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Fair Games and Full Completeness for Multiplicative Linear Logic without the MIXRule
, 1993
"... We introduce a new category of finite, fair games, and winning strategies, and use it to provide a semantics for the multiplicative fragment of Linear Logic (mll) in which formulae are interpreted as games, and proofs as winning strategies. This interpretation provides a categorical model of mll wh ..."
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Cited by 40 (4 self)
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We introduce a new category of finite, fair games, and winning strategies, and use it to provide a semantics for the multiplicative fragment of Linear Logic (mll) in which formulae are interpreted as games, and proofs as winning strategies. This interpretation provides a categorical model of mll which satisfies the property that every (historyfree, uniformly) winning strategy is the denotation of a unique cutfree proof net. Abramsky and Jagadeesan first proved a result of this kind and they refer to this property as full completeness. Our result differs from theirs in one important aspect: the mixrule, which is not part of Girard's Linear Logic, is invalidated in our model. We achieve this sharper characterization by considering fair games. A finite, fair game is specified by the following data: ffl moves which Player can play, ffl moves which Opponent can play, and ffl a collection of finite sequences of maximal (or terminal) positions of the game which are deemed to be fair. N...
Explicit Substitution Internal Languages for Autonomous and *Autonomous Categories
 In Proc. Category Theory and Computer Science (CTCS'99), Electron
, 1999
"... We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. We show tha ..."
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Cited by 7 (2 self)
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We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. We show that the eight equality and three commutation congruence axioms of the autonomous type theory characterise autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek [12]: the equality of maps in any autonomous category freely generated from a discrete graph is decidable. 1 Introduction In this paper we introduce a family of type theories which can be regarded as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the standard simplytyped calculus with surjective pairing is...