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Memoization in typedirected partial evaluation
 PROCEEDINGS OF THE 2002 ACM SIGPLAN/SIGSOFT CONFERENCE ON GENERATIVE PROGRAMMING AND COMPONENT ENGINEERING, NUMBER 2487 IN LECTURE NOTES IN COMPUTER SCIENCE
, 2002
"... We use a code generator—typedirected partial evaluation— to verify conversions between isomorphic types, or more precisely to verify that a composite function is the identity function at some complicated type. A typed functional language such as ML provides a natural support to express the function ..."
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Cited by 14 (6 self)
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We use a code generator—typedirected partial evaluation— to verify conversions between isomorphic types, or more precisely to verify that a composite function is the identity function at some complicated type. A typed functional language such as ML provides a natural support to express the functions and typedirected partial evaluation provides a convenient setting to obtain the normal form of their composition. However, offtheshelf typedirected partial evaluation turns out to yield gigantic normal forms. We identify that this gigantism is due to redundancies, and that these redundancies originate in the handling of sums, which uses delimited continuations. We successfully eliminate these redundancies by extending typedirected partial evaluation with memoization capabilities. The result only works for pure functional programs, but it provides an unexpected use of code generation and it yields ordersofmagnitude improvements both in time and in space for type isomorphisms.
Remarks on isomorphisms in typed lambda calculi with empty and sum types
 In Proc. of the 17 th Annual IEEE Symposium on Logic in Computer Science (LICS’02
, 2002
"... Tarski asked whether the arithmetic identities taught in high school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question for the language of arithmetic expressions using a constant for the number one and the operations of product and exponenti ..."
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Cited by 10 (2 self)
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Tarski asked whether the arithmetic identities taught in high school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question for the language of arithmetic expressions using a constant for the number one and the operations of product and exponentiation is affirmative, and the complete equational theory also characterises isomorphism in the typed lambda calculus, where the constant for one and the operations of product and exponentiation respectively correspond to the unit type and the product and arrow type constructors. This paper studies isomorphisms in typed lambda calculi with empty and sum types from this viewpoint. We close an open problem by establishing that the theory of type isomorphisms in the presence of product, arrow, and sum types (with or without the unit type) is not finitely axiomatisable. Further, we observe that for type theories with arrow, empty and sum types the correspondence between isomorphism and arithmetic equality generally breaks down, but that it still holds in some particular cases including that of type isomorphism with the empty type and equality with zero. 1
A Linear Logical View of Linear Type Isomorphisms
, 1999
"... . The notion of isomorphisms of types has many theoretical as well as practical consequences, and isomorphisms of types have been investigated at length over the past years. Isomorphisms in weak system (like linear lambda calculus) have recently been investigated due to their practical interest ..."
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Cited by 9 (2 self)
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. The notion of isomorphisms of types has many theoretical as well as practical consequences, and isomorphisms of types have been investigated at length over the past years. Isomorphisms in weak system (like linear lambda calculus) have recently been investigated due to their practical interest in library search. In this paper we give a remarkably simple and elegant characterization of linear isomorphisms in the setting of Multiplicative Linear Logic (MLL), by making an essential use of the correctness criterion for Proof Nets due to Girard. 1 Introduction and Survey The interest in building models satisfying specific isomorphisms of types (or domain equations) is a long standing one, as it is a crucial problem in the denotational semantics of programming languages. In the 1980s, though, some interest started to develop around the dual problem of finding the domain equations (type isomorphisms) that must hold in every model of a given language. Alternatively, one could say tha...
Keeping sums under control
, 2004
"... In a recent paper [31], I presented with Marcelo Fiore and Roberto Di Cosmo a new normalisation tool for the λcalculus with sum types, based on the technique of normalisation by evaluation, and more precisely on techniques developped by Olivier Danvy for partial evaluation, using control operators. ..."
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In a recent paper [31], I presented with Marcelo Fiore and Roberto Di Cosmo a new normalisation tool for the λcalculus with sum types, based on the technique of normalisation by evaluation, and more precisely on techniques developped by Olivier Danvy for partial evaluation, using control operators. The main characteristic of this tool is that it produces a result in a canonical form we introduced. That is to say: two βηequivalent terms will be normalised into (almost) identical terms. It was not the case with the traditional algorithm, which could even lead to an explosion of the size of code. This canonical form is an ηlong βnormal form with constraints, which capture the definition of ηlong normal form for the λcalculus without sums, and reduces drastically the ηconversion possibilities for sums. The present paper recall the definition of these normal forms and the normalisation algorithm, and shows how it is possible to use these tools to solve a problem of characterization of type isomorphisms. Indeed, the canonical form allowed to find the complicated counterexamples we exhibited in another work [6], that proves that type isomorphisms in the λcalculus with sums are not finitely axiomatisable. What’s more, when proving that these terms are isomorphisms, the new partial evaluation algorithm avoids an explosion of the size of the term that arises with the old one.