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Elementary explicit types and polynomial time operations
, 2008
"... This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polyn ..."
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This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polynomial time. The systems considered are natural extensions of the firstorder applicative theories introduced in
Weak theories of operations and types
"... This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywor ..."
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This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywords: Proof theory, Feferman’s explicit mathematics, applicative theories, higher types, types and names, partial truth, feasible operations 1
Adventures in time and space
 33th ACM Symposium on Principles of Programming Languages
, 2006
"... Abstract. This paper investigates what is essentially a callbyvalue version of PCF under a complexitytheoretically motivated type system. The programming formalism, ATR, has its firstorder programs characterize the polynomialtime computable functions, and its secondorder programs characterize ..."
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Abstract. This paper investigates what is essentially a callbyvalue version of PCF under a complexitytheoretically motivated type system. The programming formalism, ATR, has its firstorder programs characterize the polynomialtime computable functions, and its secondorder programs characterize the type2 basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATRtypes are confined to levels 0, 1, and 2.) The type system comes in two parts, one that primarily restricts the sizes of values of expressions and a second that primarily restricts the time required to evaluate expressions. The sizerestricted part is motivated by Bellantoni and Cook’s and Leivant’s implicit characterizations of polynomialtime. The timerestricting part is an affine version of Barber and Plotkin’s DILL. Two semantics are constructed for ATR. The first is a pruning of the naïve denotational semantics for ATR. This pruning removes certain functions that cause otherwise feasible forms of recursion to go wrong. The second semantics is a model for ATR’s time complexity relative to a certain abstract machine. This model provides a setting for complexity recurrences arising from ATR recursions, the solutions of which yield secondorder polynomial time bounds. The timecomplexity semantics is also shown to be sound relative to the costs of interpretation on the abstract machine. 1.
A New Characterization of Mehlhorn's Polynomial Time Functionals (Extended Abstract)
 PROCEEDINGS OF THE 32ND ANNUAL IEEE SYMPOSIUM FOUNDATIONS OF COMPUTER SCIENCE
, 1991
"... A type 1 function is a total mapping from N to N. We will denote the set of all such functions by N N. A type 2 functional is a total mapping from ( N N) k \Theta N l to N, for some k; l. More specifically, we will call a mapping of this sort a functional with rank (k; l). For type 1 fu ..."
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A type 1 function is a total mapping from N to N. We will denote the set of all such functions by N N. A type 2 functional is a total mapping from ( N N) k \Theta N l to N, for some k; l. More specifically, we will call a mapping of this sort a functional with rank (k; l). For type 1 functions, there is a well established notion of computational feasibility. Namely a function is feasible if it is computable in polynomial time on a Turing machine. More specifically, a function f is poly time if there is a TM M and a polynomial p such that for all x, M with input x computes f(x) and runs in time p(n), where n = jxj, and for x 2 N, jxj denotes the length of the binary notation of x, that is dlog(x + 1)e. This notion of f...
Complexity and Intensionality in a Type1 Framework for Computable Analysis
 Computer Science Logic: 19th International Workshop, CSL 2005, 14th Annual Conference of the EACSL
"... Abstract. Implementations of real number computations have largely been unusable in practice because of their very bad performance, especially in comparison to floating point arithmetic implemented in hardware. This performance problem is to a very large extent due to the type2 nature of the comput ..."
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Abstract. Implementations of real number computations have largely been unusable in practice because of their very bad performance, especially in comparison to floating point arithmetic implemented in hardware. This performance problem is to a very large extent due to the type2 nature of the computable analysis frameworks usually employed. This problem can be overcome by employing a type1 approach. This paper presents such an approach and deals with properties of it that have not been well studied before, namely the introduction of complexity measures for type1 representations of real functions and ways to define intensional functions, i.e. functions that may return different real numbers for the same real argument given in different representations. 1
On the Proof Theory of Applicative Theories
 PHD THESIS, INSTITUT FÜR INFORMATIK UND ANGEWANDTE MATHEMATIK, UNIVERSITÄT
, 1996
"... ..."
Resourcebounded Continuity and Sequentiality for Typetwo Functionals
"... Devices]: Complexity Measures and Classesrelations among complexity classes; F.1.1 [Computation by Abstract Devices]: Models of Computationrelations between models; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logiccomputability theory General Terms: Theory, Algorithms Ad ..."
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Devices]: Complexity Measures and Classesrelations among complexity classes; F.1.1 [Computation by Abstract Devices]: Models of Computationrelations between models; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logiccomputability theory General Terms: Theory, Algorithms Additional Key Words and Phrases: Higherorder complexity, sequential computation, decision trees 1.
Feasible Functionals And Intersection Of Ramified Types
, 2003
"... We show that the basic feasible functions of Cook and Urquhart's BFF [8,9] are precisely the functionals definable in a natural system of ramified recurrence that uses type intersection (for tiervariants of a common type). This further confirms the stability of BFF as a notion of computational ..."
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We show that the basic feasible functions of Cook and Urquhart's BFF [8,9] are precisely the functionals definable in a natural system of ramified recurrence that uses type intersection (for tiervariants of a common type). This further confirms the stability of BFF as a notion of computational feasibility in higher type. It also suggests the potential usefulness of typeintersection restricted to sortvariants of a common type.
Game semantics approach to sequential complexity
"... We propose a notion of size and complexity for strategies for a class of sequential games. This applies in particular to strategies for pcf, which leads to a notion of complexity for higherorder functions, as well as a class of polynomial time computable higherorder functions. ..."
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We propose a notion of size and complexity for strategies for a class of sequential games. This applies in particular to strategies for pcf, which leads to a notion of complexity for higherorder functions, as well as a class of polynomial time computable higherorder functions.