Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifier-free) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes pspace-complete. We also establish membership in np for the multiplicative fragment, np-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for certain fragments of noncommutative propositional linear logic. 1 Introduction Linear logic, introduced by Girard [14, 18, 17], is a refinement of classical logic which may be derived from a Gentzen-style sequent calculus axiomatization of classical logic in three steps. The resulting sequent system Lincoln@CS.Stanford.EDU Department of Computer Science, Stanford University, Stanford, CA 94305, and the Computer Science Labo...

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Investigations into the Relevant and Paraconsistent model theory of rstorder arithmetic have provided interesting new methods and results which have revived the interest in Hilbert's program. The attempt to develop Strict Finitist Mathematics using G. Priest's Collapsing lemma to nitize innite models is an example. In the investigation of some systems of Relevant Logics, another nitization procedure is used to solve positively their decision problem and to prove the nite model property for these systems. Some results related to the procedure used in these investigations show that Hilbert's ideal cannot be entirely fullled or that it must be reinterpreted. 1

Abstract. Well quasi-orders (wqo’s) are an important mathematical tool for proving termination of many algorithms. Under some assumptions upper bounds for the computational complexity of such algorithms can be extracted by analyzing the length of controlled bad sequences. We develop a new, self-contained study of the length of bad sequences over the product ordering of N n, which leads to known results but with a much simpler argument. We also give a new tight upper bound for the length of the longest controlled descending sequence of multisets of N n, and use it to give an upper bound for the length of controlled bad sequences in the majoring ordering of sets of tuples. We apply this upper bound to obtain complexity upper bounds for decision procedures of automata over data trees. In both cases the idea is to linearize bad sequences, i.e. transform them into a descending one over a well-order for which upper bounds can be more easily handled. 1