Curry-Howard Research Articles

Call-by-name, call-by-value, call-by-need and the linear lambda calculus

Girard described two translations of intuitionistic logic into linear logic, one where A→ B maps to (!A)⊸B and another where it maps to !(A⊸B). We detail the action of these translations on terms and show that the first corresponds to a call-by-name calculus, while the second corresponds to call-by-value. We further show that if the target of the translation is taken to be an affine calculus, where! controls contraction but weakening is allowed everywhere, then the second translation corresponds to a call-by-need calculus, as recently defined by Ariola, Felleisen, Maraist, Odersky and Wadler. Thus the different calling mechanisms can be explained in terms of logical translations, bringing them into the scope of the Curry–Howard isomorphism. Our results extend neatly to translations of extensions for recursion in the call-by-name and call-by-value calculi, and in general to extensions for products and for the corresponding untyped systems.

Type fixpoints

Positive recursive (fixpoint) types can be added to the polymorphic (Church-style) lambda calculus λ2 (System F ) in several different ways, depending on the choice of the elimination operator. We compare several such definitions and we show that they fall into two equivalence classes with respect to mutual interpretability by means of beta-eta reductions. Elimination operators for fixpoint types are thus classified as either "iterators" or "recursors". This classification has an interpretation in terms of the Curry-Howard correspondence: types of iterators and recursors can be seen as images of induction axioms under different dependency-erasing maps. Systems with recursors are beta-eta equivalent to a calculus λ2U of recursive types with the operators Fold : σ[μα.σ/α]←μα.σ and Unfold : μα.σ←σ[μα.σ/α], where the composition Unfold or Fold reduces to identity.It is known that systems with iterators can be defined within λ2, by means of beta reductions. We conjecture that systems with recursors can not. In this paper we show that the system λ2U does not have such a property. For this we study the notion of polymorphic type embeddability (via (beta) left-invertible terms) and we show that if a type σ is embedded into another type τ then τ must be of depth at least equal to the depth of σ.

A Resource Sensitive Interpretation of Lexical Functional Grammar

This paper investigates whether the fundamental linguistic insights and intuitions of Lexical Functional Grammar (LFG), which is usually presented as a ’’constraint-based‘‘ linguistic theory, can be reformulated in a ’’resource sensitive‘‘ framework using a substructural modal logic. In the approach investigated here, LFG‘s f-descriptions are replaced with expressions from a multi-modal propositional logic (with permutation and possibly limited contraction). In effect, the feature structure ’’unification‘‘ basis of LFG‘s f-structures is replaced with a very different resource based mechanism. It turns out that some linguistic analyses that required non-monotonic devices in LFG (such as the ’’constraint equations‘‘ in the Andrews (1982) analysis of Icelandic) can be straightforwardly expressed in the framework presented here. Moreover, a Curry–Howard correspondence between proofs in this logic and λ-terms provides a semantic interpretation as a by-product of the process of showing syntactic well-formedness.

ON THE CONSERVATIVITY OF LEIBNIZ EQUALITY

We embed a first order theory with equality in the Pure Type System λMON2 that is a subsystem of the well-known type system λPRED2. The embedding is based on the Curry-Howard isomorphism, i.e. → and ∀ coincide with → and Π. Formulas of the form [Formula: see text] are treated as Leibniz equalities. That is, [Formula: see text] is identified with the second order formula ∀ P. P(t1)→ P(t2), which contains only →'s and ∀'s and can hence be embedded straightforwardly. We give a syntactic proof — based on enriching typed λ-calculus with extra reduction steps — for the equivalence between derivability in the logic and inhabitance in λMNO2. Familiarity with Pure Type Systems is assumed.

Computational types from a logical perspective

Moggi's computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus also arises naturally as the term calculus corresponding (by the Curry–Howard correspondence) to a novel intuitionistic modal propositional logic. We give natural deduction, sequent calculus and Hilbert-style presentations of this logic and prove strong normalisation and confluence results.

Monad as modality

In 1989, Eugenio Moggi proposed a categorical framework for program semantics based on the notion of a strong monad. He showed that various kinds of computation can be modeled in his framework. On the other hand, strong monads are not suited for the categorical semantics of traditional modal logics. According to these observations, Moggi thought that the CurryHoward correspondence would not hold between programs and constructive proofs in modal logics. However, contrary to his view, we can show that proofs in a certain kind of modal logics are actually considered as programs. In this paper, first we shall introduce the notion of an ℓ-strong monad which is a generalization of strong monads. Using this new notion, we can generalize Moggi's semantics-preserving soundness and completeness with respect to his equational logic. Next we shall show that ℓ-strong monads give a sound and complete semantics of a constructive version of S4 modal logic. Finally, we present a method to extract a monad-based imperative functional program from a proof in the modal logic. Interestingly, this method can also be understood in terms of ℓ-strong monads.

