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Practical Program Extraction from Classical Proofs

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

doi:10.1016/j.entcs.2005.11.071
fatcat:hxkhify4efeitlkrjc5czgvhpu
*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*. ...##
###
Refined program extraction from classical proofs

2002
*
Annals of Pure and Applied Logic
*

In this paper we develop a re ned method of

doi:10.1016/s0168-0072(01)00073-2
fatcat:muws4ifhffcvfgum6e7ydexklq
*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. ...##
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Getting results from programs extracted from classical proofs

2004
*
Theoretical Computer Science
*

We present a new method to

doi:10.1016/j.tcs.2004.03.006
fatcat:xq4tbhjxv5ah5jpyzuubufdocu
*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*. ...##
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Light Dialectica Program Extraction from a Classical Fibonacci Proof

2007
*
Electronical Notes in Theoretical Computer Science
*

We demonstrate

doi:10.1016/j.entcs.2006.10.050
fatcat:ozqg6y3sbne4xc3xpycrd32iea
*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*. ...##
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The greatest common divisor: A case study for program extraction from classical proofs
[chapter]

1996
*
Lecture Notes in Computer Science
*

Here we treat that example as a case study for

doi:10.1007/3-540-61780-9_60
fatcat:ohln4bijzrbzhndjx6gfg5vbaq
*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. ...##
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Proofs-as-Programs in Computable Analysis

2009
*
Electronic Communications of the EASST
*

In Computer Science this idea is known as the "

doi:10.14279/tuj.eceasst.23.332
dblp:journals/eceasst/Berger09
fatcat:nihg2u2vobbmhgxto2k7vzq7wy
*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 ...##
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APPLICATIONS OF INDUCTIVE DEFINITIONS AND CHOICE PRINCIPLES TO PROGRAM SYNTHESIS
[chapter]

2005
*
From Sets and Types to Topology and Analysis
*

We describe two methods of

doi:10.1093/acprof:oso/9780198566519.003.0008
fatcat:n4z453pujrenfgte7abt4tkjhi
*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 ...##
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Towards the animation of proofs – testing proofs by examples

2002
*
Theoretical Computer Science
*

Logically,

doi:10.1016/s0304-3975(00)00350-9
fatcat:lhtwroijgza4xmjfsh5lyox7qi
*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. ...##
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Towards the theory of programming in constructive logic
[chapter]

1990
*
Lecture Notes in Computer Science
*

We develop an approach to the theory of

doi:10.1007/3-540-52592-0_78
fatcat:wy7miwvy5jgmxdpzkw5mmhlcly
*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. ...##
###
Typed realizability for first-order classical analysis

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 ...

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

1992
*
Lecture Notes in Computer Science
*

We use this interpretation for motivating a constructive procedure, based on reduction rules, for

doi:10.1007/3-540-55251-0_1
fatcat:ik225pjisbhgthrfgtivnapyz4
*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,.. ...##
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Extracting Herbrand trees from Coq
[article]

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 ...

##
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PX: A computational logic

1991
*
Science of Computer Programming
*

The principal purpose of PX is the

doi:10.1016/0167-6423(91)90043-w
fatcat:ltyu3phw6ndthdjcjcukss6yei
*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 ...##
###
A computational interpretation of open induction

2004
*
Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.
*

Via modified realizability we obtain a new direct method for

doi:10.1109/lics.2004.1319627
dblp:conf/lics/Berger04
fatcat:aihxty33a5evzetc57jlzwbiz4
*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 § ! ...##
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Implementing program extraction from CL1-proofs
[article]

2013
*
arXiv
*
pre-print

In this paper, we present a

arXiv:1311.6542v1
fatcat:7b3dlped3zaozgwj54lfpbc6wq
*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. ...
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