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Games and Strategies inside Elementary Logic

Johan van Benthem, Amsterdam & StanfordJanuary 2000

to appear in proceedings 7th

Asian logic Conference, Taiwan, June 1999

1 Logic and GamesConnections between logic and games go back to Antiquity, and the first systematicinterest in rules of rational debate. Argumentation is a kind of game, and improvingone’s skills in winning arguments draws many students to the lecture halls of logic.Well-defined ‘logic games’ have been around since the 1950s, when Paul Lorenzengave his pioneering analysis of argumentation as a two-person game between theproponent and opponent of some thesis under discussion – where validity of a thesismeans that its proponent has a winning strategy against any opponent. By now, thereare many logic games, for purposes of proof, semantic evaluation (Hintikka), modelcomparison (Ehrenfeucht), or model construction (Hodges), and the literature on thesubject is growing quickly. Moreover, games related to logic games have becomewide-spread in computer science. Despite this growing interest, the game-theoreticapproach has not become an established ‘perspective’ in logic, the way ‘semantics’or ‘proof theory’ are considered major modes of viewing validity and other logicalkey notions. The received opinion has it that logic games are a didactic tool, a usefulhandmaiden to syntax and semantics without independent interest. Moreover, theyare mostly a façon de parler, as there are no substantial connections with the deepermathematics of game theory as developed by Von Neuman/Morgenstern and Nash.

Our aim in this paper is to argue for a deeper connection between logic and games.Our reason for this lies in current ‘logical dynamics’, broadening the scope of logicto a general theory of information-handling processes among cognitive agents. Oneoften uses process models from computer science for these purposes (cf. ELD), butgames are certainly a richer model, which comes with vivid intuitions concerninginteraction of rational agents. In this paper, we shall leave this ideology aside, andproceed from the most basic logic games, pointing out what questions they involve,properly viewed. The latter caveat is important. We must take games seriously aslogical objects in their own right. Even the inventors of well-known logic games, ortheir well-known mathematical analyzers do not seem to love the games for theirown sake. They often assimilate logic games (i.e., dynamic activities) with assertions(i.e., static propositions) about their winning strategies, trying to make them fitstandard logic in a way that obscures new insights that might emanate from them.Our contribution in this respect involves no deep results – we rather try to change thereader’s perspective, showing how familiar logical things can become very new inthis light. This paper has succeeded if you find the experience worth-while! (Note 1.)

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More concretely, in Sections 2 – 4 we review the most familiar types of logic game.We show how these exemplify issues of more general game-theoretic significance.In Section 5 we discuss the first set of these: the choice of natural game operations,a proper notion of game equivalence, and the resulting basic game algebra. Section 6returns to first-order predicate logic, and re-analyzes its structure in this new light.In particular, ‘first-order validity’ becomes a mixture of a decidable game-theoreticcore plus extras. Finally, in Section 7, we briefly consider another basic issue: therole of strategies as a unifying notion across logic. Thus, logic games become muchmore than didactic tools: they affect our view of the architecture of logic.

2 Semantic Evaluation

2.1 Evaluation gamesSuppose two parties disagree about a statement in some model M under discussion:Verifier V claims it is true, Falsifier F that it is false. 'Evaluation games' describetheir moves of defense and attack – with a schedule driven by the statement at issue:

atoms test to determine who winsdisjunction A∨B V chooses which disjunct to playconjunction A∧B F chooses which conjunct to playnegation ¬A role switch between the two players,

play continues with respect to Aexistential quantifiers ∃x A(x) V picks an object d, after which

play continues with respect to A(d)universal quantifiers ∀x A(x) the same, but now for F

E.g., consider a universe with just two objects s, t . Here is the game of perfectinformation for the logical formula ∀x ∃y x≠y, pictured as a tree of possiblemoves, with the scheduling read from top to bottom:

F x:= s x:= tV V

y:= s y:= t y:= s y:= t loseV winV winV loseV

Falsifier starts, Verifier must respond. There are 4 possible plays, with 2 wins foreach player. (Another view of this particular game is as ‘matching pennies’.)Evaluation games for more complex formulas can be quite attractive in teachingelementary logic to students. Finally, note that a formula itself is just an abstract‘game form’. It becomes a real game only when a particular model M is chosen,which specifies the possible quantifier moves, and the outcomes for atomic tests.

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2.2 Truth and Verifier’s winning strategiesIn the preceding example, the players are not evenly matched. Evidently, V canalways win: after all, she is defending the truth of the matter… More precisely,she has a winning strategy, a pattern of behaviour that will guarantee a certainoutcome, in this case: ‘winning”. By contrast, F has no winning strategy, sinceV may always thwart him. But neither does he have a ‘losing strategy’: he cannotforce V to finish him off… Thus, players’ powers of determining the outcomes ofa game may be quite different. (Note 2.) Here is the fundamental connection:

Proposition A statement A is true in model M if and only if Verifierhas a winning strategy for A’s evaluation game played in M.

Proof The proof is a direct induction on formulas, whose steps exhibit the closeanalogy between the logical operators and ways of combining strategies. Wemention two cases. (a) If A∨B is true, then at least one of A, B is true – say, A .By the inductive hypothesis, V has a winning strategy σ for A. But then she hasa winning strategy for the game A∨B whose first move is ‘choose left’, and thenfollows up with σ . (b) If A∧B is true, then both A, B are true, and hence bythe inductive hypothesis, V has winning strategies σ , τ for A, B, respectively.But then, the combination of these two is a winning strategy for her in the wholegame A∧B: if in the first move, F chooses left, V plays σ , if F goes right, Vplays τ . Finally, both arguments (a), (b) are easily converted. QED

Thus, we have our first fundamental connection between a key notion in logic(‘truth’) and one in game theory (‘strategy’). We will broaden this as we continue.

2.3 First links with game theory: determinacy, game equivalenceThere are deeper aspects to the above Proposition. In particular, logical laws nowacquire game-theoretic import! E.g., consider classical Excluded Middle A∨¬A ,often taken to be a triviality. That Verifier has a winning strategy for this statementin every model means she can choose to play either A as Verifier, or ¬A asVerifier, i.e.: A as Falsifier, and still have a winning strategy for the remainder.But this is another way of expressing a well-known notion from game theory:

Fact Excluded Middle A∨¬A expresses determinacy of evaluation games:i.e., one of the two players must always have a winning strategy.

This is just the tip of an iceberg. Evaluation games are determined because of afew very general features. In the first decade of this century, Zermelo proved thefollowing result, whose main notions will be intuitively clear:

Theorem All two-player games that are zero-sumand have finite depth are determined.

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The proof is again a simple induction, this time on the depth of trees, which caneven be turned into an easy bottom-up algorithm for determining the playerhaving the winning strategy at any given node, at least in principle. Zermelo wasmainly concerned with finite-depth games like chess, where his result implies thatone of the two players has a non-losing strategy. Much stronger results thanZermelo’s have been found since, witness our next subsection. (Note 3.)

But logical laws also encode even more basic issues about games, which do notdepend on determinacy at all. To see this, consider a simple propositional law likethe Distribution of conjunction over disjunction. The two finite game trees in thefollowing figure correspond to the two propositional formulas A∧(B∨C) and(A∧B)∨(A∧C), that are logically equivalent. (Note 4.) Does this mean that thepictured games are ‘the same’ in some natural sense? We will return to this issuelater on – but the reader may want to check that indeed,

both players have the same powers of achieving outcomes in both trees:

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A 1

B C

1

2 2

A B A C

Of course, the intuitive notion of ‘power’ is not very precise. One can havepowers w.r.t. achieving outcomes, or also intermediate states – and moreover,powers may refer to internal game states, or to external effects of a game. Indeed,the general theme here, which has no generally accepted solution in the game-theoretic literature, is game equivalence and its mathematical laws. Distributionseems one valid principle of ‘etiquette’ for extensive games, telling us whichchange in ‘precedence’ are admissible without affecting players’ powers. Gameequivalence and its laws will be taken up in greater generality in Section 5.

2.4 Variations: infinite gamesBy contrast, current evaluation games for more complex languages than first-orderpredicate logic may go on forever, and determinacy becomes a more subtle issue.This technical issue again has a clear practical repercussion. Short-term disputesand debates may have finite depth – if only by stipulation of a limit to repetitions(as happens in chess). Such games are like special-purpose programs that mustterminate. But there are also general-purpose programs such as operating systems,whose task is to ensure that short-term tasking can go on indefinitely. When wemove to this higher level, and consider macro-games like ‘language use’ (the‘operating system’ of cognition), infinite patterns and strategies become essential.

