Epistemic ATL with Perfect Recall, Past and Strategy Contexts

Epistemic ATL with Perfect Recall, Past and Strategy Contexts Dimitar P. Guelev1 and Catalin Dima2 1 2 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences gelevdp@math.bas.bg Laboratory of Algorithms, Complexity and Logic, Université Paris Est-Créteil France dima@univ-paris12.fr Abstract. We propose an extension to epistemic ATL with perfect recall, past, and distributed knowledge by strategy contexts and demonstrate the strong completeness of a Hilbert-style proof system for its (.U.)-free subset. Introduction Alternating time temporal logic (ATL, [2,3]) was introduced as a reasoning tool for the analysis of strategic abilities of coalitions in extensive multiplayer games with temporal winning conditions. Systems of ATL in the literature vary on their restrictions on the players’ information on the game state, which may be either complete or incomplete (imperfect), and the players’ ability to keep full record of the past, which is known as perfect recall [11,17]. The informal reading of the basic game-theoretic (cooperation) construct Γ ϕ of ATL is the members of coalition Γ can cooperate to enforce temporal condition ϕ regardless of the actions of the rest of the players. Every player is either committed to the objective ϕ, or free to obstruct it. This restiction is overcome in Strategy Logic (SL, [6]), where propositional LTL language is combined with a predicate language interpreted over a domain of strategies to enable flexible quantification over strategies. LTL formulas are evaluated at the unique paths which are determined by dedicated parameter lists of strategies. For instance, assuming just two players 1 and 2, 1(pUq) translates into the SL formula ∃x∀y(pUq)(x, y), where (x, y) indicates evaluating (pUq) at the path determined by 1 and 2 following strategies x and y, respectively. This translation is not invertible in general and ATL is not expressively complete wrt SL. Some practically interesting properties which cannot be written in ATL for this reason are given in [5]. To enable the expression of such properties, ATL was extended by strategy contexts in various ways [20,5,15,21]. Strategy contexts are assignments of strategies to some of the players which the rest of the players can safely assume to be followed. All of the works [20,15,21] are about strategy contexts in ATL with complete information. To facilitate reasoning about games with incomplete information, ATL was extended with epistemic operators [18,11]. Such combinations can be viewed as extending temporal logics of M. Fisher et al. (Eds.): CLIMA 2012, LNAI 7486, pp. 77–93, 2012. c Springer-Verlag Berlin Heidelberg 2012  78 D.P. Guelev and C. Dima knowledge (cf. e.g [7]) in the way ATL extends computational tree logic CTL. A study of the system of epistemic linear- and branching-time temporal logics (without the game-theoretic modalities) which arise from the various possible choices can be found in [19,10]. In this work we embark on the study of an extension of epistemic ATL with perfect recall and past by strategy contexts. Our extension to the language of ATL is different from those in [15,21] but brings the same expressive power for the case of complete information. The language extension we chose has facilitated upgrading our axiomatic system for epistemic ATL with perfect recall from [9] to include strategy contexts by making only the obvious changes. Following [9], the semantics in this paper is based on the variant from [13,14] of interpreted systems, which are known from the study of knowledge-based programs [7]. The main result in the paper is the completeness of our proof system for the ”basic” subset of epistemic ATL with past, perfect recall and strategy contexts. This subset excludes the iterative constructs Γ (.U.), Γ ✸ and Γ ✷, but includes the past operators ⊖ and (.S.), and following [9] again, the operator DΓ of distributed knowledge. The future subset of the system can be viewed as an extension of Coalition Logic [16] as well. The system is compact. This enabled us to prove strong completeness, i.e., that an arbitrary consistent set of formulas is also satisfiable. The proof system includes axioms for temporal logic (cf. e.g. [12]), epistemic modal logic with distributed knowledge (cf. e.g. [7]), appropriately revised ATL-specific axioms and rules from the axiomatization of ATL with complete information in [8] and from the extension of ATL by strategy contexts proposed in [20], and some axioms from our previous work [9]. Structure of the Paper. After preliminaries on interpreted systems we introduce our logic. We briefly review the related logics from [20,15,21] and give a satisfaction preserving translation between our proposed logic and that from [15]. In the subsequent sections we present our proof system for the basic subset of the logic and demonstrate its completeness. 