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Provable Isomorphisms of Types
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 1990
"... A constructive characterization is given of the isomorphisms which must hold in all models of the typed lambda calculus with surjective pairing. By the close relation between closed Cartesian categories and models of these calculi, we also produce a characterization of those isomorphisms which hold ..."
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Cited by 39 (8 self)
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A constructive characterization is given of the isomorphisms which must hold in all models of the typed lambda calculus with surjective pairing. By the close relation between closed Cartesian categories and models of these calculi, we also produce a characterization of those isomorphisms which hold in all CCC's. By the correspondence between these calculi and proofs in intuitionistic positive propositional logic, we thus provide a characterization of equivalent formulae of this logic, where the definition of equivalence of terms depends on having "invertible" proofs between the two terms. Rittri (1989), on types as search keys in program libraries, provides an interesting example of use of these characterizations.
On the Unification Problem for Cartesian Closed Categories (Extended Abstract)
 IN PROCEEDINGS, EIGHTH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1989
"... Cartesian closed categories (CCC's) have played and continue to play an important role in the study of the semantics of programming languages. An axiomatization of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce and Longo leads to seven e ..."
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Cited by 19 (5 self)
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Cartesian closed categories (CCC's) have played and continue to play an important role in the study of the semantics of programming languages. An axiomatization of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce and Longo leads to seven equalities. We show that the unification problem for this theory is undecidable, thus settling an open question. We also show that an important subcase, namely unification modulo the linear isomorphisms, is NPcomplete. Furthermore, the problem of matching in CCC's is NPcomplete when the subject term is irreduc...
The maximality of the typed lambda calculus and of cartesian closed categories
 Publ. Inst. Math. (N.S
"... From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here ..."
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Cited by 18 (1 self)
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From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson.
Retrieving Library Functions By Unifying Types Modulo Linear Isomorphism
, 1992
"... An improved method to retrieve a library function via its Hindley/Milner type is described. Previous retrieval systems have identified types that are isomorphic in any Cartesian closed category (CCC), and have retrieved library functions of types that are either isomorphic to the query, or have ..."
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Cited by 18 (0 self)
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An improved method to retrieve a library function via its Hindley/Milner type is described. Previous retrieval systems have identified types that are isomorphic in any Cartesian closed category (CCC), and have retrieved library functions of types that are either isomorphic to the query, or have instances that are. Sometimes it is useful to instantiate the query too, which requires unification modulo isomorphism. Although unifiability modulo CCCisomorphism is undecidable, it is decidable modulo linear isomorphism, that is, isomorphism in any symmetric monoidal closed (SMC) category. We argue that the linear isomorphism should retrieve library functions almost as well as CCCisomorphism, and we report experiments with such retrieval from the Lazy ML library. When unification is used, the system retrieves too many functions, but sorting by the sizes of the unifiers tends to place the most relevant functions first. R'esum'e Ce papier pr'esente une nouvelle m'ethode pour la re...
A Logic of Subtyping
, 1996
"... The relation of inclusion between types has been suggested by the practice of programming, as it enriches the polymorphism of functional languages. We propose a simple (and linear) calculus of sequents for subtyping as logical entailment. This allows us to derive a complete and coherent approach to ..."
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Cited by 14 (4 self)
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The relation of inclusion between types has been suggested by the practice of programming, as it enriches the polymorphism of functional languages. We propose a simple (and linear) calculus of sequents for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subtyping from a few, logically meaningful, sequents. In particular, transitivity and antisymmetry will be derived from elementary logical principles, which stresses the power of sequents and Gentzenstyle proof methods. Proof techniques based on cutelimination will be at the core of our results. 1 Introduction 1.1 Motivations, Theories and Models In recent years, several extensions of core functional languages have been proposed to deal with the notion of subtyping; see, for example, [CW85, Mit88, BL90, BCGS91, CMMS91, CG92, PS94, Tiu96, TU96]. These extensions were suggested by the practice of programming in computer science. In particular, they were inspired by the notion of inheritance...
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.
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 11 (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...
Efficient Type Matching
, 2001
"... Palsberg and Zhao [14] presented an O(n²) time algorithm for matching two recursive types. In this paper, we present an O(n log n) algorithm for the same problem. Our algorithm works by reducing the type matching problem to the wellunderstood problem of finding a sizestable partition of a graph. ..."
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Cited by 10 (3 self)
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Palsberg and Zhao [14] presented an O(n²) time algorithm for matching two recursive types. In this paper, we present an O(n log n) algorithm for the same problem. Our algorithm works by reducing the type matching problem to the wellunderstood problem of finding a sizestable partition of a graph. Our result may help improve systems, such as Polyspin and Mockingbird, that are designed to facilitate interoperability of software components. We also discuss possible applications of our algorithm to Java. Issues related to subtyping of recursive types are also discussed.
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
Coherence and Transitivity of Subtyping as Entailment
, 1996
"... The relation of inclusion between types has been suggested by the practice of programming as it enriches the polymorphism of functional languages. We propose a simple (and linear) sequent calculus for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subty ..."
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Cited by 10 (3 self)
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The relation of inclusion between types has been suggested by the practice of programming as it enriches the polymorphism of functional languages. We propose a simple (and linear) sequent calculus for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subtyping from a few, logically meaningful sequents. In particular, transitivity and antisymmetry will be derived from elementary logical principles.