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1997, Studia Logica
AI
This paper explores the interplay between lambda calculus and intuitionistic linear logic through the lens of the Curry-Howard isomorphism. It emphasizes how this relationship can be used to model computational environments specifically focusing on resource management. By analyzing operational semantics and normalization properties, the study shows that linear logic can enhance the understanding of functional programming.
Theoretical Computer Science, 1989
Linear Logic, we concisely write LL, has been introduced recently by Jean Yves Girard in Theoretical Computer Science ~0 (1987). Born from the semantics of second order lambda calculus, LL is more expressive than traditional logic (both classical and intuitionistic). Due to the absence of structural rules and to a partict:!ar treatment of negation, which is denoted by ~, proofs in LL do not have a "directional character". The constructive meaning of a proof of A-, B is a function mapping all proofs of A into proofs of B; in LL A-~B has a similar meaning, but B±-oA ± represents the same formula and has the same proofs: so one of such proofs can map a proof of A into one of B as well as a proof of B x into one of.4±~ In this paper we are interested in the multiplicative second order subsystem L,:~* of linear logic; we introduce a calculus (called z-calculus) whose terms are canonical represent: ~ions of proofs. The aim of the calculus is to give a be~ter comprehension of the computational aspects of the process of cut-elimination. We prove that the z-calculus obeys strong normalization and the Church-Rosser properties.
Lecture Notes in Computer Science
It is surprising that despite the simplicity of its syntax, the λ-calculus hosts a large body of notation, abbreviations, naming conventions, etc. Our aim, as far as the notation throughout this work is concerned, is to remain consistent, clear and unambiguous as much as possible. By and large we adhere to the notation set down in Church's Calculi of Lambda Conversion [4], and Barendregt's The Lambda Calculus, Its Syntax and Semantics [1]. There are however instances where our notation diverges. Most notably, we avoid abbreviating the names of combinators. Thus, for example, we use Succ Church to denote the λ-term computing the successor function on Church numerals, instead of Church's S + , 1 etc. A variety of typefaces are used in denoting logical and mathematical entities. In general, we strive for consistency, unless prevented from doing so by existing, well-established notation. The following table illustrates the various typefaces used throughout this work, and the context in which they appear: Typeface Usage within this Document Math Italics variables, λ-terms used locally Roman Bold λ-terms used throughout this work San Serif sets, relations, trees, syntactic objects Fraktur Gothic: Fraktur arithmetic functions Greek: α, β, γ,. .. rules, well-known combinators Blackboard bold: N, Q, Z,. .. well-known sets of numbers CALLIGRAPHIC systems of equations Our introduction to the λ-calculus consists of five sections, which cover the following topics, in order: Syntax, reduction, λ-definability, fixed-points, and bases. By no stretch of the imagination can this selection of topics be considered a complete survey of the main areas of the λ-calculus. The most striking omissions include models and types. This introduction is, however, sufficient for its intended purposes, which are, as stated earlier, to make this work as self-contained as possible, and to establish a common language with the reader, in terms of which, the ideas contained in this work can be conveyed. The sections on syntax and reduction cover the λ-calculus as a formalism for describing computation. Section 4 covers λ-definability, which concerns itself with representing logical and mathematical entities in the λ-calculus. Section 5 covers fixed points. The treatment is not complete, but is sufficient for our purposes. Section 6 concludes the introduction to the λ-calculus with a treatment of bases. Bases in the λ-calculus are analogous, in many ways, to bases in linear algebra.
The λ-calculus was first published in , with the aim of providing a foundation for logic which would be an alternative to Russell's type theory or Zermelo's set theory. The theory is based on the concept of functions rather than sets and it is about functions as rules 1 rather than as graphs 2 . Almost immediately after publication a contradiction was found in it. Church revised the theory a year later, the inconsistency of which was discovered by two students of Church, Stephen . Discouraged in his foundational project, Church gave a consistent subtheory, known as λI -calculus in ].
Journal of Functional Programming, 2009
We present an extension of the λ(η)-calculus with a case construct that propagates through functions like a head linear substitution, and show that this construction permits to recover the expressiveness of ML-style pattern matching. We then prove that this system enjoys the Church–Rosser property using a semi-automatic ‘divide and conquer’ technique by which we determine all the pairs of commuting subsystems of the formalism (considering all the possible combinations of the nine primitive reduction rules). Finally, we prove a separation theorem similar to Böhm's theorem for the whole formalism.
Bulletin of Symbolic Logic, 1997
One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.
2007
We introduce a natural deduction formulation for the Logic of Proofs, a refinement of modal logic S4 in which the assertion□ A is replaced by [[s]] A whose intended reading is “s is a proof of A”. A term calculus for this formulation yields a typed lambda calculus λ I that internalises intensional information on how a term is computed. In the same way that the Logic of Proofs internalises its own derivations, λ I internalises its own computations. Confluence and strong normalisation of λ I is proved.