1 Preliminaries C on interpreted systems. An interpreted system is In this paper we define ATLDP iR defined with respect to some given finite set Σ = {1, . . . , N } of players, and a set of propositional variables (atomic propositions) AP . There is also an environment e ∈ Σ. In the sequel we write Σe for Σ ∪ {e}. Definition 1 (interpreted systems). An interpreted system for Σ and AP is a tuple of the form Li : i ∈ Σe , I, Act i : i ∈ Σe , t, V  where:  Li , i ∈ Σe , are nonempty sets of local states; LΓ stands for L i , Γ ⊆ Σe ; i∈Γ I ⊆ LΣe is a nonempty set of initial global states;  Act i , i ∈ Σe , are nonempty sets of actions; Act Γ stands for Act i ; i∈Γ t : LΣe × Act Σe → LΣe is a transition function; V ⊆ LΣe × AP is a valuation of the atomic propositions. Epistemic ATL with Perfect Recall, Past and Strategy Contexts 79 The elements of LΣe are called global states. For every i ∈ Σe and l′ , l′′ ∈ LΣe such that li′ = li′′ and le′ = le′′ the function t is required to satisfy (t(l′ , a))i = (t(l′′ , a))i . In the literature, interpreted systems also have a protocol Pi : Li → P(Act i ) for every i ∈ Σe . Pi (l) is the set of actions which are available to i at local state l. We assume the same sets of actions to be available to agents at all states for the sake of simplicity. For the rest of the paper in our working definitions we assume the considered interpreted system IS to be clear from the context and its components to be named as above. Definition 2 (global runs). Given an n (Act Σe LΣe )n is a run of length |r| = n, if all all runs of  j k< n. We denote  nthe set of <n R (IS) by R (IS) and R (IS) and k<n ≤ ω, r = l0 a0 l1 a1 . . . ∈ LΣe l0 ∈ I and lj+1 = t(lj , aj ) for length n by Rn (IS). We denote R≤n (IS), respectively. We write k≤n Rfin (IS) and R(IS) for R<ω (IS) and R≤ω (IS), respectively. Given m, k < ω such that m ≤ k ≤ |r|, we write r[m..k] for lm am . . . ak−1 lk . We write R[m..k] for {r[m..k] : r ∈ R} in case the lengths of the runs in R ⊂ R(IS) are at least k. Runs of length n < ω are indeed sequences of 2n + 1 states and actions. Definition 3 (local states, local runs and indiscernibility of runs). Given an l ∈ L and Γ ⊆ Σe , we write lΓ for li : i ∈ Γ ; aΓ ∈ Act Γ is defined similarly for a ∈ Act Σe , and indeed for a ∈ Act ∆ with arbitrary ∆ such that Γ ⊆ ∆ ⊆ Σe . Sometimes we write lΓ (aΓ ) just in order to emphasize that the index set of l (a) is Γ . Given r = l0 a0 . . . ∈ R(IS), we write rΓ = lΓ0 a0Γ . . . for the corresponding local run of Γ . Given r′ , r′′ ∈ R(IS) and n ≤ |r′ |, |r′′ |, we write r′ ∼nΓ r′′ if |r ′ | rΓ′ [0..n] = rΓ′′ [0..n] and r′ ∼Γ r′′ for the conjunction of r′ ∼Γ r′′ and |r′ | = |r′′ |. Obviously ∼nΓ and ∼Γ are equivalence relations on R(IS). We denote {r′ ∈ R(IS) : r′ ∼Γ r} by [r]Γ . Sequences of the form r∅ consist of s and [r]∅ is the class of all runs of length |r|. Definition 4 (joins of vectors of actions). Given two vectors ai = ai,j : j ∈ Γi , i = 1, 2, such that Γ1 , Γ2 ⊆ Σe and Γ1 ∩ Γ2 = ∅, we write a1 ∪ a2 for the vector indexed by Γ1 ∪ Γ2 with action (a1 ∪ a2 )j being either a1,j or a2,j , depending on whether j ∈ Γ1 or j ∈ Γ2 . Definition 5 (strategies and outcomes). A strategy for i ∈ Σe is a function of type {ri : r ∈ Rfin (IS)} → Act i . We write S(Γ ) for the set of the vectors of strategies with one strategy for every member of Γ in them. We apply the notation introduced for vectors of actions in Definitions 3 and 4 to vectors of strategies as well. Given s ∈ S(Γ ) and r ∈ Rfin (IS), we write out(r, s) for the set {r′ = l0 a0 . . . ∈ Rω (IS) : r′ [0..|r|] = r, aji = si (r{i} [0..j]) for all i ∈ Γ, j ≥ |r|} 80 D.P. Guelev and C. Dima of the possible outcomes of r whenΓ follow s from time |r| on. Given an X ⊂ out(r, s). Rfin (IS), we write out(X, s) for r∈X Definition 6 (indiscernibility of strategy vector sequences). Given s′ , s′′ ∈ S(Σe ), we write s′ ∼Γ s′′ if s′Γ = s′′Γ . Given two sequences s′ = s′0 . . . s′n , s′′ = s′′0 . . . s′′n ∈ (S(Σe ))n+1 , we write s′ ∼Γ s′′ if s′k ∼Γ s′′k for k = 0, . . . , n. Definition 7 (strategy revision). Given Γ ⊆ Σe , s′ , s′′ ∈ S(Γ ), and an n < ω, we write s′ △n s′′ for the vector of strategies which is defined by the case distinction:  ′ si (r{i} ), if |r| < n; (s′ △n s′′ )i (r) = s′′i (r{i} ), if |r| ≥ n. Definition 8 (consistency of strategy vector sequences). A sequence s = s0 , . . . , sn ∈ S(Σe )n+1 is consistent, if sk (r) = sk+1 (r) for all r ∈ R<k (IS) and all k < n. In words, s is consistent, if, for k > 0, sk+1 returns the same vectors of actions as sk for runs of length up to k − 1. The reason to require sk−1 (r) = sk (r) only C if |r| ≤ k − 1 is that, according to the definition of |= in ATLDP below, for iR k |r| ≥ k, the values of s (r) represent the context strategies to be followed from step k on and these strategies are subject to revision. 2 Epistemic ATL with Perfect Recall, Past and Strategy C ) Contexts (ATLDP iR C ATLDP has an additional parameter ∆ to its game-theoretic operator to desiR ignate the set of the players whose behaviour is assumed to be as described in the strategy context. As it becomes clear below, having a cooperation modality with such a parameter facilitates the use of appropriate variants of the axioms and rules for ATLDP iR from [9]. In Section 3 we explain that this form of the cooperation modality has the same expressive power as (an appropriately defined incomplete-information variant of) the cooperation modalities from [15]. Definition 9 (syntax). Here follows a BNF for the syntax of formulas in C ATLDP and the intended informal reading of the connectives: iR ϕ, ψ ::= ⊥ | p | (ϕ ⇒ ψ) | logical falsehood, atomic proposition, implication ⊖ϕ | ϕ one step ago (ϕSψ) | ψ either now, or some time ago and ϕ has been true ever since ψ held last; | Γ know ϕ; DΓ ϕ Γ | ∆ ◦ ϕ | Γ can enforce ϕ in one step, provided that ∆ follow their current strategies; Γ | ∆(ϕUψ) | Γ can enforce reaching a ψ-state along a path of ϕ-states, provided that ∆ follow their current strategies; [[Γ | ∆]](ϕUψ) | Γ cannot prevent reaching a ψ-state along a path of ϕ-states, unless ∆ give up their current strategies. Epistemic ATL with Perfect Recall, Past and Strategy Contexts 81 Γ | ∆ and [[Γ | ∆]] are well-formed only if Γ ∩ ∆ = ∅. We write Var(ϕ) for the set of the atomic propositions which occur in ϕ. Note that we do not introduce dedicated notation for individual knowledge. Below it becomes clear that Ki can be written as D{i} . C Definition 10 (modelling relation of ATLDP ). The relation IS, s, r |= ϕ is iR fin defined for r ∈ R (IS), a consistent strategy vector sequence s = s0 , . . . , s|r| ∈ S(Σe )|r|+1 , and formulas ϕ, by the clauses: IS, s, r |= ⊥; IS, s, l0 a0 . . . an−1 ln |= p iff V (ln , p) for atomic propositions p; IS, s, r |= ϕ ⇒ ψ iff either IS, s, r |= ϕ or IS, s, r |= ψ; IS, s, r |= DΓ ϕ iff (∀r ′ ∈ [r]Γ )(s′ ∈ S(Σe ))(s′ ∼Γ s implies IS, s′ , r ′ |= ϕ); IS, s, r |= Γ | ∆θ iff |r| (∃s′ ∈ S(Γ ))(∀s′′ ∈ S(Σe \ (Γ ∪ ∆)))(∀s′′′ ∈ S(Σe )|r| )(∀r ′ ∈ out([r]Γ ∪∆ , s′ ∪ s∆ )) |r| ′′ ′ ′′′ ′′′ ′′′|r| |r| ′ △ (s ∪ s∆ ∪ s )), r , |r| |= θ); (s ∼Γ ∪∆ s implies IS, s · (s IS, s, r |= [[Γ | ∆]]θ iff IS, s, r |= Γ | ∆¬θ; IS, s, r |= ⊖ϕ iff |r| > 0 and IS, s[0..|r| − 1], r[0..|r| − 1] |= ϕ;   IS, s[0..n − k], r[0..n − k] |= ψ and . iff (∃k ≤ |r|) (∀u < k)IS, s[0..n − u], r[0..n − u] |= ϕ IS, s, r |= (ϕSψ) In the clauses for Γ | ∆θ and [[Γ | ∆]]θ above, θ stands for a possibly negated ◦ϕ or (ϕUψ). We use an auxiliary form of |= to define the satisfaction of θ, which, being an LTL formula, takes an infinite run and a position in it to interpret. Given an r ∈ Rω (IS) and a k < ω, IS, s, r, k |= ◦ϕ iff IS, s,  r[0..k + 1] |= ϕ;  IS, s, r[0..k + m] |= ψ and IS, s, r, k |= (ϕUψ) iff (∃m) ; (∀n < m)IS, s, r[0..k + n] |= ϕ IS, s, r, k |= ¬θ iff IS, s, r, k |= θ. Validity of formulas in an entire interpreted system and on the class of all inC , is defined as satisfaction at all terpreted systems, that is, in the logic ATLDP iR 0-length runs in the considered interpreted system, and at all the 0-length runs in all interpreted systems, respectively. In the definition of |=, we use sequences of strategy vectors and not simply strategy vectors as the strategy context, in order to enable the interpretation of the past operators ⊖ and (.S.). This complication of the form of |= is inevitable because the interpretation of Γ | ∆◦ allows the strategy context to be revised, and it is necessary to be able to revert to contexts from before such revisions for the correct interpretation of the past operators. The semantics of the fuC can be defined with s being just a vector of strategies in ture subset of ATLDP iR 82 D.P. Guelev and C. Dima |=. Then the satisfaction condition for IS, s, r |= Γ | ∆θ can be given the following simpler form: (∃s′ ∈ S(Γ ))(∀s′′ ∈ S(Σe \ (Γ ∪ ∆)))(∀s′′′ ∈ S(Σe ))(∀r′ ∈ out([r]Γ ∪∆ , s′ ∪ s∆ )) (s′′′ ∼Γ ∪∆ s implies IS, s′′′ △|r| (s′ ∪ s∆ ∪ s′′ ), r′ , |r| |= θ) Abbreviations. ⊤, ¬, ∨, ∧ and ⇔ are used to abbreviate formulas written with ⊥ and ⇒ in the common way. The abbreviations below are specific to ATL and other temporal and epistemic logics: I ⇋ ¬⊖⊤ −ϕ ⇋ (⊤Sϕ) ✸ 3 −¬ϕ ⊟ϕ ⇋ ¬✸ PΓ ϕ ⇋ ¬DΓ ¬ϕ [[Γ | ∆]] ◦ ϕ ⇋ ¬Γ | ∆ ◦ ¬ϕ. Related Work Next we give a brief account of the systems ATLsc , BSIL and ATLES from [15,21,20], respectively. An extension of ATL∗ by strategy contexts can be found in [1] too, where the authors focus mainly on modelling issues, and not technical results. We show that ATLsc from [15] admits a satisfaction preserving C . ATLsc , BSIL and ATLES were originally introduced translation into ATLDP iR for alternating transition systems and concurrent game structures. To assert the C semantical compatibility with ATLDP and, for the sake of brevity, we spell iR C , all these systems out their semantics on interpreted systems. Unlike ATLDP iR have complete information semantics. However, complete information can be straightforwardly modelled in interpreted systems by assigning the same local state space L1 = . . . = LN = Le to each i ∈ Σe and restricting the reachable states to be in the diagonal of L1 × . . . × LN × Le . ATL with strategy contexts (ATLsc , [15]) has state formulas ϕ and path formulas ψ. Their syntax can be given by the BNFs ϕ ::= ⊥ | p | (ϕ ⇒ ϕ) | ·Γ ·ψ | ·Γ ·ϕ and ψ ::= ¬ψ | ◦ϕ | (ϕUϕ) Satisfaction has the form IS, ρ, r |= ϕ with r ∈ Rfin (IS) for state formulas and IS, ρ, r |= ψ with r ∈ Rω (IS) for path formulas. In both cases ∆ ⊆ Σ and ρ ∈ S(∆). The clauses about |= for the ATLsc -specific operators are as follows: IS, ρ, r |= ·Γ ·ψ iff (∃s ∈ S(Γ ))(∀r ′ ∈ out(r, ρdomρ\Γ ∪ s))IS, ρdomρ\Γ ∪ s, r ′ |= ϕ; IS, ρ, r |= ·Γ ·ϕ iff IS, ρdomρ\Γ , r |= ϕ. Thanks to the presence of strategy contexts and the possibility to combine ·.· with ¬ in ATLsc , ATLsc and ATL∗sc , where path formulas can have arbitrary combinations of boolean connectives, have the same expressive power. C The translation t∆ below, maps from ATLsc state formulas to ATLDP foriR mulas. The auxiliary parameter ∆ ⊆ Σ is the domain of the reference context. Epistemic ATL with Perfect Recall, Past and Strategy Contexts 83 For the sake of brevity (·.·) stands for either [·.·] or ·.· in the translation clauses. The meaning of ((.)) is similar, wrt .