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Infinite evaluation games arise with first-order logic with fixed-point operators,defining smallest/greatest fixed-points of monotone operators defined by positiveformulas. (Cf. Ebbinghaus-Flum for the first-order case, Stirling for the modal‘μ–calculus’, Barwise & Moss for an ode to greatest fixed-points, co-algebra andco-induction.) In games for this language, Verifier and Falsifier play according tothe earlier rules when the outermost logical operator is a first-order one. But theymust also have an instruction for dealing with, say, smallest fixed-point formulas

[μP,x. φ(P)](d)

Here the formula defining the next game becomes automatically φ(P)](d) – withthis understanding that, whenever further play hits an atom of the form Pe , noevaluation takes place (P is a ‘bound predicate variable), but the fixed-pointformula [μP,x. φ(P)](e) is substituted back in. (Note 5.) As a result legitimate runsof the game may indeed be infinite. The somewhat technical winning conventionfor these games says essentially (details are not relevant here) that

DefinitionAn infinite run is a win for Verifier if the fixed-point subformulain the highest syntactic position in the original formula whichhas been called infinitely often is a greatest fixed-point.

Falsifier wins if that highest recurring subformula is a smallest fixed-point. E.g.,consider the smallest fixed-point definition for transitive closure of a relation R :

μP, xy• (Rxy ∨ ∃z (Pxz ∧ Rzy))

To show this for objects d , e, Verifier need not show directly that Rde , but maychoose the second disjunct, take some f for z , and now claim that Rfe . But shemay only do that finitely often – otherwise, she has not made her point, and loses.The case of greatest fixed-point definitions is dual: Falsifier must ‘put up’ in somefinite number of cycles, while it would be OK for Verifier to keep the cycle going.

PropositionA formula in first-order fixed-point logic is true if and only ifVerifier has a winning strategy in the game just described.

See Doets 1999 for details of the argument, which is no longer a simple induction.Again, there is a more general game-theoretic result in the background here,having to do with determinacy. First, we cite a key result for infinite games:

Gale-Stewart TheoremInfinite games in which at least one player hasan open set of winning runs are determined.

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Definition An open set of runs O is one with the following property:every infinite sequence in O has a finite initial segment Xsuch that all runs of the game sharing X are in O .

Thus, the winning condition is based only on what happened during some finiteinitial play. We will see conditions of this kind with other logic games in thispaper. The above condition is not of this kind, however, as it essentially involveswhat happens over the whole infinite sequence. But it does fall under a strongerresult called Martin’s Theorem, stating that “All Borel games are determined”,where the winning conditions lie in the so-called Borel Hierarchy of sets. Withnon-Borel winning conventions, games can become non-determined. (Note 6.)

Infinite evaluation games are a conservative extension of our first-order games.But even for the predicate-logical language, genuine variations are possible. E.g.,in contrast to the above object-picking moves, one can consider moves thatwithdraw objects from a domain without replacement – or even moves thatdrastically change a domain or an interpretation function, by adding a subtractingindividuals and facts. (Some real games drastically affect their playing field...)Such variations mix the process of evaluation, leaving a model undisturbed, withmodel construction, the pure form of which we will consider in the next section.

Even further connections between logic and games can be found here. Notably,what are the natural operations that form new games out of old (cf. Section 5)?But first we turn to other logic games, to broaden our budget of examples.

3 Model Construction

3.1 Tableau gamesEvaluation and model comparison games start from given models. But what if weallow moves that create new objects, rather than select existing ones? Consider theSatisfiability Problem: given some formula, to determine whether it has a model.(Note 7.) The task here is to build a model satisfying the constraints expressed bythe given formula. Here is a model construction game between two players:Builder and Critic. The following format is a game version of the well-knownmethod of semantic tableaux, explained in elementary textbooks (Hodges 1977).At each stage of the game, there are two finite boxes of formulas: YES containsthe formulas that must be made true by Builder, NO those to be made false. (Note

8.) Play proceeds in rounds, each of which starts with Critic scheduling someformula for treatment. Some follow-up moves may be considered automatic:

if ¬A sits in one box, it changes to A in the otherif A&B sits in YES, it is replaced by A, B separatelyif Α∨Β sits in NO, it is replaced by A, B separately

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These 'simplify tasks' in an obvious sense. Other scheduled formulas requiredeliberate actions by players:

(a) Disjunctions in YES and conjunctions in NO are a choice by Builder.(b) Existential formulas ∃xφ in TRUE must be replaced by φ(d)

where d is some new object chosen by Builder.(c) For existential formulas ∃xφ in NO, Critic mentions some object

already in the domain under construction so far, and puts φ(d)in the NO box, without replacing ∃xφ .

(d) Dual rules hold for universal quantifiers, with infinite repetition for Criticnow on the YES side.

As to the winning convention, a stage is a loss for Builder if some formula occursin both boxes: obviously, no model construction can result in a formula being bothtrue and false. Builder wins a run if no such loss occurs at any stage. Examples ofthis game occur in any semantic tableau for a predicate-logical satisfiabilityproblem. Here are some striking differences with our evaluation games.

(i) The schedule of moves is not enforced by any specific formula, as it was inSection 2: tasks may be selected more freely. (ii) The roles of players areasymmetric, leaving room for various procedural conventions concerning theirrights and duties. (iiii) The game may be set up in alternative ways withoutaffecting outcomes. (Note 9.) (iv) Runs of the model construction may be infinite,as some satisfiable first-order formulas have only infinite models, with infiniteconstruction. (Note 10.) Still, the above winning condition yields an open set forCritic, whence the Gale-Stewart Theorem applies: first-order model constructiongames are determined: either Builder or Critic has a winning strategy.

3.2 Adequacy theorem: models and proofs as strategiesA semantic tableau is a finitely branching tree of 'sequents' of formulas YES•NOwhich can have both finite branches (ending in wins or losses for Builder) andinfinite ones. The latter phenomenon suggests inevitable complexity, and indeedthe SAT problem for first-order logic is undecidable, by Church's Theorem.Without strict constraints on scheduling, there may be infinite branches whereCritic stupidly repeats the same attacks, letting Builder get by with shoddyconstruction. Such branches are not informative, as no model is constructed, whileits existence is not refuted either. We might impose stricter scheduling – but wecan also bring in strategies, that have to deal with the strongest possible counter-play anyway. (Note 11.) Now comes the usual completeness result about tableaux:

Adequacy Theorem The following two assertions are equivalent:(a) The set of formulas {A1, .., Ak , ¬B1, …, ¬Bm} is satisfiable(b) Builder has a winning strategy in the construction game

starting with the A's put in YES and the B's in NO

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Proof The proof is like the usual one. From (a) to (b), any satisfying model willgive Builder all the information he needs to answer Critic's challenges. Inparticular, all disjunctions are decided by the model, which guides his choices.Conversely, take Builder’s winning strategy, and let it respond to Critic's strongestplay in terms of scheduling formulas, making sure that each possible challengeoccurs. The result is a branch (finite or infinite) without losses for Builder, whichinduces a model M consisting of all objects introduced along the branch. Then asimple induction on formulas shows that each formula in a YES box on the branchis true in M, while each formula in a NO box is false. QED

But this argument really yields more information, namely (Note 12.):

Fact We can exhibit an explicit correspondence between(a) winning strategies for Builder,(b) models for the given formulas.

But Builder’s strategies are not the only relevant items. Given the asymmetrybetween the players, there is an independent logical interest to (possible) winningstrategies for Critic. Now, as we noticed before, the model construction game isdetermined, by the Gale-Stewart theorem: and so, if Builder does not have awinning strategy, Critic has one. (Note 13.) What does the latter do? In this case,there is no model for the initial formulas, whence we have the valid consequence:

A1 & … & Ak |= B1 ∨…∨ Bm

Consider a game tree where Critic uses strict scheduling in his strategy. Then,each run ends in some finite stage where Builder loses. Thus, the game tree has noinfinite branches. Now we apply another general mathematical result: (Note 14.)

König's LemmaEach finitely branching infinite tree has an infinite branch.

Applying this result to our finitely branching game tree without infinite branches,it must be finite! Thus winning strategies for Critic are associated with finiteobjects, the 'closed tableaus'. In particular, in these construction games, Critic canmake sure the game ends in some fixed finite number of rounds. (Note 15.) Thisreally shows a well-known fact: closed tableaus correspond to proofs of validsequents A1 & … & Ak |= B1 ∨…∨ Bm . In all, we have one more useful

Fact We can exhibit an explicit correspondence between(a) winning strategies for Critic,(b) proofs for the initial sequent.

