Valentin Goranko The Basic Algebra of Game Equivalences Abstract. We give a complete axiomatization of the identities of the basic game algebra valid with respect to the abstract game board semantics. We also show that the additional conditions of termination and determinacy of game boards do not introduce new valid identities. En route we introduce a simple translation of game terms into plain modal logic and thus translate, while preserving validity both ways, game identities into modal formulae. The completeness proof is based on reduction of game terms to a certain 'minimal canonical form', by using only the axiomatic identities, and on showing that the equivalence of two minimal canonical terms can be established from these identities. Keywords: game operations, game algebra, game identities, axiomatization, completeness, modal logic 1. Introduction The relationships between logic and games go back to the ancient Greek philosophy, and have been explored in modern times by a number of logicians and computer scientists (see [1] and [3] for details and further references). Parikh has initiated a formal logical study of games by introducing the Game Logic in [5]. Recently this idea has been advanced in [1] and systematically developed for coalition games in [4]. In particular, an algebraic approach to the study of the games structure and equivalence between games emerged from [5] and was further developed in [1], where the problem of establishing the complete axiomatization of the valid identities of the basic game algebra was raised. Here we give a solution to that problem. The paper is organized as follows. In Section 2 we introduce the syntax and semantics of the basic algebra of games in terms of abstract game boards and in Section 3 we give an axiomatization of its valid identities. In Section 4 we define canonical forms of game terms and show that every game term is provably equivalent to a minimal canonical one. In Section 5 we introduce a translation of game terms and identities to plain modal logic and show that it preserves validity of game identities. The converse preservation of validity Special Issue on Game Logic and Game Algebra Edited by Marc Pauly and Rohit Parikh Studia Logica 75: 221–238, 2003. c© 2003 Kluwer Academic Publishers. Printed in the Netherlands. 222 V. Goranko is proved in Section 6 where we also establish other technical results used in the completeness proof presented in Section 7. The last Section 8 includes some further results and concluding remarks, where we show that restriction of the semantics to determined and terminating games does not introduce new valid identities. We also discuss the complexity of the validity problem and the relations between game algebras and logics. 2. Basic algebra of games We consider two-player games of a most general type. The game language GL consists of: • a set of atomic games Gat = {ga}a∈A; • game operations: ∨,d , ◦. For technical convenience, we include an 'idle' atomic game ι = g0 in Gat. Definition 1. Game terms: • Every atomic game is a game term. • If G,H are game terms then Gd, G ∨H and G ◦H are game terms. Besides, we define G ∧H := (Gd ∨Hd)d. Intuitively, the operations d,∨,∧, ◦ mean respectively dualization (swapping the two players' roles), choice of first player, choice of second player, and composition of games. The algebra of game terms will be denoted by GA. Atomic games and their duals will be called literals. Models for GL are game boards: 〈 S, {ρia}a∈A;i=1,2 〉 where S is a set of states and ρia ⊆ S×P (S) are atomic forcing relations satisfying the following forcing conditions: • upwards monotonicity (MON): for any s ∈ S and X ⊆ Y ⊆ S, if sρiaX then sρiaY ; • consistency of the powers (CON): for any s ∈ S,X ⊆ S, if sρ1aX then not sρ2a(S −X) and (hence) likewise with 1 and 2 swapped. We also consider the following optional conditions: • termination of the games (FIN): for any s ∈ S, sρiaS. This is of a less imperative nature, since some games may go on forever and never reach an outcome state. Game boards satisfying that condition will be called terminating and the class of terminating game boards will be denoted by FIN. The Basic Algebra of Game Equivalences 223 • determinacy (DET): sρ2a(S−X) iff not sρ 1 aX. Game boards satisfying this condition will be called determined and the class of determined game boards will be denoted by DET. The