|.. t∆ (⊥) ⇋ ⊥, t∆ (p) ⇋ p, t∆ (ϕ1 ⇒ ϕ2 ) ⇋ t∆ (ϕ1 ) ⇒ t∆ (ϕ2 ) t∆ (·Γ ·¬ψ) ⇋ ¬t∆ ([·Γ ·]ψ), t∆ ([·Γ ·]¬ψ) ⇋ ¬t∆ (·Γ ·ψ) t∆ ((·Γ ·) ◦ ϕ) ⇋ ((Γ | ∆ \ Γ )) ◦ tΓ ∪∆ (ϕ) t∆ ((·Γ ·)(ϕ1 Uϕ2 )) ⇋ ((Γ | ∆ \ Γ ))(tΓ ∪∆ (ϕ1 )UtΓ ∪∆ (ϕ2 )) t∆ (·Γ ·ϕ | ∆) ⇋ t∆\Γ (ϕ) An induction on the construction of formulas shows that, for any ATLsc state C tdomρ (ϕ) where s formula ϕ, IS, ρ, r |=ATLsc ϕ is equivalent to IS, s, r |=ATLDP iR stands for any sequence of strategy vectors that is consistent with r and features ρ as the strategy assignment to the members of domρ in its last member. The C : translation can be inverted on the future subset of ATLDP iR t−1 (Γ | ∆ϕ) ⇋ ·Σ \ ∆· ·Γ ·t−1 (ϕ). Basic strategy-interaction logic (BSIL, [21]) language includes state formulas ϕ, path formulas ψ and tree formulas θ: ϕ ::= ⊥ | p | (ϕ ⇒ ϕ) | Γ θ | Γ ψ θ ::= ⊥ | (θ ⇒ θ) | +Γ θ | +Γ ψ ψ ::= ◦ϕ | (ϕUϕ) | (ϕWϕ) The modelling relation has the form IS, ρ, l |= ϕ for state and tree ϕ, and IS, ρ, r |= ψ for path ψ, where l is a (global) state, r ∈ Rω (IS), and ρ is a strategic context. The clauses for Γ  and +Γ  with a path argument formula ψ are as follows: IS, ρ, l |= Γ ψ iff (∃s ∈ S(Γ ))(∀r ∈ out(l, s))IS, s, r |= ψ; IS, ρ, l |= +Γ ψ iff (∃s ∈ S(Γ ))(∀r ∈ out(l, ρdomρ\Γ ∪ s))IS, ρdomρ\Γ ∪ s, r |= ψ The clauses for tree argument formulas are similar. BSIL admits a translation C into an appropriate ∗ -extension of ATLDP . iR ATL with explicit strategies (ATLES , [20]) extends the syntax of . by subscripting it with mappings ρ of subsets of Σ to finite syntactical descriptions of strategies called strategy terms. In our notation, ρ denote elements of S(domρ) and the clause about |= for Γ ρ ◦ is: IS, r |= Γ ρ ◦ ϕ iff (∃s ∈ S(Γ \ domρ))(∀r′ ∈ out(r, s ∪ ρ))IS, r′ [0..|r| + 1] |= ϕ. The clauses for Γ ρ (ϕUψ) and Γ ρ ✷ϕ follow the same pattern. Unlike ATLsc , C , an ATLES formula may have several (freely occurring) fixed BSIL and ATLDP iR strategy context terms for each player. There appears to be no obvious way to reconcile this with the semantics of the other systems. 84 4 D.P. Guelev and C. Dima C A Proof System for Basic ATLDP iR C C Basic ATLDP is the subset of ATLDP without . | .(.U.) and [[. | .]](.U.): iR iR ϕ, ψ ::= ⊥ | p | (ϕ ⇒ ψ) | ⊖ϕ | (ϕSψ) | DΓ ϕ | Γ | ∆ ◦ ϕ Along with all propositional tautologies and the rule Modus Ponens (M P ), our system includes the following axioms and rules: The epistemic operator D. (KD ) DΓ (ϕ ⇒ ψ) ⇒ (DΓ ϕ ⇒ DΓ ψ) (TD ) DΓ ψ ⇒ ψ (4D ) DΓ ψ ⇒ DΓ DΓ ψ (5D ) (Mono D ) DΓ ψ ⇒ DΓ ∪∆ ψ (ND ) ¬DΓ ψ ⇒ DΓ ¬DΓ ψ ϕ DΓ ϕ (INT D ) (DΓ \∆ (p ⇒ ϕ) ∧ D∆ (¬p ⇒ ϕ)) ⇒ ψ DΓ ∪∆ ϕ ⇒ ψ The past modalities ⊖ and (.S.) (K⊖ ) ⊖(ϕ ⇒ ψ) ⇒ (⊖ϕ ⇒ ⊖ψ) (⊖⊥) ¬⊖⊥ (FP (.S.) ) (ϕSψ) ⇔ ψ ∨ (ϕ ∧ ⊖(ϕSψ)) (Fun ⊖ ) ⊖¬ϕ ⇒ ¬⊖ϕ (Mono ⊖ ) ϕ⇒ψ ⊖ϕ ⇒ ⊖ψ (N⊟ ) ϕ ⊟ϕ General ATL axioms and rules (. | . ◦ ⊥) ¬Γ | ∆ ◦ ⊥ (. | . ◦ ⊤) Γ | ∆ ◦ ⊤ (S) Γ ′ \ Γ ′′ | ∆′  ◦ ϕ ∧ Γ ′′ | ∆′′  ◦ ψ ⇒ Γ ′ ∪ Γ ′′ | ∆′ ∪ ∆′′  ◦ (ϕ ∧ ψ) Γ ′ \ Γ ′′ | ∆′  ◦ (p ⇒ ϕ) ∧ Γ ′′ | ∆′′  ◦ (¬p ⇒ ϕ) ⇒ ψ Γ ′ ∪ Γ ′′ | ∆′ ∪ ∆′′  ◦ ϕ ⇒ ψ ϕ⇒ψ (Mono .|.◦ ) Γ | ∆ ◦ ϕ ⇒ Γ | ∆ ◦ ψ (INT .|.◦ ) Committed versus neutral players (See Lemma 1 from [20].) (W HW ) Γ | Ψ ∪ ∆ ◦ ϕ ⇒ Γ ∪ Ψ | ∆ ◦ ϕ Interactions between ⊖, (.S.), . | .◦ and D. (D◦ ) Γ | ∆ ◦ ϕ ⇔ DΓ ∪∆ Γ | ∆ ◦ ϕ Γ | ∆ ◦ ϕ ⇔ Γ | ∆ ◦ DΓ ∪∆ ϕ (P R) ⊖DΓ ϕ ⇒ DΓ ⊖ϕ (. | . ◦ ⊖) Γ | ∆ ◦ (⊖ϕ ∧ ψ) ⇔ DΓ ∪∆ ϕ ∧ Γ | ∆ ◦ ψ ⊖∅ | ∆ ◦ ϕ ⇒ ∅ | ∆ ◦ ⊖ϕ (DI ) DΓ I ∨ DΓ ¬I Epistemic ATL with Perfect Recall, Past and Strategy Contexts 85 The rules INT .|. ◦ and INT D require p ∈ Var(ϕ) ∪ Var(ψ). Note that instances of S are well-formed only if (Γ ′ ∪ Γ ′′ ) ∩ (∆′ ∪ ∆′′ ) = ∅. 5 Completeness of the Proof System We fix the vocabulary AP for the rest of this section and denote the set of all the C basic ATLDP formulas built using variables from AP by L. We write Φ ⊢MP ϕ iR C for the derivability of ϕ from the premises Φ, the theorems of ATLDP and M P iR as the only proof rule. Auxiliary Propositional Variables and Formulas. Given Γ ⊆ Σ and i ∈ Σ, we write Γ <i for the set Γ ∩ {1, . . . , i − 1}. Given the formulas DΓ ψ and Γ | ∆ ◦ ψ, we introduce the auxiliary variables qi,DΓ ψ , i ∈ Γ <max Γ , and qi, Γ |∆ ◦ψ , i ∈ (Γ ∪ ∆)<max(Γ ∪∆) and use them to construct the formulas pi,DΓ ψ ⇋ qi,DΓ ψ ∧  ¬qj,DΓ ψ , i ∈ Γ <max Γ , and pmax Γ,DΓ ψ ⇋ j∈Γ <i  ¬qj,DΓ ψ . j∈Γ <max Γ Obviously these formulas satisfy ⊢  pi,DΓ ψ and ⊢ ¬(pi,DΓ ψ ∧ pj,DΓ ψ ) for i = j. i∈Γ We put pmax Γ,DΓ ψ ⇋ ⊤ in case |Γ ∪ ∆| = 1. We use pi,DΓ ψ to construct the formulas Di,Γ ψ ⇋ Di (pi,DΓ ψ ⇒ ψ), i ∈ Γ. The formulas pi, Γ |∆ ◦ψ , i ∈ Γ ∪∆, are written similarly in terms of the variables qi, Γ |∆ ◦ψ , i ∈ (Γ ∪ ∆)<max(Γ ∪∆) . We use pi, Γ |∆ ◦ψ to construct the formulas  i | ∅ ◦ (pi, Γ |∆ ◦ψ ⇒ ψ), for i ∈ Γ ; i, Γ | ∆ ◦ ψ ⇋ ∅ | i ◦ (pi, Γ |∆ ◦ψ ⇒ ψ), for i ∈ ∆. Given a set of formulas x written in AP , we write x for the set x ∪ {Di,Γ ψ : i ∈ Γ, DΓ ψ ∈ x} ∪ {i, Γ | ∆ ◦ ψ : Γ | ∆ ◦ ψ ∈ x, i ∈ Γ ∪ ∆}. Lemma 1. Let x be a consistent set of formulas written in AP . Then x is consistent too. The proof of this lemma is similar to that of Lemma 12 from [9] and involves the rules INT D and INT .