Lambda Calculus and Intuitionistic Linear Logic

The introduction of Linear Logic extends the Curry-Howard Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment with an explicit control about the management of resources.

Intuitionism — Counting its Blessings. Preliminary version

A brief survey of the impact of intuitionistic logic and mathematics on modern practice is presented. The main influence is via the so-called “BHK interpretation” (or “proof interpretation”). By somewhat relaxing the conditions on the notion of proof the familiar notions of ‘realizability’, ‘Curry-Howard isomorphism’ etc. are motivated. Also some attention is paid to the characteristic process of replacing traditional notions by strong positive ones.

A Typed Pattern Calculus

The theory of programming with pattern-matching function definitions has been studied mainly in the framework of first-order rewrite systems. We present a typed functional calculus that emphasizes the strong connection between the structures of whole pattern definitions and their types. In this calculus, type-checking guarantees the absence of runtime errors caused by non-exhaustive pattern-matching definitions. Its operational semantics is deterministic in a natural way, without the imposition of ad hoc solutions such as clause order or “best fit”. In the spirit of the Curry–Howard isomorphism, we design the calculus as a computational interpretation of the Gentzen sequent proofs for the intuitionistic propositional logic. We prove the basic properties connecting typing and evaluation: subject reduction and strong normalization. We believe that this calculus offers a rational reconstruction of the pattern-matching features found in successful functional languages.

A Temporal-Logic Approach to Binding-Time Analysis

<p>The Curry-Howard isomorphism identifies proofs with typed lambda-<br />calculus terms, and correspondingly identifies propositions with<br />types. We show how this isomorphism can be extended to relate<br />constructive temporal logic with binding-time analysis. In particular,<br />we show how to extend the Curry-Howard isomorphism<br />to include the ("next") operator from linear-time temporal<br />logic. This yields the simply typed lambda-calculus which we prove<br />to be equivalent to a multi-level binding-time analysis like those<br />used in partial evaluation.</p><p><br />Keywords: Curry-Howard isomorphism, partial evaluation, binding-time analysis, temporal logic.</p>

Second Order Isomorphic Types: A Proof Theoretic Study on Second Order λ-Calculus with Surjective Pairing and Terminal Object

We investigate invertible terms and isomorphic types in the second order lambda calculus extended with surjective pairs and terminal (or Unit ) type. These two topics are closely related: on one side, the study of invertibility is a necessary tool for the characterization of isomorphic types; on the other hand, we need the notion of isomorphic types to study the typed invertible terms. The result of our investigation is twofold: we give a constructive characterization of the invertible terms, extending previous work by Dezani and Bruce-Longo, and a decidable equational theory of the isomorphisms of types which hold in all models of the calculus, which is a conservative extension to the second order case of the results previously achieved for the case of first order typed calculi. Via the Curry-Howard correspondence, this work also provides a decision procedure for strong equivalence of formulae in second order intuitionistic positive propositional logic, that is suitable to search equivalent proofs in automated deduction systems.

Call-by-name, Call-by-value, Call-by-need, and the Linear Lambda Calculus

Girard described two translations of intuitionistic logic into linear logic, one where A → B maps to (!A) –○ B, and another where it maps to !(A –○ B). We detail the action of these translations on terms, and show that the first corresponds to a call-by-name calculus, while the second corresponds to call-by-value. We further show that if the target of the translation is taken to be an affine calculus, where ! controls contraction but weakening is allowed everywhere, then the second translation corresponds to a call-by-need calculus, as recently defined by Ariola, Felleisen, Maraist, Odersky and Wadler. Thus the different calling mechanisms can be explained in terms of logical translations, bringing them into the scope of the Curry-Howard isomorphism.

Proofs as processes

The main purpose of this short paper is to serve as an introduction to the following paper, “On the rr-calculus and linear logic”, by Gianluigi Bellin and Philip Scott. The circumstances from which it arises are as follows. In June 1991 I gave a lecture on “Proofs as processes” at the symposium held at Tel-Aviv University to celebrate Boris Trakhtenbrot’s 70th birthday [6]. Material from this lecture also appeared in lectures subsequently given at the International Category Theory Meeting in Montreal (June 1991) and the London Mathematical Society Symposium on Category Theory in Computer Science in Durham (July 1991). The material was also presented in my tutorial lecture on linear logic given at the International Logic Programming Symposium in San Diego in October 1991. The lecture notes for these talks, particularly the tutorial lecture, were quite widely circulated; however, I did not write up a paper for publication, for reasons shortly to be explained. My point of departure in this work was the “propositions as types” paradigm (encompassing such notions as “Curry-Howard isomorphism”, realizability, functional interpretation and BHK semantics) familiar from proof theory and typed functional programming (see e.g. [9]). In this paradigm, we have the correspondences