forcing relations ρiι of the idle game ι have a fixed interpretation: sρiιX iff s ∈ X. Compositions of idle literals (ι or ι d) will be called idle game terms. Given a game board, the atomic forcing relations are extended to forcing relations {ρiG}G∈G;i=1,2 for all game terms, following the recursive definitions given in [1]: • sρ1 Gd X iff sρ2GX; • sρ2 Gd X iff sρ1GX; • sρ1G1∨G2X iff sρ 1 G1 X or sρ1G2X; • sρ2G1∨G2X iff sρ 2 G1 X and sρ2G2X; • sρ1G1◦G2X iff there exists Z such that sρ 1 G1 Z and zρ1G2X for each z ∈ Z; • sρ2G1◦G2X iff there exists Z such that sρ 2 G1 Z and zρ2G2X for each z ∈ Z. The meaning1 of sρiGX is: "Player i has a strategy to play the game G so that if an outcome state is attained, it is in X." Proposition 2. Each forcing condition propagates over all forcing relations. Proof. Routine check. Note that the cases for (FIN) and (DET) use (MON). It is easy to see that all idle terms have the same forcing relations as ι. 3. Axiomatization of the algebra of games 3.1. Inclusions and identities of game terms Definition 3. Let G1 and G2 be game terms and B a game board. • G1 is i-included in G2 on B for i = 1, 2, denoted G1 ⊆i G2, if ρiG1 ⊆ ρ i G2 . • G1 is included in G2 on B, denoted B |= G1  G2 if G1 ⊆1 G2 and G2 ⊆2 G1 on B. 1This is the 'partial correctness' style of interpreting forcing relations. Alternatively, they can be interpreted like 'total correctness' statements: "Player i has a strategy to play the game G so that an outcome state is attained and it is in X." 224 V. Goranko • G1 and G2 are equivalent on B, denoted B |= G1 = G2 if they are assigned the same forcing relations in B. • Further, G1 is included in G2, denoted G1  G2 if B |= G1  G2 for every game board B. Then we also say that G1  G2 is a valid term inclusion, denoted by |= G1  G2. • Respectively, G1 and G2 are equivalent, denoted G1 ∼ G2 if they are equivalent on every game board, i.e. G1 = G2 is a valid term identity, also denoted by |= G1 = G2. Analogous notation will be used for validity in a class of game boards, e.g. DET |= G1 = G2 will mean that G1 = G2 is valid in every determined game board. Note that G1 ∼ G2 iff G1  G2 and G2  G1. Actually,  can be reduced to ∼ in the well-known lattice-theoretic fashion: Proposition 4. G1  G2 iff G1 ∨G2 ∼ G2 iff G1 ∧G2 ∼ G1. 3.2. The axioms of the algebra of games The main goal of this paper is to make precise and confirm the conjecture of [1] that the following term equivalences provide a complete axiomatization of the game algebra: 1. Double dualization: G ∼ Gdd; 2. The usual identities for ∨ in distributive lattices: idempotency, commutativity, associativity. 3. Absorption: G1 ∨ (G1 ∧G2) ∼ G1. 4. Distributivity: G1 ∨ (G2 ∧G3) ∼ (G1 ∨G2) ∧ (G1 ∨G3). 5. Associativity of ◦. 6. Distribution of d over ◦ : (G1 ◦G2) d ∼ Gd1 ◦G d 2. 7. Left-distribution for ∨ and ◦ : (G1 ∨G2) ◦G3 = (G1 ◦G3)∨ (G2 ◦G3). 8. Right-distributive inclusion: G1 ◦G2  G1 ◦ (G2 ∨G3). (According to Prop. 4, this is equivalent to an identity). 9. The extras for ι: multiplicative unit: G◦ι ∼ ι◦G ∼ G and self-duality: ι ∼ ιd. We denote the set of all these identities by GAι, and the set of those not involving ι by GA. Note that the respective identities for ∧, as well as the dual absorption, distributivity, left-distribution for ∧ and ◦, right-distributive inclusion G1 ◦ The Basic Algebra of Game Equivalences 225 (G2∧G3)  G1◦G2, and the two De Morgan's laws for ∨,∧ and d easily follow from the definition of ∧ and GAι in the equational logic for the algebra of games, which includes the standard set of derivation rules reflecting the fact that ∼ is a congruence in the algebra of games. Proposition 5. All identities in GAι are valid. Proof. Routine verification. Theorem 6. Every valid term identity of the game algebra can be derived from GAι in the standard equational logic. The proof of this theorem will be presented in the last section, and meanwhile we will build up the necessary machinery and will obtain auxiliary results for it. Remark. We note that ι can be omitted from the language together with its axioms, and the remaining axiom system GA will remain complete for the reduced language. The proof of this follows the same line as the one presented here, with a little technical and notational overhead due to the absence of ι. 4. Canonization of game terms Definition 7. Canonical game terms are defined recursively as follows: • ι is a