Soundness and Completeness Proofs by Coinductive Methods

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation t...

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Bibliographic Details
Published in:Journal of automated reasoning Vol. 58; no. 1; pp. 149 - 179
Main Authors: Blanchette, Jasmin Christian, Popescu, Andrei, Traytel, Dmitriy
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.01.2017
Springer Nature B.V
Springer Verlag
Subjects:
Artificial Intelligence
Automated reasoning
Completeness
Computer Science
Derivation
Flavours
Logic
Logic in Computer Science
Mathematical Logic and Formal Languages
Mathematical Logic and Foundations
Nesting
Specification and description languages
Specifications
Symbolic and Algebraic Manipulation
Trees
Gentian systems
First-order logic
Isabelle/HOL
Soundness
Completeness
Codatatypes
Proof assistants
Lazy evaluation
ISSN:0168-7433, 1573-0670
Online Access:Get full text
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Summary:We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL’s recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL’s new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion.
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ISSN:0168-7433
1573-0670
DOI:10.1007/s10817-016-9391-3