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Multilanguage Hierarchical Logics (or: How We Can Do Without Modal Logics)
, 1994
"... MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of first order languages, each language containing names for the language below, and propose them as an alternative to ..."
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Cited by 177 (47 self)
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MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of first order languages, each language containing names for the language below, and propose them as an alternative to modal logics. The motivations of our proposal are technical, epistemological and implementational. From a technical point of view, we prove, among other things, that the set of theorems of the most common modal logics can be embedded (under the obvious bijective mapping between a modal and a first order language) into that of the corresponding ML systems. Moreover, we show that ML systems have properties not holding for modal logics and argue that these properties are justified by our intuitions. This claim is motivated by the study of how ML systems can be used in the representation of beliefs (more generally, propositional attitudes) and provability, two areas where modal logics have been extensively used. Finally, from an implementation point of view, we argue that ML systems resemble closely the current practice in the computer representation of propositional attitudes and metatheoretic theorem proving.
Experiments with Proof Plans for Induction
 Journal of Automated Reasoning
, 1992
"... The technique of proof plans, is explained. This technique is used to guide automatic inference in order to avoid a combinatorial explosion. Empirical research is described to test this technique in the domain of theorem proving by mathematical induction. Heuristics, adapted from the work of Boye ..."
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Cited by 94 (32 self)
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The technique of proof plans, is explained. This technique is used to guide automatic inference in order to avoid a combinatorial explosion. Empirical research is described to test this technique in the domain of theorem proving by mathematical induction. Heuristics, adapted from the work of Boyer and Moore, have been implemented as Prolog programs, called tactics, and used to guide an inductive proof checker, Oyster. These tactics have been partially specified in a metalogic, and the plan formation program, clam, has been used to reason with these specifications and form plans. These plans are then executed by running their associated tactics and, hence, performing an Oyster proof. Results are presented of the use of this technique on a number of standard theorems from the literature. Searching in the planning space is shown to be considerably cheaper than searching directly in Oyster's search space. The success rate on the standard theorems is high. Keywords Theorem prov...
A rational reconstruction and extension of recursion analysis
 Proceedings of the Eleventh International Joint Conference on Artificial Intelligence
, 1989
"... The focus of this paper is the technique of recur8\on analysis. Recursion analysis is used by the BoyerMoore Theorem Prover to choose an appropriate induction schema and variable to prove theorems by mathematical induction. A rational reconstruction of recursion analysis is outlined, using the tech ..."
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Cited by 26 (13 self)
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The focus of this paper is the technique of recur8\on analysis. Recursion analysis is used by the BoyerMoore Theorem Prover to choose an appropriate induction schema and variable to prove theorems by mathematical induction. A rational reconstruction of recursion analysis is outlined, using the technique of proof plans. This rational reconstruction suggests an extension of recursion analysis which frees the induction suggestion from the forms of recursion found in the conjecture. Preliminary results are reported of the automation of this rational reconstruction and extension using the CLAMOyster system.
Higher Order Function Synthesis Through Proof Planning
, 2001
"... . The close association between higher order functions (HOFs) and algorithmic skeletons is a promising source of automatic parallelisation of programs. An approach to synthesising HOFs from functional programs through proof planning is presented, and its realisation in Clam is discussed. 1. Introdu ..."
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Cited by 3 (0 self)
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. The close association between higher order functions (HOFs) and algorithmic skeletons is a promising source of automatic parallelisation of programs. An approach to synthesising HOFs from functional programs through proof planning is presented, and its realisation in Clam is discussed. 1. Introduction 1.1. Higher Order Functions Pure functional languages, satisfying the ChurchRosser property of evaluation order independence, have long been proposed as a basis for parallel programming. Thus, Wegner[Weg71] observed in 1971: Note that [the ChurchRosser theorem] essentially states that lambda expressions can be evaluated by asynchronous multiprocesssing applied in arbitrary order to local subexpressions. page 185 Early work on functional parallelism focussed on reduction of Curry combinators [Tur79], lifted automatically from functional programs, but these proved of too low granularity for efficient parallel evaluation [Sto84]. Somewhat more success has obtained from parallel red...
From Metalevel Tactics to Objectlevel Programs
"... The paper describes a variant of MartinLof type theory extended by the principle of type induction, as introduced in the oyster2 theorem proving system. The system has a procedural metalanguage in which a variety of search strategies have been written. Theorem proving in the system yields obj ..."
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The paper describes a variant of MartinLof type theory extended by the principle of type induction, as introduced in the oyster2 theorem proving system. The system has a procedural metalanguage in which a variety of search strategies have been written. Theorem proving in the system yields objectlevel functional programs that implement procedures associated with the theorem that is proved. We demonstrate the application of type induction in proving the existence of a partial decision procedure for a fragment of type theory inside the theory itself. As a result of that proof, we obtain an objectlevel program that implements such a partial decision procedure, ie given a type theoretic formula x as input, it returns either a proof of x (formally, an element of the type x), or a refutation of x (formally, an element of the type x \Gamma! void) or a distinguished value indicating that no decision has been made. Initially an interactive proof of the existence of such a proce...
Theorem Proving and Program Synthesis with Oyster
 In Proceedings of the IMA Unified Computation Laboratory
, 1990
"... MartinLof type theory provides a formal framework for the construction of verified programs, both specified and written in the type theory. We describe an implementation of the type theory that aims to provide an environment for software engineering using this approach. We illustrate this by des ..."
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MartinLof type theory provides a formal framework for the construction of verified programs, both specified and written in the type theory. We describe an implementation of the type theory that aims to provide an environment for software engineering using this approach. We illustrate this by describing the synthesis of a simple evaluator for arithmetic expressions in the system. 1 Introduction There is currently much interest in providing tools for computeraided reasoning about programs. This can be done using a general purpose theorem prover. For example, the Boyer and Moore theorem prover [1] provides reasoning within the quantifierfree predicate calculus, and programs can be shown correct by formulating properties of the programming language in question within this logic. More recently various logical framework systems have been proposed, where a variety of logics can be treated uniformly. Hence reasoning using program logics can be automated, as in [2]. Another approach i...
La Deduzione Automatica
, 1994
"... Scopo di questo articolo e` dare una panoramica introduttiva alla deduzione automatica, mettendo in evidenza obiettivi, differenze e similitudini di alcuni fra i piu` importanti approcci al problema. ..."
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Scopo di questo articolo e` dare una panoramica introduttiva alla deduzione automatica, mettendo in evidenza obiettivi, differenze e similitudini di alcuni fra i piu` importanti approcci al problema.