canonical term. • Let {Gik|k ∈ Ki, i ∈ I} be a finite non-empty family of canonical terms and {gik|k ∈ Ki, i ∈ I} be a family of literals such that gik can be an idle literal only if Gik is an idle term. Then ∨ i∈I ∧ k∈Ki gik ◦ Gik is a canonical term. Remark. Any (or all) index sets I,Ki above can be singletons. Nevertheless, the respective disjunctions/conjunctions remain in place. The only essential use of the idle term ι here is to facilitate this canonical presentation of game terms and to provide a convenient base for structural induction on canonical terms. Its use can be circumvented at the cost of minor technical complications, though. Proposition 8. Every game term G is equivalent, provably in GAι, to a canonical game term. 226 V. Goranko Proof. First we prove by induction on canonical terms that the dual of a canonical term is equivalent to a canonical term. The case of ι is trivial since ιd ∼ ι. Let G = Hd where H = ∨ i∈I ∧ k∈Ki hik ◦ Hik and the claim holds for the canonical terms Hik. Then H d ∼ ∧ i∈I ∨ k∈Ki hdik ◦H d ik which, using the distributive laws for ∨ and ∧ and using the inductive hypothesis for the Hik's, converts into an equivalent canonical term. Now, we prove the main claim by induction on the length of arbitrary terms. The atomic case: g ∼ ∨∧ g ◦ ι. The case of duals was done above. The case G = G1 ∨G2 is almost trivial, using ι ∼ ι ◦ ι, if necessary. The remaining case G = G1 ◦G2 is treated by induction on G1, assuming that G2 is canonical. If G1 is a literal, G1 ◦G2 can be written as ∨∧ G1 ◦G2 where ∨ and ∧ are over singletons, so it is canonical. The inductive step for G1 = ∨ i∈I ∧ k∈Ki gik ◦ Gik is enabled by the left-distributive laws for ◦, pushing it inside the ∨ and ∧ , followed by the associativity of ◦ which eventually reduces the case to all Gik ◦G2 which are covered by the inductive hypothesis. This proof also outlines an algorithm for canonizing game terms which can be easily made precise. Remark. Canonical game terms impose a periodic structure on games: every game is a composition of one or several rounds, each consisting of: • a choice of player I, • followed by a choice of player II, • followed by an atomic game by one of the players (depending of the sign of the literal). Of course, some of these choices may be vacuous, when only one disjunct or conjunct is available to choose from, but still the 'ritual' is strictly followed. Definition 9. Two canonical terms G,H are isomorphic, denoted G ' H, if one can be obtained from the other by means of successive permutations of conjuncts (resp. disjuncts) within the same ∧ 's (resp. ∨ 's) in subterms. In other words, isomorphic terms are the same, up to the order of the conjuncts and disjuncts. Term isomorphism is the intermediate syntactic notion between identity and semantic equivalence ∼, which we will eventually prove equivalent to the latter. In fact, isomorphism of terms can be replaced by genuine identity at the cost of introducing a linear ordering on literals and terms and applying it to order the ∧ 's and ∨ 's in the definition of canonical terms. The Basic Algebra of Game Equivalences 227 Proposition 10. Isomorphic terms are equivalent, provably in GAι. Proof. Easy. Definition 11. We define recursively embedding of canonical terms, denoted by  as follows: • ι  ι; • Auxiliary notions: if g, h are literals andG,H are canonical terms, g◦G embeds into h ◦H iff g = h and G  H; a conjunction ∧ k∈K gk ◦Gk embeds into a conjunction ∧ m∈M hm ◦Hm if for every m ∈M there is some k ∈ K such that gk ◦Gk  hm ◦Hm. • Let G = ∨ i∈I ∧ k∈Ki gik ◦Gik and H = ∨ j∈J ∧ m∈Mj hjm ◦Hjm. Then G  H iff every disjunct of G embeds into some disjunct of H. Proposition 12. If G,H are canonical terms and G  H then G  H is provable in GAι. To be precise, then GAι ` G ∨H = H. Proof. Double induction on G and H, using, inter alia, the right-distributive inclusions. Thus, embedding of terms is the syntactic counterpart of inclusion. Definition 13. Minimal canonical terms: • ι is a minimal canonical term. • Let G = ∨ i∈I ∧ k∈Ki gik ◦ Gik be a canonical term where all Gik are minimal canonical. Then G is minimal canonical if: 1. ιd does not occur in G. 2. None of gik is ι unless Gik is ι. 3. No conjunct occurring in a conjunction ∧ k∈K gik ◦ Gik embeds into another conjunct from the same conjunction. 