Sequent combinators: a Hilbert system for the lambda calculus

2000 ◽  
Vol 10 (1) ◽  
pp. 1-79 ◽  
Author(s):  
HEALFDENE GOGUEN ◽  
JEAN GOUBAULT-LARRECQ
Keyword(s):  
Computer Science ◽  
Programming Languages ◽  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Lambda Calculus ◽  
First Order ◽  
Uniform Approach ◽  
Explicit Substitutions ◽  
Suitable Framework ◽  
Proof Terms

This paper introduces Hilbert systems for λ-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 λ-calculus for type theories and programming languages. Two calculi are introduced here. The calculus SKIn captures λ-calculus reduction faithfully, is confluent even in the presence of meta-variables, is normalizing but not strongly normalizing in the typed case, and standardizes. The sub-calculus SKInT captures λ-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 first confluent, first-order calculus to capture λ-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.

scholarly journals Arithmetic Formulated Relevantly

2021 ◽  
Vol 18 (5) ◽  
pp. 154-288
Author(s):  
Robert Meyer
Keyword(s):  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Relevant Logic ◽  
Peano Arithmetic ◽  
Critical Problem ◽  
First Order ◽  
Order Form ◽  
Relevant Implication ◽  
Relevant Logics ◽  
Natural Way

The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that (1) it is trivial that relevant arithmetic is absolutely consistent, but (2) classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under (1), I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, of course, is exactly the formula whose unprovability was sought in the Hilbert program for proving arithmetic consistent. Under (2), I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment.

scholarly journals Using Vampire in Soundness Proofs of Type Systems

10.29007/22x6 ◽  
2018 ◽  
Author(s):  
Sylvia Grewe ◽  
Sebastian Erdweg ◽  
Mira Mezini
Keyword(s):  
Programming Languages ◽  
Type System ◽  
Lambda Calculus ◽  
Type Systems ◽  
Current Approach ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Domain Specific ◽  
Automated Proof

Type systems for programming languages shall detect type errors in programs before runtime. To ensure that a type system meets this requirement, its soundness must be formally verified. We aim at automating soundness proofs of type systems to facilitate the development of sound type systems for domain-specific languages.Soundness proofs for type systems typically require induction. However, many of the proofs of individual induction cases only require first-order reasoning. For the development of our workbench Veritas, we build on this observation by combining automated first-order theorem provers such as Vampire with automated proof strategies specific to type systems. In this paper, we describe how we encode type soundness proofs in first-order logic using TPTP. We show how we use Vampire to prove the soundness of type systems for the simply-typed lambda calculus and for parts of a typed SQL. We report on which parts of the proofs are handled well by Vampire, and what parts work less well with our current approach.

scholarly journals Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic

1998 ◽  
Vol 63 (3) ◽  
pp. 869-890 ◽  
Author(s):  
Wil Dekkers ◽  
Martin Bunder ◽  
Henk Barendregt
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Lambda Calculus ◽  
Preceding Paper ◽  
Predicate Calculus ◽  
Combinatory Logic ◽  
First Order ◽  
Two Systems

AbstractIllative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions.

Quantifier elimination and parametric polymorphism in programming languages

1992 ◽  
Vol 2 (2) ◽  
pp. 213-226 ◽  
Author(s):  
Harry G. Mairson
Keyword(s):  
Programming Languages ◽  
Lambda Calculus ◽  
Quantifier Elimination ◽  
Type Inference ◽  
Cut Elimination ◽  
Standard Interpretation ◽  
Inference Problem ◽  
First Order ◽  
Typed Lambda Calculus ◽  
Parametric Polymorphism

AbstractWe present a simple and easy-to-understand explanation of ML type inference and parametric polymorphism within the framework of type monomorphism, as in the first order typed lambda calculus. We prove the equivalence of this system with the standard interpretation using type polymorphism, and extend the equivalence to include polymorphic fixpoints. The monomorphic interpretation gives a purely combinatorial understanding of the type inference problem, and is a classic instance of quantifier elimination, as well as an example of Gentzen-style cut elimination in the framework of the Curry-Howard propositions-as-types analogy.

