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Predicative Recursion and Computational Complexity
, 1992
"... The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct r ..."
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Cited by 45 (3 self)
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The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct reference to polynomials, time, or even computation. Complexity classes characterized in this way include polynomial time, the functional polytime hierarchy, the logspace decidable problems, and NC. After developing these "resource free" definitions, we apply them to redeveloping the feasible logical system of Cook and Urquhart, and show how this firstorder system relates to the secondorder system of Leivant. The connection is an interesting one since the systems were defined independently and have what appear to be very different rules for the principle of induction. Furthermore it is interesting to see, albeit in a very specific context, how to retract a second order statement, ("inducti...
A New Characterization Of Type 2 Feasibility
, 1996
"... . K. Mehlhorn introduced a class of polynomial time computable operators in order to study poly time reducibilities between functions. This class is defined using a generalization of A. Cobham's definition of feasibility for type 1 functions to type 2 functionals. Cobham's feasible functions are equ ..."
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Cited by 36 (6 self)
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. K. Mehlhorn introduced a class of polynomial time computable operators in order to study poly time reducibilities between functions. This class is defined using a generalization of A. Cobham's definition of feasibility for type 1 functions to type 2 functionals. Cobham's feasible functions are equivalent to the familiar poly time functions. We generalize this equivalence to type 2 functionals. This requires a definition of the notion `poly time in the length of type 1 inputs'. The proof of this equivalence is not a simple generalization of the proof for type 1 functions; it depends on the fact that Mehlhorn's class is closed under a strong form of simultaneous limited recursion on notation, and requires an analysis of the structure of oracle queries in time bounded computations. Key words. type 2 computability, polynomial time, notational recursion, oracle Turing machine AMS subject classifications. 68Q05,68Q15,03D65,03D20 1. Introduction. A type 1 function is a mapping from N to ...
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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Cited by 6 (1 self)
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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...
Parallel computable higher type functionals (Extended Abstract)
 In Proceedings of IEEE 34th Annual Symposium on Foundations of Computer Science, Nov 35
, 1993
"... ) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different way ..."
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Cited by 4 (4 self)
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) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different ways. Recursiontheoretic, prooftheoretic and machinetheoretic characterizations are given for various classes, providing evidence of their naturalness. In this section, we motivate the approach of our work. In proof theory, primitive recursive functionals of higher type were introduced in Godel's Dialectica [13] paper, where they were used to "witness" the truth of arithmetic formulas. For instance, a function f witnesses the formula 8x9y\Phi(x; y), where \Phi is quantifierfree, provided that 8x\Phi(x; f(x)); while a type 2 functional F witnesses the formula 8x9y8u9v\Phi(x; y; u; v), provided that 8x8u\Phi(x; f(x); u; F (x; f(x); u)): Godel's formal system T is a variant of the finit...
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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Cited by 3 (3 self)
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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.
Feasibly Continuous TypeTwo Functionals
, 1997
"... A wellknown theorem of typetwo recursion theory states that a functional is continuous if and only if it is computable relative to some oracle. We show that a feasible analogue of this theorem holds, using techniques originally developed in the study of Boolean decision tree complexity. 1 Introduc ..."
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Cited by 1 (0 self)
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A wellknown theorem of typetwo recursion theory states that a functional is continuous if and only if it is computable relative to some oracle. We show that a feasible analogue of this theorem holds, using techniques originally developed in the study of Boolean decision tree complexity. 1 Introduction Typetwo computability theory deals with the computability of functionals, which take functions and numbers as input, and produce numbers as output. A surprising and pleasing aspect of typetwo computability is its close connections with topology on Baire space. Notions of relative typetwo computability (that is, computability with respect to some oracle,) can be characterized using purely topological notions. In particular, a typetwo functional is computable relative to an oracle if and only if it is continuous. While the theory of typetwo computability has been widely successful, relatively little work has been done on the development of a complexity theory for typetwo functionals...
On Type2 Complexity Classes
 Proceedings of the Third International Workshop on Implicit Computational Complexity
, 2001
"... There are now a number of things called "highertype complexity classes." The most promenade of these is the class of basic feasible functionals [CU93, CK90], a fairly conservative highertype analogue the (type1) polynomialtime computable functions. There is however currently no satisfactory gene ..."
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There are now a number of things called "highertype complexity classes." The most promenade of these is the class of basic feasible functionals [CU93, CK90], a fairly conservative highertype analogue the (type1) polynomialtime computable functions. There is however currently no satisfactory general notion of what a highertype complexity class should be. In this paper we propose one such notion for type2 functionals and begin an investigation of its properties. The most striking di#erence between our type2 complexity classes and their type1 counterparts is that, because of topological constrains, the type2 classes have a much more ridged structure. Example: It follows from McCreight and Meyer's Union Theorem [MM69] that the (type1) polynomialtime computable functions form a complexity class (in the strict sense of Definition 1 below). The analogous result fails for the class of type2 basic feasible functionals. 1.