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23
Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 219 (28 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
Complete sequent calculi for induction and infinite descent
 Proceedings of LICS22
, 2007
"... This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing induct ..."
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Cited by 25 (8 self)
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This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cutfree complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (nonwellfounded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple casesplit rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cutfree complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
The Inverse Method
, 2001
"... this paper every formula is equivalent to a formula in negation normal form ..."
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Cited by 13 (1 self)
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this paper every formula is equivalent to a formula in negation normal form
Predicate logic with sequence variables and sequence function symbols
 Proc. of the 3rd Int. Conference on Mathematical Knowledge Management. Vol. 3119 of LNCS
, 2004
"... Abstract. We describe an extension of firstorder logic with sequence variables and sequence functions. We define syntax, semantics and inference system for the extension so that Completeness, Compactness, LöwenheimSkolem, and Model Existence theorems remain valid. The obtained logic can be encoded ..."
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Cited by 11 (6 self)
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Abstract. We describe an extension of firstorder logic with sequence variables and sequence functions. We define syntax, semantics and inference system for the extension so that Completeness, Compactness, LöwenheimSkolem, and Model Existence theorems remain valid. The obtained logic can be encoded as a special ordersorted firstorder theory. We also define an inductive theory with sequence variables and formulate induction rules. The calculus forms a basis for the topdown systematic theory exploration paradigm. 1
A systematic presentation of quantified modal logics
 University of Edinburgh
, 2002
"... this paper is an attempt at providing a systematic presentation of Quantified Modal Logics (with constant domains and rigid designators). We present a set of modular, uniform, normalizing, sound and complete labelled sequent calculi for all QMLs whose frame properties can be expressed as a finite se ..."
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Cited by 5 (3 self)
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this paper is an attempt at providing a systematic presentation of Quantified Modal Logics (with constant domains and rigid designators). We present a set of modular, uniform, normalizing, sound and complete labelled sequent calculi for all QMLs whose frame properties can be expressed as a finite set of firstorder sentences with equality. We first present CQK, a calculus for the logic QK, and then we extend it to any such logic QL. Each calculus, called CQL, is modular (obtained by adding rules to CQK), uniform (each added rule is clearly related to a property of the frame), normalizing (frame reasoning only happens at the top of the proof tree) and Kripkesound and complete for QL. We improve on the existing literature on the subject (mainly, [21]) by extending the class of logics for which such a presentation is given, and by giving a new proof of soundness and completeness.
Smart matching
"... One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or ..."
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Cited by 4 (4 self)
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One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or isomorphisms), are usually implicit in the mathematical discourse, and it would be highly desirable to obtain a similar behaviour in interactive provers. The paper describes the superpositionbased implementation of this feature inside the Matita interactive theorem prover, focusing in particular on the so called smart application tactic, supporting smart matching between a goal and a given result.
Herbrand's Theorem And Equational Reasoning: Problems And Solutions
 In Bulletin of the European Association for Theoretical Computer Science
, 1996
"... this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the socalled simultaneous rigid Euni cation problem. In the second part, we survey recent result on th ..."
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Cited by 4 (2 self)
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this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the socalled simultaneous rigid Euni cation problem. In the second part, we survey recent result on the simultaneous rigid Euni cation problem
Automating Type Soundness Proofs via Decision Procedures and Guided Reductions
 In 9th International Conference on Logic for Programming Artificial Intelligence and Reasoning, volume 2514 of LNCS
, 2002
"... Operational models of fragments of the Java Virtual Machine and the .NET Common Language Runtime have been the focus of considerable study in recent years, and of particular interest have been specifications and machinechecked proofs of type soundness. In this paper we aim to increase the level of ..."
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Cited by 4 (0 self)
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Operational models of fragments of the Java Virtual Machine and the .NET Common Language Runtime have been the focus of considerable study in recent years, and of particular interest have been specifications and machinechecked proofs of type soundness. In this paper we aim to increase the level of automation used when checking type soundness for these formalizations. We present a semiautomated technique for reducing a range of type soundness problems to a form that can be automatically checked using a decidable firstorder theory. Deciding problems within this fragment is exponential in theory but is often efficient in practice, and the time required for proof checking can be controlled by further hints from the user. We have applied this technique to two case studies, both of which are type soundness properties for subsets of the .NET CLR. These case studies have in turn aided us in our informal analysis of that system.
Lean Induction Principles for Tableaux
, 1997
"... . In this paper, we deal with various induction principles incorporated in an underlying tableau calculus with equality. The induction formulae are restricted to literals. Induction is formalized as modified closure conditions which are triggered by applications of the ffirule. Examples dealing wit ..."
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Cited by 3 (1 self)
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. In this paper, we deal with various induction principles incorporated in an underlying tableau calculus with equality. The induction formulae are restricted to literals. Induction is formalized as modified closure conditions which are triggered by applications of the ffirule. Examples dealing with (weak forms of) arithmetic and strings illustrate the simplicity and usability of our induction handling. We prove the correctness of the closure conditions and discuss possibilities to strengthen the induction principles. 1 Introduction The use of induction principles in various forms certainly is an important and prominent topic in Automated Deduction, as witnessed, e.g., by [15, 6, 1] and quite recently in [17]. The complexity of the problem  both, in terms of proof search and formulation of appropriate deduction mechanisms  is well known. Our aim is to demonstrate that various forms of induction can be incorporated into classical free variable tableaux in an elegant way. We were...