Mechanical Procedure for Proof Construction via Closed Terms in Typed lambda Calculus (original) (raw)
Mechanical procedure for proof construction via closed terms in typed λ calculus
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.
Mechanical procedure for proof construction via closed terms in typed ? calculus
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 2 calculus. The soundness and completeness of this method are proved.
A Prolog-based Proof Tool for Type-Theory TA-lambda and Implicational Intuitionistic-Logic
EPiC series in computing, 2018
Studies on type theory have brought numerous important contributions to computer science. In this paper we present a GUI-based proof tool that provides assistance in constructing deductions in type theory and validating implicational intuitionistic-logic formulas. As such, this proof tool is a testbed for learning type theory and implicational intuitionistic-logic. This proof tool focuses on an important variant of type theory named TA λ , especially on its two core algorithms: the principal-type algorithm and the type inhabitant search algorithm. The former algorithm finds a most general type assignable to a given λ-term, while the latter finds inhabitants (closed λ-terms in β-normal form) to which a given type can be assigned. By the Curry-Howard correspondence, the latter algorithm provides provability for implicational formulas in intuitionistic logic. We elaborate on how to implement those two algorithms declaratively in Prolog and the overall GUI-based program architecture. In our implementation, we make some modification to improve the performance of the principal-type algorithm. We have also built a web-based version of the proof tool called λ-Guru.
On the Correspondence Between Proofs and lambda-Terms
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).
J-Calc: A typed lambda calculus for Intuitionistic Justification Logic
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.
Constructive Logics. Part I: A Tutorial on Proof Systems and Typed lambda-Calculi.
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 ).
On the Algebraic Foundation of Proof Assistants for Intuitionistic Type Theory
Lecture Notes in Computer Science, 2008
An algebraic presentation of Martin-Löf's intuitionistic type theory is given which is based on the notion of a category with families with extra structure. We then present a type-checking algorithm for the normal forms of this theory, and sketch how it gives rise to an initial category with families with extra structure. In this way we obtain a purely algebraic formulation of the correctness of the type-checking algorithm which provides the core of proof assistants for intuitionistic type theory.
Lambda Calculus and Intuitionistic Linear Logic
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.
A note on the proof theory the ?II-calculus
Studia Logica, 1995
The AH-calculus, a theory of first-order dependent function types in Curry-Howard-de Bruijn correspondence with a fragment of minimal first-order logic, is defined as a system of (]inearized) natural deduction. In this paper, we present a Gentzen-style sequent calculus for the AH-calculus and prove the cut-elimination theorem. The cut-elimination result builds upon the existence of normal forms for the natural deduction system and can be considered to be analogous to a proof provided by Prawitz for first-order logic. The type-theoretic setting considered here elegantly illustrates the distinction between the processes of normalization in a natural deduction system and cutelimination in a Gentzen-style sequent calculus. We consider an application of the cut-free calculus, via the subformula property, to proof-search in the All-calculus. For this application, the normalization result for the natural deduction calculus alone is inadequate, a (cut-free) calculus with the subformula property being required.
A typed calculus based on a fragment of linear logic
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.
The λ Calculus and the Unity of Structural Proof Theory
Theory of Computing Systems / Mathematical Systems Theory, 2009
In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato’s calculus. It is a calculus with modus ponens and primitive substitution; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a “multiary” calculus, because “applicative terms” may exhibit a list of several arguments. But the combination of “multiarity” and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: normalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.
A Type-Theoretic Framework for Formal Reasoning with Different Logical Foundations
Lecture Notes in Computer Science, 2007
A type-theoretic framework for formal reasoning with different logical foundations is introduced and studied. With logic-enriched type theories formulated in a logical framework, it allows various logical systems such as classical logic as well as intuitionistic logic to be used effectively alongside inductive data types and type universes. This provides an adequate basis for wider applications of type theory based theorem proving technology. Two notions of set are introduced in the framework and used in two case studies of classical reasoning: a predicative one in the formalisation of Weyl's predicative mathematics and an impredicative one in the verification of security protocols.
THE JUDGEMENT CALCULUS FOR INTUITIONISTIC LINEAR LOGIC: PROOF THEORY AND SEMANTICS
Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1992
In this paper we propose a new set of rules for a judgement calculus, i.e. a typed lambda calculus, based on Intuitionistic Linear Logic; these rules ease the problem of definiag a suitable mathematical semantics. A proof of the canonical form theorem for this new system is given: it assures, beside the consistency of the calculus, the termination of the evaluation process of every well-typed element. The defmition of the mathematical semantics and a completeness theorem, that turns out to be a representation theorem, follow. This semantics is the basis to obtain a semantics for the evaluation process of every 1991 MSC: 03B20, O3B40
Proof-Assistants Using Dependent Type Systems
2001
this article we will not attempt to describe all the dierent possible choicesof type theories. Instead we want to discuss the main underlying ideas, with a specialfocus on the use of type theory as the formalism for the description of theoriesincluding proofs
Theorem proving for untyped constructive -calculus: implementation and application
Logic Journal of IGPL, 2001
This paper presents a theorem prover for a highly intensional logic, namely a constructive version of property theory [25] (this language essentially provides a combination of constructive first-order logic and the λ-calculus). The paper presents the basic theorem prover, which is a higher-order extension of Manthey and Bry's model generation theorem prover for first-order logic ; considers issues relating to the compile-time optimisations that are often used with first-order theorem provers; and shows how the resulting system can be used in a natural language understanding system.
