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Algorithms: A quest for absolute definitions
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTurin ..."
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Cited by 19 (9 self)
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y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTuring thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.
On Sets, Types, Fixed Points, and Checkerboards
 In Pierangelo Miglioli, Ugo Moscato, Daniele Mundici, and Mario Ornaghi, editors, Theorem Proving with Analytic
, 1996
"... Most current research on automated theorem proving is concerned with proving theorems of firstorder logic. We discuss two examples which illustrate the advantages of using higherorder logic in certain contexts. For some purposes type theory is a much more convenient vehicle for formalizing mathema ..."
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Cited by 8 (2 self)
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Most current research on automated theorem proving is concerned with proving theorems of firstorder logic. We discuss two examples which illustrate the advantages of using higherorder logic in certain contexts. For some purposes type theory is a much more convenient vehicle for formalizing mathematics than axiomatic set theory. Even theorems of firstorder logic can sometimes be proven more expeditiously in higherorder logic than in firstorder logic. We also note the need to develop automatic theoremproving methods which may produce proofs which do not have the subformula property. 1. Introduction In some ways it appears that the field of automated deduction is ahead of its time. We have increasingly good methods for proving theorems, but the hardware available is still not adequate for many of the problems to which we would like to apply our techniques. However, this will change. Radically new computers based on exotic technologies such as DNA computing, quantum computing, and ...
Combining and Automating Classical and NonClassical Logics in Classical HigherOrder Logics
 ANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCE (PREFINAL VERSION)
"... Numerous classical and nonclassical logics can be elegantly embedded in Church’s simple type theory, also known as classical higherorder logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security, and logics for spatial reasoning. Furthermor ..."
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Cited by 1 (1 self)
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Numerous classical and nonclassical logics can be elegantly embedded in Church’s simple type theory, also known as classical higherorder logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security, and logics for spatial reasoning. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Offtheshelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about embedded logics and logics combinations. In this article we focus on combinations of (quantified) epistemic and doxastic logics and study their application for modeling and automating the reasoning of rational agents. We present illustrating example problems and report on experiments with offtheshelf higherorder automated theorem provers.
Computability and analysis: the legacy of Alan Turing
, 2012
"... For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a par ..."
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For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a particular geometric