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32
Some Connections between Bounded Query Classes and NonUniform Complexity
 In Proceedings of the 5th Structure in Complexity Theory Conference
, 1990
"... This paper is dedicated to the memory of Ronald V. Book, 19371997. ..."
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Cited by 70 (22 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
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 an ..."
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Cited by 49 (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 functio ..."
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Cited by 41 (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 ...
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 30 (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) ...
Computational Complexity and Induction for Partial Computable Functions in Type Theory
 In Preprint
, 1999
"... An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in ..."
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Cited by 12 (7 self)
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An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in type theory, and consequently such concepts are not available in applications of type theory where they are greatly needed. It is also not clear how to provide a practical and adequate account in programming logics based on set theory. This paper provides a practical theory supporting all these concepts in the setting of constructive type theories. We first introduce an extensional theory of partial computable functions in type theory. We then add support for intensional reasoning about programs by explicitly reflecting the essential properties of the underlying computation system. We use the resulting intensional reasoning tools to justify computational induction and to define computationa...
Semantics vs. Syntax vs. Computations  Machine Models For Type2 . . .
 JOURNAL OF COMPUTER AND SYSTEM SCIENCE
, 1997
"... This paper investigates analogs of the KreiselLacombeShoenfield Theorem in the context of the type2 basic feasible functionals. We develop a direct, polynomialtime analog of effective operation in which the time boundingon computations is modeled after Kapron and Cook's scheme for their bas ..."
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Cited by 10 (0 self)
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This paper investigates analogs of the KreiselLacombeShoenfield Theorem in the context of the type2 basic feasible functionals. We develop a direct, polynomialtime analog of effective operation in which the time boundingon computations is modeled after Kapron and Cook's scheme for their basic polynomialtime functionals. We show that if P = NP, these polynomialtime effective operations are strictly more powerful on R (the class of recursive functions) than the basic feasible functions. We also consider a weaker notion of polynomialtime effective operation where the machines computing these functionals have access to the computations of their procedural parameter, but not to its program text. For this version of polynomialtime effective operations, the analog of the KreiselLacombeShoenfield is shown to holdtheir power matches that of the basic feasible functionals on R.
Complexity Theory for Operators in Analysis
"... We propose a new framework for discussing computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only by approximation. The key idea is to use a certain class of string functions, which we call regular functions, as ..."
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We propose a new framework for discussing computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only by approximation. The key idea is to use a certain class of string functions, which we call regular functions, as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An important advantage of using regular functions is that we can define their size in the way inspired by highertype complexity theory. This enables us to talk about computation on regular functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP and PSPACEcompleteness under suitable manyone reductions. Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. The latter two cannot be represented succinctly using existing approaches based on infinite sequences, so ours is the first treatment of them. As an interesting example, the task of numerical algorithms for solving the initial value problem of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results could only state how complex the solution can be. We now reformulate them to show that the operator itself is polynomialspace complete.
Gaplanguages and logtime complexity classes
 THEORETICAL COMPUTER SCIENCE
, 1997
"... This paper shows that classical results about complexity classes involving “delayed diagonalization” and “gap languages,” such as Ladner’s Theorem and Schöning’s Theorem and independence results of a kind noted by Schöning and Hartmanis, apply at very low levels of complexity, indeed all the way dow ..."
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Cited by 9 (6 self)
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This paper shows that classical results about complexity classes involving “delayed diagonalization” and “gap languages,” such as Ladner’s Theorem and Schöning’s Theorem and independence results of a kind noted by Schöning and Hartmanis, apply at very low levels of complexity, indeed all the way down in Sipser’s logtime hierarchy. This paper also investigates refinements of Sipser’s classes and notions of logtime reductions, following on from recent work by Cai, Chen, and others.
A Tight Relationship between Generic Oracles and Type2 Complexity Theory
, 1997
"... We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct. ..."
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Cited by 7 (1 self)
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We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct.
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...