Thus, the single notion of strategy for two players in one single game may 'unify'very different logical information: in our case, models and proofs. (Note 16.)

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3.3 Variations: other procedures, decidable once-only gamesStrategies are also proofs in Lorenzen dialogue games. These also have a tableau-like character, but they differ in some crucial respects. First, players are bothcommitted to certain statements. Second,

intuitionistic, not classical logic results.

There are also different procedural conventions – but note that in our constructiongames, too, the roles of the two players are very different. Thus, somepeculiarities of Lorenzen games also emerge in our semantic alternative. We leavethe matter of a precise comparison of the two kinds of game for further study.

Even with construction games, simple variations raise hard questions. Considermoves for false existential quantifiers where only one challenge can be made (forone object, or all available ones): ∃xφ is not copied along. This is a natural finite-depth game. But what does it measure? Critic’s winning positions are a

decidable first-order logic without a Contraction Rule

for identical formulas. Does it have a natural semantics? Next, taking an idea fromSection 5, we could add a new game operator !φ to the language, creating specialformulas φ that can be attacked an arbitrary number of times. (Note 17.)

Construction games do not have the same natural operations as the evaluationgames of Section 2. In particular, having a formula A&B in the TRUE box doesnot correspond to some choice made by Critic once and for all at the beginning:it rather signals a kind of ‘parallel’ game conjunction: both conjuncts have to betaken care of during the construction. But the main new point of Section 3 for ushas been the importance of strategies as logical objects in their own right.

4 Model ComparisonPerhaps the most widely taught logic game are Ehrenfeucht-Fraïssé games. Onecan measure the expressive power of a language as follows: how much power ofdiscrimination does it have between given two models M, N ? The fine-structureof this can be measured by the following games. (Note 18.)

4.1 Back-and-forth gamesEhrenfeucht-Fraïssé games test finite 'degrees of potential isomorphism’. Thereare two players, Spoiler (S) and Duplicator (D). S claims the two models aredifferent, D that they are similar. They agree to play and disagree about this oversome finite number of k rounds. Each round proceeds as follows:

S chooses one of the models, and picks an object d in its domain.D then chooses an object e in the other model, and the pair(d, e) is added to the current list of matched objects.

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At the end of the k rounds, the total matching obtained is inspected. If it is apartial isomorphism (Note 19), D wins; otherwise, S wins the game. Here are twoexamples, illustrating games and winning strategies.

I 3-arrow cycle versus 4-arrow cycle

1 2 i j

3 l k

Round 1 S chooses 1 in M D chooses i in NRound 2 S chooses 2 in M D chooses j in NRound 3 S chooses 3 in M D chooses k in N

S has won, because this match is not a partial isomorphism. But S can do better: there is a winning stategy in 2 rounds, starting with i in N, and then choosing k in the next round. No such pattern occurs in M, so D is bound to lose.

II Z (integers with < ) versus Q (rational numbers with < )These two linear orders have different properties: one is dense, the other discrete.

….. -1 0 +1 … Z------------------------------------------------------- Q

0 1/3 ↑ 1/5

D has a winning strategy here for the game over 2 rounds. But Scan win this comparison game in 3 rounds. Here is a typical play:

Round 1 S chooses 0 in Z D chooses 0 in QRound 2 S chooses 1 in Z D chooses 1/3 in QRound 3 S chooses 1/5 in Q D chooses any object: and loses.

These games are determined again by Zermelo’s Theorem, being of finite depth.Of course, players may play badly and lose, even if they have a winning strategy.Spoiler wins by exploiting some 'definable difference'. For instance, in II, this isthe first-order formula for density of < : ∀x∀y (x<y → ∃z (x<z & z<y)).

4.2 Distinctions as spoiler’s strategiesThe key fact about the role and use of Ehrenfeucht-Fraïssé games is the followingresult, using the quantifier depth of a first-order formula, measuring the longestnesting of quantifiers occurring inside it (e.g., density had depth 3):

Adequacy Theorem For all k, M, N, the following are equivalent:(a) D has a winning strategy in the k-round game between M and N(b) M, N agree on all first-order sentences up to quantifier depth k .

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The proof of this is by induction, and can be found in any good textbook. What isobserved less often is the following. As in our preceding section, the inductiveargument establishes something extra – which is stated most easily for Spoiler.

Fact There exists an effective correspondence between(a) winning strategies for S in the k-round game, and(b) first-order sentences φ of quantifier depth k

with M|= φ , not N |= φ.

Proof From (b) to (a). Every formula A of quantifier depth n defines auniform (though not necessarily effective) winning strategy for S in an n-roundgame between arbitrary models. Each round n-k starts with a match betweenlinked objects chosen so far which differ on some subformula B of A ofquantifier depth n-k . By Boolean analysis, Spoiler finds some existentialsubformula ∃x•C of B with C of quantifier depth n-k-1 on which the twomodels disagree. Spoiler's next choice is a witness in that model of the two where∃x•B holds. From (a) to (b). Each winning strategy for S induces adistinguishing formula of proper depth. To obtain this, let S make his first choiced in model M , and start with an existential quantifier for that object. Ourformula will be true in M , and false in N . We know that each choice of D for acorresponding object e in N gives a winning position for S in all remaining n-1round games starting from d–e . By induction, these induce distinguishingformulas of depth n-1 . By the Finiteness Lemma (Note 20), only finitely manysuch formulas are available. Some of these will start with 'their' first quantifier inM (say A1, .., Ar) - others in N (say B1, .., Bs). The total distinguishing formulais the M-true ∃x• (A1 & .. & Ar & –B1 & ... & –Bs). QED

With construction games in Section 3, strategies for both players corresponded toindependently baptized logical objects. Here, we saw the same for Spoiler. It is abit harder to say, however, what ‘objects’ of independent logical interestcorrespond to Duplicators’s winning strategies. One might call them ‘analogies’.

Finally, the finite limit can be pushed upwards. One can play model comparisongame forever, saying simply that D wins an infinite run if she has maintained apartial isomorphism all along the way. In this manner, we get a game in which

winning strategies for D correspond precisely to so-calledpotential isomorphisms between the two models. (Note 21.)

As the winning rule for S is open, the Gale-Stewart Theorem applies, and

even infinite model comparison games are determined.

Here again, while the notion of winning strategy remains the same across games,their logical interpretation may differ. In infinite Ehrenfeucht games,

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Duplicator’s winning strategies are precisely the potential isomorphism.

This time, Spoiler’s winning strategies lack an obvious pre-existing counterpart.They are ways of blocking each attempt at a potential isomorphism at some finitestage – but not necessarily ‘finite methods’ for doing this (cf. Note 15).

One attraction of Ehrenfeucht games is their flexibility. By varying the rules, onecan investigate and characterize a wide variety of logical languages. (Note 22.)

4.3 Relating strategies across games: ‘E = H2’Logic games often involve recurrence of the same idea – such as ‘back and forth’between the two quantifiers. We present one example in technical detail, showinga systematic connection between comparison games and the evaluation games ofSection 3. Moves in evaluation games and moves in comparison games seemsimilar. Indeed, the Adequacy Theorem for Ehrenfeucht games suggests a link.Here is a simple observation which seems new, relating strategies across the twotypes of game, both taken in their finite versions played with respect to twomodels M, N . First-order differences φ of quantifier depth k between M, Ncorresponded to winning strategies for Spoiler in the k-round Ehrenfeucht gamebetween M, N. Suppose that M|=φ, N |=¬ φ . This induces a winning strategy forVerifier in a Hintikka game G(φ, M) plus one for Falsifier in G(φ, N). We cancorrelate all this directly without mentioning the formulas essentially:

Proposition There exists an effective correspondence between(a) winning strategies for Spoiler in the n-round comparison game(b) pairs of winning strategies for Verifier and Falsifier in some

n-round evaluation game, played in opposite models.

Proof From (b) to (a). We say that an H-pair of depth n consists of a formulaφ of quantifier depth n plus a winning strategy σ for V in the φ-game in oneof the models, and a winning strategy τ for F in the φ-game in the other model.We sketch the way to 'merge' σ, τ . Spoiler looks at the two evaluation games.Suppose V wins φ in M , and F wins φ in N . Without loss of generality,

formulas can be assumed to be constructed from atoms withnegations, disjunctions, and existential quantifiers only.

If φ is a negation ¬ψ , Spoiler switches to the obvious strategies for F and Vw.r.t. ψ . (This is all 'internal computation': the opponent in the comparison gamedoes not see any action as an effect yet.) If φ is a disjunction ψ∨ξ , Spoiler useshis V-strategy in the one model to choose a disjunct. His F-strategy in the othermodel will also win against that disjunct. Proceeding in this way, the formula isbroken down until an existential subformula ∃xψ is reached. Spoiler then useshis V-strategy σ in the model where it lives, say M , to pick a witness d .

