Finite Model Theory and Descriptive Complexity

ly, we can describe a two-player game with positional winning conditions by a directed game graph G = (V, V0, V1, E), with a partioning V = V0 ∪ V1 of the nodes into positions where Player 0 moves and positions where Player 1 moves. The possible moves are described by the edge relation E ⊆ V × V . We call w a successor of v if (v, w) ∈ E, and we denote the set of all successors of v by vE. To decribe the winning conditions, we adopt the convention that Player σ loses at positions v ∈ Vσ where no moves are possible. (Alternatively, one could explicitly include in the game description the sets S0, S1 of winning terminal positions for each player.) A play of G is a path v0, v1, . . . formed by the two players starting from a given position v0. Whenever the current position vn belongs to Vσ, Player σ chooses a move to a successor vn+1 ∈ vnE; if no move is available, then Player σ has lost the play. If this never occurs, the play goes on infinitely and the winner has to be established by a winning condition on infinite plays. For the moment, let us say that infinite plays are won by neither of the players. A strategy for a player is a function defining a move for each situation in a play where she has to move. Of particular interest are positional strategies, which do not depend on the history of the play, but only on the current position. Hence, a positional strategy for Player σ in G is a (partial) function f : Vσ → V which indicates a choice (v, f(v)) ∈ E for positions v ∈ Vσ . A play v0, v1, . . . is consistent with a positional strategy f for Player σ if vn+1 = f(vn) for all vn ∈ Vσ. A strategy for a player is winning from position v0 if she wins every play starting from v0 that is consistent with that strategy. We say that a strategy is winning on a setW if it is winning from each position inW . The winning regionWσ for Player σ is the set of positions from which she has a winning strategy. A game is well-founded if all its plays are finite. Note that a modelchecking game G(A, ψ(a)) for a first-order formula ψ has a finite game graph if, and only if, A is finite, but it is well-founded in all cases. In general, however, games with finite game graphs need not be well-founded. A game is determined if, from each position, one of the players has a winning strategy, i.e. if W0 ∪ W1 = V . Well-founded games are always 4 We shall later introduce games with more interesting winning conditions for infinite plays. 130 3 Finite Model Theory and Descriptive Complexity determined, and so are large classes of more general games (such as games in the Borel hierarchy; see [82, 96]). We denote by Game the strategy problem for games with finite game graphs and positional winning conditions, i.e. Game = {(G, v) : Player 0 has a winning strategy in G from position v}. It is obvious that the Game problem can be solved in polynomial time. Denote byW σ the set of positions from which Player σ has a strategy to win the game in at most n moves. Then W 0 σ = {v ∈ V1−σ : vE = ∅} is the set of winning terminal positions for Player σ, and we can compute the sets W σ inductively by using W σ := {v ∈ V0 : vE ∩W σ 6= ∅} ∪ {v ∈ V1 : vE ⊆W σ } until W σ =W n σ . To see that Game can actually be solved in linear time, a little more work is necessary. The following algorithm is a variant of depth-first search, and computes the entire winning sets for both players in time O(|V |+ |E|). Theorem 3.1.2.Winning regions of finite games can be computed in linear time. Proof. We present an algorithm that computes, for each position, which player, if any, has a winning strategy for the game starting at that position. During the computation three arrays are used: • win[v] contains either 0 or 1, indicating which player wins, or ⊥ if we do not know yet, or if none of the players has a winning strategy from v; • P [v] contains the predecessors of v; and • n[v] is the number of those successors for which win[v] = ⊥. A linear-time algorithm for the Game problem Input: A game G = (V, V0, V1, E) forall v ∈ V do (∗ 1: initialization ∗) win[v] := ⊥ P [v] := ∅ n[v] := 0 enddo forall (u, v) ∈ E do (∗ 2: calculate P and n ∗) P [v] := P [v] ∪ {u} n[u] := n[u] + 1 enddo 3.1 Definability and Complexity 131 forall v ∈ V0 (∗ 3: calculate win ∗) if n[v] = 0 then Propagate(v, 1) forall v ∈ V1 if n[v] = 0 then Propagate(v, 0) return win end procedure Propagate(v, σ) if win[v] 6= ⊥ then return win[v] := σ (∗ 4: mark v as winning for Player σ ∗) forall u ∈ P [v] do (∗ 5: propagate change to predecessors ∗) n[u] := n[u]− 1 if u ∈ Vσ or n[u] = 0 then Propagate(u, σ) enddo end The heart of this algorithm is the procedure Propagate(v, σ) which is called any time we have found that Player σ has a winning strategy from position v. Propagate(v, σ) records this fact and investigates whether we are now able to determine the winning player for any of the predecessors of v. This is done by applying the following rules: • If the predecessor u belongs to Player σ, then this player has a winning strategy from u by moving to position v. • If the predecessor u belongs to the opponent of Player σ, if win[u] is undefined, and if the winning player has already been determined for all successors w of u, then win[w] = σ for all of those successors, and hence Player σ wins from u regardless of the choice of her opponent. Since parts 4 and 5 of the algorithm are reached only once for each position v, the inner part of the loop in part 5 is executed at most ∑ v |P [v]| = |E| times. Therefore the running time of the algorithm is O(|V |+ |E|). The correctness of the value assigned to win[v] is proved by a straightforward induction on the number of moves in which the corresponding player can ensure that she wins. Note that the positions satisfying n[v] = 0 in part 3 are exactly those without outgoing edges even if n[v] is modified by Propagate. Game is known to be a PTIME-complete problem (see [57]). This remains the case for strictly alternating games, where E ⊆ V0×V1∪V1×V0. Indeed, any game can be transformed into an equivalent strictly alternating one by introducing for each move (u, v) ∈ Vσ × Vσ a new node e ∈ V1−σ and by replacing the move (u, v) by two moves (u, e) and (e, u). The Game problem (sometimes also called the problem of alternating reachability) is a general combinatorial problem that reappears in different guises in many areas. To illustrate this by an example, we shall now show that the satisfiability problem for propositional Horn formulae is essentially the same problem as Game. 132 3 Finite Model Theory and Descriptive Complexity Satisfiability for Horn Formulae It is well known that Sat-Horn, the satisfiability problem for propositional Horn formulae, is • PTIME-complete [57], and • solvable in linear time [36, 68]. Using the Game problem, we can obtain very simple proofs for both results. Indeed, Game and Sat-Horn are equivalent under log–lin reductions, i.e. reductions that are computable in linear time and logarithmic space. The reductions are so simple that we can say that Game and Sat-Horn are really the same problem. Theorem 3.1.3. Sat-Horn is log–lin equivalent to Game. Proof. Game ≤log−lin Sat-Horn. Given a finite game graph G = (V, V0, V1, E), we can construct in time O(|V | + |E|) a propositional Horn formula ψG consisting of the clauses u ← v for all edges (u, v) ∈ E with u ∈ V0, and the clauses u ← v1 ∧ · · · ∧ vm for all nodes u ∈ V1, where uE = {v1, . . . , vm}. The minimal model of ψG is precisely the winning set W0 for Player 0. Hence v ∈W0 if the Horn formula ψG ∧ (0← v) is unsatisfiable. Sat-Horn ≤log−lin Game: Given a Horn formula ψ(X1, . . . , Xn) =