Lambda-free Logical Frameworks

Special Issue Workshop on Intuitionistic Modal Logic and Applications 2013

In this paper we offer a system J-Calc that can be regarded as a typed λ-calculus for the {→, ⊥} fragment of Intuitionistic Justification Logic. We offer different interpretations of J-Calc, in particular, as a two phase proof system in which we proof check the validity of deductions of a theory T based on deductions from a stronger theory T and computationally as a type system for separate compilations. We establish some first metatheoretic results.

Information & Computation, 2003

Intersection types discipline allows to define a wide variety of models for the type free lambda-calculus, but the Curry-Howard isomorphism breaks down for this kind of type systems. In this paper we show that the correspondence between types and suitable logical formulas can still be recovered appealing to the fact that there is a strict connection between the semantics for lambda-calculus induced by the intersection types and a Kripke style semantics for modal and relevant logics. Indeed, we present a modal logic hinted by the analysis of the sub-typing relation for intersection types, and we show that the deduction relation for such a modal system is a conservative extension of the relation of sub-typing. Then, we define a Kripke-style semantics for the formulas of such a system, present suitable sequential calculi, prove a completeness theorem and give a syntactical proof of the cut elimination property. Finally, we define a decision procedure for theorem-hood and we show that it yields the finite model property and cut-redundancy.

Journal of Automated Reasoning - JAR, 1988

In this paper is presented an algorithm for constructing natural deduction proofs in the propositional intuitionistic and classical logics according to the analogy relating intuitionistic propositional formulas and natural deduction proofs, respectively, to types and terms of simple type theory. Proofs are constructed as closed terms in the simple typed 2 calculus. The soundness and completeness of this method are proved.

Studia Logica - An International Journal for Symbolic Logic, 1997

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. This paper introduces a typed functional language Λ! and a categorical model for it. The terms of Λ! encode a version of natural deduction for Intuitionistic Linear Logic such that linear and non linear assumptions are managed multiplicatively and additively, respectively. Correspondingly, the terms of Λ! are built out of two disjoint sets of variables. Moreover, the λ-abstractions of Λ! bind variables and patterns. The use of two different kinds of variables and the patterns allow a very compact definition of the one-step operational semantics of Λ!, unlike all other extensions of Curry-Howard Isomorphism to Intuitionistic Linear Logic. The language Λ! is Church-Rosser and enjoys both Strong Normalizability and Subject Reduction. The categorical model induces operational equivalences like, for example, a set of extensional equivalences. The paper presents also an untyped version of Λ! and a type assignment for it, using formulas of Linear Logic as types. The type assignment inherits from Λ! all the good computational properties and enjoys also the Principal-Type Property.

Logical Methods in Computer Science, 2008

We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. We show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. Finally we describe a simple set-theoretic semantics.

Logical Methods in Computer Science, 2011

We define a logical framework with singleton types and one universe of small types. We give the semantics using a PER model; it is used for constructing a normalisation-by-evaluation algorithm. We prove completeness and soundness of the algorithm; and get as a corollary the injectivity of type constructors. Then we give the definition of a correct and complete type-checking algorithm for terms in normal form. We extend the results to proof-irrelevant propositions. 1998 ACM Subject Classification: F.4.1. CC Creative Commons 2 A. ABEL, T. COQUAND, AND M. PAGANO

Journal of Automated Reasoning, 1988

In this paper is presented an algorithm for constructing natural deduction proofs in the propositional intuitionistic and classical logics according to the analogy relating intuitionistic propositional formulas and natural deduction proofs, respectively, to types and terms of simple type theory. Proofs are constructed as closed terms in the simple typed λ calculus. The soundness and completeness of this method are proved.

Technical Reports (CIS), 1993

The correspondence between natural deduction proofs and ��-terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) ��-terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed ��-calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors���,��,+,���,��� and���(falsity)(with or without ��-like rules).

Electronic Notes in Theoretical Computer Science, 2008

The type theory λP corresponds to the logical framework LF. In this paper we present λH, a variant of λP where convertibility is not implemented by means of the customary conversion rule, but instead type conversions are made explicit in the terms. This means that the time to type check a λH term is proportional to the size of the term itself. We define an erasure map from λH to λP , and show that through this map the type theory λH corresponds exactly to λP : any λH judgment will be erased to a λP judgment, and conversely each λP judgment can be lifted to a λH judgment. We also show a version of subject reduction: if two λH terms are provably convertible then their types are also provably convertible. 1 Thanks to Thorsten Altenkirch for the suggestion to use John-Major equality in our system.

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 ).