Banach-Mazur Games on Graphs

We survey determinacy, definability, and complexity issues of Banach-Mazur games on finite and infinite graphs. Infinite games where two players take turns to move a token through a directed graph, thus tracing out an infinite path, have numerous applications in different branches of mathematics and computer science. In the usual format, the possible moves of the players are given by the edges of the graph; in each move a player takes the token from its current position along an edge to a next position. In Banach-Mazur games the players instead select in each move a path of arbitrary finite length rather than just an edge. In both cases the outcome of a play is an infinite path. A winning condition is thus given by a set of infinite paths which is often specified by a logical formula, for instance from S1S, LTL, or first-order logic. Banach-Mazur games have a long tradition in descriptive set theory and topology, and they have recently been shown to have interesting applications also in computer science, for instance for planning in nondeterministic domains, for the study of fairness in concurrent systems, and for the semantics of timed automata. It turns out that Banach-Mazur games behave quite differently than the usual graph games. Often they admit simpler winning strategies and more efficient algorithmic solutions. For instance, Banach-Mazur games with ω-regular winning conditions always have positional winning strategies, and winning positions for finite Banach-Mazur games with Muller winning condition are computable in polynomial time. 1 Banach-Mazur Games Game playing is a powerful metaphor that fits situations in which interaction between autonomous agents plays a central role. Indeed, numerous problems in computer science and other fields can be understood, mathematically treated, and solved in terms of appropriate mathematical models of games. There is of course a large variety of game models, leading to vastly different mathematical theories of games. A prominent class of games, which is particularly useful for problems such as the synthesis and verification of interactive systems (with non-terminating behaviour and ongoing interaction between system and environment), or for the evaluation of fixed point logics and other important specification formalisms, are infinite games, where two players take turns to move a token through a directed graph thus tracing out an infinite path. The objectives of the players are given by suitable properties of infinite paths, often specified by logical formulae, for instance frommonadic second order logic (S1S), linear-time temporal logic (LTL), or first-order logic (FO). Some central mathematical questions concerning such games are: Which games are determined (in the sense that from each position, exactly one player has a winning strategy)? How to compute winning positions? Are there optimal strategies, and if so, what is their complexity and how to compute them efficiently? Howmuch knowledge of the play history is necessary to compute an optimal next action? In what logical formalisms can we define winning positions and winning strategies? And so on. c © Grädel; licensed under Creative Commons License-NC-ND FSTTCS 2008 IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science http://drops.dagstuhl.de/opus/volltexte/2008/1768 GRÄDEL FSTTCS 2008 365 These questions are not just of theoretical interest. They are in fact standard design and verification problems (of interactive systems) in purified form. For background on such methodologies, based on the interplay between logic, automata, and games, see e.g. [8]. In the usual format of infinite games on graphs, the possible moves of the players are given by the edges of the graph; in each move a player takes the token from its current position along an edge to a next position. Here we study a different variant of graph games where, in each move, the players select a path of arbitrary finite length rather than just an edge. We call these games Banach-Mazur games on graphs. DEFINITION 1. A Banach-Mazur game BM(G, v,Win) is given by an a directed graph G = (V, E) without terminal nodes, an initial position v ∈ V, and a winning condition Win ⊆ Paths(G, v) where Paths(G, v) ⊆ V denotes the set of infinite paths through G that start at node v. The game BM(G, v,Win) is played by two players, called Player 0 and Player 1. In the opening move, Player 0 selects a finite, non-empty path x0 from v through G. The players take turns, extending in each move the finite path x0x1 . . . xm−1 played so far by a new segment xm (which again has to be a non-empty and finite path). In an infinite number of moves, the players thus trace out an infinite path π ∈ Paths(G, v). Player 0 wins the play, if π ∈ Win, otherwise Player 1 wins. In somewhat different forms, Banach-Mazur games have been extensively studied in descriptive set theory (see [13, Chapter 6] or [14, Chapter 8.H]) and topology (see e.g. [21]). In their original variant (see [15, pp. 113–117]), the winning condition is a set W of real numbers; in the first move, one of the players selects an interval d1 on the real line, then her opponent chooses an interval d2 ⊂ d1, then the first player selects a further refinement d3 ⊂ d2 and so on. The first player wins if the intersection ⋂ n∈ω dn of all intervals contains a point ofW, otherwise her opponent wins. A similar game can be played on any topological space. Let V be a family of subsets of a topological space X such that each V ∈ V contains a non-empty open subset of X, and each nonempty open subset of X contains an element V ∈ V . In the Banach-Mazur game defined on X,V with winning winning condition W ⊆ X, the players take turns to choose sets V0 ⊃ V1 ⊃ V2 ⊃ . . . in V , and Player 0 wins the play if ⋂ n<ω Vn ∩Win 6= ∅. We refer to [21] for a survey on topological games and their applications to set-theoretical topology. Notice that Banach-Mazur games on graphs are just a special case of this general topological setting. Indeed, the set Paths(G, v) of infinite paths through G from v is a topological space whose basic open sets are O(x), the sets of infinite prolongations of some finite path x ∈ FinPaths(G, v). Thus, when a player prolongs the finite path x played so far to a new path xy, she reduces the set of possible outcomes of the play from O(x) to O(xy), and she wins an infinite play x0x1 . . . if, and only if ⋂ n<ω O(x0 . . . xn−1) ∩Win 6= ∅. Applications of Banach-Mazur games. Banach-Mazur games on graphs have recently appeared in several application areas in computer science. Pistore and Vardi used a variation of Banach-Mazur games for planning in nondeterministic domains [20]. In their scenario, the desired infinite behaviour of a system, which should be enforced by a plan, is specified by formulae in linear temporal logic LTL. It is assumed that the outcome of actions may be 366 BANACH-MAZUR GAMES ON GRAPHS nondeterministic. Hence a plan does not have only one possible execution path in the planning domain, but an execution tree. Between weak planning (some possible execution path satisfies the specification) and strong planning (all possible outcomes are consistent with the specification), there is a spectrum of intermediate cases such as strong cyclic planning: every possible partial execution of the plan can be extended to an execution reaching the desired goal. In this context, planning can be modelled by a game between a friendly player E and a hostile player A selecting the outcomes of nondeterministic actions. The game is played on the execution tree of the plan, and the question is whether the friendly player E has a strategy to ensure that the outcome (a path through the execution tree) satisfies the given LTL-specification. In contrast to the general scenario of Banach-Mazur games, the main interest here are games with finitely many alternations between players. Pistore and Vardi show that the planning problems in this context can be solved by automata-based methods in 2EXPTIME. Banach-Mazur games appear also in the characterisation of fair behaviour in concurrent systems. There are many different notions of fairness. A very convincing one [23] defines a fairness property in a transition system as a set of (infinite) runs that is topologically large (co-meager). This is equivalent to say that, in an associated Banach-Mazur game, the first player (the scheduler) has a winning strategy to ensure fairness. It is a consequence of the positional determinacy of Banach-Mazur games with ω-regular winning conditions (see Theorem 17 below) that, on