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Algorithmic Game Semantics
 In Schichtenberg and Steinbruggen [16
, 2001
"... Introduction SAMSON ABRAMSKY (samson@comlab.ox.ac.uk) Oxford University Computing Laboratory 1. Introduction Game Semantics has emerged as a powerful paradigm for giving semantics to a variety of programming languages and logical systems. It has been used to construct the first syntaxindependen ..."
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Cited by 54 (3 self)
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Introduction SAMSON ABRAMSKY (samson@comlab.ox.ac.uk) Oxford University Computing Laboratory 1. Introduction Game Semantics has emerged as a powerful paradigm for giving semantics to a variety of programming languages and logical systems. It has been used to construct the first syntaxindependent fully abstract models for a spectrum of programming languages ranging from purely functional languages to languages with nonfunctional features such as control operators and locallyscoped references [4, 21, 5, 19, 2, 22, 17, 11]. A substantial survey of the state of the art of Game Semantics circa 1997 was given in a previous Marktoberdorf volume [6]. Our aim in this tutorial presentation is to give a first indication of how Game Semantics can be developed in a new, algorithmic direction, with a view to applications in computerassisted verification and program analysis. Some promising steps have already been taken in this
Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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Cited by 36 (0 self)
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
Characterizations of the Basic Feasible Functionals of Finite Type (Extended Abstract)
 Feasible Mathematics: A Mathematical Sciences Institute Workshop
, 1990
"... Stephen A. Cook and Bruce M. Kapron Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 1 Introduction Functionals are functions which take natural numbers and other functionals as arguments and return natural numbers as values. The class of "feasible" functional ..."
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Cited by 27 (6 self)
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Stephen A. Cook and Bruce M. Kapron Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 1 Introduction Functionals are functions which take natural numbers and other functionals as arguments and return natural numbers as values. The class of "feasible" functionals of finite type was introduced in [6] via the typed lambda calculus, and used to interpret certain formal systems of arithmetic: systems capturing the notion of "feasibly constructive proof" (we equate feasibility with polynomial time computability) . Here we name the functionals of [6] the basic feasible functionals and justify the designation by presenting results which include two programming style characterizations of the class. We also give examples of both feasible and infeasible functionals, and argue that the notion plays a natural role in complexity theory. Type 2 functionals take numbers and ordinary numerical functions as arguments. When these argument functions are 01 valued (i.e. sets) ...
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 12 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
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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Cited by 5 (1 self)
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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
"... ..."
Inductive Definition and Domain Theoretic Properties of Fully Abstract Models for PCF and PCF+
 LOGICAL METHODS IN COMPUTER SCIENCE 3(3:7), 1–50 (2007)
, 2007
"... A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the sevent ..."
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Cited by 1 (1 self)
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A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given levelbylevel in the finite type hierarchy. To prove full abstraction and nondcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are nonomegacomplete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omegacompleteness), “naturally” continuous (with respect to existing directed “pointwise”, or “natural” lubs) and also “naturally” omegaalgebraic and “naturally” bounded complete—appropriate generalisation of the ordinary notions of domain theory to the case of nondcpos.
Speedup Theorems in Type2 Computation Using Oracle Turing Machines
"... Abstract. A classic result known as the speedup theorem in machineindependent complexity theory shows that there exist some computable functions that do not have best programs for them [2, 3]. In this paper we lift this result into type2 computation. Although the speedup phenomenon is essentially ..."
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Abstract. A classic result known as the speedup theorem in machineindependent complexity theory shows that there exist some computable functions that do not have best programs for them [2, 3]. In this paper we lift this result into type2 computation. Although the speedup phenomenon is essentially inherited from type1 computation, we observe that a direct application of the original proof to our type2 speedup theorem is problematic because the oracle queries can interfere the speed of the programs and hence the cancellation strategy used in the original proof is no longer correct at type2. We also argue that a type2 analog of the operator speedup theorem [15] does not hold, which suggests that this curious speedup phenomenon disappears in highertyped computation beyond type2. The result of this paper adds one more evidence to support the general type2 complexity theory under the framework proposed in [14, 12, 13] as a reasonable setup. 1