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Practical Program Extraction from Classical Proofs

Yevgeniy Makarov
2006 Electronical Notes in Theoretical Computer Science  
Since C is similar to call/cc operator in Scheme and SML/NJ, this opens a possibility of extracting programs in these languages from classical proofs.  ...  We illustrate the usefulness of this extension with a complete example of program extraction.  ...  and y extracted from the proof.  ... 
doi:10.1016/j.entcs.2005.11.071 fatcat:hxkhify4efeitlkrjc5czgvhpu

Refined program extraction from classical proofs

Ulrich Berger, Wilfried Buchholz, Helmut Schwichtenberg
2002 Annals of Pure and Applied Logic  
In this paper we develop a re ned method of extracting reasonable and sometimes unexpected programs from classical proofs.  ...  This case study also demonstrates that the program extracted from the classical proof, albeit correct, may w ell be in need of further optimization.  ...  Now let A := kG(n k), and M X be obtained from M by replacing every occurrence of ? by X.  ... 
doi:10.1016/s0168-0072(01)00073-2 fatcat:muws4ifhffcvfgum6e7ydexklq

Getting results from programs extracted from classical proofs

C. Raffalli
2004 Theoretical Computer Science  
We present a new method to extract from a classical proof of ∀x(I [x] → ∃y(O[y] ∧ S[x; y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate.  ...  Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs and this is veriÿed on a nontrivial example: a proof of Dickson's lemma.  ...  The program had to be adjusted to do the check in the same order as the term extracted from the proof.  ... 
doi:10.1016/j.tcs.2004.03.006 fatcat:xq4tbhjxv5ah5jpyzuubufdocu

Light Dialectica Program Extraction from a Classical Fibonacci Proof

Mircea-Dan Hernest
2007 Electronical Notes in Theoretical Computer Science  
We demonstrate program extraction by the Light Dialectica Interpretation (LDI) on a minimal logic proof of the classical existence of Fibonacci numbers.  ...  Although syntactically different, it also has the same computational complexity as the original program yielded by the refined A-translation from the undistorted input classical Fibonacci proof.  ...  Introduction There has been quite some work in the last years in the field of program extraction from classical proofs.  ... 
doi:10.1016/j.entcs.2006.10.050 fatcat:ozqg6y3sbne4xc3xpycrd32iea

The greatest common divisor: A case study for program extraction from classical proofs [chapter]

U. Berger, H. Schwichtenberg
1996 Lecture Notes in Computer Science  
Here we treat that example as a case study for program extraction from classical proofs. We apply H.  ...  Friedman's Atranslation [3] followed by a modified realizability interpretation to extract a program from this proof.  ...  -There is an interesting phenomenon which may occur if we extract a program from a classical proof which uses the least element principle.  ... 
doi:10.1007/3-540-61780-9_60 fatcat:ohln4bijzrbzhndjx6gfg5vbaq

Proofs-as-Programs in Computable Analysis

Ulrich Berger
2009 Electronic Communications of the EASST  
In Computer Science this idea is known as the "proofs-as-programs paradigm" or "Curry-Howard correspondence".  ...  We present examples from computable analysis showing that this paradigm not only works in principle, but can be used to automatically synthesise practically relevant certified programs.  ...  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


Ulrich Berger, Monika Seisenberger
2005 From Sets and Types to Topology and Analysis  
We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles.  ...  Both approaches are explained by means of a case study on Higman's Lemma and its well-known classical proof due to Nash-Williams.  ...  The program extraction from Nash-Williams' classical proof via A-translation described in section 3 could be extended in a straightforward way to a corresponding classical proof of Kruskal's Theorem, even  ... 
doi:10.1093/acprof:oso/9780198566519.003.0008 fatcat:n4z453pujrenfgte7abt4tkjhi