finite graphs, ω-regular fairness properties coincide with ωregular properties that are probabilistically large under positive Markov measures. Hence , any ω-regular fairness property has probability one under randomised scheduling. As a further consequence, one can use results about finite Markov chains for checking whether a finite system is fairly correct with respect to LTL or ω-regular specifications. Finally, Banach-Mazur games have recently been used to describe the semantics of timed automata [1, 2]. Timed automata are an important model for verification, but for many purposes, its idealizedmathematical features such as infinite precision, instantaneous events lead to violations of specifications due to unlikely sequences of events. Therefore alternative semantics for the satisfaction of LTL specifications have been proposed, based on probability or on topological largeness, to rule out unlikely runs. By means of BanachMazur games, it has been established, that the two semantics coincide. Here we study Banach-Mazur games on graphs, and focus on the above-mentioned central mathematical questions, such as determinacy, the structure and algorithmic properties of winning strategies, and the definability of winning regions. Acknowledgement. This survey is based on joint research with Dietmar Berwanger and Stephan Kreutzer. 2 Topology and determinacy For any arena (G, v) of a Banach-Mazur game, the space Paths(G, v) is endowed with a topology whose basic open sets are O(x), the sets of infinite prolongations of some finite path x ∈ FinPaths(G, v). A set X ⊆ Paths(G, v) is open if it is a union of basic open sets O(x), i.e., if X = W ·V ∩ Paths(G, v) for some setW ⊆ V∗. A tree T ⊆ FinPaths(G, v) is a GRÄDEL FSTTCS 2008 367 set of finite paths that is closed under prefixes. It is easily seen that X ⊆ Paths(G, v) is closed (i.e., the complement of an open set) if, and only if, it is the set of infinite branches of some tree T, denoted X = [T]. Notice that Paths(G, v) itself is a closed set in the space V, the set of all infinite sequences on V. The class of Borel sets is the closure of the open sets under countable union and complementation. Borel sets form a natural hierarchy of classes Ση for 1 ≤ η < ω1, whose first levels are Σ1 (or G), the collection of all open sets, Π 0 1 (or F), the closed sets, Σ 0 2 (or Fσ), the countable unions of closed sets, and Π2 (or Gδ), the countable intersections of open sets. In general, Πη contains the complements of the Σ 0 η-sets, Σ 0 η+1 is the class of countable unions of Πη-sets, and Σ 0 λ = ⋃ η<λ Σ 0 η for limit ordinals λ. We recall that a set X in a topological space is dense, if its intersection with every (basic) non-empty open set is non-empty. LEMMA 2. For any strategy g of Player 1 in a Banach-Mazur game on a graph (G, v), the set Plays(g) of all plays that are consistent with g is a countable intersection of dense open sets. PROOF. Clearly, Plays(g) = ⋂ n∈ω Playsn(g) where Playsn(g) is the set of all plays that may arise if Player 1moves according to g during her first nmoves. Obviously, Playsn(g) is open. But it is also dense, since every finite path x can be used by Player 0 as her opening move, so there must be a prolongation of x in Playsn(g), which means that O(x) ∩ Playsn(g) 6= ∅. Notice that, if X ⊆ Paths(G, v) is a dense open set, then any finite path x has a finite prolongation xy such that O(xy) ⊆ X. In a topological sense, the dense open sets are large sets, and so is any countable intersection of such. Hence, by any strategy in a Banach-Mazur game, Player 1 can exclude only a topologically small set of plays. This means that she can only have a winning strategy if the set Win of winning plays for Player 0 is small, and her own set of winning plays, Paths(G, v) \W, is large. For strategies of Player 0, the situtation is slightly different, since she starts the play. Hence, for any strategy f of Player 0, Plays( f ) ⊆ O(x) where x is the opening move by f . After the first move, the remaining game is one where the role of the players have been switched (i.e. Player 1 now moves first). By the same argument as in the previous lemma we infer that the set of plays consistent with a strategy of Player 0 is large inside some basic open set of plays. LEMMA 3. For any strategy f of Player 