Conference Paper

Games on Strings with a Limited Order Relation.

DOI: 10.1007/978-3-540-92687-0_12 Conference: Logical Foundations of Computer Science, International Symposium, LFCS 2009, Deerfield Beach, FL, USA, January 3-6, 2009. Proceedings
Source: DBLP


In this paper, we show how Ehrenfeucht-Fraïssé games can be successfully exploited to compare (finite) strings. More precisely, we give necessary and sufficient conditions for Spoiler/Duplicator to win games played on finite structures with a limited order relation, that lies in between the successor relation and the usual (linear) order relation, and a finite number of unary predicates. On the basis of such conditions, we outline a polynomial (in the size of the input strings) algorithm to compute the "remoteness" of a game and to determine the optimal strategies/moves for both players.

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    ABSTRACT: We show that deciding the winner of the r-moves Ehrenfeucht-Fraïssé game on two finite structures A and B, over any fixed signature Σ that contains at least one binary and one ternary relation, is PSPACE complete. We consider two natural modifications of the EF game, the one-sided r-moves EF game, where the spoiler can choose from the first structure A only, and therefore the duplicator wins only if B satisfies all the existential formulas of rank at most r that A satisfies; and the k-alternations r-moves EF game (for each fixed k), where the spoiler can choose from either structure, but he can switch structure at most k times, and therefore the duplicator wins iff A and B satisfy the same first order formulas of rank at most r and quantifier alternation at most k (defined in the paper). We show that deciding the winner in both the one-sided EF game and the k-alternations EF game is also PSPACE complete.
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    ABSTRACT: In this paper we initiate the study of Ehrenfeucht-Fraïssé games for some standard finite structures. Examples of such standard structures are equivalence relations, trees, unary relation structures, Boolean algebras, and some of their natural expansions. The paper concerns the following question that we call Ehrenfeucht-Fraïssé problem. Given n ∈ ω as a parameter, two relational structures A\mathcal A and B\mathcal B from one of the classes of structures mentioned above, how efficient is it to decide if Duplicator wins the n-round EF game Gn(A,B)G_n(\mathcal A,\mathcal B) ? We provide algorithms for solving the Ehrenfeucht-Fraïssé problem for the mentioned classes of structures. The running times of all the algorithms are bounded by constants. We obtain the values of these constants as functions of n.
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