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The history and concept of computability
 in Handbook of Computability Theory
, 1999
"... We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in th ..."
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We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in their present roles, and how
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
"... ..."
How to compare the power of computational models
 In Computability in Europe 2005: New Computational Paradigms
, 2005
"... Abstract. We argue that there is currently no satisfactory general framework for comparing the extensional computational power of arbitrary computational models operating over arbitrary domains. We propose a conceptual framework for comparison, by linking computational models to hypothetical physica ..."
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Abstract. We argue that there is currently no satisfactory general framework for comparing the extensional computational power of arbitrary computational models operating over arbitrary domains. We propose a conceptual framework for comparison, by linking computational models to hypothetical physical devices. Accordingly, we deduce a mathematical notion of relative computational power, allowing the comparison of arbitrary models over arbitrary domains. In addition, we claim that the method commonly used in the literature for “strictly more powerful” is problematic, as it allows for a model to be more powerful than itself. On the positive side, we prove that Turing machines and the recursive functions are “complete ” models, in the sense that they are not susceptible to this anomaly, justifying the standard means of showing that a model is “hypercomputational.” 1
Explicit Substitutions and All That
, 2000
"... Explicit substitution calculi are extensions of the lambdacalculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In ..."
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Explicit substitution calculi are extensions of the lambdacalculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In this paper we explore new developments on two of the most successful styles of explicit substitution calculi: the lambdasigma and lambda_secalculi.
A Formalization of the ChurchTuring Thesis for StateTransition Models
"... Abstract. Our goal is to formalize the ChurchTuring Thesis for a very large class of computational models. Specifically, the notion of an “effective model of computation ” over an arbitrary countable domain is axiomatized. This is accomplished by modifying Gurevich’s “Abstract State Machine ” postu ..."
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Abstract. Our goal is to formalize the ChurchTuring Thesis for a very large class of computational models. Specifically, the notion of an “effective model of computation ” over an arbitrary countable domain is axiomatized. This is accomplished by modifying Gurevich’s “Abstract State Machine ” postulates for statetransition systems. A proof is provided that all models satisfying our axioms, regardless of underlying data structure—and including all standard statetransition models—are equivalent to (up to isomorphism), or weaker than, Turing machines. To allow the comparison of arbitrary models operating over arbitrary domains, we employ a quasiordering on computational models, based on their extensionality. LCMs can do anything that could be described as “rule of thumb ” or “purely mechanical”.... This is sufficiently well established that it is now agreed amongst logicians that “calculable by means of an LCM” is the correct accurate rendering of such phrases. 1
Historical Projects in Discrete Mathematics and Computer Science
"... A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itse ..."
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A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development
Typed SelfInterpretation by Pattern Matching
"... Selfinterpreters can be roughly divided into two sorts: selfrecognisers that recover the input program from a canonical representation, and selfenactors that execute the input program. Major progress for staticallytyped languages was achieved in 2009 by Rendel, Ostermann, and Hofer who presented ..."
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Selfinterpreters can be roughly divided into two sorts: selfrecognisers that recover the input program from a canonical representation, and selfenactors that execute the input program. Major progress for staticallytyped languages was achieved in 2009 by Rendel, Ostermann, and Hofer who presented the first typed selfrecogniser that allows representations of different terms to have different types. A key feature of their type system is a type:type rule that renders the kind system of their language inconsistent. In this paper we present the first staticallytyped language that not only allows representations of different terms to have different types, and supports a selfrecogniser, but also supports a selfenactor. Our language is a factorisation calculus in the style of Jay and GivenWilson, a combinatory calculus with a factorisation operator that is powerful enough to support the patternmatching functions necessary for a selfinterpreter. This allows us to avoid a type:type rule. Indeed, the types of System F are sufficient. We have implemented our approach and our experiments support the theory.
A Rulebased Approach to the Implementation of Evaluation Strategies Mircea Marin
 PETCU D., NEGRU V., ZAHARIE D., JEBELEAN T., Eds., Proceedings of SYNASC 2004
"... We describe a new methodology to program evaluation strategies, which relies on some advanced features of the rulebased programming language #Log. We illustrate how our approach works for a number of important evaluation strategies. 1 ..."
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We describe a new methodology to program evaluation strategies, which relies on some advanced features of the rulebased programming language #Log. We illustrate how our approach works for a number of important evaluation strategies. 1
The ChurchTuring Thesis as an Immature Form of the ZuseFredkin Thesis (More Arguments in Support of the “Universe as a Cellular Automaton” Idea)
"... In [1] we have shown a strong argument in support of the "Universe as a computer " idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only "some kind of computer", but also a concrete computational model known as a ..."
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In [1] we have shown a strong argument in support of the "Universe as a computer " idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only "some kind of computer", but also a concrete computational model known as a "cellular automaton".
A Staging Calculus and its Application to the Verification of Translators
, 1993
"... We develop a calculus in which the computation steps required to execute a computer program can be separated into discrete stages. The calculus, denoted 2 , is embedded within the pure untyped calculus. The main result of the paper is a characterization of sucient conditions for conuence for terms ..."
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We develop a calculus in which the computation steps required to execute a computer program can be separated into discrete stages. The calculus, denoted 2 , is embedded within the pure untyped calculus. The main result of the paper is a characterization of sucient conditions for conuence for terms in the calculus. The condition can be taken as a correctness criterion for translators that perform reductions in one stage leaving residual redexes over for subsequent computation stages. As an application of the theory, we verify the correctness of a macro expansion algorithm. The expansion algorithm is of some interest in its own right since it solves the problem of desired variable capture using only the familiar capture avoiding substitutions. 1 Introduction The calculus is widely used as a metalanguage for programming language semantics because most of the complexities of real programming languages can be modeled by relatively simple and wellunderstood aspects of the calculus. ...