Optimization complexity of linear logic proof games

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Abstract

A class of linear logic proof games is developed, each with a numeric score that depends on the number of preferred axioms used in a complete or partial proof tree. The complexity of these games is analyzed for the NP-complete multiplicative fragment (MLL) extended with additive constants and the PSPACE-complete multiplicative, additive fragment (MALL) of propositional linear logic. In each case, it is shown that it is as hard to compute an approximation of the best possible score as it is to determine the optimal strategy. Furthermore, it is shown that no efficient heuristics exist unless there is an unexpected collapse in the complexity hierarchy.

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1

Work supported under NSF Grant CCR-9224858 and ONR Grant N00014-95-C-0168.

2

Partially supported by NSF Grants CCR-9303099 and CCR-9629754.

3

Partially supported by NSF Grant CCR-94-00907, by ONR Grant N00014-92-J-1916, and by a Centennial Research Fellowship from the American Mathematical Society.