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This model M and object d are his opening move in the first round of the E-game.

Note that what remains for him still is a winning strategy σ- for ψ in M afterthis first move. Now, let Duplicator respond with any object e in the other modelN . This choice can be seen as a move by Verifier in the evaluation game for∃xψ in N . We know that Falsifier still has a winning strategy τ- for ψ in Nafter this first move. So, proceed by induction: we still have an H-pair of depthn-1 , which can be merged into a follow-up winning strategy for Spoiler in the(n-1)-round comparison game between M and N . The total effect is an n-roundS-strategy. This argument easily yields an algorithm for Spoiler’s computation.

From (a) to (b). This direction seems harder, as we have to ‘decompose’ oneobject: Spoiler’s winning strategy, into two separate ones that must form a goodH-pair. One proof of this follows our earlier construction of a 'difference formulaof depth n' from an S-strategy. This formula induces two evaluation strategieseffectively. Let us describe 'splitting' of a comparison strategy directly. Considerany winning strategy for S in the n-round comparison game between twomodels M, N . In the first move, S chooses, say, model M and object d .

Our desired formula will then start with an existential quantifier,and V has the winning strategy in M .

Let Duplicator make any response e in N . We know that Spoiler still has an(n-1)-round winning strategy in the two expanded models (M , d) , (N , e) .Inductively, we can find H-pairs of depth n-1 for each choice e that Duplicatormakes. By the Finiteness Lemma, only finitely many logically non-equivalentformulas can be involved in these pairs. Then,

One over-all existential quantification over a suitable conjunctionof formulas of depth n-1 defines our desired H-pair of depth n .

In particular, if it is V who has the winning strategy of a relevant H-pair φ inthe model M , put itself in the conjunction; otherwise, put its negation. QED

(See Note 23 for some comments and qualifications concerning this result.) OurProposition is just one structural connection between different logic games. Thereare many other interesting examples of this phenomenon. For instance, the naturalproof of the ‘Stage-Comparison Theorem’ for inductive definitions (Moschovakis197x), involves a combination of two evaluation games played by switchingbetween evaluation games in two models, and seeing which one ends first. Moresophisticated versions occur in descriptive set theory. Logic games are connected,and the precise constructions going from one to the other remain to be understood.

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5 Logical Structure of GamesAfter this round of logic games, we have a better view of the potential of gamesfor performing cognitive tasks. We now turn to general issues across all of these.

5.1 Games versus statementsAt the outset, we make a simple point. ‘Adequacy theorems’ are misleading inthat they suggest the only purpose of the game is to set up a correspondence withsome prior classical notion. Moreover, they suggest that the meaning of a game isfully captured by the assertion that some player has a winning strategy in it. Butthis is a reactionary prejudice, and worse: a conceptual confusion. Games aredynamic activities, not static statements. We have to distinguish carefully betweengames and assertions about them. (Note 24.) This simple distinction has manyrepercussions. E.g., Hintikka’s well-known work on evaluation games waversbetween formulas as games and formulas as statements. Therefore, a question like“does A imply B?” is taken as unproblematic – whereas the real challenge goesunnoticed: finding a good notion of ‘implication’ between games (i.e., activities)is highly non-trivial! Once we see this, the general question becomes

which aspects of the game setting need to be first class logical citizens?

A full calculus should include (a) games, and (b) assertions about them – whichcan be done on the model of dynamic logic in computer science. (Note 25.) Buteven this is not enough. Given our discussion so far, (c) strategies should beincluded as first-class citizens as well. And then, there is still a bias which needsto be removed. Strategies need not be directed merely toward ‘winning’, they canserve any purpose. Thus we need (d) statements about their effects. This format isneeded if one wants to formalize the usual elementary arguments about games andstrategies, say, the proof of Zermelo’s Theorem. It becomes even more imperativewhen we want to formalize players’ expectations leading to a Nash Equilibrium.We are not going to develop all this in detail, but do note the general format:

in game G , strategy σ for player i achievesa set of outcomes satisfying proposition A

Reasoning with such a format will involve explicit rules such as the following(where we use ‘V’, ‘F’ to denote the two players; cf. Section 6 for details):

if strategy σ for V achieves outcomes satisfying A in game G ,and strategy τ for V achieves outcomes satisfying B in game H,then strategy <σ, τ> for V achieves a set of outcomes satisfyingΑ∨Β in the game G∧H (where the other player F starts).

Note how this removes the usual existential quantifiers (“having a strategy”),hiding all this information, instead displaying concrete manipulations.

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5.2 Game operations: sequential and parallelThe Boolean algebra example in Section 2 showed that even propositional logiccontains an algebra of ‘equivalent games’. The exact calculus that emerges willdepend on two decisions. First, we must define our conception of games in termsof some underlying semantic invariance (more on this below). Next, we mustdetermine which operations we wish to consider. Our evaluation games had

choice (∨, ∧), dual (¬)

for propositional logic, which involved only choices on the way toward finaloutcomes (atomic test games, or arbitrary further games). In predicate logic, onealso has intermediate actions, viz. object-picking for quantifiers, which need to be‘glued in’ by means of

sequential composition (;)

Thus, the evaluation game ∃x Px is really the sequential composition ∃x ; (Px)?(‘First pick an object, then test it for some property’.) We shall look at calculi forthese operations in a few subsections from now. But these sequential operationsare certainly not the complete natural repertoire for games. For instance, considerconjunction of games. One way of playing this was Boolean choice ∧ : player Fchooses at the start, and afterwards only one of the two games is played. But if wewant to play both games we need rather composition ; : first play one, then theother (notice the order dependence). But even this conjunction may not be quiteright in many situations. Many academics indeed play sequential composition(first 'career', then 'happy family'), whereas they later wish they had played aparallel game composition: making progress in both. Various natural parallelcomposition operations || exist in process theories for concurrency, and also inthe literature on logic games, e.g. for linear logic. (Note 26.)

Excursion. One example beyond what we considered so far is parallel disjunctionof games G+G', where one plays episodes from each game, resuming where oneleft off. (Think of interleaved play of “attending the department meeting” and“reading one’s favourite novel”.) Essential here is not just the initial choicebetween G and G' , but rather, which player is allowed the further initiative inswitching between these subgames. A popular example from the literature is this."If I can play a chess game once as Black and once as White, copying the movesof my opponent, I can produce two identical runs of both games, one of which willbe a win for me, or both are draws. This way I can play against Kasparov, andnever come out a loser." Thus, for parallel (as opposed to Boolean) gamedisjunction, Excluded Middle holds, even when the underlying games are notdetermined! Even so, parallel disjunction does not satisfy all classical propertiesof Boolean ∨ . In particular, it is not idempotent. E.g., G+G is not the same gameas G , because I can use your moves against you in the first, but not in the second.

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But the space of game operations is much richer still, even for logic games. E.g.,Section 3 showed how semantic tableaux involve parallel, rather than sequentialoperations – and 4.3 showed how comparison games can be analysed asinterleaved combinations of evaluation games in the two models being compared.And the latter game combination has not yet been studied at all. Another newcandidate is the ‘repetition’ operation on construction games from Section 3.3.

Basic open question: What are the natural game operations?

5.3 Game equivalenceThe following issue came up in connection with evaluation games. ‘When are twogames the same?’ This asks for the basic invariance defining the field. To beprecise, one would have to define first which representation of games one wantsto use: trees, or some other format. We will leave this open, thinking of trees in astandard game-theoretic sense. Instead, we briefly discuss some possibleinvariances, drawing upon the discussion in van Benthem “Torino”. (Note 27.)

We note at the outset that there is no generally accepted answer to the following

Basic open question: What are the natural game equivalences?