|. ◦. Lemma 2 (customized Lindenbaum lemma). Let ≺ be a well-ordering of L and let x be a consistent subset of L. Then there exists a consistent set x′ ⊇ x which is maximal in L and is such that for any initial interval Φ ⊂ L of L, ≺ the consistency of x ∪ Φ entails Φ ⊆ x′ . Proof. We construct the ascending (transfinite) sequence xϕ , ϕ ∈ L, of consistent subsets of L indexed by the elements of L by induction on the well-ordering ≺. Let ϕ0 be the least element of L. Then xϕ0 is x ∪ {ϕ0 } in case x ∪ {ϕ0 } is consistent; otherwise it is just x. Similarly, for all non-limit ϕ, given that ϕ′ is xϕ ∪ {ϕ′ } in case xϕ ∪ {ϕ′ } is consistent and the successor of ϕ in ≺, xϕ′ is   xϕ has ϕψ . A direct check shows that x′ = otherwise. For limit ϕ, xϕ = ψ≺ϕ the desired property. ϕ∈L 86 D.P. Guelev and C. Dima In the sequel, given x and a well-ordering a of L, we denote a fixed maximal consistent set (MCS) with the above property by x + a. Next we build a interpreted system IS = Li : i ∈ Σe , I, Act i ∈ Σe , t, V  C which is canonical in the sense adopted in modal logic. for basic ATLDP iR Definition 11 (global states, local states). Let W to be the set of all the maximal consistent sets of formulas in the vocabulary AP . Given w ∈ W and Γ ⊆ Σ, we put DΓ (w) = {ϕ : DΓ ϕ ∈ w}, Li = {D{i} (w) : w ∈ W } for i ∈ Σ, and Le = W. Given w ∈ W , we write lw for the state D{1} (w), . . . , D{N } (w), w. Below it becomes clear that all reachable states in IS have this form. We work with the MCS w instead of the respective tuples lw wherever this is more convenient. Note that the environment component of lw is w itself. Definition 12 (valuation and initial states). We put V (w, p) ↔ p ∈ w and I = {lw : w ∈ W, I ∈ w}. Definition 13 (indiscernibility of states in terms of MCS). Given Γ ⊆ Σe , two states w, v ∈ W are Γ -similar, written w ∼Γ v, if DΓ (w) = DΓ (v). The following lemma shows that w ∼Γ v is equivalent to lw ∼Γ lv in the sense of Definition 3. Lemma 3 (Γ -similarity in terms of Γ ’s distributed knowledge). Let w, v ∈ W , Γ ⊆ Σ. Then w ∼Γ v iff D{i} (w) = D{i} (v) for all i ∈ Γ . Proof. (←): Let w ∼Γ v and DΓ ϕ ∈ DΓ (w). Then Di,Γ ϕ ∈ w for every i ∈ Γ . Then, by  4D and Mono D, DΓ Di,Γ ϕ ∈ v for every i ∈ Γ too. Now S5 reasoning Di,Γ ⇒ DΓ ϕ. Hence DΓ ϕ ∈ DΓ (v). pi,DΓ ϕ entail and ⊢ i∈Γ i∈Γ (→): Let Di ϕ ∈ w. Then, by S5 reasoning, w ⊢MP DΓ Di ϕ, whence, by DΓ (w) = DΓ (v), v ⊢MP DΓ Di ϕ and, finally Di ϕ ∈ v. Hence D{i} (w) ⊆ D{i} (v). The symmetrical inclusions are proved similarly. Definition 14 (actions). An action for player i ∈ Σ is either the symbol d or a tuple of the form Φ, Γ, ∆ such that Γ, ∆ ⊆ Σ are disjoint, i ∈ Γ , and Φ is a consistent set of formulas. An environment action is a well-ordering of L. Player action Φ, Γ, ∆ is represents the player’s contribution to achieving all the objectives from Φ simultaneously as a member of Γ , provided that ∆ act as described in the context. Action d indicates choosing to follow the strategy from the context. Allowing infinite sets Φ of objectives in actions is necessary because MCS may contain infinitely many formulas of the form ∅ | ∆ ◦ ϕ. Definition 15 (the past of a state). Given a w ∈ W , we write Θw for the set {⊖θ : θ ∈ w}. Epistemic ATL with Perfect Recall, Past and Strategy Contexts 87 The formulas from Θw hold at states which can be reached from lw in one step. Environment actions complement the construction of successor states. Let x consist of the formulas to be satisfied due to the player actions performed at state lw . Then environment action ae comlements x ∪ Θw to an MCS description of the successor state (x ∪ Θw ) + ae . Lemma 2 entails that any MCS x′ ⊇ x ∪ Θw has the form (x ∪ Θw ) + ae for some appropriate ae . Definition 16 (effectiveness of coalitions). Let a ∈ Act Σe and w ∈ W . We write ∆a for the set {i ∈ Σ : ai = d}. Coalition Γ ⊆ Σ is effective in a wrt w, if (1) Γ ⊆ Σ \ ∆a , ∆ ⊆ ∆a ; assuming that ai = Φi , Γi , ∆i , i ∈ Γ , (2) Γ = Γi for all  i ∈ Γ; (3) Γi | ∆i  ◦ Φ′ ∈ w for all finite Φ′ ⊆ Φi ; (4) ∆i = ∆j for all i, j ∈ Γ . Coalition Γ is effective in state w, iff its objectives are achievable in w. Different coalitions which are effective in the same state cannot overlap. Definition 17 (aw ). Let the coalitions which are effective in a ∈ Act Σe wrt w ∈ W be Γ1 , . . . , Γk . Let Υ = Γ1 ∪ . . . ∪ Γk . We define aw ∈ Act Σe by the clause ⎧ if i = e; ae , ⎪ ⎪ ⎨ if i ∈ Υ ; Φi , Γi , ∆i , aw,i = w}, {i}, ∅, if i ∈ ∆a ; {ψ : ∅ | i ◦ ψ ∈ ⎪ ⎪ ⎩ ∅, {i}, ∅, otherwise. The vector of actions aw is a revision of a according to the plausibility of the actions from a in w. In aw , players i who participate in effective coalitions are described as acting to achieve the common objective of their coalitions; players who follow their respective strategies from the context are described as acting to achieve whatever consequences these actions have according to w, and players who neither participate in effective coalitions, nor follow the context strategies, are described as acting in singleton coalitions {i} to achieve nothing. Consequently, all the coalitions in aw are effective wrt w: Lemma 4 (effectiveness of coalitions in aw ). Assuming the  notation from Definition 17, if i ∈ Σ and aw,i = Φ, Γ, ∆, then Γ | ∆ ◦ Φ′ ∈ w for all finite Φ′ ⊂ Φ. Proof. The lemma follows immediately for i ∈ Υ and i ∈ Σ \ (Υ ∪ ∆a ).Players i ∈ ∆a appear in the singleton coalitions {i} in aw , and we have | i◦ Φ′ ∈ w  ∅ ′ ′ for all finite Φ ⊂ Φ. By Axiom W HW this entails i | ∅ ◦ Φ ∈ w. As it becomes clear below, a and aw cause the same transitions from w. The notation aw is introduced to avoid lengthy explanations about treating the various sorts of actions separately. Definition 18 (transition function). Let w ∈ W , a  ∈ Act Σe andaw =  Φi , Γi , ∆i  : i ∈ Σ ∪ ae . Then t(lw , a) = lv where v = Φi ∪ Θw + ae , i∈Σ 88 D.P. Guelev and C. Dima i.e., v is the extension of the set of all the objectives which can be simultaneously achieved by the coalitions which are effective in a wrt w and the formulas which describe lw ’s past, to an MCS by ae , as in Lemma 2.  