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A symbolic shortest path algorithm for computing subgame-perfect Nash equilibria

Góngora P. A, Rosenblueth D. A.

International Journal of Applied Mathematics and Computer Science

2015

Vol. 25, no. 3

577--596

Consider games where players wish to minimize the cost to reach some state. A subgame-perfect Nash equilibrium can be regarded as a collection of optimal paths on such games. Similarly, the well-known state-labeling algorithm used in model checking can be viewed as computing optimal paths on a Kripke structure, where each path has a minimum number of transitions. We exploit these similarities in a common generalization of extensive games and Kripke structures that we name “graph games”. By extending the Bellman–Ford algorithm for computing shortest paths, we obtain a model-checking algorithm for graph games with respect to formulas in an appropriate logic. Hence, when given a certain formula, our model-checking algorithm computes the subgame-perfect Nash equilibrium (as opposed to simply determining whether or not a given collection of paths is a Nash equilibrium). Next, we develop a symbolic version of our model checker allowing us to handle larger graph games. We illustrate our formalism on the critical-path method as well as games with perfect information. Finally, we report on the execution time of benchmarks of an implementation of our algorithms.

A Multiple-Clause Folding Rule Using Instantiation and Generalization

Rosenblueth D.A.

Fundamenta Informaticae

2006

Vol. 69, nr 1,2

219-249

A program-transformation system is determined by a repertoire of correctness-preserving rules, such as folding and unfolding. Normally, we would like the folding rule to be in some sense the inverse of the unfolding rule. Typically, however, the folding rule of logic program transformation systems is an inverse of a limited kind of unfolding. In many cases this limited kind of folding suffices. We argue, nevertheless, that it is both important and possible to extend such a folding so as to be able to fold the clauses resulting from any unfolding of a positive literal. This extended folding rule allows us to derive some programs underivable by the existing version of this rule alone. In addition, our folding rule has applications to decompilation and reengineering, where we are interested in obtaining high-level program constructs from low-level program constructs. Moreover, we establish a connection between logic program transformation and inductive logic programming. This connection stems from viewing our folding rule as a common extension of the existing multiple-clause folding rule, on the one hand, and an operator devised in inductive logic programming, called ``intra-construction,'' on the other hand. Hence, our folding rule can be regarded as a step towards incorporating inductive inference into logic program transformation. We prove correctness with respect to Dung and Kanchanasut's semantic kernel.