Our aim in this section is just to make a connection with standard logical validitiesshowing their game-theoretic import – and at the same time suggest a systematicapproach of game-theoretic interest. First, in thinking about equivalence, one caneither focus on the activity structure of games (‘play’), or only on their outcomes.(“Are you in it for the game, or just for the marbles?” is the Dutch expression.)The former line is like thinking about process equivalences in computer science,such as ‘bisimulation’ (van Benthem ELD). The latter line thinks only of the‘powers’ of players, mentioned already in Section 2. Here is a definition, inspiredby work of Parikh. (Note 28.) In general, players in a game may not be able to forceunique outcomes. But they can force certain sets of outcome states (regardless ofhow the other plays), satisfying certain properties. This motivates the following

Definition (forcing relations)ρGi sX player i has a strategy for playing game G

whose resulting states are always in the set X

For instance, consider the case of propositional distribution pictured in Section 2.The forcing relations for the two players V, F between the top node and sets offinal nodes (outcomes) are the same in both games:

for F {A}, {B, C}for V {A, B}, {A, C}

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Given any game tree, we can compute these forcing relations in an obviousinductive manner. If player i is to move at node s , then she can force exactlythose sets of outcomes that are forceable from at least one of his successor nodes.The other players can only force unions ∪x Yx of sets which he can force atthese successor nodes x . (Note 29.) Here are the corresponding recursive clauses.We assume that forcing relations are closed under supersets:

if ρGi x, Y and Y' contains Y, then ρGi x, Y'

Then we get the following natural recursive clauses for computing forcingrelations for both players under the basic game operations:

Propositionρ1 G∨G' x, Y iff ρ1 G x, Y or ρ1 G' x, Y

ρ2 G∨G' x, Y iff ∃Z, Ζ': ρ2 G x,Z and ρ2 G' x, Z' and Y=Z∪Z'

ρ1 ¬G x, Y iff ρ2 G x, Y

ρ2 ¬G x, Y iff ρ1 G x, Yρ1 G•G' x, Y iff ∃Z: ρ1 G x, Z & ∀z∈Z ρ1 G' z, Yρ2 G•G' x, Y iff ∃Z: ρ2 G x, Z & ∀z∈Z ρ2 G' z, Y

Remark Often, all games are determined, now in the sense that for each set Y ,either 1 can force Y, of 2 can force W–Y. Then we just define these relations forplayer 1 , simplifying the dual clause to ρ1 ¬G x, Y ⇔ not ρ1 G x, W–Y .

Now, to define the corresponding game equivalence, we assume that the forcingrelations ρGi are given somehow in the two games under consideration.

Definition A power bisimulation between two games G, G’ is abinary relation E between game states in G, G' satisfying(1) Atomic Harmony:

if x E y, then x, y satisfy the same proposition letters.(2) Zigzag Clauses for each player i :

(2a) if x E y and ρGi x, U, then there exists a set Vwith ρG'i y, V and ∀v∈V ∃u∈U u E v; (2b) vice versa.

If the two games being compared have the same space of outcomes, we can justcompare outcome sets in the power relations. Basically then, if player i can forcean outcome to lie in set A in one game, there exists a subset B of A so that shecan force the other game to end up inside B . The above definition takes care ofthe fact that outcome sets may only be ‘comparable’ through some simulation E .Forcing bisimulation fits with our earlier intuitions. Additional evidence is (a) itsnice characterization in terms of modal game logic (cf. van Benthem 1999), and(b) its similarity to Spoiler/Duplicator rounds in Ehrenfeucht games for monotone

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generalized quantifiers (as pointed out by Scott Weinstein, p.c.). But once again,there may be other natural notions of game equivalence, for instance, bringing incomparisons between players’ strategies explicitly. (Note 30.)

Excursion Comparison games between gamesThe themes in this paper are all related. As in Section 4, we can use gamesthemselves to compare games! Let Spoiler and Duplicator play over some finitenumber k of rounds. For power bisimulation, with a current match between statesx – y , Spoiler chooses some set U with ρi x, U and Duplicator has to respondwith a set V such that ρi y, V . Still in that round, Spoiler chooses a state v∈V ,and Duplicator must respond with a state u∈U . Then u–v becomes the newcurrent match. An Adequacy Theorem ties this up with game-logical differencesof operator depth k . Thus, comparison games answer a question of fine-structure:

how similar are two given games?

Finally, here is how the concerns of Sections 5.3, 5.2 tie up.

Definition A game operation O is safe for bisimulation E if,whenever A E A’, B E B’ (where we think of a bisimulation Erelating the top nodes of the games), then O(A,B) E O(A’, B’) .

All sequential operations mentioned in 5.2 are safe for power bisimulations.

5.4 Game algebraCalculi of valid principles show the behaviour of a game semantics most vividly.Our first observation is this.

Fact Identifying games under power bisimulation makes all laws of classicalBoolean Algebra valid except those involving the constants 0, 1.

This can be shown by a straightforward computation over the game trees for termsin any axiom list for Boolean Algebra. (Note 31.) Thus the valid principles are: (a)distributivity of ∧ over ∨ and vice versa as formulated in Section 2, (b)commutativity, associativity, and idempotence of ∨ and ∧ , (c) De Morgan lawsfor ¬, ∧, ∨, as well as (d) the two laws of Absorption:

A∧ (A∨B) = A A∨ (A∧B) = A

Typically non-valid in general are Excluded Middle and Non-Contradiction, aswell as principles such as A∧ (B∨¬B) = A . This may seem strange, given that wetreated Excluded Middle as a ‘logical law’ in Section 2. But the point is this – aswe shall elaborate later. In evaluation games, the atomic proposition letters standfor very specific games: one-step test games. But in general games, they just stand

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for arbitrary games, and A∨¬A is no longer a valid principle in any obvioussense. One gets a choce of playing A as any one of its two players, but this maynot even yield a winning strategy (when A is non-determined). Likewise, it is notobvious at all that, for general games A, B, A is ‘the same game’ as A∧ (B∨¬B) .And the power relation computation bears this out: no identity is enforced.

Next, looking also at composition of games by computing pairs of power relationsfor putative identities, we find the following additional laws:

G ; (G' ; G'') = (G ; G') ; G'' associativity(G ∨ G') ; G'' = (G ; G'') ∨ (G' ; G'') left-distribution¬ (G ; G') = ¬G ; ¬G' dualization

These laws are reminiscent of Relational Algebra (reading ; as composition),though game negation over composition behaves quite idiosyncratically. Furthervalid identities are derivable from these by algebraic manipulation, such as

(G ∧ G') ; G'' = (G ; G'') ∧ (G' ; G'')

But unlike Relational Algebra, game algebra lacks right distribution:

G ; (G' ∨ G'') = (G ; G') ∨ (G ; G'')

Here is a counter-example in our evaluation games: substitute some universalquantifier game for G , and get a clearly invalid first-order game equivalence.The general reason is that, from one given starting state, game G may end indifferent states, each of which may have either G’ or G’’ for a succcesfulcontinuation, without there being any uniform choice doing the same job. Thus,we define Basic Game Algebra as follows:

for ¬, ∧, ∨ all Boolean laws minus those involving 0, 1for ; associativity, left-distribution, and dualization.

We summarize our observations so far in the following

TheoremBasic game algebra is sound for equality of power relations.

Proof Here is a formal procedure. Starting from two relation symbols ρ1a xY ,ρ2a xY for each basic game expression, one can write power relations for bothplayers in complex games using the earlier recursion clauses. It is easily checkedthat these expressions are logically equivalent for both game terms in thementioned laws. E.g., for propositional distribution between games, this followsby Boolean propositional distribution applied to these power relation formulas.Similarly, the validity of left-distribution follows by ordinary predicate logic. Thistranslation is illustrated best by two earlier-mentioned non-valid principles:

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(a) A ∧ (B∨¬B) = A

On the left, our procedure gives ρ1A xY ∧ (ρ1B xY ∨ ρ2B xY) for player 1’spower relation. But this is not equivalent to ρ1A x Y – since ρ1B xY ∨ ρ2B xY isby no means a tautology. E.g., it fails in the game of Section 3.1, where neitherplayer can force the singleton outcome set {x=s, y=s}.

(b) G ; (G' ∨ G'') = (G ; G') ∨ (G ; G'')

On the left, for player 1, we get ∃Z: ρ1 G xZ & ∀z∈Z (ρ1 G' zY∨ ρ1 G'’ zY) .But this is obviously not equivalent to what we get for 1 on the right-hand side,viz. ∃Z: ρ1 G xZ & ∀z∈Z ρ1 G' zY ∨ ∃Z: ρ1 G xZ & ∀z∈Z ρ1 G'’ zY . QED

The failure of right-distribution is reminiscent of process algebras in computerscience, with somewhat similar causes. (Note 32.) Converting soundness, we

Conjecture Basic game logic based on forcing bisimulation is preciselybasic process algebra enriched with the stated negation laws.

Many other algebraic laws emerge once we add further (parallel) game operations.Indeed, systems of linear logic encode many such game equivalences.

6 Deconstructing Predicate LogicThe preceding discussion has been at a level of abstraction going much beyondconcrete logic games. Returning to the logic setting, we see a new perspective.For instance, note that on our current view, evaluation games for predicate logicare really compounds of two kinds of atomic game:

object picking, and fact testing,

formed by means of the operations of

choice, dualization and composition.