Singleton, Union, and Intersection Types for Program Extraction

Two types theories, ATT and ATTT, are introduced. ATT is an impredicative type theory closely related to the polymorphic type theory of implicit typing of MacQueen et al . ((1986), Inform. and Control 71 , 95-130). ATTT is another version of ATT that extends the Girard-Reynolds second order lambda calculus. ATT has notions of intersection, union, and singleton types. ATTT has a notion of refinement types as in the type system for ML by Freeman and Pfenning ((1991), in "ACM SIGPLAN ′91," ACM Press), plus intersection and union of refinement types and singleton refinement types. We will show how singleton, union, and intersection types serve for development of programs without unnecessary codes via a variant of the Curry-Howard isomorphism. More exactly, they give a way to write types as specifications of programs without the unnecessary codes which are inevitable in the usual Curry-Howard isomorphism.

Intersection Types as Logical Formulae

The aim of this paper is to investigate, in the Curry-Howard isomorphism approach, a logical characterization for the intersection-type discipline First a novel formulation of the intersection type inference for combinatory logic is presented, such that it is equivalent to the original version of the system, while the intersection operator is no longer dealt with as a proof-theoretical connective Then a Hilbert-style axiomatization is defined and proved to totally parallel lntersecOon-denvability, in such a way that inhabited intersection-types are all and only the provable formulae in the logic system

Computational interpretations of linear logic

We study Girard's linear logic from the point of view of giving a concrete computational interpretation of the logic, based on the Curry—Howard isomorphism. In the case of Intuitionistic linear logic, this leads to a refinement of the lambda calculus, giving finer control over order of evaluation and storage allocation, while maintaining the logical content of programs as proofs, and computation as cut-elimination. In the classical case, it leads to a concurrent process paradigm with an operational semantics in the style of Berry and Boudol's chemical abstract machine. This opens up a promising new approach to the parallel implementation of functional programming languages; and offers the prospect of typed concurrent programming in which correctness is guaranteed by the typing.

From Petri nets to linear logic

Linear logic has recently been introduced by Girard as a logic of actions that seems well suited for concurrent computation. In this paper, we establish a systematic correspondence between Petri nets, linear logic theories, and linear categories. Such a correspondence sheds new light on the relationships between linear logic and concurrency, and on how both areas are related to category theory. Categories are here viewed as concurrent systems the objects of which are states, and the morphisms of which are transitions. This is an instance of the Lambek-Lawvere correspondence between logic and category theory that cannot be expressed within the more restricted framework of the Curry-Howard correspondence.

A Selected Bibliography on Constructive Mathematics, Intuitionistic Type Theory and Higher Order Deduction

THIERRY COQUAND AND G$RARD HUETINRIA, Domaine de Voluceau,78150 Rocquencourt, France(Received 2 April 1985)IntroductionWe present a selection of articles dealing with constructive mathematics, from the pointof view of the computer-aided synthesis of provably correct computer programs. Amongthe topics covered we find:-typed A-calculus, where types obey the Curry-Howard isomorphism betweenpropositions and proofs-higher-order intuitionistic logic-computer-aided proof-checking and theorem-proving in type theory-functional programming languages.There would be no point in attempting a complete bibliography of such aninterdisciplinary area. Our aim is rather to provide the researcher with an up-to-dateselection of work relevant to the current stream of research on de Bruijn's Automathlanguages, Girard's second-order types, and Martin-Lof's intuitionistic type theory .List of PapersAndrews, P. B. (1965). Transfinite Type Theory. Amsterdam: North-Holland.Andrews, P. B. (1971). Resolution in Type Theory. J. Symbolic Logic 36, 3, 414-432.Andrews, P. B., Miller, D. A., Cohen, E. L. & Pfenning, F. (1983). Automating higher-order logic.Deptof Math., University Carnegie-Mellon.Backhouse, R. (1984). Algorithm development in Martin-Lof's type theory. Colchester: University ofEssex.Backhouse, R. & Khamiss, A. (1984). The while-rule in Martin-Lofs theory of types. Colchester:University of Essex.Barendregt, H. (1980). The Lambda-Calculus: Its Syntax and Semantics. Amsterdam: North-Holland.Benabou, J. (1983). Fibered categories and the foundations of naive category theory . Paris: UniversiteParis Nord.Bishop, E. (1967). Foundations of Constructive Analysis. New-York: McGraw-Hill.Bishop, E. (1970). Mathematics as a numerical language. In: (Myhill, J., Kino, A., Vesley, R. E.,eds) Intuitionism and Proof Theory, pp. 53-71 . Amsterdam: North-Holland.Bohm, C. & Berarducci, A. (1983). Toward an Automatic Synthesis of Polyrnorphic Typed LambdaTerms. Unpublished manuscript.0747-7171/85/030323+06 $03.00/0 ©1985 Academic Press I c. (L d ) Ltd.