相关论文: Analysis and Extension of Omega-Rule
Buchholz' Omega-rule is a way to give a syntactic, possibly ordinal-free proof of cut elimination for various subsystems of second order arithmetic. Our goal is to understand it from an algebraic point of view. Among many proofs of cut…
In a functional calculus, the so called \Omega-rule states that if two terms P and Q applied to any closed term <i>N</i> return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the…
We describe a method for inverting Gentzen's cut-elimination in classical first-order logic. Our algorithm is based on first computign a compressed representation of the terms present in the cut-free proof and then cut-formulas that realize…
Cut-elimination is the bedrock of proof theory with a multitude of applications from computational interpretations to proof analysis. It is also the starting point for important meta-theoretical investigations including decidability,…
In recent years, G\"odel's ontological proof and variations of it were formalized and analyzed with automated tools in various ways. We supplement these analyses with a modeling in an automated environment based on first-order logic…
In this paper we propose a semantics in which the truth value of a formula is a pair of elements in a complete Boolean algebra. Through the semantics we can unify largely two proofs of cut-eliminability (Hauptsatz) in classical second order…
Cut-elimination is the bedrock of proof theory. It is the algorithm that eliminates cuts from a sequent calculus proof that leads to cut-free calculi and applications. Cut-elimination applies to many logics irrespective of their semantics.…
This paper presents a cut-elimination proof for the logic $LG^\omega$, which is an extension of a proof system for encoding generic judgments, the logic $\FOLDNb$ of Miller and Tiu, with an induction principle. The logic $LG^\omega$, just…
Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $\mathbf{LK}$, several other proofs have been proposed. Even though the…
We consider a certain class of infinitary rules of inference, called here restriction rules, using of which allows us to deduce complete theories of given models. The first instance of such rules was the $\omega$-rule introduced by Hilbert,…
In previous work, an attempt was made to apply the schematic CERES method [8] to a formal proof with an arbitrary number of {\Pi} 2 cuts (a recursive proof encapsulating the infinitary pigeonhole principle) [5]. However the derived…
Several theorems about the equivalence of familiar theories of reverse mathematics with certain well-ordering principles have been proved by recursion-theoretic and combinatorial methods (Friedman, Marcone, Montalban et al.) and with…
In this paper we present a constructive proof of cut elimination for a system of full second order logic with the structural rules absorbed and using sets instead of sequences. The standard problem of the cutrank growth is avoided by using…
For relational monadic formulas (the L\"owenheim class) second-order quantifier elimination, which is closely related to computation of uniform interpolants, projection and forgetting - operations that currently receive much attention in…
The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable…
Shoenfield's completeness theorem (1959) states that every true first order arithmetical sentence has a recursive $\omega$-proof encodable by using recursive applications of the $\omega$-rule. For a suitable encoding of Gentzen style…
The first-order theory of finite and infinite trees has been studied since the eighties, especially by the logic programming community. Following Djelloul, Dao and Fr\"uhwirth, we consider an extension of this theory with an additional…
Recently data trees and data words have received considerable amount of attention in connection with XML reasoning and system verification. These are trees or words that, in addition to labels from a finite alphabet, carry data values from…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…
We present a combination of raising, explicit variable dependency representation, the liberalized delta-rule, and preservation of solutions for first-order deductive theorem proving. Our main motivation is to provide the foundation for our…