The definition of t above relies on the fact that Φi ∪ Θw is consistent. To i∈Σ  realise that, assume the contrary. Then there exist some finite Φ′ ⊆ Φi and i∈Σ  ′  ′ ′ Θ ⊂ w such that ⊢ Φ ⇒ ⊖¬ Θ . By the  monotonicity  of′ Σ | ∅◦ and Axiom . | . ◦ ⊖, this entails⊢ Σ | ∅ ◦ Φ′ ⇒ DΣ ¬ Θ . By Axioms S   Γi | Γi | ∆i  ◦ (Φi ∩ Φ′ ) ⇒ Σ | ∅ ◦ Φ′ . Hence ⊢ and W HW , ⊢ i∈Σ i∈Σ   ∆i  ◦ (Φi ∩ Φ′ ) ⇒ DΣ¬ Θ′ , which is a contradiction because Θ′ ⊂ w and, by Lemma 4, Γi | ∆i  ◦ (Φi ∩ Φ′ ) ∈ w for all i ∈ Σ. We define t only on states of the form lw . The set of these states contains I and is closed under t. Hence the definition of t on other states is irrelevant. Definitions 11, 12, 14 and 18 give a complete description of the interpreted system IS. Below we prove that if r ∈ Rfin (IS) and s ∈ S(Σe )n+1 is consistent with s, then, for any ϕ ∈ L, IS, s, r |= ϕ iff ϕ ∈ w where lw is the last state of r. Definition 19 (extracting strategic context from MCS). Given a w ∈ R0 (IS), we define aw ∈ Act Σ by putting aw,i = {ϕ : ∅|i ◦ ϕ ∈ w}, {i}, ∅. We define the vector of strategies sIS ∈ S(Σ) by putting, given an arbitrary r = w0 a0 . . . a|r|−1 w|r| ∈ Rfin (IS), sIS (r) = aw|r| . The strategies sIS are built according to the working of the transition function along runs in which the players act as described in the context. They are memoryless, i.e., determined by the last state of the argument run. Note that we extract strategic context from w, which contains the explicit descriptions i, Γ | ∆ ◦ ψ of the contribution of individual coalition members i ∈ Γ to the achievement of goals of their respective coalitions Γ . According to our definition, local runs are sequences of the form ri = D{i} (w0 )a0i D{i} (w1 )a1i . . . ak−1 D{i} (wk ) . . . . i To realise that the strategies sIS,i are determined from the local run of player i, note that ∅|{i} ◦ ϕ ∈ w is equivalent to ∅|{i} ◦ ϕ ∈ v for v such that D{i} (v) = D{i} (w) due to the Axioms D◦ . Definition 20 (consistency between runs and strategy vector sequences). Let n < ω. Run r = w0 a0 . . . an−1 wn ∈ Rn (IS) and strategy vector sequence s = s0 , . . . , sn ∈ (S(Σe ))n+1 are consistent, if s is consistent (in the sense of Definition 8) and ak = sn (r[0..k]), k = 0, . . . , n − 1. Note that the restrictions on sk which follow from the consistency between r and s for k < n are implied by the consistency of s as a sequence of strategy vectors. The following lemma states that if w ∈ W and DΓ (w) is consistent with some arbitrary formula ϕ, then indeed DΓ (w) ⊢MP PΓ ϕ. Lemma 5. Let Γ ⊆ Σ, w ∈ W , ϕ ∈ L and let w be consistent with ϕ. Then DΓ (w) ⊢MP PΓ ϕ. Epistemic ATL with Perfect Recall, Past and Strategy Contexts 89 Proof. Since w is maximal consistent, either PΓ ϕ ∈ w or DΓ ¬ϕ ∈ w. The latter is impossible as it would entail DΓ (w) ⊢MP ¬ϕ. By S5 reasoning, PΓ ϕ ∈ w entails DΓ PΓ ϕ ∈ w. The latter formula appears in DΓ (w) as well. Hence, by S5 reasoning again, DΓ (w) ⊢MP PΓ ϕ. Lemma 6. Let w, v ∈ W , Γ ⊆ Σ and DΓ (w) ⊆ DΓ (v). Then DΓ (w) = DΓ (v). Proof. Let ϕ ∈ L be such that DΓ ϕ ∈ w. Then DΓ PΓ ¬ϕ ∈ w by Lemma 5 and S5 reasoning. By DΓ (w) ⊆ DΓ (v), this entails PΓ ¬ϕ ∈ v. Hence DΓ ϕ ∈ v. The following lemmata are the parts of the inductive proof of the Truth Lemma C below (Theorem 1) about the various ATLDP modalities. iR Lemma 7 (PΓ ). Let n < ω, r = w0 a0 . . . an−1 wn ∈ Rn (IS), let s = s0 , . . . , sn ∈ n+1 (S(Σe )) be consistent with r. Let Φ be a set of formulas such that PΓ Φ′ ∈ n w for all finite Φ′ ⊆ Φ. Then there exist an r′ = v 0 b0 . . . bn−1 v n ∈ Rfin (IS) and a sequence s′ = s′0 , . . . , s′n ∈ (S(Σe ))n+1 such that s′ is consistent with r′ , r′ ∼Γ r, s′ ∼Γ s, and Φ ⊆ v n . Proof. Induction If r ∈ R0 (IS), then I ∈ w0 and, by S5 reasoning and  ′ on n. 0 DI , PΓ (I ∧ Φ ) ∈ w for all finite Φ′ ⊆ Φ. By S5 reasoning this entails that Φ ∪ {I} ∪ {DΓ ψ : DΓ ψ ∈ w0 } is consistent. Now Lemma 6 entails that we can choose v 0 to be any MCS which contains the latter set and put s′0 = sIS . Next we prove the lemma for |r| = n+1 assuming that it holds for |r| = n. Let Θ = {θ : DΓ (wn+1 ) ∪ Φ ⊢MP ⊖θ}. Assume that DΓ (wn ) ∪ Θ is inconsistent for ′ n ′ the sake of contradiction. ⊂Θ  ′ Then there  ′ exist some finite D ⊂ D Γ (w′ ) and Θ n+1 ). such that ⊢ DΓ ⊖ D ⇒ ⊖¬ Θ . By Axiom P R, DΓ ⊖ D ∈ DΓ (w Hence ¬ Θ′ ∈ Θ, which entails that DΓ (wn+1 ) ∪ Φ is inconsistent. This means  that there exists a finite Φ′ ⊆ Φ such that DΓ (wn+1 ) ⊢MP DΓ ¬ Φ′ , which is and consequently, because of a contradiction. Hence DΓ (wn ) ∪ Θ is consistent,  the closedness of Θ under conjunction, {PΓ Θ′ : Θ′ ⊂fin Θ} ⊂ wn . By the inductive hypothesis, there exist an r′′ = v 0 b0 . . . bn−1 v n ∈ Rfin (IS) and a sequence s′′ = s′′0 , . . . , s′′n ∈ (S(Σe ))n+1 such that s′′ is consistent with r′′ , r′′ ∼Γ r[0..n], s′′ ∼Γ s[0..n], and Θ ⊆ v n . Next we define s′n s′n+1 ∈ S(Σe ) so that s′ = s′′ [0..n − 1] · s′n , s′n+1 is consistent with r′ = r′′ bn v n+1 ∈ Rn+1 (IS) where bn = s′n (r′′ ), v n+1 = t(v n , bn ), s′ ∼Γ s, DΓ (v n+1 ) = DΓ (wn+1 ) and Φ ⊆ v n+1 . Given g ∈ Rfin (IS), we put ⎧ ′ n−1 si (g), if |g| < n, for all i ∈ Σe , ⎪ ⎪ ⎪ n ⎪ s (g) if |g| = n, and i ∈ Γ, ⎪ ⎪ ⎨ i i, ∅, {i}, ∅ if |g| = n, and i ∈ Σ \ Γ, s′i n (g) = any well-ordering of L in which ⎪ ⎪ ⎪ ⎪ DΓ (wn+1 ) ∪ Φ forms an initial interval, if |g| = n, and i = e, ⎪ ⎪ ⎩ sIS,i (g), if |g| ≥ n. For s ′n+1 we put s ′n+1 (g) =  s′′n (g), if |g| < n + 1, sIS (g), if |g| ≥ n + 1. 