Intuitionistic fixed point logic

Berger, Ulrich; Tsuiki, Hideki

doi:10.1016/j.apal.2020.102903

Cited by 18 publications

(103 citation statements)

References 66 publications

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“…Note that although the logic is based on intuitionistic logic a fair amount of classical logic is available. For example, any disjunction-free formula that is classically true may be admitted as an axiom (Berger and Tsuiki, 2021).…”

Section: Extracting Digital Trees From Co-inductive Proofsmentioning

confidence: 99%

Computing with continuous objects: a uniform co-inductive approach

Spreen¹

2021

Math. Struct. Comp. Sci.

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A uniform approach to computing with infinite objects like real numbers, tuples of these, compacts sets and uniformly continuous maps is presented. In the work of Berger, it was shown how to extract certified algorithms working with the signed digit representation from constructive proofs. Berger and the present author generalised this approach to complete metric spaces and showed how to deal with compact sets. Here, we unify this work and lay the foundations for doing a similar thing for the much more comprehensive class of compact Hausdorff spaces occurring in applications. The approach is of the same computational power as Weihrauch’s Type-Two Theory of Effectivity.

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Section: Extracting Digital Trees From Co-inductive Proofsmentioning

confidence: 99%

Computing with continuous objects: a uniform co-inductive approach

Spreen¹

2021

Math. Struct. Comp. Sci.

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“…PRAWF: An Interactive Proof System for Program Extraction [22] (by U. Berger, O. Petrovska, H. Tsuiki) This paper presents an interactive proof system dedicated to program extraction from proofs. The implementation uses an updated version of IFP presented in [24]. This paper gives an overview of the prototype implementation and explains its use through several case studies.…”

Section: Optimized Program Extraction For Induction and Coinductionmentioning

confidence: 99%

“…This enables extraction of programs from proofs avoid-ing the admissibility condition, allowing proofs that use unrestricted strictly positive inductive and coinductive definitions. This improvement was introduced [24], where the corresponding soundness proof was outlined. The present thesis includes a proof of this Soundness Theorem worked out in detail.…”

Section: Optimized Program Extraction For Induction and Coinductionmentioning

confidence: 99%

“…However, to be completely confident that no cases were omitted, the aim was to find an alternative approach that would allow lifting this restriction. Fortunately, whilst working on [24], Hideki Tsuiki discovered the workaround through introduction of the intermediary system IFP'. This system is an exact copy of the original IFP with the (co)induction rules redefined.…”

Section: Main Contributionsmentioning

confidence: 99%

“…Taking this into account, the Soundness Theorem has been redefined and instead of IFP it uses the link between IFP' and RIFP. The initial soundness proof was drafted by the authors in [24], although it was not worked out in detail. After the paper was released to the public, I verified that the Soundness Theorem in this new formulation works.…”

Section: Main Contributionsmentioning

confidence: 99%

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Enhanced Realizability Interpretation for Program Extraction

Petrovska¹

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This thesis presents Intuitionistic Fixed Point Logic (IFP), a schema for formal systems aimed to work with program extraction from proofs. IFP in its basic form allows proof construction based on natural deduction inference rules, extended by induction and coinduction. The corresponding system RIFP (IFP with realiz-ers) enables transforming logical proofs into programs utilizing the enhanced re-alizability interpretation. The theoretical research is put into practice in PRAWF1, a Haskell-based proof assistant for program extraction.

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Extracting total Amb programs from proofs

Berger

Tsuiki

2022

Lecture Notes in Computer Science

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We present a logical system CFP (Concurrent Fixed Point Logic) that supports the extraction of nondeterministic and concurrent programs that are provably total and correct. CFP is an intuitionistic first-order logic with inductive and coinductive definitions extended by two propositional operators, $$B|_{A}$$ B | A (restriction, a strengthening of implication) and $${\mathbf {\downdownarrows }}(B)$$ ⇊ ( B ) (total concurrency). The source of the extraction are formal CFP proofs, the target is a lambda calculus with constructors and recursion extended by a constructor Amb (for McCarthy’s amb) which is interpreted operationally as globally angelic choice and is used to implement nondeterminism and concurrency. The correctness of extracted programs is proven via an intermediate domain-theoretic denotational semantics. We demonstrate the usefulness of our system by extracting a nondeterministic program that translates infinite Gray code into the signed digit representation. A noteworthy feature of our system is that the proof rules for restriction and concurrency involve variants of the classical law of excluded middle that would not be interpretable computationally without Amb.

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Intuitionistic fixed point logic

Cited by 18 publications

References 66 publications

Computing with continuous objects: a uniform co-inductive approach

Computing with continuous objects: a uniform co-inductive approach

Enhanced Realizability Interpretation for Program Extraction

Extracting total Amb programs from proofs

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