0 in a Banach-Mazur game, Plays( f ) is a countable intersection of dense open subsets of O(x), where x is the opening move by f . The observations that we made on the set of plays that are consistent with strategies in Banach-Mazur games give a quite precise characterisation, in term of topological notions, of the games for which Players 0 and 1 have winning strategies. A set in a topological space is nowhere dense if it is not dense in any open set or, equivalently, if its complement contains a dense open set. A set is meager (or topologicaly small) if it is a union of countably many nowhere dense sets, and co-meager (or topologically large) if its complement is meager. A topological space is called a Baire space if no non-empty set is both open and meager, or equivalently, if any countable intersection X = ⋂ n<ω Xn of dense open sets Xn is dense. The spaces Paths(G, v) are Baire spaces since, for any finite path x, 368 BANACH-MAZUR GAMES ON GRAPHS we find an infinite extension xy0y1 · · · ∈ X by choosing, for each n, a finite prolongation xy0 . . . yn of xy0 . . . yn−1 such that O(xy0 . . . yn) ⊆ Xn. In Baire spaces a set is co-meager if, and only if, it contains a dense Π2 set. Hence we have shown that, in Banach-Mazur games, Plays(g) is co-meager for every strategy g of Player 1 and Plays( f ) is co-meager in some basic open set for every strategy f of Player 0. Conversely, for any meager set W ⊆ Paths(G, v), Player 1 has a strategy g such that Plays(G) ∩W = ∅. Indeed, if W = ⋃ n<ω Xn with Xn nowhere dense, then in her n-th move, Player 1 prolongs the path constructed so far to a path xn such that O(xn) ∩ Xn = ∅ which is always possible since the complement of Xn contains a dense open set. Clearly every play consistent with this strategy avoids W. Analogously, for every set that is comeager in some basic open set, Player 0 has a strategy f such that Plays( f ) ⊆ W. Our observations are summarized by the Banach-Mazur-Theoremwhich gives a precise characterisation of the games where Player 0 or Player 1 has a winning strategy. THEOREM 4.[Banach-Mazur] (1) Player 1 has a winning strategy for the game BM(G, v,Win) if, and only if, Win ⊆ Paths(G, v) is meager. (2) Player 0 has a winning strategy for BM(G, v,Win) if, and only if, there exists a finite path x ∈ FinPaths(G, v) such that O(x) \Win is meager in Paths(G, v) (i.e., Win is co-meager in some basic open set). This result appears, in different terms, in the Scottish Book [15, Problem 43] where it is mentioned as a conjecture due to Mazur, with an addendum by Banach, dated August 4, 1935 saying that “Mazur’s conjecture is true”. The Banach-Mazur-Theorem was published for the first time by Mycielski, Świerczkowski, and Zieba [18], without proof; the first published proof is due to Oxtoby [19]. ¿From Theorem 4 we easily get strong results on determinacy of Banach-Mazur games. COROLLARY 5. Every Banach-Mazur game BM(G, v,Win) such thatWin ⊆ Paths(G, v) has the Baire property is determined. Recall that a set X in a topological space has the Baire property if its symmetric difference with some open set is meager. Since Borel sets have the Baire property, it follows that Banach-Mazur games are determined for Borel winning conditions. Standard winning conditions used in computer science applications (in particular the ω-regular winning conditions) are contained in very low levels of the Borel hierarchy. A converse to Corollary 5 in terms of specific games does not hold. Indeed one can construct determined games with winning conditions of arbitrary complexity by combining a trivial game won by Player 0 with an arbitrarily complex game in such a way that Player 0 can avoid the complicated part. A more interesting question is whether one can prove a converse for winning conditions that guarantee determinacy in the following sense. LetW ⊆ C be a set of infinite words on some alphabet C. On every graph G = (V, E) equipped with a function Ω : V → C, the set W defines a winning condition Ω−1(W) := {π ∈ Paths(G, v) : Ω(π) ∈ W}. We then say that W guarantees determinacy for Banach-Mazur games if all games with a winning condition Ω−1(W) are determined. GRÄDEL FSTTCS 2008 369 We can link the Baire property with the determinacy of Banach-Mazur game in the following class-wise sense. THEOREM 6. For