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

**ABSTRACT** 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.10/2006: pages 159-170; - [Show abstract] [Hide abstract]

**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.06/2007: pages 293-309; - [Show abstract] [Hide abstract]

**ABSTRACT:**Ehrenfeucht-Fraïssé games are commonly used as a method to measure the expressive power of a logic, but they are also a flexible tool to compare structures. To exploit such a comparison power, explicit conditions characterizing the winning strategies for both players must be provided. We give a necessary and sufficient condition for Duplicator to win games played on finite structures with a successor relation and a finite number of unary predicates. This structural characterization suggests an algorithmic approach to the analysis of games, which can be used to compute the “remoteness” of a game and to determine the optimal moves for both players, that is, to derive algorithms for Spoiler and Duplicator that play optimally. We argue that such an algorithmic solution may be used in contexts where the “degree of similarity” between two structures must be measured, such as the comparison of biological sequences.Logic for Programming, Artificial Intelligence, and Reasoning, 12th International Conference, LPAR 2005, Montego Bay, Jamaica, December 2-6, 2005, Proceedings; 01/2005

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