reference I am able to nd), it was concluded that \As far as questions of informational independence are concerned, the situation is within limits of fpo. quanti cation theory." All told, I hope to have demonstrated that this is not the case, by way of proposing several IF epistemic logics for concepts that go beyond linear and even partially ordered representations. References [1] Carlson, L. and ter Meulen, A. (1979) Informational independence in intensional contexts, 61{72 in Saarinen, E. et al., (eds.), Essays in Honour of Jaakko Hintikka. Dordrecht: D. Reidel. [2] Geach, P. (1967) Intentional identity, Journal of Philosophy 74, 309{344. [3] Hintikka, J. (1992) Independence-friendly logic as a medium of knowledge representation and reasoning about knowledge, 258{265 in Ohsuga, S. et al., (eds.), Information Modelling and Knowledge Bases III: Foundations, Theory and Applications. Amsterdam: IOS Press.


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[4] Hintikka, J. (1996) Knowledge acknowledged: knowledge of propositions vs. knowledge of objects, Philosophy and Phenomenological Research 56, 251{275. [5] Hintikka, J. (1996) The Principles of Mathematics Revisited, Cambridge: Cambridge University Press. [6] Hintikka, J. and Sandu, G. (1997) Game-theoretical semantics, 361{410 in van Benthem, J. and ter Meulen, A. (eds.), Handbook of Logic and Language, Amsterdam: Elsevier. [7] Hodges, W. (1997) Compositional semantics for a language of imperfect information, Logic Journal of the IPGL 5, 539{563. [8] Hodges, W. (1997) Some strange quanti ers, 51-65 in Mycielski, J. et al., (eds.), Structures in Logic and Computer Science, Lecture Notes in Computer Science

1261, Berlin: Springer. [9] Krynicki, M. and Lachlan, A. H. (1979) On the semantics of Henkin quanti er, Journal of Symbolic Logic 44, 184{199. [10] Krynicki, M. and Vaananen, J. (1989) Henkin and function quanti ers, Annals of Pure and Applied Logic 43, 273-292. [11] Pietarinen, A. (1998) Imperfect Information in Epistemic Logic, 223{234 in Kruij-Korbayova, I., (ed.), Proceedings of the Third ESSLLI Student Session, the 10th European Summer School of Logic, Language and Information, Saarbrucken, Germany. School of Cognitive and Computing Sciences, University of Sussex, Falmer, Brighton BN1 9QH, U.K.

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