Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. This accounting is made possible by removing the ‘structural’ rules of contraction and weakening, adding a modal operator and adding finer versions of the propositional connectives. Linear logic has fundamental logical interest and applications to computer science, particularly to Petri nets, concurrency, storage allocation, garbage collection and the control structure of logic programs. In addition, there is a direct correspondence between polynomial-time computation (...) and proof normalization in a bounded form of linear logic. In this paper we show that unlike most other propositional 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 fragments of noncommutative propositional linear logic. (shrink)

Mitchell, J.C. and E. Moggi, Kripke-style models for typed lambda calculus, Annals of Pure and Applied Logic 51 99–124. The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripke-style models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. To those familiar with Kripke models of (...) modal or intuitionistic logics, Kripke lambda models are likely to seem adequately ‘semantic’. However, when viewed as cartesian closed categories, they do not have the property variously referred to as concreteness, well- pointedness or having enough points. While the traditional lambda calculus proof system is not complete for Henkin models that may have empty types, we prove strong completeness for Kripke models. In fact, every set of equations that is closed under implication is the theory of a single Kripke model. We also develop some properties of logical relations over Kripke structures, showing that every theory is the theory of a model determined by a Kripke equivalence relation over a Henkin model. We discuss cartesian closed categories but present the main definitions and results without the use of category theory. (shrink)

§ 1. Introduction. Perhaps the most surprising recent development in complexity theory is the discovery that the class NP can be characterized using a form of randomized proof checker that only examines a constant number of bits of the “proof” that a string is in a language [6, 5, 31, 3, 4]. More specifically, writing ∣x∣ for the length of a string x, a language L in the class NP of languages recognizable in Nondeterministic polynomial time is traditionally given by (...) a polynomial p and a polynomial-time predicate P such that a string x is in L iff there is some string y satisfying P, where ∣y∣ ≤ p. Intuitively, we can think of a string y as a possible proof that x ϵ L, with the predicate P some kind of proof checker that distinguishes good proofs from bad ones. A string x is therefore in a language L ϵ NP if there is a short proof that x ϵ L, and not in L otherwise. The surprising discovery is that the deterministic proof checker that reads the entire input and proof can be replaced by a probabilistic verifier that on input x and possible proof y, where y is presented in a certain way, flips at most O coins and reads at most a constant number of bits of x and y. Obviously, a probabilistic verifier that does not read the whole proof will sometimes make mistakes. However, the surprising and essentially non-intuitive mathematical fact is that for each language L in NP, it is possible to find a polynomial q and verifier V flipping a logarithmic number of coins and reading a constant number of bits such that, for any x ϵ L, there exists y with ∣y∣ ≤ q and with V accepting with probability 1 and, for x ∉ L, there is no y with ∣y∣ ≤ q and with V accepting with probability ≥ 1/4. This idea canalsobeextended to PSPACE [10, 9], using a game that alternates between two players instead of a proof presented by a “single player.”. (shrink)