every class Γ ⊆ P(C) the following are equivalent. (1) All winning conditionsW ∈ Γ guarantee determinacy for Banach-Mazur. (2) All setsW ∈ Γ have the Baire property. PROOF. IfW ⊆ C has the Baire property then so has Ω−1(W), for all functions Ω : V → C that label the nodes of a graph G = (V, E) with elements of C. Thus, by Corollary 5 W guarantees determinacy. For the converse, suppose that W ⊆ C does not have the Baire property. To construct a non-determined game, let G(C) be the complete directed graph on C itself (and let Ω be the identity function on C). We do not useW directly as a winning condition, but modify it as follows. Let S := {x ∈ C∗ : O(x) \W is meager} and let Z be the symmetric difference ofW with the open set Y = ⋃ x∈S O(x). We claim that the Banach-Mazur game on G(C) with winning condition Z is not determined. Since Z is the symmetric difference of W with an open set, it cannot be meager (otherwiseW would have the Baire property), hence Player 1 does not have a winning strategy. So suppose that Player 0 has a winning strategy. This can only happen if Z is co-meager in some basic open set O(x). For x ∈ S, this is impossible since O(x) ∩ Z = O(x) \W is meager. Hence x ∈ C∗ \ S. But then O(x) ∩ Y = ∅. Otherwise we would have some y ∈ S such that O(x) ∩ O(y) 6= ∅, which means that O(x) ⊆ O(y) or O(y) ⊆ O(x). In either case, since O(y) ∩ Z = O(y) \W is meager, Z cannot be co-meager in O(x). Now, since O(x) ∩ Y = ∅, we have O(x) ∩ Z = O(x) ∩W, and if this set were comeager in O(x) then x ∈ S, a contradiction. Thus, none of the players has a winning strategy. A specific example of a non-determined Banach-Mazur game can be obtained by modifying a well-known construction on the basis of ultrafilters. Let G2 be the complete directed graph with vertices 0,1, and for any setU ⊆ P(ω), let letWU be the set of infinite sequences x0x1x2 · · · ∈ {0, 1} ω such that {n : xn = 0} ∈ U. An ultrafilter over ω is a set U ⊆ P(ω) that does not contain ∅, that includes with any set also all its supersets, with any two sets also their intersection, and such that for any set x ⊆ ω either x ∈ U or ω \ x ∈ U. An ultrafilter is free if it contains all co-finite sets. As a consequence, it does not contain any finite set. The Boolean Prime Ideal Theorem (a weak form of the Axiom of Choice) implies that free ultrafilters exist. PROPOSITION 7. If U is a free ultrafilter, then the Banach-Mazur game on G2 with winning conditionWU is not determined. PROOF. Without loss of generality, we may assume that Player σ plays in each move a finite word in σ+. Hence the game is equivalent to the game where the players play a strictly increasing sequence a0 < a1 < a2 < . . . and Player 0 wins the resulting infinite play if, and only if, the set [0, a0) ∪ [a1, a2) ∪ [a3, a4) ∪ . . . belongs to U. Assume that Player 0 has a winning strategy f which maps any increasing sequence a0 < a1 < · · · < a2n−1 of even length to a2n = f (a0a1 . . . a2n−1) > a2n−1. We consider two intertwined counter-strategies of Player 1, essentially forcing Player 0 to simultaneously 370 BANACH-MAZUR GAMES ON GRAPHS perform two plays against herself. In reply to the first move a0, Player 1 selects an arbitrary a1 > a0 and then sets up the two plays as follows: In the first one she replies to a0 by a1 and waits for the answer a2 = f (a0a1) by Player 0. She then uses a2 as her own reply to a0 in the second play and gets the answer a3 = f (a0a2) by Player 0, which she now uses as her next move in the first play. There Player 0 responds by a4 = f (a0a1a2a3) which is again used by Player 1 as her answer to a0a2a3 in the second play. And so on. In this way, the two infinite plays result in sequences a0 < a1 < a2 < . . . and a0 < a2 < a3 < . . . . Since Player 0 plays with her winning strategy in both plays. it follows that X = [0, a0) ⋃ n∈ω[a2n+1, a2n+2) ∈ U, but also X ′ = [0, a0] ∪ ⋃ n>0[a2n, a2n+1) ∈ U. By closure under intersection, it follows that X ∩X′ = [0, a0) ∈ U. ButU is a free ultrafilter, so it cannot contain a finite set. It follows by the same argument that Player 1 cannot have a winning strategy. 