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Hereditarily Sequential Functionals
 In Proceedings of the Symposium on Logical Foundations of Computer Science: Logic at St. Petersburg, Lecture notes in Computer Science
, 1994
"... In order to define models of simply typed functional programming languages being closer to the operational semantics of these languages, the notions of sequentiality, stability and seriality were introduced. These works originated from the definability problem for PCF, posed in [Sco72], and the full ..."
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Cited by 58 (0 self)
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In order to define models of simply typed functional programming languages being closer to the operational semantics of these languages, the notions of sequentiality, stability and seriality were introduced. These works originated from the definability problem for PCF, posed in [Sco72], and the full abstraction problem for PCF, raised in [Plo77]. The presented computation model, forming the class of hereditarily sequential functionals, is based on a game in which each play describes the interaction between a functional and its arguments during a computation. This approach is influenced by the work of Kleene [Kle78], Gandy [Gan67], Kahn and Plotkin [KP78], Berry and Curien [BC82, Cur86, Cur92], and Cartwright and Felleisen [CF92]. We characterize the computable elements in this model in two different ways: (a) by recursiveness requirements for the game, and (b) as definability with the schemata (S1) (S8), (S11), which is related to definability in PCF. It turns out that both definitio...
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 48 (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
Mathematically Strong Subsystems of Analysis With Low Rate of Growth of Provably Recursive Functionals
, 1995
"... This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of ..."
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Cited by 34 (21 self)
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This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifierfree choice ACqf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non standard' axiom F  which does not hold in the full settheoretic model but in the strongly majorizable functionals): From a proof GnA # +ACqf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full settheoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +ACqf+F  proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +ACqf+# for suitable #. 1
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 33 (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
General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
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Cited by 31 (18 self)
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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
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" functionals of finit ..."
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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) ...
Correspondence between Operational and Denotational Semantics
 Handbook of Logic in Computer Science
, 1995
"... This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational ..."
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This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational semantics of PCF induced by an interpretation; (standard) Scott model, adequacy, weak adequacy and its proof (by a computability predicate) Domain Theory up to SFP and Scott domains; non full abstraction of the standard model, definability of compact elements and full abstraction for PCFP (PCF + parallel or), properties of orderextensional (continuous) models of PCF, Milner's model and Mulmuley's construction (excluding proofs) Additional topics (time permitting): results on pure simplytyped lambda calculus, Friedman 's Completeness Theorem, minimal model, logical relations and definability, undecidability of lambda definability (excluding proof), dIdomains and stable functions Homepa...
A Combinatory Algebra for Sequential Functionals of Finite Type
 University of Utrecht
, 1997
"... It is shown that the type structure of finitetype functionals associated to a combinatory algebra of partial functions from IN to IN (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from IN to IN), is isomorphic ..."
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Cited by 21 (2 self)
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It is shown that the type structure of finitetype functionals associated to a combinatory algebra of partial functions from IN to IN (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from IN to IN), is isomorphic to the type structure generated by object N (the flat domain on the natural numbers) in Ehrhard's category of "dIdomains with coherence", or his "hypercoherences". AMS Subject Classification: Primary 03D65, 68Q55 Secondary 03B40, 03B70, 03D45, 06B35 Introduction PCF , "Godel's T with unlimited recursion", was defined in Plotkin's paper [16]. It is a simply typed calculus with a type o for integers and constants for basic arithmetical operations, definition by cases and fixed point recursion. More importantly, there is a special reduction relation attached to it which ensures (by Plotkin's "Activity Lemma") that all PCF definable highertype functionals have a sequential, i.e. nonparal...