In particular, then, quantifiers are no longer logical operations on a par with theBooleans, but atomic games that compose with any game that follows! If nothingelse, this is a radical departure from classical syntax. Moreover, the operationalsuperstructure of first-order evaluation games is completely general: it has nothingto do with evaluation per se, and may occur in any game. This suggests a newfine-sructure to ‘predicate-logical validity’. One expects general game-validitiesin the core, surrounded by laws that reflect more specific aspects of the particularatomic games that are special to the system. We discuss a few aspects of this.

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6.1 Propositional logicConsider first basic propositional logic without negations. In line with ourdiscussion of Section 5. 4, we have the following propositional core. (Note 33.)

Completeness TheoremA propositional equivalence is valid in the sense of equal power relationsiff it follows from the laws of Boolean Algebra minus those with 0, 1 .

Proof We saw already that the latter principles are sound for power equivalence.Conversely, consider any two propositional formulas A, B whose correspondinggames have the same power relations. Using only valid principles – in particular,De Morgan and Boolean distribution, we can transform these into power-equivalent distributive normal form games A’, B’ . Moreover, using theAbsorption laws, we may assume that no disjuncts have sets of literals that areincluded one way or the other. Now, the information about equal power relationsamounts to this (keeping in mind that power relations are closed under supersets):

any set of literals in one disjunct of A’ includessuch a set of literals of B; and vice versa.

Given the non-inclusions on either side, this means that the two normal formshave the same sets of disjuncts, and we see that they are the same. QED

Remark Distributive normal forms themselves have an interesting interpretationin terms of games. One player moves just once, then the other player does. Thismeans that the first player hands all information about her whole strategy to theother in one step, without any loss in power over the outcome. This is reminiscentof ‘strategic forms’ of games. (Note 34.) Handing things over in reverse order canalso occur in logic games. E.g., Barwise & van Benthem 1999 have a version ofEhrenfeucht games where Duplicator starts the round, not Spoiler, and hands a setof ordered pairs consisting of all possible choices for Spoiler plus her response tothem. This does reveal her ‘tactics’ though not her longer-term strategy. Spoilernow just chooses one of these. This is like a colleague of mine, who at departmentmeetings likes to say “Now you will say A, and I will then say B, or you mightsay C, and I will then say D, etc.”. running through all your possible moves.

Laws for negation are easy to add to the above. Essentially, these are justIdempotence plus De Morgan. One can still do the above normal form argument,even though no ‘truth table simplifications’ with constant truth values are allowed.But this outcome also shows an expressive weakness of just propositional logic asa game calculus. Consider this evident fact concerning power relations:

ρ1G x,Y → ¬ρ2G x,–Y if player 1 can force some outcome set, then player 2 cannot force its complement.

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To state this fact, we need a richer language that can talk about both games andoutcomes, like that of Parikh 1981. Then, we could also state the converse, whichexpresses the special case that the game G is determined.

Digression There is also a converse direction, from games to logical formulas.(a) The finite-depth games in Zermelo’s Theorem rewrite to propositionalformulas with proposition letters “win”, “lose” to indicate wins and losses forplayer 1 at the bottom, while finite conjunctions and disjunctions modelsuccessive players’ choices in an obvious manner. The proof of the theorem thensimplifies that propositional formula to one atom: either “win” or “lose”, usinggeneral propositional validities plus two special principles ‘win ∧ lose ↔ lose’,‘win ∨ lose ↔ win’ .(b) Another example are the game trees in Osborne & Rubinstein 1994, Chapter11, which discusses equivalence of extensive games. Some of the equivalencetransformations given there are literally propositional moves, such as distribution.(Cf. van Benthem 2000 for further discussion of this connection.)

6.2 Fine-structure of predicate-logical validitiesProceeding from more general to more specific, we can distinguish the followinglevels in predicate-logical validities. Recall our format stated at the beginning.

Level1 First, there is a most general game logic of these operations, describedin Section 5. This calculus may also be viewed as follows. Consider any standardpredicate-logical validity written in our format. Define

schematic validity which allows any substitution for atomic actions (i.e., both quantifiers and facts) and demand truth for all the results.

This is a much stricter notion of validity than the usual one. For instance, not eventhe familiar right-distribution law remains valid:

∃x (Px ∨ Qx) ↔ ∃xPx ∨ ∃xQx

is refuted by a substitution of ∀x for ∃x . On the other hand, witness our earlierconjecture, this base calculus has a good chance of being decidable. (Note 35.)

Level 2 Next comes an intermediate level of validity, identifying asemantic feature of the existential quantifier validating the above right-distributionlaw:

Distributivity of the atomic action ∃x : Verifier can alwaysachieve one single outcome with it, starting from any state.

This singleton feature blocks the earlier counter-example to right-distributivity.

Other examples of distributive atomic games are atomic tests, defined as follows:

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with a test game (A)? in state s , Verifier achieves {s} if A is true at s,while Falsifier achieves nothing (empty forcing relation at s) ;when A is false at s, these stipulations reverse.

One can set up game calculi determining the special additional effects of havingjust distributive atomic actions (see Pauly 1998, 1999). With the full repertoire offirst-order game operations, these are complex, as dualization loses distributivity.(The non-distributive game ∀x equals ¬(∃x) .) Without duals, a much simplerdecidable calculus results, which corresponds to studying implications betweenpredicate-logical formulas formed using only

atoms, conjunctions, disjunctions, and existential quantifiers.

In general game terms, at this level, expressions have a normal form consisting of

disjunctions of conjunctions of compositions of atoms and their negations.

Level 3 Finally, predicate logic reflects special features of the atomic gamesthat form the system. For instance, unlike games in general, both are ‘idempotent’:

(A)? ; (A)? = (A)?∃x ; ∃x = ∃x

These features, too, influence logical validities – but we do not pursue this here.Instead, we discuss one final point.

Digression Skolem normal formsIt is often said that Skolem forms are the typical expression of the game-theoreticperspective on first-order logic. Skolem functions are the strategies for Verifier.This is true, but also misleading. What is stated by an equivalence like

∀x ∃y Rxy ↔ ∃f ∀x Rxf(x) ?

First, this concerns Verifier‘s strategies only, even though the game ∀x ∃y Rxyis symmetric with respect to its players. Also, the Skolem form is a statement, nota game. Can we read it as a game as well? In that case, we are considering a newtype of game, where one player hands his whole strategy to the other, and waitsfor the result. This is a bit like propositional distributive forms. (Note 36.) Indeed, itsuggests a general operation of strategic lifting for any game, where playersprepare their strategies beforehand, and then offer choices between these. Thismakes sense far beyond logic games.

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This concludes out discussion of standard predicate logic, viewed as a calculus ofevaluation games, and the resulting new look at the fine-structure of validities.(Note 37.)7 Calculus of StrategiesArguments in Section 6 become more informative by adding explicit strategies.So far, these have been mostly hidden under existential quantifiers, witness theearlier forcing statements – an ‘∃-sickness' which pervades much of logic (cf. thepreference for ‘provability’ over ‘proof’, ‘interpretability’ over ‘interpretation’).This short section provides some ingredients for a cure.

Calculi for combining strategies occur in the literature on linear logic, on theanalogy of proof terms that get combined in a type-theoretical framework. But inthe general perspective of this paper, such strategy combination is totally general,though its interpretation may vary according to what the strategies stand for(proofs, models, analogies, etc.). First, consider this simple sequent derivation fora propositional validity, whose steps are well-known valid inference rules:

Α ⇒ Α Β ⇒ ΒΑ, Β ⇒ Α Α, Β ⇒ Β C ⇒ C

Α, Β ⇒ Α∧Β A, C ⇒ CΑ, Β ⇒ (Α∧Β) ∨ C A, C ⇒ (Α∧Β) ∨ C

Α, Β ∨ C ⇒ (Α∧Β) ∨ CΑ ∧ (Β ∨ C) ⇒ (Α∧Β) ∨ C

Here is a corresponding (as yet uninterpreted!) form indicating strategies:

x:Α ⇒ x:Α y:Β ⇒ y:Β x:Α, y:Β ⇒ x:Α x:Α, y:Β ⇒ y:Β z:C ⇒ z:C

x:Α, y:Β ⇒ (x, y):Α∧Β x:A, z:C ⇒ z:C x:Α, y:Β ⇒ <L, (x, y)>:(Α∧Β) ∨ C x:A, z:C ⇒ <R, z>:(Α∧Β) ∨ C x:Α, u:(Β∨C) ⇒ if head(u)=L then <x, tail(u)> else tail(u): (Α∧Β) ∨ C v: Α ∧ (Β∨C) ⇒ if head((v)2)=L then <(v)1, tail((v)2)> else tail((v)2): (Α∧Β) ∨ C

Such an annotated derivation may be read in different ways. One can see thevariables and terms as standing for proofs, and the various operations introducedthen encode what goes on as logical operations get added. But proofs are just oneinstance of strategies, and we can also read the tree as a construction of strategies.