90 D.P. Guelev and C. Dima By construction, s′ is consistent (as a sequence of strategy vectors), s′ ∼Γ s, and s′ is consistent with r′′ . We need to prove that DΓ (v n+1 ) = DΓ (wn+1 ) and Φ ⊆ v n+1 . Note that, by S5-reasoning, DΓ (wn+1 ) ⊂ v n+1 n entails DΓ (wn+1 ) = DΓ (v n+1 ). This is so, because wn+1 is a MCS, whence, if DΓ ψ ∈ v n+1 \ wn+1 , then, DΓ ¬DΓ ψ ∈ wn+1 follows by negative introspection from ¬DΓ ψ ∈ wn+1 . The latter entails DΓ ¬DΓ ψ ∈ v n+1 , which, by T, entails ¬DΓ ψ ∈ v n+1 , and this would contradict the consistency of v n+1 . Hence we only need to prove that Φ ⊆ v n+1 . By the definition of the transition function t, this would follow from  Φi ∪ {⊖θ : θ ∈ v n } ∪ DΓ (wn+1 ) ∪ Φ where Φi are the sets the consistency of i∈Γ n of formulas occurring in bvn ,Γ . This boils down to the consistency of {⊖θ : θ ∈ v n } ∪ DΓ (wn+1 ) ∪ Φ, because v n ∼Γ wn entails that an has the same effective Γi ⊆ Γ wrt w as bn wrt v n , and therefore Φi , i ∈ Γ , are the same in anwn ,Γ .  Φi is equivalent Hence, by the definition of wn+1 = t(wn , an ) and D◦ , ψ ∈ i∈Γ to DΓ ψ ∈ DΓ (wn+1 ). Now let us assume that {⊖θ : θ ∈ v n } ∪ DΓ (wn+1 ) ∪ Φ is inconsistent for the sake of contradiction. Then there exist some finitely many θ1 , . . . , θk ∈ v n such that DΓ (wn+1 )∪Φ ⊢MP ◦¬(θ1 ∧. . . ∧θk ). This is impossible by the choice of v n to be a superset of Θ = {θ : DΓ (wn+1 ) ∪ Φ ⊢MP ⊖θ}. Lemma 8 (DΓ ). Let n < ω, r, r′ ∈ Rn (IS). Let r = w0 a0 . . . an−1 wn and r′ = v 0 b0 . . . bn−1 v n . Then r′ ∼Γ r implies ϕ ∈ v n . Proof. By the definition of r′ ∼Γ r, DΓ ϕ ∈ DΓ (wn ) = DΓ (v n ) ⊆ v n , whence ϕ ∈ v n follows by TD .  Lemma 9 (. | .◦). Let w ∈ W , Ψ ⊂ Land Γ | ∆ ◦ Ψ ′ ∈ w for all finite Ψ ′ ⊆ Ψ . Let a′Γ = {pi, Γ |∆ ◦Ψ ′ ⇒ Ψ ′ : Ψ ′ ⊆fin Ψ }, {i}, ∅ : i ∈ Γ  Then Ψ ⊆ t(w, a′Γ ∪ aw,∆ ∪ a′′Σe \(Γ ∪∆) ), where aw,∆ is the restriction of aw as introduced in Definition 19, for all a′′Σe \(Γ ∪∆) ∈ Act Σe \(Γ ∪∆) . Proof. Obviously Γ is effective in a′Γ ∪ aw,∆ ∪ a′′Σ\(Γ ∪∆) wrt w. Lemma 10 ([[. | .]]◦). Let n < ω, r = w0 a0 . . . an−1 wn ∈ Rn (IS), Γ, ∆ ⊆ Σ, Γ ∩ ∆ = ∅. Let s = s0 , . . . , sn ∈ (S(Σe ))n+1 be consistent with r. Let gΓ ∈ Act Γ ′ be such that ψ ∈ t(v n , gΓ ∪ s′n (v n )∆ ∪ gΣ ) for all r′ = v 0 b0 . . . bn−1 v n ∈ e \(Γ ∪∆) n ′ ′ n+1 which are consistent with r′ and R (IS) such that r ∼Γ ∪∆ r, all s ∈ (S(Σe )) ′ ′ satisfy s ∼Γ ∪∆ s, and all gΣe \(Γ ∪∆) ∈ Act Σe \(Γ ∪∆) . Then Γ | ∆ ◦ ψ ∈ wn . Proof. Consider an r′ as in the lemma. Let gi′ = ∅, {i}, ∅ for i ∈ Σ \(Γ ∪∆), Let ′ ′ a = gΓ ∪ s′n (v n )∆ ∪ gΣ\(Γ ∪∆) ∪ ge . Let Φ be the union of all the sets of formulas in awn , i.e., the actions of the coalitions which are effective in a wrt wn and, consequently, wrt any v n which is Γ ∪ ∆-similar to wn . Then, by the definition of t, t(v n , a) = (Φ ∪ Θvn ) + ge′ . Assume that the least element of ge′ is ¬ψ. Then, Θvn ⊢MP ψ. By since ψ ∈ t(v n , a) and by the construction of (Φ ∪ Θvn ) + ge′ , Φ ∪ ′ n such that Φ ⊢MP ⊂ Θ Θvn ⇒ ψ. Let compactness, there exists a finite Θ n v v  Θv′ n = {⊖θ1 , . . . , ⊖θk }. Then Θv′ n is equivalent to ⊖(θ1 ∧ . . . ∧ θk ). The latter Epistemic ATL with Perfect Recall, Past and Strategy Contexts 91 formula is in Θvn . Hence, for every v n which is the last state of an r′ as described in the lemma, there exists a ξ ∈ L such that ⊖ξ ∈ Θvn and Φ ⊢MP ⊖ξ ⇒ ψ. Let Ξ n = {ξ ∈ L : Φ ⊢MP ⊖ξ ⇒ ψ, ⊖ξ ∈ Θvn for some v n ∈ W such that there exists an r′ ∈ R(IS), r′ ∼Γ ∪∆ r with v n as its last state}. A direct check shows that Ξ n is closed under disjunction. Assume that {¬ξ : ξ ∈ Ξ n } ∪ DΓ ∪∆ (wn ) is consistent for the sake of contradiction. Then, by Lemma 5, DΓ ∪∆ (wn ) ⊢MP PΓ ∪∆ ¬ξ for every ξ ∈ Ξ n . SinceΞ n is closed under disjunction, we can conclude that DΓ ∪∆ (wn ) ⊢MP ¬ξ for any finite Ξ n′ ⊂ Ξ n . By Lemma 7, this entails that there PΓ ∪∆ ξ∈Ξ n ′ exist an r′ ∼Γ ∪∆ r and an s′ ∼Γ ∪∆ s such that s′ is consistent with r′ and {¬ξ : ξ ∈ Ξ n } ⊆ v n where v n is the last state of r′ . This contradicts the consistency of v n since, as we established above, for every v n with the specified properties, there exists a ξ ∈ Ξ n such that ξ ∈ v n as well. Hence {¬ξ : ξ ∈ Ξ n′ } ∪ DΓ ∪∆ (wn )is inconsistent. By compactness, this entails that there exists DΓ ∪∆ (wn ) ⊢MP Ξ n′ for some finite Ξ n′ ⊂ Ξ n . Now, having in mind that Φ ⊢MP ⊖ξ ⇒ ψ for each of the ξs in Ξ n′ , we can conclude that Φ ∪ {⊖δ : δ ∈ ′ ′ n DΓ ∪∆ (wn )} ⊢MP ψ. Hence  ′ there exist some finite Φ ⊆ Φ and D ⊂ DΓ ∪∆ (w ) ′ such that  ⊢ Φ ∧ ⊖ D ⇒ ψ. By the monotonicity of Γ | ∆◦, ⊢ Γ | ∆◦ ( Φ′ ∧⊖ D′ ) ⇒ Γ | ∆◦ ψ. Now, by Axiom . | .◦ ⊖, the latter entails n ⊢ Γ | ∆ ◦ ( Φ′ ) ∧ DΓ ∪∆ D′ ⇒ Γ′ | ∆n◦ ψ. For ′any v as innthe lemma, the definition Γ | ∆ ◦ ( Φ ) ∈ v , and D ⊂ DΓ ∪∆ (w ) = DΓ ∪∆ (v n )  of′ Φ entails n entails D ∈ v . Hence, finally, Γ | ∆ ◦ ψ ∈ v n as well. Lemma 11 (⊖ and (.S.)). Let n, r ∈ Rn (IS) and s ∈ (S(Σe ))n+1 be as in the previous lemmata. Then, ◦n I ∈ wn and, for any two formulas ϕ, ψ ∈ L, ⊖ϕ ∈ wn iff n > 0 and ϕ ∈ wn−1 , and (ϕSψ) ∈ wn iff there exists a k ≤ n such that ψ ∈ wk and ϕ ∈ wk+1 , . . . , wn . C We omit the proof of this lemma as it contains nothing specific to ATLDP . We iR n n only note that the fact ◦ I ∈ w guarantees that the presence of (.S.) does not affect the compactness of the system, despite that (.S.) is an iterative operator. Theorem 1 (truth lemma). Let n < ω, r ∈ Rn (IS), r = w0 a0 . . . an−1 wn , and let s ∈ S(Σe )n+1 be consistent with r. Then IS, r |= ϕ iff ϕ ∈ wn . Proof. Induction on the construction