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136
Structured Calculational Proof
, 1996
"... We propose a new format for writing proofs, which we call structured calculational proof. The format is similar to the calculational style of proof already familiar to many computer scientists, but extends it by allowing large proofs to be hierarchically decomposed into smaller ones. In fact, struc ..."
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Cited by 16 (9 self)
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We propose a new format for writing proofs, which we call structured calculational proof. The format is similar to the calculational style of proof already familiar to many computer scientists, but extends it by allowing large proofs to be hierarchically decomposed into smaller ones. In fact, structured calculational proof can be seen as an alternative presentation of natural deduction. Natural deduction is a well established style of reasoning which uses hierarchical decomposition to great effect, but which is traditionally expressed in a notation that is inconvenient for writing calculational proofs. The hierarchical nature of structured calculational proofs can be used for proof browsing. We comment on how browsing can increase the value of a proof, and discuss the possibilities offered by electronic publishing for the presentation and dissemination of papers containing browsable proofs. Note: This paper is also available as Australian National University Joint Computer Science Tec...
2011): Nominal terms and nominal logics: from foundations to metamathematics
 In: Handbook of Philosophical Logic
"... ABSTRACT: Nominal techniques concern the study of names using mathematical semantics. Whereas in much previous work names in abstract syntax were studied, here we will study them in metamathematics. More specifically, we survey the application of nominal techniques to languages for unification, rew ..."
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Cited by 15 (9 self)
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ABSTRACT: Nominal techniques concern the study of names using mathematical semantics. Whereas in much previous work names in abstract syntax were studied, here we will study them in metamathematics. More specifically, we survey the application of nominal techniques to languages for unification, rewriting, algebra, and firstorder logic. What characterises the languages of this chapter is that they are firstorder in character, and yet they can specify and reason on names. In the languages we develop, it will be fairly straightforward to give firstorder ‘nominal ’ axiomatisations of namerelated things like alphaequivalence, captureavoiding substitution, beta and etaequivalence, firstorder logic with its quantifiers—and as we shall see, also arithmetic. The formal axiomatisations we arrive at will closely resemble ‘natural behaviour’; the specifications we see typically written out in normal mathematical usage. This is possible because of a novel namecarrying semantics in nominal sets, through which our languages will have namepermutations and termformers that can bind as primitive builtin features.
Are Tableaux an Improvement on TruthTables? CutFree proofs and Bivalence
, 1992
"... We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance bet ..."
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Cited by 15 (0 self)
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We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cutfree proofs and the Principle of Bivalence. Finally we discuss some ways in which this principle can be built into a tableaulike method without affecting its "analytic" nature. 1 Introduction The truthtable method, introduced by Wittgenstein in his Tractatus LogicoPhilosophicus, provides a decision procedure for propositional logic which is immediately implementable on a machine. However this timehonoured method is usually mentioned only to be immediately dismissed because of its incurable inefficiency. The wellknown tableau method (which is closely related to Gentzen's cutfree sequent calculus) is commonly regarded as a "shortcut" in testing the logical validity of complex propositions...
Semantic cut elimination in the intuitionistic sequent calculus
 Typed Lambda Calculi and Applications, number 3461 in Lectures
, 2005
"... Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to ..."
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Cited by 15 (10 self)
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Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to extend the cut elimination result to other intuitionistic deduction systems, in particular to deduction modulo provided the rewrite system verifies some properties. We also give an example of rewrite system for which cut elimination holds but that doesn’t enjoys proof normalization.
A Brief History of Natural Deduction
 HISTORY AND PHILOSOPHY OF LOGIC
, 1999
"... Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Jaskowski in 1934. This article traces the development of natural dedu ..."
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Cited by 14 (0 self)
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Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Jaskowski in 1934. This article traces the development of natural deduction from the view that these founders embraced to the widespread acceptance of the method in the 1960s. I focus especially on the different choices made by writers of elementary textbooks  the standard conduits of the method to a generation of philosophers  with an eye to determining what the `essential characteristics’ of natural deduction are.
The Deduction Rule and Linear and Nearlinear Proof Simulations
"... ... that a Frege proof of n lines can be transformed into a treelike Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus treelike systems simulate Frege systems with proof lengths bounded by O(n log n). ..."
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Cited by 12 (5 self)
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... that a Frege proof of n lines can be transformed into a treelike Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus treelike systems simulate Frege systems with proof lengths bounded by O(n log n).
Occurrences in Debugger Specifications
 In Proceedings of the ACM SIGPLAN'91 Conference on Programming Language Design and Implementation
, 1991
"... We describe formal manipulations of programming language semantics that permit execution animation for interpreters. We first study the use of occurrences in the calculus and we describe an implementation of the notion of residuals. We then describe applications in the development of interpreters f ..."
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Cited by 11 (1 self)
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We describe formal manipulations of programming language semantics that permit execution animation for interpreters. We first study the use of occurrences in the calculus and we describe an implementation of the notion of residuals. We then describe applications in the development of interpreters for the lazy calculus and the parallel language Occam. 1. Introduction Formal descriptions of programming language semantics have already been shown to yield executable specifications of interpreters for these languages [MiniML], [Esterel]. However, while the obtained interpreters have the clear advantage of being "certified" implementations, they lack a nice user interface for the very reason that the definition only deals with semantic values. An interpreter can be tranformed into a debugging tool by adding tracing, profiling, or control of execution functionalities. For us, an execution trace is a list of basic instruction calls that describes a history of execution, a profile is a list...
What does it mean to say that logic is formal
, 2000
"... Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and ..."
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Cited by 10 (0 self)
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Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topicneutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this
A Certified Compiler for an Imperative Language
, 1998
"... This paper describes the process of mechanically certifying a compiler with respect to the semantic specification of the source and target languages. The proofs are performed in type theory using the Coq system. These proofs introduce specific theoretical tools: fragmentation theorems and general in ..."
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Cited by 10 (3 self)
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This paper describes the process of mechanically certifying a compiler with respect to the semantic specification of the source and target languages. The proofs are performed in type theory using the Coq system. These proofs introduce specific theoretical tools: fragmentation theorems and general induction principles.
Semantic Verification of Web Sites Using Natural Semantics
 PROCEEDINGS OF THE 6TH RIAO CONFERENCE  CONTENTBASED MULTIMEDIA INFORMATION ACCESS
, 2000
"... The huge amount of information and knowledge available on the Web leads to the fact that it is more and more difficult to manage this information. Two different ways are commonly explored: giving a syntactical structure to Web sites, and annotating their content to facilitate Web mining. In this pap ..."
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Cited by 9 (1 self)
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The huge amount of information and knowledge available on the Web leads to the fact that it is more and more difficult to manage this information. Two different ways are commonly explored: giving a syntactical structure to Web sites, and annotating their content to facilitate Web mining. In this paper we explore a different approach inherited from software engineering: specifying the semantics of Web sites, allowing semantic verifications that will help both the conception and the maintenance of Web sites. To achieve this goal, we have experimented with the application of Natural Semantics (traditionally used to specify the semantics of programming languages) to Web sites specification and verification.