We present a process semantics for the purely additive fragment of linear logic in which formulas denote protocols and (equivalence classes of) proofs denote multi-channel concurrent processes. The polycategorical model induced by this process semantics is shown to be equivalent to the free polycategory based on the syntax (i.e., it is full and faithfully complete). This establishes that the additive fragment of linear logic provides a semantics of concurrent processes. Another property of this semantics is that it gives a canonical representation of proofs in additive linear logic.

It is known that the strongly stable functions which arise in the semantics of PCF can be realized by sequential algorithms, which can be considered as deterministic strategies in games associated to PCF types. Studying the connection between strongly stable functions and sequential algorithms, two dual classes of hypercoherences naturally arise: the parallel and serial hypercoherences. The objects belonging to the intersection of these two classes are in bijective correspondence with the so-called "serial-parallel" graphs, that can essentially be considered as games. We show how to associate to any hypercoherence a parallel hypercoherence together with a projection onto the given hypercoherence and present some properties of this construction. Intuitively, it makes explicit the computational time of a hypercoherence.

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. lincoln@csl.sri.com SRI International Computer Science Laboratory, Menlo Park CA 94025 USA. Work supported under NSF Grant CCR-9224858 and ONR Grant N00014-95-C-0168. y mitchell@cs.stanford.edu WWW: http://theory.stanford.edu/people/jcm/home.html Department of Computer Science, Stanford University, Stanford, CA 94305-9045 USA. Partiall...

Proof search in linear logic is known to be difficult: the provability of propositional linear logic formulas is undecidable. Even without the modalities, multiplicative-additive fragment of propositional linear logic, mall, is known to be pspace-complete, and the pure multiplicative fragment, mll, is known to be np-complete. However, this still leaves open the possibility that there might be proof search heuristics (perhaps involving randomization) that often lead to a proof if there is one, or always lead to something close to a proof. One approach to these problems is to study strategies for proof games. A class of linear logic proof games is developed, each with a numeric score that depends on the number of certain preferred axioms used in a complete or partial proof tree. Using recent techniques for proving lower bounds on optimization problems, the complexity of these games is analyzed for the fragment mll extended with additive constants and for the fragment mall. It ...

) Patrick D. Lincoln y John C. Mitchell z Andre Scedrov x 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 jxj 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 polynomialtime predicate P such that a string x is in L iff there is some string y satisfying P (x; y) , where jyj p(jxj) . Intuitively, we can think of a string y as a possible proof that x 2 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 2 np if there is a short proof that x 2 L , and not in L otherwise. The surprising discovery is that the determi...