scholarly journals The Impact of the Lambda Calculus in Logic and Computer Science

1997 ◽  
Vol 3 (2) ◽  
pp. 181-215 ◽  
Author(s):  
Henk Barendregt
Keyword(s):  
Computer Science ◽  
Programming Languages ◽  
Functional Programming ◽  
Lambda Calculus ◽  
The Other ◽  
Functional Programming Languages ◽  
Other Hand ◽  
The One ◽  
The Impact

AbstractOne 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.

scholarly journals Constructive Game Logic

2020 ◽  
pp. 84-111 ◽  
Author(s):  
Brandon Bohrer ◽  
André Platzer
Keyword(s):  
Natural Deduction ◽  
Operational Semantics ◽  
Dynamic Logic ◽  
Program Logic ◽  
Game Logic ◽  
First Order ◽  
Constructive Proofs ◽  
Winning Strategies ◽  
Theoretical Results ◽  
Proof Terms

AbstractGame Logic is an excellent setting to study proofs-about-programs via the interpretation of those proofs as programs, because constructive proofs for games correspond to effective winning strategies to follow in response to the opponent’s actions. We thus develop Constructive Game Logic, which extends Parikh’s Game Logic (GL) with constructivity and with first-order programs à la Pratt’s first-order dynamic logic (DL). Our major contributions include: 1. a novel realizability semantics capturing the adversarial dynamics of games, 2. a natural deduction calculus and operational semantics describing the computational meaning of strategies via proof-terms, and 3. theoretical results including soundness of the proof calculus w.r.t. realizability semantics, progress and preservation of the operational semantics of proofs, and Existential Properties on support of the extraction of computational artifacts from game proofs. Together, these results provide the most general account of a Curry-Howard interpretation for any program logic to date, and the first at all for Game Logic.

scholarly journals On the correspondence between nested calculi and semantic systems for intuitionistic logics

2020 ◽  
Author(s):  
Tim Lyon
Keyword(s):  
Intuitionistic Logic ◽  
Structural Rules ◽  
First Order ◽  
Constant Domains ◽  
The Relationship ◽  
Intuitionistic Logics

Abstract This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics.

scholarly journals Shaping the foundations of programming languages

2021 ◽  
Vol 64 (6) ◽  
pp. 120
Author(s):  
Leah Hoffmann
Keyword(s):  
Science Education ◽  
Computer Science ◽  
Programming Languages ◽  
Computer Science Education ◽  
The Future ◽  
Turing Award

ACM A.M. Turing Award recipients Alfred Aho and Jeffrey Ullman discuss their early work, the 'Dragon Book,' and the future of 'live' computer science education.

scholarly journals On Correspondence between Selective CPS Transformation and Selective Double Negation Translation

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 385
Author(s):  
Hyeonseung Im
Keyword(s):  
Programming Languages ◽  
Intuitionistic Logic ◽  
Reference Point ◽  
Classical Logic ◽  
Proof System ◽  
Double Negation

A double negation translation (DNT) embeds classical logic into intuitionistic logic. Such translations correspond to continuation passing style (CPS) transformations in programming languages via the Curry-Howard isomorphism. A selective CPS transformation uses a type and effect system to selectively translate only nontrivial expressions possibly with computational effects into CPS functions. In this paper, we review the conventional call-by-value (CBV) CPS transformation and its corresponding DNT, and provide a logical account of a CBV selective CPS transformation by defining a selective DNT via the Curry-Howard isomorphism. By using an annotated proof system derived from the corresponding type and effect system, our selective DNT translates classical proofs into equivalent intuitionistic proofs, which are smaller than those obtained by the usual DNTs. We believe that our work can serve as a reference point for further study on the Curry-Howard isomorphism between CPS transformations and DNTs.

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