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Classical Program Extraction in the Calculus of Constructions [chapter]

Alexandre Miquel
Lecture Notes in Computer Science  
We show how to extract classical programs expressed in Krivine λc-calculus from proof-terms built in a proof-irrelevant and classical version of the calculus of constructions with universes.  ...  Finally, we extend the extraction scheme to a primitive data-type of natural numbers in a way which preserves the whole compatibility with the classical realisability interpretation of second-order arithmetic  ...  Introduction Program extraction has been a major concern from the early development of the calculus of constructions (CC) [3] to its more recent extensions [12, 16] implemented in proof assistants  ... 
doi:10.1007/978-3-540-74915-8_25 fatcat:kuh6lyjvlre7jm563sdyy3wple

Towards the theory of programming in constructive logic [chapter]

A. A. Voronkov
1990 Lecture Notes in Computer Science  
We develop an approach to the theory of extracting programs from proofs based on constructive semantics of the first order formulas called consTrucTive TruLb. The underlying ideas are discussed.  ...  Using this notion of truth we define an appropriate notion of constructive calculus.  ...  the main ideas and problems The extraction of programs from proofs or programming in constructive logic is based on the idea that under some restrictions proofs can be considered as programs.  ... 
doi:10.1007/3-540-52592-0_78 fatcat:wy7miwvy5jgmxdpzkw5mmhlcly

Mechanically Verified Calculational Abstract Interpretation [article]

David Darais, David Van Horn
2015 arXiv   pre-print
Our proof "by calculus" closely follows the original paper-and-pencil proof and supports the extraction of a verified static analyzer.  ...  Calculational abstract interpretation, long advocated by Cousot, is a technique for deriving correct-by-construction abstract interpreters from the formal semantics of programming languages.  ...  Acknowledgments We gratefully acknowledge the Colony Club in Washington, DC for providing a fruitful environment in which to do this research.  ... 
arXiv:1507.03559v1 fatcat:a2ho4ehqyjaqrprr5nujpaj7ri

Extending the Calculus of Constructions with Tarski's fix-point theorem [article]

Yves Bertot
2006 arXiv   pre-print
We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions in the calculus of inductive constructions.  ...  We claim that the extended framework makes it possible to reason about terminating and non-terminating computations and we show that common facilities of the calculus of inductive construction, like program  ...  Acknowledgments Benjamin Werner and Hugo Herbelin played a significant role in understanding what form of the axioms of classical logic provide safe extensions of the calculus of constructions.  ... 
arXiv:cs/0610055v1 fatcat:bvuamb63wvhknndm4zgvuxk6au

Page 5756 of Mathematical Reviews Vol. , Issue 95j [page]

1995 Mathematical Reviews  
In this respect, it stands in contrast to other frameworks for program extraction, such as the polymorphic lambda-calculus (of J. Girard) or the calculus of constructions (of T. Coquand and G. Huet).  ...  (For the newcomer to the field of program extraction, we remark that the Curry-Howard correspondence does not work for classical first-order logic.)  ... 

Programming and Proving with Classical Types [chapter]

Cristina Matache, Victor B. F. Gomes, Dominic P. Mulligan
2017 Lecture Notes in Computer Science  
We present example closed µML programs with classical tautologies for types, including some inexpressible as closed programs in the original λµ-calculus, and some example tactic-driven µTP proofs of classical  ...  In this work, we use the λµ-calculus as the foundation for a system of proof terms for classical first-order logic.  ...  We thank the anonymous referees and Peter Sewell for their helpful comments.  ... 
doi:10.1007/978-3-319-71237-6_11 fatcat:2dz6qias4vbmpp3qggznclk3ee

Proofs-as-Programs in Computable Analysis

Ulrich Berger
2009 Electronic Communications of the EASST  
Since the work of Brouwer, Kolmogorov, Goedel, Kleene and many others we know that constructive proofs have computational meaning.  ...  In Computer Science this idea is known as the "proofs-as-programs paradigm" or "Curry-Howard correspondence".  ...  The interesting aspect of this example is that the program is extracted from the classical i.e. non-constructive proof by Nash-Williams [NW63] using a version of Friedman's A-translation [Fri78] that  ... 
doi:10.14279/tuj.eceasst.23.332 dblp:journals/eceasst/Berger09 fatcat:nihg2u2vobbmhgxto2k7vzq7wy