Stanford Encyclopedia of Philosophy, 2016
Intuitionistic type theory (also constructive type theory or Martin-Lof type theory) is a formal logical system and philosophical foundation for constructive mathematics. It is a full-scale system which aims to play a similar role for constructive mathematics as Zermelo-Fraenkel Set Theory does for classical mathematics. It is based on the propositions-as-types principle and clarifies the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic. It extends this interpretation to the more general setting of intuitionistic type theory and thus provides a general conception not only of what a constructive proof is, but also of what a constructive mathematical object is. The main idea is that mathematical concepts such as elements, sets and functions are explained in terms of concepts from programming such as data structures, data types and programs. This article describes the formal system of intuitionistic type theory and its semantic foundations.
On Proofs and Types in Second Order Logic
2015
In my dissertation I address some questions concerning the proof theory of second order logic and its constructive counterpart, System F (Girard 1971). These investigations follow two distinct (though historically related) viewpoints in proof theory, which are compared throughout the text: on the one side, the proof theoretic semantics tradition inaugurated by Dummett and Prawitz (Prawitz 1971, Dummett 1991), focusing on the analysis of the inferential content of proofs; on the other side, the interactionist tradition arising from Kleene's realizability (Kleene 1945) and the Tait/Girard reducibility technique (Tait 1967, Girard 1971), which interprets proofs as untyped programs and focuses, rather, on the behavioral content of proofs, i.e. the way in which they interact through the cut-elimination algorithm. A distinction is made between the issues of justifying and understanding ("explaining why" and "explaining how", as in Girard 2000) impredicative reasoning, i.e. between non elementary results like the Hauptsatz and the combinatorial analysis of proofs, seen as programs, i.e. recursive objects. As for justifi cation, an epistemological analysis of the circularity involved in the second order Hauptsatz is developed; it is shown that the usual normalization arguments for second order logic do not run into the vicious circularity claimed by Poincaré and Russell, but involve a diff erent, epistemic, form of circularity. Still, this weaker circularity makes justifi cation, in a sense, pointless; in particular, some examples of inconsistent higher order theories admitting epistemically circular normalization arguments are discussed. As for the explanation issue, a constructive and combinatorial (i.e. independent from normalization) analysis of higher order order quantifi cation is developed along two directions, with some related technical results. The fi rst direction arises from the parametric and dinatural interpretations of polymorphism (Reynolds 1983, Girard-Scott-Scedrov 1992), which provide a clear mathematical meaning to Carnap's defense of impredicative quanti fication (Carnap 1983). In particular, the violation of the parametric condition leads to paradoxes which are often ignored in the philosophical literature (with the exception of Longo-Fruchart 1997). The analysis of the combinatorial content of these interpretations leads to a 1-completeness theorem (every normal closed -term in the universal closure of a simple type is typable in simple type theory), which connects the interactionist and the inferential conceptions of proof. The second direction follows the analysis of the typing conditions of the -terms associated with intuitionistic second order proofs. To the \vicious circles" in the proofs there correspond recursive (i.e. circular) speci fications for the types of the -terms. The geometrical structure of these vicious circles is investigated (following Lechenadec 1989, Malecki 1990, Giannini - Ronchi Della Rocca 1991), leading to a combinatorial characterization of typability in some inconsistent extension of System F: since, as Girard's paradox shows, a typable term need not be normalizing, one is indeed naturally led to consider not normalizing theories. Such investigations go in the direction both of a mathematical understanding of the structure generated by the vicious circles of impredicative theories and of the development of a proof-theoretic analysis of potentially incorrect or uncertain proofs.
Proof-term synthesis on dependent-type systems via explicit substitutions
Theoretical computer science, 2001
Typed-terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the Curry-Howard isomorphism relates proof-trees with typed-terms. The proofs-as-terms principle can be used to verify the validity of a proof by type checking the-term extracted from the complete proof-tree. In this paper we present a proof synthesis method for dependent-type systems where typed open terms are built incrementally at the same time as proofs are done. This way, every construction step, not just the last one, may be type checked. The method is based on a suitable calculus where substitutions as well as meta-variables are ÿrst-class objects.