4. No disjunct in G embeds into another disjunct of G. Thus, minimal canonical terms are systematically 'minimized' canonical terms. Proposition 14. Every term G can be reduced to an equivalent minimal canonical term c(G) and this can be done provably in GAι. Proof. First, transform G to a canonical term, which then can be pruned down to an equivalent minimal canonical one, using the identities in GAι (e.g. right-distribution inclusions and the absorption laws for 3 and 4). 228 V. Goranko From now on, our strategy towards proving Theorem 6 will be to show that two minimal canonical terms are equivalent iff they are isomorphic. Presumably, that can be done entirely within game semantics. Instead, we introduce and use for the purpose a simple translation of game terms and identities to modal logic, which will make our task essentially easier. 5. Translation of the algebra of games to modal logic Here we introduce a translation of GL into plain modal logic. This translation, naturally, resembles Parikh's translation of the language GL∗ extending GL with game iteration (∗) into μ-calculus, but is simpler, computationally lighter and easier to use. In particular, we will use it in the next section to construct counter-models to invalid game equivalences, since Kripke models are rather more transparent, flexible and easier to deal with than game boards. To begin with, we consider the modal language ML comprising: • a set of atomic variables V = V ∪ {q} where V = {pa}a∈A and q /∈ V is an auxiliary variable. • the usual modal connectives: ∨,∧,¬,,♦, where ♦ will be regarded as an abbreviation for ¬¬. Some terminology and notation: • Substitution φ(ψ/q) : ψ is substituted for all occurrences of the variable q in φ. • A dual of a modal formula φ with respect to the variable q is φdq = ¬φ(¬q). Note that (φdq) d q ≡ φ. • Furthermore, we will often treat modal formulae as set operators in the standard sense, and thus, given a formula φ(q), Kripke model M = 〈S,R, V 〉 and a set X ⊆ S we will allow ourselves the sloppiness of writing φ(X) assuming its natural meaning, viz. that q has been evaluated as X. 5.1. The translation: All duals of modal formulas used in the translation will be with respect to q, so we can safely omit the subscript. Likewise, all substitutions will be of the type φ(ψ/q), which hereafter we will simply write as φ(ψ). With every game term G we associate a modal formula m(G) as follows: • m(ι) = q; The Basic Algebra of Game Equivalences 229 • m(ga) = ♦(pa → q) for any non-idle atomic game ga, a ∈ A; • m(G1 ∨G2) = m(G1) ∨m(G2); • m(Gd) = (m(G))d, also denoted by md(G). • m(G1 ◦G2) = m(G1)(m(G2)). Note that: • Every formula m(G), being positive in q, is monotone in q. • m(gda) is equivalent to ♦(pa ∧ q). Hereafter we will simply consider these equal. • m(G1 ∧G2) = m(G1) ∧m(G2). • md(G1 ◦G2) = m d(G1)(m d(G2)). Example. • m(g1 ◦ (g2 ∨ g3)) = ♦(p1 → (♦(p2 → q) ∨ ♦(p3 → q))); • m((g2 ∨g3)◦g1) = ♦(p2 → ♦(p1 → q))∨♦(p3 → ♦(p1 → q)) = m((g2 ◦ g1) ∨ (g3 ◦ g1)). • m(((g1 ◦ g2)∨ g1) d ◦ g3)) = ♦(p1 ∧♦(p2 ∧♦(p3 → q)))∧♦(p1 ∧ ♦(p3 → q)). 5.2. Preservation of validity The main result regarding this translation is: Theorem 15. For any game terms G,H, if the game inclusion G  H is valid on all determined game boards then |= m(G) → m(H). Proof. By contraposition, suppose M,u 2 m(G) → m(H) for some model M with a domain S and state u ∈ S. Then we define a game board BM = 〈 S, {ρia}a∈A;i=1,2 〉 as follows. For every X ⊆ S and s ∈ S : sρ1aX iff M, s  m(ga)(X), and sρ2aX iff M, s  m d(ga)(X). Lemma 16. BM is a determined game board. Proof of the lemma. The condition MON is immediate from the monotonicity of m(G) in q. For CON and DET, notice that M, s 2 m(ga)(X) iff M, s  ¬m(ga)(X) i.e. M, s  m d(ga)(¬X). Lemma 17. For every s ∈ S, X ⊆ S and term D : sρ1DX iff M, s |= m(D)(X), sρ2DX iff M, s |= m d(D)(X). 230 V. Goranko Proof of the lemma. Structural induction on D. For atomic games this holds by definition. The cases D = Dd1 and D = D1 ∨D2 are straightforward. Let D = D1 ◦D2 and suppose sρ1DX. Then sρ 1 D1 Z for some Z ⊆ S such that zρ1D2X for each z ∈ Z. Then M, s |= m(D1)(Z) and Z ⊆ V (m(D2)(X)), so by monotonicity M, s |= m(D1)(m(D2)(X)), i.e. M, s |= m(D1 ◦D2)(X). The case of sρ2DX is quite analogous, modulo the duality, but we'll do it nevertheless: let sρ2D1Z for some Z ⊆ S such that zρ 2 D2 X for each z ∈ Z. Then M, s |= md(D1)(Z) and Z ⊆ V (m d(D2)(X)), so by monotonicity M, s |= md(D1)(m d(D2)(X)), i.e. M, s |= m d(D1 ◦D2)(X). Conversely, letM, s |= m(D1◦D2)(X), henceM, s |= m(D1)(m(D2)(X)). Then Z = V (m(D2)(X)) is such that sρ 1 D1 Z and zρ1D2X for each z ∈ Z, by the inductive hypothesis. Therefore, sρ1D1◦D2X. Likewise for sρ 2 D1◦D2 X. This completes the induction and the proof of the lemma. Finally, recall that M,u |= m(G) and M,u 2 m(H). Let X = V (q). Then uρ1GX while ¬uρ 1 HX, so BM 2 G  H. Corollary 18. For any game terms G,H, if DET |= G = H then |= m(G) ↔ m(H). 