Proof Theory

Generalizing proofs in monadic languages

We introduce a first order extension of GL, called ML 3 , and develop its proof theory via a proxy cut-free sequent calculus GLTS. We prove the highly nontrivial result that cut is a derived rule in GLTS, a result that is unavailable in other known first-order extensions of GL. This leads to proofs of weak reflection and the related conservation result for ML 3 , as well as proofs for Craig’s interpolation theorem for GLTS. Turning to semantics we prove that ML 3 is sound with respect to arithmetical interpretations and that it is also sound and complete with respect to converse well-founded and transitive finite Kripke models. This leads us to expect that a Solovay-like proof of arithmetical completeness of ML 3 is possible.

/pdf/on-the-proof-theory-of-a-first-order-extension-of-gl-4zvscv784t.pdf

On the proof-theory of a first-order extension of GL

http://ceur-ws.org/Vol-1287/lanmr2014_paper_3.pdf

How Many Times do We Need an Assumption to Prove a Tautology in Minimal Logic? Examples on the Compression Power of Classical Reasoning

Proof systems for propositional logic called simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule, are introduced. Upper bounds are given on the lengths of proofs in these systems compared to lengths in Frege proof systems. As an application, a near-linear simulation of the propositional Gentzen sequent calculus by Frege proofs is presented. It is shown that a general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege systems where by 'nearly linear' is meant that the ratio of proof lengths is O( alpha (n)), where alpha is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, and hence a Frege proof system can simulate the propositional sequent calculus with proof lengths bounded by O(n alpha (n)). As a technical tool, the serial transitive closure problem is introduced. Given a directed graph and a list of closure edges in the transitive closure of the graph, the problem is to derive all the closure edges. A nearly linear bound is given on the number of steps in such a derivation when the graph is treelike. >

On the deduction rule and the number of proof lines

1 Tobias Nipkow revised and extended the chapter on HOL. Markus Wenzel made numerous improvements. Philippe de Groote wrote the first version of the logic LK and contributed to ZF. Tobias Nipkow developed HOL, LCF and Cube. Philippe Noël and Martin Coen made many contributions to ZF. Martin Coen developed Modal with assistance from Rajeev Goré.

/pdf/isabelle-s-object-logics-tw2z650urq.pdf

Proof Theory

Citations

Generalizing proofs in monadic languages

On the proof-theory of a first-order extension of GL

How Many Times do We Need an Assumption to Prove a Tautology in Minimal Logic? Examples on the Compression Power of Classical Reasoning

On the deduction rule and the number of proof lines

Isabelle’s Object-Logics

Related Papers (5)

Untersuchungen über das logische Schließen. II

Introduction to Metamathematics

The collected papers of Gerhard Gentzen

Basic proof theory

Linear logic