3 Determinacy by simple strategies In general, strategies can be very complicated as they may depend on the entire history of a play. However, there are interesting classes of games that are determined via relatively simple winning strategies. We will discuss several kinds of restricted strategies: (1) Decomposition invariant strategies are strategies that depend only on the finite path that has been produced so far, and not on its decomposition into the moves of the players. Thus, a decomposition invariant strategy is a function assigning to each finite path a finite prolongation. We will show that, whenever a player has a winning strategy in a Banach-Mazur game, then she also has one that is decomposition invariant. (2) Positional strategies (also called memoryless strategies) depend only on the current position, and not on the history of the play. On a game graph G = (V, E) a positional strategy is a function f : V → V∗ assigning to every node v a finite path f (v) ∈ FinPaths(G, v). It is easy to find determined games that require non-positional winning stategies, but we will prove that all Banach-Mazur games with ω-regular objectives are determined via positional winning strategies. (3) More generally, strategies with memory M depend on the history of the play in a restricted way, via a memory structure M, consisting of a set of memory locations and an update function that changes the memory location as the play proceeds. Strategies with a finite memory structure can be implemented by a finite automaton. We will show that, for Banach-Mazur games, finite memory structures are irrelevant in the sense that winning strategies with finite memory can always be transformed into positional winning strategies. This is in sharp contrast to the usual graph games where already quite simple ω-regular winning conditions (such as, in particular, Muller conditions) lead to games that are determined by finite-memory strategies , but not by positional ones. 3.1 Decomposition invariant strategies GRÄDEL FSTTCS 2008 371 DEFINITION 8. A decomposition invariant strategy in a Banach-Mazur game on a graph (G, v) is a function f : FinPaths(G, v) → FinPaths(G, v) such that x ≤ f (x) for all x. THEOREM 9. Every Banach-Mazur game that is determined is also determined via a decomposition invariant strategy. PROOF. Suppose that Player 1 has a winning strategy for the game BM(G, v,Win). Then Win = ⋃ n<ω Xn with Xn nowhere dense. This means that the complement of each Xn contains a dense open setYn. Hence there exists a function gn assigning to each finite path y a prolongation gn(y) such that O(gn(y)) ⊆ Yn. We define a decomposition-invariant strategy g as follows. Given a finite path x ∈ FinPaths(G, v), there are only finitely many n < ω such that gn(y) ≤ x for some y ∈ FinPaths(G, v). Take the minimal n such that this is not the case and set g(x) = gn(x). It remains to show that g is a winning strategy for Player 1. Let π be any infinite play that is consistent with g. For every n < ω there exists a prefix y such that gn(y) < π. Hence π ∈ Yn for all n, which means that π is won by Player 1. The argument for Player 0 is analogous 3.2 Positional determinacy To start, we present a simple example of a Banach-Mazur game that is determined, but does not admit a positional winning strategy. Example 10 Let G2 be the completely connected directed graph with nodes 0 and 1, and let the winning condition for Player 0 be the set of infinite sequences with infinitely many initial segments that contain more ones than zeros. Clearly, Player 0 has a winning strategy for this game, but not a positional one. Note that this winning condition is on the Π2-level of the Borel hierarchy. As we show next, this is the lowest level with such an example. PROPOSITION 11. If Player 0 has a winning strategy for a Banach-Mazur game with a winning conditionWin ∈ Σ2, then she also has a positional winning strategy. PROOF. Suppose that Player 0 has awinning strategy f for the Banach-Mazur game BM(G, v,Win) such that Win is a countable union of closed sets. We have Win = ⋃ n<ω[Tn] where each Tn ⊆ FinPaths(G, v) is closed under prefixes. Further, we can assume that the winning strategy f is decomposition invariant. We claim that, in fact, Player 0 can win with one move, i.e. there is a finite path x such that O(x) ⊆ Win. We construct this move by induction. Let x1 be the initial path chosen by Player 0 according to f . Let i ≥ 1 and suppose that we have already constructed a finite path xi 6∈