6. Some technical results First, some useful remarks. • Since K is complete for the class of irreflexive tree-like Kripke models, every non-valid translation of a game inclusion or identity can be refuted in a model rooted at a state s without predecessors. Note that any re-evaluation of variables at s in such a model will not affect the truth or falsity at s of any m(G), except m(ι), when the truth of q is altered, because all occurrences of other variables in these formulae are in the scope of modal operators. • Let F∗ = 〈S∗, R∗〉 where S∗ = {∗, y, z}, R∗ = {(∗, y), (y, z), (z, z)}. Then the Kripke model M+ = 〈S∗, R∗, V+〉 , where V+(q) = {∗, z} satisfies all m(G) at its root ∗, while the model M− = 〈S∗, R∗, V−〉 , where V−(q) = ∅ and V−(pa) = {z} for all a ∈ A, falsifies all m(G) at ∗. These models can be freely grafted on an irreflexive leaf of any model (taking care of q, if necessary). Here's our main technical lemma: Lemma 19. Let G and H be minimal canonical terms. The following are equivalent: The Basic Algebra of Game Equivalences 231 • G  H. • (♠) There is a disjunct ∧ k∈K gik ◦Gik in G such that every disjunct in H contains a conjunct hjmj ◦Hjmj not including any of the conjuncts gik ◦Gik for k ∈ K. • There is a finite (tree-like) Kripke model M and a state s ∈ M such that: M, s  m(G); M, s 2 m(H); and s has no predecessors in M. Proof. We prove all equivalences by double induction on the structure of G and H. The case when both of them are ι is vacuous, so suppose otherwise and let G = ∨ i∈I ∧ k∈Ki gik ◦ Gik, H = ∨ j∈J ∧ m∈Mj hjm ◦ Hjm where the claim holds for all pairs of Gik's and Hjm's. If one of G and H is ι we represent it as ∨∧ ι ◦ ι. 1) Let G  H. Then there is a game board B = 〈 S, {ρia}a∈A;i=1,2 〉 such that either G 1 H or H 2 G on B. 1.1) Suppose G 1 H. Then there is a state s and a disjunct (the choice of player I) ∧ k∈Ki gik ◦ Gik such that every conjunct gik ◦ Gik enables him to achieve some outcome X from s which he cannot force on H, so every disjunct ∧ m∈Mj hjm ◦Hjm in H contains a conjunct hjmi ◦Hjmi which lacks the power for player I to force an outcome X. Thus, none of the terms hjmi ◦Hjmi , j ∈ J, includes any of gik ◦Gik, k ∈ Ki. 1.2) Suppose H 2 G. Then player II can force some outcome X in H which she cannot force in G, so every disjunct ∧ m∈Mj hjm ◦ Hjm (possible choice of I) in H contains a conjunct (the reply of II) hjmi ◦ Hjmi which contains (s,X) in the forcing relation for II, while this is not the case for G, so some disjunct ∧ k∈Ki gik ◦Gik is such that no term gik ◦Gik in it contains (s,X) in its forcing relation for II, hence none of gik◦Gik, k ∈ Ki is included in any of hjmi ◦Hjmi , j ∈ J . Thus, in either case (♠) holds. 2) Suppose (♠). Note that there can be at most one idle term amongst all {gik ◦Gik|k ∈ Ki} and {hjmi ◦Hjmi |j ∈ J}. We will build a Kripke model M which will satisfy all {m(gik ◦Gik)|k ∈ Ki}, and hence m(G), while none of {m(hjmi ◦Hjmi)|j ∈ J}, hence it will falsify m(H). M will be rooted at some state s with no predecessors, which is needed for the inductive hypothesis because models like this will be grafted at their roots on larger models as the induction goes on. Depending on the signs of the literals gik, k ∈ Ki and hjmi , j ∈ J, the set of all these terms splits into the following subsets: • T À = {tα ◦Dα|α ∈ A} whose translations must be true at s; • TB = {t d β ◦Dβ |β ∈ B} whose translations must be true at s; 232 V. Goranko • TΓ = {tγ ◦Dγ |γ ∈ Γ} whose translations must be false at s; • T∆ = {t d δ ◦Dδ|δ ∈ ∆} whose translations must be false at s. • Possibly, Tι = {ι ◦ ι}. The terms tα, tβ, tγ , tδ above are non-idle atoms. Let pα, pβ, pγ , pδ be their corresponding variables in the modal translation. Thus, we have to satisfy at s simultaneously the following sets of formulae: • FA = {♦(pα → m(Dα))|α ∈ A}, • FB = {♦(pβ ∧m(Dβ))|β ∈ B}, • FΓ = {♦(pγ ∧ ¬m(Dγ))|γ ∈ Γ}, • F∆ = {♦(pδ → ¬m(Dδ))|δ ∈ ∆}. • Possibly, Fι = {q} or Fι = {¬q}, depending on whether there is an idle term in {gik ◦Gik|k ∈ Ki} or {hjmi ◦Hjmi|j ∈ J} respectively. We build the model M = 〈W,R, V 〉 as follows: W = {s} ∪ (A ∪ ∆) ∪ ((A ∪ ∆) × (B ∪ Γ)) ∪W ′, where A,B,Γ,∆ are the index sets above, which will form the 'carcass' of the model, and W ′ will be sub-models satisfying/falsifying the m(D)′s, which will be grafted on the carcass accordingly (see further). • For better readability, in what follows the elements of a product X×Y will be denoted as xy, for x ∈ X, y ∈ Y. R = {(s, x)|x ∈ A ∪∆}∪{(x, xy)|x ∈ A ∪∆, y∈ B ∪ Γ}∪R ′ where R′ will be the union of the inherited relations from the grafted sub-models. The rest of the model and the valuation V will be defined as follows: • Every state αβ , for α ∈ A, β ∈ B must satisfy pα → m(Dα) and pβ ∧m(Dβ). For that, we set pβ true at αβ and graft a copy of M+ at αβ. • Every state αγ , for α ∈ A, γ ∈ Γ must satisfy pα → m(Dα) and pγ∧¬m(Dγ). If α 6= γ we set pα false and pγ true at αγ and graft a copy of M− at αγ . If tα = tγ then Dα  Dγ (for, otherwise, tα◦Dα  tγ◦Dγ , which contradicts (♠)), hence by the inductive hypothesis there is a model Mαγ rooted at some u such that Mαγ , u |= m(Dα) while Mαγ , u 2 m(Dγ). Then we set pα true and graft a copy of Mαγ at αγ . • Every state δβ must satisfy pβ ∧m(Dβ) and (pδ → ¬m(Dδ). This case is treated analogously to the previous one. • Every state δγ must satisfy pγ ∧ ¬m(Dγ) and pδ → ¬m(Dδ). For that we set pγ true and graft a copy of M− at δγ . The Basic Algebra of Game Equivalences 233 • Finally, s ∈ V (q) iff q ∈ Fι. This completes the description of M. It is immediate from the construction that M, s will satisfy all formulae in FA ∪ FB ∪ FΓ ∪ F∆ and hence M, s  m(G), while M, s 2 m(H). 3) If M, s  m(G), while M, s 2 m(H) then, by Theorem 15, G  H. This completes the circle of equivalences and the induction step. Corollary 20. For any game terms G,H: 1. |= m(G) → m(H) iff G  H is a valid game inclusion. 2. |= m(G) ↔ m(H) iff G = H is a valid game identity. Proof. One direction of (1) is by Th. 15. For the other, suppose |= m(G) → m(H). We can assume that G,H are minimal canonical, again due to Th. 15, so G  H by Lemma 19. (2) follows immediately from (1). Corollary 21. |= G = H iff DET |= G = H. Given a Kripke model M and a state s, T (M, s, u) will denote the model obtained from M by adding two new states u, v such that uRv and vRs. Lemma 22. Let G,H be any terms and g, h be non-idle literals. Then g ◦G  h ◦H iff g = h and G  H. Proof. One direction is obvious. The other we prove by contraposition, assuming g 6= h or G  H and using the modal translation. Case 1: g 6= h. We falsify m(g ◦ G) → m(h ◦H) at the root of a model constructed by cases as follows: 1.1) g = ga, h = gb, a 6= b for some atoms ga, gb. Take a copy of T (M−, ∗, u) and set pa false and pb true at ∗. 1.2) g = ga, h = g d b . Take a copy of F∗ and set pa and pb false at z. 1.3) g = gda , h = gb. Take a copy of T (M+, ∗, u), add a new successor s to v and graft a copy of M− at s, setting pa true at ∗ and pb true at s, so ♦(pa ∧ m(G)) is true at u, while pb → m(H) is false at s, hence (pa → m(H)) is false at v, so ♦(pa → m(H)) is false at u. 1.4) g = gda, h = g d b , a 6= b. Take a copy of T (M+, ∗, u) and set pa true and pb false at ∗. Case 2: g = h and G  H. We can assume that G and H are minimal canonical. Let M, s 2 m(G) → m(H) (by Lemma 19) and suppose g = ga 234 V. Goranko or g = gda. Then setting pa true at s will falsify m(g ◦ G) → m(h ◦ H) at u in T (M, s, u). Thus, in each case we have shown that g ◦G  h ◦H. Lemma 23. 1. g ◦G  ι ◦ ι iff g is an idle literal and G  ι. 2. ι ◦ ι  g ◦G iff g is an idle literal and ι  G. Proof of the non-trivial directions: If g is non-idle then m(g ◦ G) → m(ι ◦ ι) is falsified at the root of T (M+, ∗, u) by setting q to false at u. Thus, if g ◦ G  ι ◦ ι then g is idle and |= m(g ◦G) → m(ι ◦ ι), hence |= m(G) → m(ι), so G  ι. Likewise, if g is non-idle then m(ι ◦ ι) → m(g ◦G) is falsified at the root of T (M−, ∗, u) by setting q to true at u. Thus, if ι ◦ ι  g ◦G then g is idle and |= m(ι) → m(G), so ι  G. 7. Proof of the completeness of GAι Lemma 24. If G,H are minimal canonical terms then G  H iff G  H. Proof. If G  H then G  H is straightforward. For the other direction we proceed by double induction on the structure of both terms. The case when both of them are ι is trivial, so