Towards the animation of proofs – testing proofs by examples

Susumu Hayashi, Ryosuke Sumitomo, Ken-ichiro Shii
2002 Theoretical Computer Science  
Logically, proof animation is merely a contrapositive of "proofs as programs", which is an application of Curry-Howard isomorphism to formal program development.  ...  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.  ...  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

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.  ...  There are essentially two groups of methods of extracting programs from proofs.  ...  Thus the algorithm for extracting programs from proofs should be based on this notion.  ... 
doi:10.1007/3-540-52592-0_78 fatcat:wy7miwvy5jgmxdpzkw5mmhlcly

Typed realizability for first-order classical analysis

Valentin Blot, Peter Selinger
2015 Logical Methods in Computer Science  
Extraction of algorithms from proofs of Π02 formulas relies on a novel implementation of Friedman's trick exploiting the control possibilities of the language.  ...  This allows to have extracted programs with simpler types than in the case of negative translation followed by intuitionistic realizability.  ...  Concerning extraction of programs from proofs, Friedman's trick is implemented through the use of an external µ-variable rather than through the replacement of the ⊥ formula by an existential statement  ... 
doi:10.2168/lmcs-11(4:22)2015 fatcat:mpy2aq4dvbdyjm2jzmjpjt2vdm

A constructive valuation interpretation for classical logic and its use in witness extraction [chapter]

Franco Barbanera, Stefano Berardi
1992 Lecture Notes in Computer Science  
We use this interpretation for motivating a constructive procedure, based on reduction rules, for extracting witnesses from proofs of ~ sentences in classical arithmetic.  ...  A constructive interpretation of classical arithmetic in terms of a Kripke-like valuation semantics of proofs is presented.  ...  ., to the formulas in classical proofs expanding any proof p of "if A 1 .... A n then A" to a proof of "if vl~t.A1,..,vn~t.An (Vl,..,Vn force A1 .... An) then 5".nlvj~t.A (the union of Vl,..  ... 
doi:10.1007/3-540-55251-0_1 fatcat:ik225pjisbhgthrfgtivnapyz4

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  ...  Notice that this transformation is intuitionistic so that any use of classical logic in the extracted realizer comes either from the proof of contradiction of the universal theory U or from the proof of  ... 
arXiv:1304.4557v1 fatcat:a2omrupl2zfcrbucwuutb22nou

PX: A computational logic

R.L. Constable
1991 Science of Computer Programming  
The principal purpose of PX is the extraction of programs from constructive proofs of their specifications.  ...  It formalizes a simple programming language similar to pure Lisp, and implements the idea that proofs are programs in the sense that it allows the mechanical extraction of programs from proofs of their  ... 
doi:10.1016/0167-6423(91)90043-w fatcat:ltyu3phw6ndthdjcjcukss6yei

A computational interpretation of open induction

U. Berger
2004 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.  
Via modified realizability we obtain a new direct method for extracting programs from classical proofs of ¡ £ ¢ ¤ -formulas using open induction.  ...  It is a classical reformulation of Nash-Williams minimal-bad-sequence argument which is used in classical proofs of Kruskal's theorem and related theorems [18, 6].  ...  By program extraction in general we mean the process of extracting from a proof of § !  ... 
doi:10.1109/lics.2004.1319627 dblp:conf/lics/Berger04 fatcat:aihxty33a5evzetc57jlzwbiz4

Implementing program extraction from CL1-proofs [article]

Meixia Qu, Ke Chen, Daming Zhu, Junfeng Luan
2013 arXiv   pre-print
In this paper, we present a program that takes a CL1-proof of an arbitrary formula F, and extract a winning strategy for F from that proof then play F using that strategy.  ...  We hope this paper would provide a starting point for further work in program extraction of the CoL-based arithmetic and other CoL-based applied systems.  ...  In this paper, we propose a program extraction from CL1-proofs which can be implemented in polynomial time.  ... 
arXiv:1311.6542v1 fatcat:7b3dlped3zaozgwj54lfpbc6wq
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