of ϕ. The cases of ϕ being ⊥, an atomic proposition, or an implication, are trivial, and we skip them. For ϕ of the forms DΓ ψ, Γ | ∆ ◦ ψ and ⊖ψ and (ψSχ) the theorem follows from Lemmata 8 and 7, 9 and 10 and Lemma 11, respectively. C ). Let x be a consisCorollary 1 (strong completeness for basic ATLDP iR tent subset of L and let I ∈ x. Then there exists an initial state l ∈ I and a vector of strategies s ∈ S(Σe ) such that IS, s, l |= ϕ for all ϕ ∈ x. Proof. Let w be any MCS such that x ⊆ w. Then Theorem 1 entails that IS, sIS , lw |= ϕ for all ϕ ∈ w. 92 D.P. Guelev and C. Dima Concluding Remarks and Future Work We have shown that extending the game-theoretic operator . of ATLDP iR by a second coalition parameter to denote players who act according to the strategic context, we obtain a language for epistemic ATL with strategic contexts which admits an axiom system that is a straightforward revision of the system for ATLDP iR from our work [9]. So far we have established the completeness of that system for a subset of ATLDP iR with strategy contexts which lacks the combinations of . with the iterative future temporal operator (.U.) and its derivatives ✸ and ✷. Taking advantage of the compactness of this subset, we have obtained a strong completeness theorem. We have established the semantical compatibility between our proposed system and the systems of ATL with strategy contexts and complete information from the literature, especially [5,15]. We intend to C , seeking to establish weak further investigate the axiomatizability of ATLDP iR completeness theorems and the decidability of validity of bigger subsets. Akcnowledgements. Dimitar P. Guelev did part of the work on this paper while visiting LACL, Université Paris Est-Créteil, in January 2012 and was supported by the EQINOCS research project ANR ANR 11 BS02 004 03 [4] and Bulgarian National Science Fund Grant DID02/32/2009. References 1. Ågotnes, T., Goranko, V., Jamroga, W.: Strategic Commitment and Release in Logics for Multi-Agent Systems (Extended Abstract). Technical report IfI-08-01, TU Clausthal (May 2008); Ågotnes, T., Goranko, V., Jamroga, W.: Strategic Commitment and Release in Logics for Multi-Agent Systems (Extended Abstract). In: Bonanno, G., Löwe, B., van der Hoek, W. (eds.) Logic and the Foundations of Game and Decision Theory. LNCS, vol. 6006, Springer, Heidelberg (2010) 2. Alur, R., Henzinger, T., Kupferman, O.: Alternating-time Temporal Logic. In: Proceedings of FCS 1997, pp. 100–109 (1997) 3. Alur, R., Henzinger, T., Kupferman, O.: Alternating-time temporal logic. Journal of the ACM 49(5), 1–42 (2002) 4. Asarin, E.: EQINOCS Research Project ANR 11 BS02 004 03, http://www.liafa.univ-paris-diderot.fr/~ eqinocs/ 5. Brihaye, T., Da Costa, A., Laroussinie, F., Markey, N.: ATL with Strategy Contexts and Bounded Memory. In: Artemov, S., Nerode, A. (eds.) LFCS 2009. LNCS, vol. 5407, pp. 92–106. Springer, Heidelberg (2008) 6. Chatterjee, K., Henzinger, T.A., Piterman, N.: Strategy logic. Information and Computation 208(6), 677–693 (2010) 7. Fagin, R., Halpern, J., Moses, Y., Vardi, M.: Reasoning about Knowledge. MIT Press (1995) 8. Goranko, V., van Drimmelen, G.: Decidability and Complete Axiomatization of the Alternating-time Temporal Logic. Theoretical Computer Science 353(1-3), 93–117 (2006) 9. Guelev, D.P., Dima, C., Enea, C.: An Alternating-time Temporal Logic with Knowledge, Perfect Recall and Past: Axiomatisation and Model-checking. Journal of Applied Non-Classical Logics 21(1), 93–131 (2011) Epistemic ATL with Perfect Recall, Past and Strategy Contexts 93 10. Halpern, J., van der Meyden, R., Vardi, M.: Complete Axiomatizations for Reasoning about Knowledge and Time. SIAM Journal on Computing 33(3), 674–703 (2004) 11. Jamroga, W., van der Hoek, W.: Agents That Know How to Play. Fundamenta Informaticae 63(2-3), 185–219 (2004) 12. Lichtenstein, O., Pnueli, A.: Propositional temporal logics: decidability and completeness. Logic Journal of the IGPL 8(1), 55–85 (2000) 13. Lomuscio, A., Raimondi, F.: Model checking knowledge, strategies, and games in multi-agent systems. In: Proceedings of AAMAS 2006, pp. 161–168. ACM Press (2006) 14. Lomuscio, A., Raimondi, F.: The Complexity of Model Checking Concurrent Programs Against CTLK Specifications. In: Baldoni, M., Endriss, U. (eds.) DALT 2006. LNCS (LNAI), vol. 4327, pp. 29–42. Springer, Heidelberg (2006) 15. Lopes, A.D.C., Laroussinie, F., Markey, N.: ATL with strategy contexts: Expressiveness and model checking. In: FSTTCS. LIPIcs, vol. 8, pp. 120–132 (2010) 16. Pauly, M.: A modal logic for coalitional power in games. Journal of Logic and Computation 12(1), 149–166 (2002) 17. Schobbens, P.Y.: Alternating-time logic with imperfect recall. ENTCS 85(2), 82–93 (2004); Proceedings of LCMAS 2003 18. van der Hoek, W., Wooldridge, M.: Cooperation, Knowledge and Time: Alternating-time Temporal Epistemic Logic and Its Applications. Studia Logica 75, 125–157 (2003) 19. van der Meyden, R., Wong, K.-S.: Complete Axiomatizations for Reasoning about Knowledge and Branching Time. Studia Logica 75(1), 93–123 (2003) 20. Walther, D., van der Hoek, W., Wooldridge, M.: Alternating-time Temporal Logic with Explicit Strategies. In: Samet, D. (ed.) TARK, pp. 269–278. ACM Press (2007) 21. Wang, F., Huang, C.-H., Yu, F.: A Temporal Logic for the Interaction of Strategies. In: Katoen, J.-P., König, B. (eds.) CONCUR 2011 – Concurrency Theory. LNCS, vol. 6901, pp. 466–481. Springer, Heidelberg (2011)