Formalizing a lazy substitution proof system for \u3bc-calculus in the Calculus of Inductive Constructions

We present a Natural Deduction proof system for the pro- positional modal \u3bc-calculus, and its formalization in the Calculus of In- ductive Constructions. We address several problematic issues, such as the use of higher-order abstract syntax in inductive sets in presence of recursive constructors, the encoding of modal (sequent-style) rules and of context sensitive grammars. The formalization can be used in the sy- stem Coq, providing an experimental computer-aided proof environment for the interactive development of error-free proofs in the \u3bc-calculus. The techniques we adopt can be readily ported to other languages and proof systems featuring similar problematic issues. \ua9 Springer-Verlag Berlin Heidelberg 1999

Ambient Calculus and its Logic in the Calculus of Inductive Constructions

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Formalizing alternating-time temporal logic in the coq proof assistant

This work presents a complete formalization of Alternating-time Temporal Logic (ATL) and its semantic model, Concurrent Game Structures (CGS), in the Calculus of (Co)Inductive Constructions, using the logical framework Coq. Unlike standard ATL semantics, temporal operators are formalized in terms of inductive and coinductive types, employing a fixpoint characterization of these operators. The formalization is used to model a concurrent system with an unbounded number of players and states, and to verify some properties expressed as ATL formulas. Unlike automatic techniques, our formal model has no restrictions in the size of the CGS, and arbitrary state predicates can be used as atomic propositions of ATL. Keywords: Reactive Systems and Open Systems, Alternating-time Temporal Logic, Concurrent Game Structures, Calculus of (Co)Inductive Constructions, Coq Proof Assistant

A machine-checked constructive metatheory of computation tree logic

This thesis presents a machine-checked constructive metatheory of computation tree logic (CTL) and its sublogics K and K* based on results from the literature. We consider models, Hilbert systems, and history-based Gentzen systems and show that for every logic and every formula s the following statements are decidable and equivalent: s is true in all models, s is provable in the Hilbert system, and s is provable in the Gentzen system. We base our proofs on pruning systems constructing finite models for satisfiable formulas and abstract refutations for unsatisfiable formulas. The pruning systems are devised such that abstract refutations can be translated to derivations in the Hilbert system and the Gentzen system, thus establishing completeness of both systems with a single model construction. All results of this thesis are formalized and machine-checked with the Coq interactive theorem prover. Given the level of detail involved and the informal presentation in much of the original work, the gap between the original paper proofs and constructive machine-checkable proofs is considerable. The mathematical proofs presented in this thesis provide for elegant formalizations and often differ significantly from the proofs in the literature.Diese Dissertation beschreibt eine maschinell verifizierte konstruktive Metatheorie von computation tree logic (CTL) und deren Teillogiken K und K*. Wir betrachten Modelle, Hilbert-Kalküle und History-basierte Gentzen-Kalküle und zeigen, für jede betrachtete Logik und jede Formel s, Entscheidbarkeit und Äquivalenz der folgenden Aussagen: s gilt in allen Modellen, s ist im Hilbert-Kalkül ableitbar und s ist im Gentzen-Kalkül ableitbar. Die Beweise bauen auf Pruningsystemen auf, welche für erfüllbare Formeln endliche Modelle und für unerfüllbare Formeln abstrakte Widerlegungen konstruieren. Die Pruningsysteme sind so konstruiert, dass abstrakte Widerlegungen zu Widerlegungen sowohl im Hilbert- als auch im Gentzen-Kalkül übersetzt werden können. Dadurch wird es möglich, die Vollständigkeit beider Systeme mit nur einer Modellkonstruktion zu zeigen. Alle Ergebnisse dieser Dissertation sind formalisiert und maschinell verifiziert mit Hilfe des Beweisassistenten Coq. In Anbetracht der Fülle an Details und der informellen Beweisführung in großen Teilen der Originalliteratur, erfordert dies teilweise tiefgreifende Veränderungen an den Beweisen aus der Literatur. Die Beweise in der vorliegenden Arbeit sind so aufgebaut, dass sie zu eleganten Formalisierungen führen

A machine-checked constructive metatheory of computation tree logic

This thesis presents a machine-checked constructive metatheory of computation tree logic (CTL) and its sublogics K and K* based on results from the literature. We consider models, Hilbert systems, and history-based Gentzen systems and show that for every logic and every formula s the following statements are decidable and equivalent: s is true in all models, s is provable in the Hilbert system, and s is provable in the Gentzen system. We base our proofs on pruning systems constructing finite models for satisfiable formulas and abstract refutations for unsatisfiable formulas. The pruning systems are devised such that abstract refutations can be translated to derivations in the Hilbert system and the Gentzen system, thus establishing completeness of both systems with a single model construction. All results of this thesis are formalized and machine-checked with the Coq interactive theorem prover. Given the level of detail involved and the informal presentation in much of the original work, the gap between the original paper proofs and constructive machine-checkable proofs is considerable. The mathematical proofs presented in this thesis provide for elegant formalizations and often differ significantly from the proofs in the literature.Diese Dissertation beschreibt eine maschinell verifizierte konstruktive Metatheorie von computation tree logic (CTL) und deren Teillogiken K und K*. Wir betrachten Modelle, Hilbert-Kalküle und History-basierte Gentzen-Kalküle und zeigen, für jede betrachtete Logik und jede Formel s, Entscheidbarkeit und Äquivalenz der folgenden Aussagen: s gilt in allen Modellen, s ist im Hilbert-Kalkül ableitbar und s ist im Gentzen-Kalkül ableitbar. Die Beweise bauen auf Pruningsystemen auf, welche für erfüllbare Formeln endliche Modelle und für unerfüllbare Formeln abstrakte Widerlegungen konstruieren. Die Pruningsysteme sind so konstruiert, dass abstrakte Widerlegungen zu Widerlegungen sowohl im Hilbert- als auch im Gentzen-Kalkül übersetzt werden können. Dadurch wird es möglich, die Vollständigkeit beider Systeme mit nur einer Modellkonstruktion zu zeigen. Alle Ergebnisse dieser Dissertation sind formalisiert und maschinell verifiziert mit Hilfe des Beweisassistenten Coq. In Anbetracht der Fülle an Details und der informellen Beweisführung in großen Teilen der Originalliteratur, erfordert dies teilweise tiefgreifende Veränderungen an den Beweisen aus der Literatur. Die Beweise in der vorliegenden Arbeit sind so aufgebaut, dass sie zu eleganten Formalisierungen führen

A Galois connection between classical and intuitionistic logics. I: Syntax

In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types - propositions and problems." We construct such a formal system QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The only new connectives ? and ! of QHC induce a Galois connection (i.e., a pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation translation of propositions into problems extends to a retraction of QHC onto QH; whereas Goedel's provability translation of problems into modal propositions extends to a retraction of QHC onto its QC+(?!) fragment, identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic modal logic, whose modality !? is a strict lax modality in the sense of Aczel - and thus resembles the squash/bracket operation in intuitionistic type theories. The axioms of QHC attempt to give a fuller formalization (with respect to the axioms of intuitionistic logic) to the two best known contentual interpretations of intiuitionistic logic: Kolmogorov's problem interpretation (incorporating standard refinements by Heyting and Kreisel) and the proof interpretation by Orlov and Heyting (as clarified by G\"odel). While these two interpretations are often conflated, from the viewpoint of the axioms of QHC neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a simplified version of Isabelle's meta-logic