Journal of Functional …, 1998
1991
In Computer Science, Lambda Calculus has been mainly used as the skeleton of functional programming languages. It has also been used as a higher order parameterization mechanism in some specification languages. In this paper we view -calculus as both the applicative structure of a ...
The purpose of this paper is to give an exposition of material dealing with constructive logic, typed λ-calculi, and linear logic. The emergence in the past ten years of a coherent field of research often named "logic and computation" has had two major (and related) effects: firstly, it has rocked vigorously the world of mathematical logic; secondly, it has created a new computer science discipline, which spans from what is traditionally called theory of computation, to programming language design. Remarkably, this new body of work relies heavily on some "old" concepts found in mathematical logic, like natural deduction, sequent calculus, and λ-calculus (but often viewed in a different light), and also on some newer concepts. Thus, it may be quite a challenge to become initiated to this new body of work (but the situation is improving, there are now some excellent texts on this subject matter). This paper attempts to provide a coherent and hopefully "gentle" initiation to this new body of work. We have attempted to cover the basic material on natural deduction, sequent calculus, and typed λ-calculus, but also to provide an introduction to Girard's linear logic, one of the most exciting developments in logic these past six years. The first part of these notes gives an exposition of background material (with some exceptions, such as "contractionfree" systems for intuitionistic propositional logic and the Girard-translation of classical logic into intuitionistic logic, which is new). The second part is devoted to more current topics such as linear logic, proof nets, the geometry of interaction, and unified systems of logic (LU ).
2000
The minimal relevant logic B+, seen as a type discipline, includes an extension of Curry types known as the intersection type discipline. We will show that the full logic B+ gives a type assignment system which produces a model of Plotkin's call-by-value λ-calculus.
Theoretical Computer Science, 2000
Church's -calculus is an enthralling object of mathematical and logical study, born in 1930 as the mathematical theory of functions as rules, and invented for foundational purposes.
Information Processing Letters, 1995
The classical notion of P-reduction in the A-calculus has an arbitrary syntactically-imposed sequentiality. A new notion of reduction fi' is defined which is a generalization of P-reduction. This notion of reduction is shown to satisfy the Church-Rosser property as well as some other fundamental properties, leading to the conclusion that this generalized notion of P'-reduction can be used in place of p-reduction without sacrificing any of the fundamental properties.
Theoretical Computer Science, 1993
A model-theoretic operation is characterised that preserves the property of being a model of typed λ-calculus. (i.e., the result of applying it to a model of typed λ-calculus is another model of typed λ-calculus.) An expression is well-typed iff the class of its models is closed under this operation.
Electronic Proceedings in Theoretical Computer Science, 2021
We propose an intersection type system for an imperative λ-calculus based on a state monad and equipped with algebraic operations to read and write to the store. The system is derived by solving a suitable domain equation in the category of ω-algebraic lattices; the solution consists of a filter-model generalizing the well known construction for ordinary λ-calculus. Then the type system is obtained out of the term interpretations into the filter-model itself. The so obtained type system satisfies the "type-semantics" property, and it is sound and complete by construction.
We introduce and study a functional language λr, having two main features. λr has the same computational power of the λ-calculus. λr enjoys the resource-awareness of the typed/typable functional languages which encode the Intuitionistic Linear Logic.
Information and Computation, 1990
2021
We propose an intersection type system for an imperative λ -calculus based on a state monad and equipped with algebraic operations to read and write to the store. The system is derived by solving a suitable domain equation in the category of ω -algebraic lattices; the solution consists of a filter-model generalizing the well known construction for ordinary λ -calculus. Then the type system is obtained out of the term interpretations into the filter-model itself. The so obtained type system satisfies the “type-semantics” property, and it is sound and complete by construction.
Mathematical Structures in Computer Science, 2000
This paper introduces Hilbert systems for -calculus, called sequent combinators, addressing many of the problems of Hilbert systems that have led to the more widespread adoption of natural deduction systems in computer science. This suggests that Hilbert systems, with their uniform approach to meta-variables and substitution, may be a more suitable framework than -calculus for type theories and programming languages. Two calculi are introduced here. The calculus SKIn captures -calculus reduction faithfully, is con uent even in the presence of meta-variables, is normalizing but not strongly normalizing in the typed case, and standardizes. The sub-calculus SKInT captures -reduction in slightly less obvious ways, and is a language of proof-terms not directly for intuitionistic logic, but for a fragment of S4 that we name near-intuitionistic logic. To our knowledge, SKInT is the rst con uent, rst-order calculus to capture -calculus reduction fully and faithfully and be strongly normalizing in the typed case. In particular, no calculus of explicit substitutions has yet achieved this goal.
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