On natural deduction in classical first-order logic: Curry–Howard correspondence, strong normalization and Herbrand's theorem

Aschieri, Federico; Zorzi, Margherita

doi:10.1016/j.tcs.2016.02.028

Cited by 12 publications

(22 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The goal is to further ease the embedding of quantum programming in traditional programming frameworks. Fourth, we are interested in developing a denotational semantics for IQu, maybe a not complete one, but suitable to tackle the equivalence between programs involving (meaningful) quantum, non-deterministic [2,3], probabilistic and reversible [20,21] aspects [4].…”

Section: Discussionmentioning

confidence: 99%

Quantum Programming Made Easy

Paolini¹,

Roversi²,

Zorzi³

2019

Electron. Proc. Theor. Comput. Sci.

Self Cite

View full text Add to dashboard Cite

We present IQu, namely a quantum programming language that extends Reynold's Idealized Algol, the paradigmatic core of Algol-like languages. IQu combines imperative programming with highorder features, mediated by a simple type theory. IQu mildly merges its quantum features with the classical programming style that we can experiment through Idealized Algol, the aim being to ease a transition towards the quantum programming world. The proposed extension is done along two main directions. First, IQu makes the access to quantum co-processors by means of quantum stores. Second, IQu includes some support for the direct manipulation of quantum circuits, in accordance with recent trends in the development of quantum programming languages. Finally, we show that IQu is quite effective in expressing well-known quantum algorithms.

show abstract

Section: Discussionmentioning

confidence: 99%

Quantum Programming Made Easy

Paolini¹,

Roversi²,

Zorzi³

2019

Electron. Proc. Theor. Comput. Sci.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The concurrent λ-calculus of [5] is typed by first-order classical logic. It resembles a session-typed λ-calculus, but supports only data transmission.…”

Section: λ`And Other Typed Concurrent λ-Calculimentioning

confidence: 99%

Par means parallel: multiplicative linear logic proofs as concurrent functional programs

Aschieri¹,

Genco

2019

Proc. ACM Program. Lang.

Self Cite

View full text Add to dashboard Cite

Along the lines of the Abramsky "Proofs-as-Processes" program, we present an interpretation of multiplicative linear logic as typing system for concurrent functional programming. In particular, we study a linear multiple-conclusion natural deduction system and show it is isomorphic to a simple and natural extension of λ-calculus with parallelism and communication primitives, called λ`. We shall prove that λ`satisfies all the desirable properties for a typed programming language: subject reduction, progress, strong normalization and confluence. * Funded by FWF grant P32080-N31. † Funded by ANR JCJC project Intuitions Bolzaniennes. linking proofs to programs, propositions to types, proof normalization to computation. Soon after, Wadler [32] introduced the session typed π-calculus CP, shown to tightly correspond to classical linear logic. LimitationsThis important research notwithstanding, it appears that there is still ground to cover toward canonical and firm foundations for concurrent computation. First of all, asynchronous communication does not appear to rest on solid foundations. Yet this communication style is easier to implement and more practical than the purely synchronous paradigm, so it is widespread and asynchronous typed process calculi have been already investigated (see [13]). Unfortunately, linear logic has not so far provided via Curry-Howard a logical account of asynchronous communication. The reason is that there is a glaring discrepancy between full cut-elimination and π-calculus reduction which has not so far been addressed. On one hand, as we shall see, linear logic does support asynchronous communication, but only through the full process of cut-elimination, which indeed makes essential use of asynchronous communication. On the other hand, the π-calculus of [9] only mimics a partial cut-elimination process that only eliminates top-level cuts. Indeed, by comparison with its type system, the π-calculus lacks some necessary computational reduction rules. Some of the missing reductions, corresponding to commuting conversions, were provided in Wadler's CP. The congruence rules that allow the extra reductions to mirror full cut-elimination, however, were rejected: in Wadler's [32] own words, "such rules do not correspond well to our notion of computation on processes, so we omit them". A set of reductions similar to that rejected by Wadler is regarded in [27] as sound relatively to a notion of "typed context bisimilarity". The notion, however, only ensures that two "bisimilar" processes have the same input/output behaviour along their main communication channel; the internal synchronization among the parallel components of the two processes is not captured and may differ significantly. Thus, the extensional flavor of the bisimilarity notion prevents it to detect the intensionally different behaviour of the related processes, that is, how differently they communicate and compute.

show abstract

Section: Markov's Principle In First-order Logicmentioning

confidence: 99%

“…The Curry-Howard correspondence we present here is by no means an ad hoc construction, only tailored for Markov's principle. It is a simple restriction of the Curry-Howard correspondence for classical first-order logic introduced in [4], where classical reasoning is formalized by the excluded middle inference rule: Γ, a : ∀x Q u : C Γ, a : ∃x ¬Q v : C EM Γ u a v : C It is enough to restrict the conclusion C of this rule to be a simply existential statement and the Q in the premises ∀x Q, ∃x ¬Q to be propositional. We shall show that the rule is intuitionistically equivalent to MP.…”

Section: Restricted Excluded Middlementioning

confidence: 99%

“…It turns out, however, that Herbrand constructivity is preserved. An intermediate logic is called Herbrand constructive if it enjoys a strong form of Herbrand's Theorem [5,4]: for every provable formula ∃α A, the logic proves as well an Herbrand disjunctionSo the Markov principle we shall interpret in this paper is MP : ¬¬∃α P → ∃α P (P propositional formula) and show that when added to intuitionistic first-order logic, the resulting system is Herbrand constructive. This is the most general form of Markov's principle that allows a significant constructive interpretation: we shall show how to non-trivially compute lists of witnesses for provable existential formulas thanks to an exception raising construct and a parallel computation operator.…”

mentioning

confidence: 99%

See 1 more Smart Citation

On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

Aschieri¹

2017

Logical Methods in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. Dummett's logic LC is intuitionistic logic extended with Dummett's axiom: for every two statements the first implies the second or the second implies the first. We present a natural deduction and a Curry-Howard correspondence for first-order and secondorder Dummett's logic. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for representing parallel computations and communication between them, and from the viewpoint of logic, Dummett's axiom. We prove that our typed calculus is normalizing and show that proof terms for existentially quantified formulas reduce to a list of individual terms forming an Herbrand disjunction.

show abstract

On natural deduction in classical first-order logic: Curry–Howard correspondence, strong normalization and Herbrand's theorem

Cited by 12 publications

References 12 publications

Quantum Programming Made Easy

Quantum Programming Made Easy

Par means parallel: multiplicative linear logic proofs as concurrent functional programs

On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

Contact Info

Product

Resources

About