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18
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 Feasible Theory for Analysis
, 1994
"... We construct a weak secondorder theory of arithmetic which includes Weak Konig's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with # b 1 graphs) of this theory are the polynomial time computable functions. It is shown that the firstorder strength of this vers ..."
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Cited by 18 (3 self)
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We construct a weak secondorder theory of arithmetic which includes Weak Konig's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with # b 1 graphs) of this theory are the polynomial time computable functions. It is shown that the firstorder strength of this version of WKL is exactly that of the scheme of collection for bounded formulae. 1
Theories With SelfApplication and Computational Complexity
 Information and Computation
, 2002
"... Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not ne ..."
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Cited by 12 (9 self)
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Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not necessarily total. It has turned out that theories with selfapplication provide a natural setting for studying notions of abstract computability, especially from a prooftheoretic perspective.
Groundwork for weak analysis
 the Journal of Symbolic Logic
, 2002
"... Abstract. This paper develops the very basic notions of analysis in a weak secondorder theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variabl ..."
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Cited by 10 (4 self)
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Abstract. This paper develops the very basic notions of analysis in a weak secondorder theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of real numbers is a real closed ordered field. In the last section of the paper, we show how to interpret the theory BTFA in Robinson’s theory of arithmetic Q. This fact entails that the elementary theory of the real closed ordered fields is interpretable in Q. §1. Introduction. The formalization of mathematics within secondorder arithmetic has a long and distinguished history. We may say that it goes back to Richard Dedekind, and that it has been pursued by, among others, Hermann Weyl, David Hilbert, Paul Bernays, Harvey Friedman, and Stephen Simpson and his students (we may also mention the insights of Georg Kreisel, Solomon Feferman,
Polynomial Time Operations in Explicit Mathematics
 Journal of Symbolic Logic
, 1997
"... In this paper we study (self)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable fu ..."
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Cited by 7 (5 self)
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In this paper we study (self)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable functions.
A ProofTheoretic Characterization of the Basic Feasible Functionals
 Theoretical Computer Science
, 2002
"... We provide a natural characterization of the type two MehlhornCookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27]. ..."
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Cited by 7 (6 self)
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We provide a natural characterization of the type two MehlhornCookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27].
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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Cited by 6 (5 self)
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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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Cited by 4 (3 self)
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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
Some notes on subword quantification and induction thereof. Typeset Manuscript
 Lecture Notes in Pure and Applied Mathematics 180
, 1996
"... The first section of this paper consists of a defense of the binary string notation for the formulation of weak theories of arithmetic which have computational significance. We defend that a stringlanguage is the most natural framework and that the usual arithmetic setting suffers from some troubles ..."
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Cited by 3 (1 self)
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The first section of this paper consists of a defense of the binary string notation for the formulation of weak theories of arithmetic which have computational significance. We defend that a stringlanguage is the most natural framework and that the usual arithmetic setting suffers from some troubles when dealing with very low complexity classes. Having introduced in the first section the theory Th − FO associated with a rather robust uniform version of the class of problems that can be decided by constant depth,polynomial size circuit families (the socalled AC 0class) we prove in the second section that the deletion of a crucial axiom from Th − FO results in a theory which is unsuitable from the computational point of view.
Basic applications of weak König’s lemma in feasible analysis
"... Abstract. In the context of a feasible theory for analysis, we investigate three fundamental theorems of analysis: the Heine/Borel covering theorem for the closed unit interval, and the uniform continuity and the maximum principles for real valued continuous functions defined on the closed unit inte ..."
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Cited by 2 (1 self)
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Abstract. In the context of a feasible theory for analysis, we investigate three fundamental theorems of analysis: the Heine/Borel covering theorem for the closed unit interval, and the uniform continuity and the maximum principles for real valued continuous functions defined on the closed unit interval. §1. The three results. The business of reverse mathematics is to investigate the logicomathematical strength of the various theorems of ordinary mathematics. This investigation is usually carried over the secondorder base theory RCA0 – a theory whose prooftheoretic strength is that of primitive recursive arithmetic. In this article, we investigate three basic theorems of analysis over a feasible base theory, i.e., a theory whose provably total functions (with appropriate graphs) are the polynomial time computable functions. Our feasible base theory is BTFA, a theory introduced by Ferreira in a paper entitled “A feasible theory for analysis ” [8]: we presuppose familiarity with the notation and results of that paper and an acquaintance with the basic features of research in reverse mathematics (as exposed in the relevant sections of chapters II, III and IV of [10]). Notice that the firstorder part of the intended model of BTFA is 2 <ω, the set of finite sequences of zeros and ones (also called binary words or strings), as opposed to the more traditional setting of the natural numbers. As it happens, we find the binary setting more perspicuous for dealing with theories concerned with subexponential classes of computational complexity. The firstorder part of a model of BTFA is denoted by W (for words). Given a formula A of the language of BTFA and x a distinguished (firstorder) variable, we say that A defines an infinite subtree of W, and write Tree∞(Ax), if ∀x∀y(A(x) ∧ y ⊆ x → A(y)) ∧∀n ∈ T ∃x(ℓ(x)=n ∧ A(x)), 1