Efficient Translation of Sequent Calculus Proofs Into Natural Deduction Proofs

Gabriel Ebner, Matthias Schlaipfer
2018 Conference on Automated Deduction  
We present a simple and efficient translation of the classical multi-succedent sequent calculus LK to natural deduction.  ...  This translation aims to produce few excluded middle inferences, and to keep the quantifier complexity of the instances of excluded middle low.  ...  To extract programs from these proofs in LK, we want an effective translation to natural deduction in order to apply established program extraction methods.  ... 
dblp:conf/cade/EbnerS18 fatcat:j46vomsdhrgp5j267md73sjcyq

Page 1071 of Mathematical Reviews Vol. , Issue 95b [page]

1995 Mathematical Reviews  
The paper treats the extraction of programs out of a proof in some constructive type theory.  ...  The specification of a program can be represented by a logical formula and the program itself can be extracted from the constructive proof of the specification.  ... 

Extracting Herbrand trees from Coq [article]

Lionel Rieg
2013 arXiv   pre-print
Classical program extraction using Krivine's classical realizability then translates this proof into a certified program that computes Herbrand trees.  ...  Software certification aims at proving the correctness of programs but in many cases, the use of external libraries allows only a conditional proof: it depends on the assumption that the libraries meet  ...  the calculus of constructions with universes enriched with classical reasoning at the level of the sort Prop of propositions.  ... 
arXiv:1304.4557v1 fatcat:a2omrupl2zfcrbucwuutb22nou

Extracting information from intermediate semiconstructive HA-systems – extended abstract

MAURO FERRARI, CAMILLO FIORENTINI, PIERANGELO MIGLIOLI
2001 Mathematical Structures in Computer Science  
Here we will concentrate our attention on semiconstructive calculi, which is a kind of calculus that is of interest in the framework of program synthesis and formal verification.  ...  In this abstract we will describe research in progress on the problem of extracting information from proofs.  ...  ND HA ++ is a uniformly semiconstructive calculus in ND PA . Hence HA ++ = Theo(ND HA ++ ) is a uniformly semiconstructive HA-system.  ... 
doi:10.1017/s0960129501003358 fatcat:pqquh7r6lra2ngxsdhggbdrvka

Proof-search in type-theoretic languages: an introduction

Didier Galmiche, David J. Pym
2000 Theoretical Computer Science  
We introduce the main concepts and problems in the theory of proof-search in type-theoretic languages and survey some speciÿc, connected topics.  ...  We do not claim to cover all of the theoretical and implementation issues in the study of proof-search in type-theoretic languages; rather, we present some key ideas and problems, starting from well-motivated  ...  Second-order systems like the Calculus of Constructions (CC) [35, 127] or AF 2 [122] , issued from studies of the system F [64] , can be used for program synthesis, i.e., to extract programs from proofs  ... 
doi:10.1016/s0304-3975(99)00169-3 fatcat:gy4nmmsupfeype23pw3jzbrvwu

Towards the animation of proofs – testing proofs by examples

Susumu Hayashi, Ryosuke Sumitomo, Ken-ichiro Shii
2002 Theoretical Computer Science  
In this paper, we introduce the notion of proof animation, which is a new application of the principle of "Curry-Howard isomorphism" to formal proof development.  ...  The motivation, aims, problems, and a prototype tool under development are presented in this paper. We also discuss possibility of "proof engineering" guided by the Curry-Howard isomorphism.  ...  This paper is a sequel to the ÿrst author's previous article [11] , which drew a negative conclusion about the idea of proofs as programs as a methodology for formal methods.  ... 
doi:10.1016/s0304-3975(00)00350-9 fatcat:lhtwroijgza4xmjfsh5lyox7qi

Computational interpretations of classical reasoning: From the epsilon calculus to stateful programs [article]

Thomas Powell
2018 arXiv   pre-print
The problem of giving a computational meaning to classical reasoning lies at the heart of logic.  ...  This article surveys three famous solutions to this problem - the epsilon calculus, modified realizability and the dialectica interpretation - and re-examines them from a modern perspective, with a particular  ...  This point of view has been explored primarily in the French style of program extraction via e.g.  ... 
arXiv:1812.05851v1 fatcat:oujik7p6jbfghkh57axpfd6d2q

Page 5731 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
The author uses Kreisel’s modified realizability interpretation to extract higher type primitive recur- sive programs (in the sense of Gédel’s calculus 7) for the fast growing hierarchy up to €9.  ...  This paper provides an interesting case study in a program extrac- tion from a constructive proof.  ... 
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