Here the major operations are

storing strategies for a player who is not to move: < , >using a strategy from a list: ( )i

computing the first recommendation of a strategy: head()as well as the remaining strategy: tail()

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making a choice dependent on some information: if then else

In earlier sections, we have seen that strategies can encode very different logicalobjects: proofs, models, analogies, etc., Thus, the above derivation can also standfor quite different things. For instance, we can also read it as a recipe for takingany winning strategy of Verifier’s in a classical evaluation game for Α ∧ (Β∨C) ,and turning it into a winning strategy for (Α∧Β) ∨ C . More generally, our point isthat classical proof-like structures, such as sequent trees or closed tableaus, maybe viewed as constructive recipes for combining strategies – and as such, theyprovide much information beyond their original purpose. (Note 38.) (Note 39.)

Nevertheless, all this is not as straightforward as it might seem. In each case, onehas to think what strategy combination means. Here is a simple concrete example.Consider the rule of our strategy calculus that goes

from premises x:1:A and y:1:B to conclusion <x, y>:1: A∧B

With proofs (for Critic) this is the well-known principle that proofs x of A and yof B may be put together canonically to form a proof of A∧B . With evaluationstrategies (for Verifier) in some fixed model, strategies are not standard model-theoretic objects at all, but rather dynamic procedures for establishing truth, whichhave hardly been studied as logical objects in their own right. Then, with modelconstruction (for Builder), one might think that the strategies are just two models,that are being put together in some way – but the issue is again more complicated.Tableau games, despite appearances, do not use the usual game constructions ofchoice at all, as the formulas in the boxes do not split up the tableau inductively.We would rather need a strategy rule for some parallel conjunction (as was notedin Section 3), to understand a conjunction in the TRUE box. Thus, the appropriatestrategy calculus for tableau games would require us to first analyze their naturaloperations. The same holds for model comparison games. There are indeed naturaloperations of ‘dual’ (let Spoiler and Duplicator switch roles) and of ‘composition’(e.g., first play a k-round game, then decide to continue with an m-round one),but the ‘initial choices’ do not occur naturally. Actually, this is only a first pass.For, ‘heterogeneous combinations’ of logic games, involving choices betweenthem, do occur naturally in applied settings. Think of a defense lawyer, who maydecide at some stage whether to attack the factual truth of some assertion made bythe prosecutor (an evaluation game w.r.t. Reality) or to be Builder, and constructsome scenario showing at least the consistency of her client’s position. (Note 40.)

Thus, our proposal is not yet precise, and it certainly has its tricky aspects.Nevertheless, we submit that it is of intererest to look at logic games in preciselythis way, with an open eye toward their deeper analogies and non-analogies. Also,we have not provided a complete strategy calculus: just one suggestive example.In line with our earlier discussion, we would eventually want to manipulate

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statements mentioning both strategies and games, but also players and outcomes.These calculi may be ‘co-inductive’, rather than ‘inductive’. (Note 41.) (Note 42.)

Finally, here is another aspect of our proposal. Calculi of strategies are a naturalgeneralization of the well-known Brouwer-Heyting-Kolmogorov interpretation ofproof-theoretic validity. The latter interprets an implication from A to B as theexistence of some transformation turning proofs for A into proofs for B . But here,we are talking about an implication between games A and B , and the existence ofa transformation turning strategies for playing A into strategies for playing B.This is more general than ‘BHK”. At the same time, it shows that the latter’s usualconstructive moral is overstated. Game implication may hold just as well betweenevaluation strategies in classical logic, witness the above example.

8 ConclusionThis paper has shown how game-theoretic analysis provides dynamics and fine-structure to standard logical notions. Of both these aspects, there is still more inreal Game Theory. There, strategies are players’ behaviours, which need to be inequilibrium to guarantee stability. Such strategies depend on more refined prefe-rences for players than the ones we have considered. Moreover, the equilibriummight only be obtained by considering repeated games, allowing players to choseactions with certain probabilities. Finally, groups of players can form coalitions toachieve things beyond their individual powers. All these additional features makesense in a broader view on elementary logic, both with models and with proofs.But that is the subject of another paper: this one ends right here…

9 Notes(1) We do not provide an exhaustive discussion of interfaces between logicand games: for further references and a broader survey, see van Benthem 1999,Hodges 1998. An original new program that sparked renewed interest: Hintikka-Sandu 1998. Modern logic/game-theory interfaces: TARK, LOFT conferences.(2) Joint powers. Even if one player has a winning strategy, outcomes maystill depend on the other, who might decide e.g., where to make his final stand.(Think of the losing party at least choosing the final battlefield with honor.)Typically, in game theory, all players are stakeholders in the final outcome.(3) As the game tree of chess is so large, it is still not known whether thisfavoured player is Black or White. Abstract existence results for strategies maystill defy the best current computers.(4) Conversely, many game trees in game theory textbooks are just anothernotation for propositional formulas. We will return to this theme in Section 6.(5) The fixed point instruction seems like a vicious circle. But with possiblydifferent objects e , there is a new pass through the game for φ(P) , which mayend in some final node that is a non-fixed-point atom. E.g., when defending thattwo objects d, e stand in the transitive closure of some relation R , Verifier may

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call the repetition step of the relevant μ–definition a finite number of times,before getting to a pair where she just chooses a testable atom R .(6) Effective descriptions of winning strategies in open, or simple Borelgames, using finite state machines and other automata, have been studied incomputer science (see survey Wolfgang Thomas).(7) This is the natural counterpart to the logical Validity Problem: determiningif a given formula holds in all models. Satisfaction calls for model construction,validity calls for proof construction.(8) Think of creating Paradise, with existential formulas in YES theforthcoming attractions, and those in NO sorts of snakes to be avoided.(9) Alternatives. (a) Builder's choices for true existentials can also beautomatic, as just some generic new object is put in. Alternatively, we can allowBuilder a genuine choice here, of some already existing object d . In tableaus, thismay make counter-examples smaller, but otherwise, there is no change inoutcomes for satisfiability. (But are the two versions equivalent as games?) (b)Critic’s challenges to false existentials might be produced automatically, evenputting in the whole lot of existing objects. (c) If we make the quantifier movesautomatic, then the only actions performed by Critic ‘’designate’ formulas: he isreally a scheduler for the construction process, and Builder just has to makechoices, while some bountiful Nature puts in the objects. This illustrates that weneed not always think of players in a logic game as antagonists: Scheduler ishelping Builder to meet his specifications, and win the rewards of his contract.(10) An example is Schütte’s Formula: the conjunction of 'transitivity,irreflexivity, plus the existence of successors' in some binary ordering < ‘ .(11) We view a tableau as a game tree, although there is a slight difference. Inour game, Critic gets to schedule a formula at each round, and he can do this invarious ways. This corresponds to a finite branching – which standard tableausavoid, taking advantage of the fact that not all these options need be investigated.The reason is that, for satisfiability, any scheduling which makes sure that allrelevant formulas get treated is all right. But there are logics where scheduling ismore crucial, and tableaus will have to investigate different reduction orders.(12) Despite the rhetoric, these arguments are non-constructive. Since modelsare non-constructive in general, so are the corresponding 'construction' strategies.Yiannis Moschovakis (p.c.) has suggested making a distinction between strategiesthemselves and what they encode, being models, or later on in this section, proofs– something which does seem preferable in the long run.(13) An independent argument for determinacy. Suppose Critic has no winningstrategy. In that case, by general reasoning in game trees, we can conclude that

Claim Builder has a strategy ensuring a run where at each stage,Critic lacks a winning strategy for the remaining game from then on.