suppose otherwise and let G = ∨ i∈I ∧ k∈Ki gik ◦Gik, H = ∨ j∈J ∧ m∈Mj hjm◦Hjm be minimal canonical terms (again, representing ι as ∨∧ ι ◦ ι) such that G  H and the claim holds for all Gik's and Hjm's, i.e. if one of these is included into another then that inclusion is embedding. Now, suppose G is not embedded into H. Then there is a disjunct ∧ k∈Ki gik ◦ Gik in G such that every disjunct ∧ m∈Mj hjm ◦Hjm in H contains a conjunct hjmi ◦Hjmi in which none of gik ◦Gik, k ∈ Ki is embedded. But that means, by the inductive hypothesis and Lemmas 22 and 23, that none of these terms is included in any of the hjmi ◦Hjmi , for j ∈ J. This is precisely the condition (♠) of Lemma 19. Therefore G  H. This completes the inductive step and the proof of the lemma. Proposition 25. If G,H are minimal canonical terms such that G  H and H  G then G ' H. Proof. Again, double induction on G,H. The case when both of them are ι is trivial. Suppose G = ∨ i∈I ∧ k∈Ki gik ◦Gik, H = ∨ j∈J ∧ m∈Mj hjm ◦Hjm be minimal canonical terms such that G  H and H  G and the claim holds for all Gik's and Hjm's. The Basic Algebra of Game Equivalences 235 Take any disjunct D from G. It embeds into some disjunct D ′ from H, which in turns embeds into some D′′ from G, so D is embedded into D′′, hence D and D′′ must coincide because G is minimal canonical. Therefore, D  D′ and D′  D, so for every conjunct C in D there is a conjunct C ′ in D′ embedded into C; and there is a conjunct C ′′ in D embedded into C ′, hence C ′′ is embedded into C. Again by minimal canonicity of G, that implies that C and C ′′ coincide, hence C  C ′ and C ′  C. Let C = t ◦ T and C ′ = t′ ◦ T ′ for some literals t and t′ and minimal canonical terms T, T ′ for which the inductive hypothesis holds. Therefore, C  C ′ and C ′  C, hence by Lemmas 22 and 23, t = t′, T  T ′ and T ′  T, so T  T ′ and T ′  T by Lemma 24, hence T ' T ′ by the inductive hypothesis. Therefore C ' C ′. Thus, every conjunct from D is isomorphic to a conjunct from D ′ and vice versa. This is a bijection because of the minimal canonicity of G and H. Hence, every disjunct from G is isomorphic to a disjunct from H and vice versa. Again, this is a bijection due to the minimal canonicity of G and H. Therefore, G ' H. Corollary 26. The minimal canonical terms G and H are equivalent iff they are isomorphic. Proof. G ∼ H iff (G  H and H  G) iff (G  H and H  G) iff G ' H. Proof of Theorem 6. Let G ∼ H and c(G), c(H) be minimal canonical terms obtained from G and H by reduction within GAι. Then c(G) ∼ c(H), hence c(G) ' c(H) by Corollary 26. Since each of the equivalences G ∼ c(G), c(G) ' c(H), c(H) ∼ H is derivable in GAι, so is G ∼ H. 8. Concluding remarks 8.1. Valid identities and game board conditions On the one hand, it can be easily verified that all axiomatic identities, and hence all valid ones, remain valid if the condition for consistency of powers (CON) is omitted.2 On the other hand, as Corollary 21 shows, determinacy of game boards does not add new valid identities of game terms. This result can be strengthened: termination can be added, too, without introducing new valid identities. 2This observation is essentially due to Yde Venema, who raised the question. 236 V. Goranko Proposition 27. |= G = H iff (DET∩FIN) |= G = H. Proof. It is sufficient to modify the proof of Lemma 19 by showing that whenever G  H for minimal canonical terms G and H, the counter-model for m(G) → m(H) can be constructed in such a way that the game board determined by it as in the proof of Theorem 15 is terminating as well, i.e. sρiaS holds for each atomic game ga (and hence for every game term). These conditions impose the following requirements on the Kripke model: • Termination for ι. It holds trivially. • sρ1aS iff M, s |= ♦(pa → >) i.e. M, s |= ♦>. This is satisfied by the current construction. • sρ2aS iff M, s |= ♦(pa ∧>) i.e. M, s |= ♦pa for each non-idle a ∈ A. To satisfy this condition we extend the construction in the proof of Lemma 19 as follows: for each α ∈ A we add one more successor, α′ to α, graft a copy of M+ at α ′, and set all pa to be true at α ′. Likewise, for each δ ∈ ∆ we add one more successor, δ ′ to ∆, graft a copy of M− at δ ′, and set all pa to be true at δ ′. That will preserve the truth at s of all formulas from FA ∪ FB ∪ FΓ ∪ F∆, while forcing all ♦pa to be true at s. Thus, every invalid inclusion G  H can be falsified in a determined and terminating game board. 