Refined program extraction from classical proofs

Berger, Ulrich; Buchholz, Wilfried; Schwichtenberg, Helmut

doi:10.1016/s0168-0072(01)00073-2

Cited by 65 publications

(74 citation statements)

References 11 publications

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“…Note that this behavior is the same as the one we observe when treating the same example using Friedman's method or its refinements [3]. Of course, this similarity is not a coincidence since Friedman's translation is actually hard-wired in Krivine's semantics (as already pointed out in [25]), and we shall come back to this point with more details in sections 7 and 8.…”

supporting

confidence: 76%

Existential witness extraction in classical realizability and via a negative translation

Miquel¹

2011

Logical Methods in Computer Science

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Abstract. We show how to extract existential witnesses from classical proofs using Krivine's classical realizability-where classical proofs are interpreted as λ-terms with the call/cc control operator. We first recall the basic framework of classical realizability (in classical second-order arithmetic) and show how to extend it with primitive numerals for faster computations. Then we show how to perform witness extraction in this framework, by discussing several techniques depending on the shape of the existential formula. In particular, we show that in the Σ 0 1 -case, Krivine's witness extraction method reduces to Friedman's through a well-suited negative translation to intuitionistic second-order arithmetic. Finally we discuss the advantages of using call/cc rather than a negative translation, especially from the point of view of an implementation.

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confidence: 76%

Existential witness extraction in classical realizability and via a negative translation

Miquel¹

2011

Logical Methods in Computer Science

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“…From a type-theoretic point of view, realizability collapses a dependently typed lambda-calculus (which Minlog's proof calculus is an instance of) to a simply typed lambda-calculus. The collapse happens on the level of atomic formulas, since τ (P t) = τ (P ) where τ (P ) is a simple type assigned to the predicate P , discarding the first-order terms t. Minlog extends program extraction to classical proofs via a refined A-translation [BBS02] or Dialectica Interpretation (see eg. [SW11]).…”

Section: Program Extractionmentioning

confidence: 99%

Minlog - A Tool for Program Extraction Supporting Algebras and Coalgebras

Berger

Miyamoto

Schwichtenberg

et al. 2011

Algebra and Coalgebra in Computer Science

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Abstract. Minlog is an interactive system which implements prooftheoretic methods and applies them to verification and program extraction. We give an overview of Minlog and demonstrate how it can be used to exploit the computational content in (co)algebraic proofs and to develop correct and efficient programs. We illustrate this by means of two examples: one about parsing, the other about exact real numbers in signed digit representation.

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“…It takes into account that most applications of functional interpretation in recent years both to concrete proofs in numerical analysis and to conservation results do not actually use terms which realize the Gödel functional interpretation but terms which majorize 6 (some) realizers. Monotone functional interpretation extracts majorizing terms which are simpler than the actual realizers produced by functional interpretation.…”

Section: Introductionmentioning

confidence: 99%

A Complexity Analysis of Functional Interpretations

Hernest¹,

Kohlenbach²

2003

BRICS

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We give a quantitative analysis of Gödel's functional interpretation and its monotone variant. The two have been used for the extraction of programs and numerical bounds as well as for conservation results. They apply both to (semi-)intuitionistic as well as (combined with negative translation) classical proofs. The proofs may be formalized in systems ranging from weak base systems to arithmetic and analysis (and numerous fragments of these). We give upper bounds in basic proof data on the depth, size, maximal type degree and maximal type arity of the extracted terms as well as on the depth of the verifying proof. In all cases terms of size linear in the size of the proof at input can be extracted and the corresponding extraction algorithms have cubic worst-time complexity. The verifying proofs have depth linear in the depth of the proof at input and the maximal size of a formula of this proof.

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Refined program extraction from classical proofs

Cited by 65 publications

References 11 publications

Existential witness extraction in classical realizability and via a negative translation

Existential witness extraction in classical realizability and via a negative translation

Minlog - A Tool for Program Extraction Supporting Algebras and Coalgebras

A Complexity Analysis of Functional Interpretations

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