Here is the mechanics of the strategy. If the move is Builder's, then Critic cannothave a winning strategy for each of his actions (otherwise Critic would have awinning strategy), so she chooses a continuation where Critic has no winning

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strategy. If the move is Critic's, no continuation can lead to a winning position forhim, and so any move is OK for Builder. This proves our Claim. QED Now consider any resulting run of such a strategy. If it had generated a loss forBuilder at some stage (by a pair of contradictory requirements), Critic would havehad a winning strategy at that stage. But he did not: therefore, all stages were safefor Builder, and hence he wins the run. Therefore, the strategy is even a winningone for Builder. This argument is really that for the Gale-Stewart Theorem.(14) The proof of König’s Lemma is easy. At successive nodes, starting fromthe root, select some successor node whose subtree is infinite. This is alwayspossible (by a pigeon-hole argument), and generates an infinite branch. QED(15) The latter outcome is not general. E.g., winning strategies for Duplicatoror Spoiler in an infinite Ehrenfeucht game need not be finite objects at all: there isno finite branching when the models are infinite – and König's Lemma does notapply. Spoiler's strategies will make every run of the game end in a finite numberof rounds, but there need not be a uniform finite bound on their length.(16) We can also design games that mix semantic evaluation and modelconstruction – as often happens in practice with a partially given situation, wherewe still have some freedom to expand by adding objects and interpretations fornew predicates. In that case, the first-order language must become richer, say withdifferent quantifiers that distinguish evaluation moves from construction moves.(17) Here is a different speculation raised by this section. We have consideredvarious 'equivalent' versions of model construction games. But, sensitised by ourearlier discussion, we ask: equivalent in precisely which game-theoretic sense?(18) An additional advantage is that this model-theoretic technique works in theabsence of compactness on non-elementary families of models. Cf. Doets 1998.(19) A partial isomorphism is simply an injective partial function (often finite)between subsets of the domains of two models, which is an isomorphism as far asits own domain and range are concerned.(20) Finiteness Lemma. Fix some set of free variables x1, .., xm. Up to logicalequivalence, there are only finitely many first-order formulas of q-depth ≤ k .This fact is shown by Boolean analysis plus a simple induction on formulas.(21) Definition of potential isomorphism.(22) Such variations are pebble games (cf. Barwise & van Benthem 1999), orgames for guarded languages (cf. Andréka, van Benthem & Németi 1998).(23) The use of good old formulas in the preceding proof may seem cheating.But do note that these merely serve as short-hand for the relevant games! Still, wedo not know how to extend this result to infinite games, and in particular, we arenot aware of any nice Ehrenfeucht game matching elementary fixed-point logic.(24) An analogy. No computer scientist would identify programs (denotingdynamic processes) with statements about their execution, such as termination.One reason for the confusion proposition/game-activity in the logic literature maybe unfortunate notation. With evaluation games, the notation ‘φ’ does doubleduty for a logical formula (taken in its traditional sense) and the associated game.(25) Cf. Harel & Kozen 1998, van Benthem 1996.

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(26) Papers on linear logic and games: Abramsky, Blass.(27) See van Benthem “Torino” for more detailed discussion of representationformats for games: game trees, process graphs, models for branching time.(28) Parikh’s 1981 dynamic game calculus. Marc Pauly’s various studies.(29) Forcing intermediate sets of nodes. Powers make sense for any state of agame, and any subsequent set of states. I might force visiting certain 'intermediatezones' in a chess game, perhaps regardless of my winning or losing.(30) Game theorists’ answer: games are equivalent when they have ‘the samestrategic form’. This turns out to be under-defined, and not very helpful.(31) In this discussion we side-step the difference between formulas (Booleanterms) as abstract game forms, and their interpretations in concrete settings.(32) References to Process Algebra.(33) When negations are added in later, this calculus is what Mike Dunn calls‘De Morgan Lattices’. Thanks to him for pointing this out at Bamboo Village.(34) DNF and strategic form of a game are not quite the same though. One cansee this by computing strategic forms for the two formulas involved in Booleandistibution. One side has 2 strategies for Falsifier, the other 4 .(35) Van Benthem 2000 proves the following. Predicate-logical evaluationgames are complete for basic game algebra, in the sense of Section 5. That is, anyabstract game identity G1=G2 which is not valid in general can be refuted bysome concrete instance where G1, G2 are predicate-logical evaluation games.This result also justifies the latter’s pride of place in our paper.(36) Even so, there remain some strange aspects to Skolem forms as strategyindications for the whole game. E.g., consider the formula

∀x ∃y ∀z Rxyz ↔ ∃f ∀x ∀z Rxf(x)z .Here, the Skolem form does not capture that, intuitively, Falsifier can make hischoice for z depending on what Verifier did with y .(37) Yiannis Moschovakis has pointed out another use for the level distinctionsthat we find for first-order validities. When analyzing Frege’s notion of a ‘sense’,one must identify sentences with some more dynamic object than a truth value,such as an ‘algorithm’ or a ‘game’. In that case, synonymy becomes a finer sievethan classical valid equivalence, and our levels might become relevant to the studyof ‘sense equivalence’.(38) For instance, the usual decomposition rules of a semantic tableau may beread as ways of combining strategies for Builder or Critic in games to strategies innew games. Thus, if Critic has winning strategies σ for X, A • Y and τ for X,B • Y, then he has a combined winning strategy (σ,τ) for X, A∨B • Y – whereasBuilder only needs one such strategy to win the new game.(39) With predicate-logical examples, one will encounter the complication ofstrategy terms having parameters. The reason is that a follow-up strategy to aquantifier may depend on the object chosen for its variable. The more generalissue underlying this, which surfaces in many logical settings, is giving a goodaccount fo composition of strategies.

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(40) Such choices may be governed by procedural conventions such as the‘burden of proof’. See van Benthem 2000B for a discussion of possible uses oflogic games in the analysis of actual legal settings.(41) The best mathematical framework for doing this need not view strategiesinductively, built from base cases by applying constructions. In infinite games, itis better to think co-inductively, viewing strategies as infinite 'streams' whose headcan be used for a first move, making the tail available for whatever may come.Think of a strategy as your doctor. You take advice (at the head), then expect himto be as ready and bright as ever (the tail) to deal with the next item in yourendless stream of complaints. For co-algebra, see Jacobs & Rutten, Baltag.(42) In the BHK-literature, there are many discussions of the relative proof-theoretic status of introduction versus elimination rules for the logical operations.In our current perspective, this may be just a methodological discussion betweenthe inductive and the co-inductive perspective. Introduction rules refer to smallestfixed points, elimination rules to greatest fixed points.

10 ReferencesS. Abramsky & R. Jagaadesan. Linear logic games, JSL.H. Andréka, J. van Benthem & I. Németi, 1998, Journal of Philosophical Logic.J. Baeten & P. Weyland, Process Algebra.A. Baltag, ‘A logic for games and strategies’, CWI Amsterdam.J. Barwise & J. van Benthem, 1999, Journal of Symbolic Logic.J. Barwise & L. Moss, 1997, Vicious Circles, CSLI Publications, Stanford.J. van Benthem, 1996, Exploring Logical Dynamics , CSLI Publications, Stanford.J. van Benthem, 1998–99, Logic and Games, www.turing.wins.uva.nl/~johan/J. van Benthem, 1999, 'When are Two Games the Same?'. Talk presented at

LOFT-III, Torino 1998. ILLC preprint series.J. van Benthem, 2000A, ‘First-Order Evaluation Games are Complete for Basic

Game Algebra’, ILLC, Amsterdam.J. van Benthem, 2000B, ‘Games and Procedure in Legal Reasoning’, to appear in

Cardozo Law Review.J. van Benthem, 2000C, ‘Hintikka Self-Applied’, to appear in L. Hahn, ed.,

Library of Living Philosophers.H.C. Doets, 1996, Basic Model Theory, CSLI Publications, Stanford.H.C. Doets, 1999, note on adequacy of fixed-point evaluation games.D. Harel & D. Kozen, 1998, Dynamic Logic.J. Hintikka, 1973, Logic, Language Games, and Information, Clarendon, Oxford.J. Hintikka and Gabriel Sandu, ‘Games of Informational Independence’.W. Hodges, 1977, Logic, Penguin Books.W. Hodges, 1998, An Invitation to Logical Games, Queen Mary's College,

London.I. Hodkinson, 1998, Model Construction Games in Algebraic Logic.B. Jacobs & J. Rutten, Introduction to Co-Algebra and Co-Induction, CWI.Y. Moschovakis, 197x, Inductive Definability over Abstract Structrures.

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Y. Moschovakis, 198x, Descriptive Set Theory.Y. Moschovakis, 1991, ‘Sense and reference as algorithm and value’, departmentof mathematics, UCLA.M. Osborne and A. Rubinstein, 1994, A Course in Game Theory, MIT Press,

Cambridge.R. Parikh, 1981, Dynamic Logic of Games.M. Pauly, ‘Logic & Games in Amsterdam, http://www.cwi.nl/~pauly/games.htmlM. Pauly, 1999, CWI/ILLC papers on dynamic game logic, and coalitions.W. Thomas, 1999, games and automata in computer science, RWTH Aachen.

Please note: references and credits still to be finalized!