8.2. On the complexity of the validity of game identities While the translation m of canonical terms to formulae of modal logic is polynomial in the size of the terms, because only literals occur on the left of compositions, in general that translation can be exponential in the size of the terms (e.g. consider the translation of (g11∨g21)◦(g12∨g22)◦ ...◦(g1n∨g2n)). However, as Venema has noted, the number of different subformulae in the resulting translation is still polynomial in the size of the term, which can be shown by a simple induction on terms. Thus, the complexity of the validity of game identities is not greater than the complexity of the validity in the basic modal logic K. Thus, we obtain the following. Proposition 28. The validity problem for identities of game terms is in PSPACE. 8.3. From game algebras to game logics. Clearly, the game algebra of a fixed language of game terms can be regarded as a fragment of the corresponding game logic. In particular, every valid The Basic Algebra of Game Equivalences 237 game identity G = H corresponds to a pair of valid formulas 〈G〉 q ↔ 〈H〉 q and [G]q ↔ [H]q (in the notation of [5] and [3]) and vice versa. The dual-free fragment of game logic (with tests) was axiomatized and proved complete in [5], by modifying appropriately the completeness proof for PDL, while the iteration-free game logic, corresponding to the game language considered here (with additional tests), has been axiomatized and proved complete in [4] using an adaptation of Parikh's proof, combined with the method of canonical models for neighbourhood semantics of modal logic. That proof, however, does not imply the present completeness result because it does not show if the derivation of every valid formula of the type 〈G〉 q ↔ 〈H〉 q or [G]q ↔ [H]q can be translated into equational logic. On the other hand, the modal translation introduced here readily extends to the iterationfree game logic, and accordingly the method of proving completeness applied here can be modified to an alternative completeness proof for that logic, by extending the notion of canonical forms to all formulas. We note that this method can also be adapted to prove completeness of modal logic itself. In fact, that was essentially done quite a while ago in [2], where the use of normal forms in modal logic was promoted. 8.4. Representing game algebras Meanwhile, Venema has strengthened in [6] the completeness result presented here by proving a representation theorem for abstract game algebras defined by the set of identities GA into game boards. As the completeness of the full Game Logic introduced in [5] is still open, it is interesting to see if the method applied here or Venema's algebraic approach can be extended to the game language with iteration and thus provide a handle to solving that problem, too. 9. Acknowledgments This work was done during my visit to the Institute for Logic, Language and Computation at the University of Amsterdam in the fall of 2000, supported by ILLC, research grant GUN 2034353 of the National Research Foundation of South Africa, and the SASOL research fund of the Faculty of Natural Sciences of Rand Afrikaans University. My interest in logics of games and in the problem solved here was inspired by Johan van Benthem's lectures and notes on his course "Logic and games" given at ILLC at that time. I am indebted to Johan, Marc Pauly, and Yde Venema for stimulating discussions, important comments, and corrections on earlier drafts. Finally, am grateful 238 V. Goranko to the anonymous referee for meticulous reading of this text and a number of corrections on its style and content. References [1] van Benthem, J., Logic and Games, Lecture notes, ILLC report X-2000-03, 2000. [2] Fine, K., 'Normal Forms in Modal Logic', Notre Dame Journal of Formal Logic, XVI:229-237, 1975. [3] Pauly, M., Game Logic for Game Theorists, CWI report INS-R0017, Sept. 2000. [4] Pauly, M., Logic for Social Software, ILLC Dissertation Series DS 2001-10, University of Amsterdam, 2001. [5] Parikh, R., 'The Logic of Games and its Applications', Annals of Discrete Mathematics, 24:111-140, 1985. [6] Venema, Y., 'Representation of Game Algebras', Studia Logica, 75:239–256, 2003. Valentin Goranko Department of Mathematics Rand Afrikaans University PO Box 524, Auckland Park 2006 Johannesburg, South Africa E-mail: vfg@na.rau.ac.za