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Games and Full Completeness for Multiplicative Linear Logic
 JOURNAL OF SYMBOLIC LOGIC
, 1994
"... We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the den ..."
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Cited by 210 (26 self)
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We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cutfree proof net. A key role is played by the notion of historyfree strategy; strong connections are made between historyfree strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass et al.
Formal Parametric Polymorphism
 THEORETICAL COMPUTER SCIENCE
, 1993
"... A polymorphic function is parametric if its behavior does not depend on the type at which it is instantiated. Starting with Reynolds's work, the study of parametricity is typically semantic. In this paper, we develop a syntactic approach to parametricity, and a formal system that embodies this appro ..."
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Cited by 122 (6 self)
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A polymorphic function is parametric if its behavior does not depend on the type at which it is instantiated. Starting with Reynolds's work, the study of parametricity is typically semantic. In this paper, we develop a syntactic approach to parametricity, and a formal system that embodies this approach, called system R . Girard's system F deals with terms and types; R is an extension of F that deals also with relations between types. In R , it is possible to derive theorems about functions from their types, or "theorems for free", as Wadler calls them. An easy "theorem for free" asserts that the type "(X)XBool contains only constant functions; this is not provable in F. There are many harder and more substantial examples. Various metatheorems can also be obtained, such as a syntactic version of Reynolds's abstraction theorem.
Sequentiality vs. Concurrency in Games and Logic
 Math. Structures Comput. Sci
, 2001
"... Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic. ..."
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Cited by 15 (0 self)
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Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.
Parametricity as a Notion of Uniformity in Reflexive Graphs
, 2002
"... data types embody uniformity in the form of information hiding. Information hiding enforces the uniform treatment of those entities that dier only on hidden information. ..."
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Cited by 11 (3 self)
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data types embody uniformity in the form of information hiding. Information hiding enforces the uniform treatment of those entities that dier only on hidden information.
The GirardReynolds isomorphism (second edition
 Theoretical Computer Science
, 2004
"... polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfi ..."
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Cited by 5 (0 self)
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polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by wellbehaved models. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. We show that the inductive naturals are exactly those values of type natural that satisfy Reynolds’s notion of parametricity, and as a consequence characterize situations in which the Girard projection followed by the Reynolds embedding is also the identity. An earlier version of this paper used a logic over untyped terms. This version uses a logic over typed term, similar to ones considered by Abadi and Plotkin and by Takeuti, which better clarifies the relationship between F2 and P2. This paper uses colour to enhance its presentation. If the link below is not blue, follow it for the colour version.
Parametricity as Isomorphism
 Theoretical Computer Science
, 1993
"... . We investigate a simple form of parametricity, based on adding "abstract" copies of preexisting types. Connections are made with the ReynoldsMa theory of parametricity by logical relations, with the theory of parametricity via dinaturality, and with the categorical notion of equivalence. Introdu ..."
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Cited by 1 (1 self)
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. We investigate a simple form of parametricity, based on adding "abstract" copies of preexisting types. Connections are made with the ReynoldsMa theory of parametricity by logical relations, with the theory of parametricity via dinaturality, and with the categorical notion of equivalence. Introduction In his fundamental paper on the notion of parametricity in connection with type theories [Rey83], John Reynolds links the notion of parametricity firmly to the notion of data abstraction. This, unlike Strachey's earlier characterization via algorithm reuse, is a needdriven analysis. We need things to be parametric because otherwise our data abstractions will no longer be abstract. In his subsequent paper with Ma [MR91], two further points are made. One is that the problems reside more at the level of parametrized types than at the level of the quantified polymorphic types, and the other is that the notion of parametricity is not absolute, but relative. The MaReynolds work produces ...
Encodings In Polymorphism, revisited
, 1992
"... We consider encodings in polymorphism with finite product types. These encodings are given in terms of Ialgebras. They have the property that all canonical terms (ground terms) are normal terms. We transplant the proof of a wellknown result to our setting and show why weak recursion is admissible. ..."
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Cited by 1 (1 self)
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We consider encodings in polymorphism with finite product types. These encodings are given in terms of Ialgebras. They have the property that all canonical terms (ground terms) are normal terms. We transplant the proof of a wellknown result to our setting and show why weak recursion is admissible. We also show how to carry out the dual encodings using the existential quantifier. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Science, Manchester University, are available by anonymous ftp from m1.cs.man.ac.uk (130.88.13.4) in the directory /pub/TR. The files are stored as PostScript, in compressed form, with the report number as filename. Alternatively, reports are available by post from The Comput...
Formal Parametric Polymorphism
 Theoretical Computer Science
, 1993
"... A polymorphic function is parametric if its behavior does not depend on the type at which it is instantiated. Starting with Reynolds's work, the study of parametricity is typically semantic. In this paper, we develop a syntactic approach to parametricity, and a formal system that embodies this appro ..."
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A polymorphic function is parametric if its behavior does not depend on the type at which it is instantiated. Starting with Reynolds's work, the study of parametricity is typically semantic. In this paper, we develop a syntactic approach to parametricity, and a formal system that embodies this approach, called system R . Girard's system F deals with terms and types; R is an extension of F that deals also with relations between types. In R , it is possible to derive theorems about functions from their types, or "theorems for free", as Wadler calls them. An easy "theorem for free" asserts that the type "(X)XBool contains only constant functions; this is not provable in F. There are many harder and more substantial examples. Various metatheorems can also be obtained, such as a syntactic version of Reynolds's abstraction theorem. 1. Explicit relations A polymorphic function is parametric if its behavior does not depend on the type at which it is instantiated [Strachey 1967] . A function that reverses lists, for example, is parametric because it does not look at the types of the elements of the lists given as inputs. There are important nonparametric polymorphic functions, such as a print function that maps values of any type to text representations. With this caveat, it can be argued that "truly" polymorphic functions are parametric, and in any case it is the parametric polymorphic functions that form the core of languages such as ML [Milner, Tofte, Harper 1989] . Reynolds's work provides a precise counterpart to the informal definition of parametricity just given [Reynolds 1983; Ma, Reynolds 1991] . Reynolds's abstraction theorem concerns a language similar to Girard 's system F [Girard, Lafont, Taylor 1989], and implies that the instances of a polymorphic function at d...