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A New RecursionTheoretic Characterization Of The Polytime Functions
 COMPUTATIONAL COMPLEXITY
, 1992
"... We give a recursiontheoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds. In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2 xy ) of Cobham. ..."
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Cited by 179 (7 self)
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We give a recursiontheoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds. In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2 xy ) of Cobham.
Functionalgebraic characterizations of log and polylog parallel time
 Computational Complexity
, 1994
"... Abstract. The main results of this paper are recursiontheoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fanin circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log ..."
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Cited by 14 (4 self)
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Abstract. The main results of this paper are recursiontheoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fanin circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log and polylog time). The present characterizations avoid the complex base functions, function constructors, and a priori size or depth bounds typical of previous work on these classes. This simplicity is achieved by extending the \tiered recursion " techniques of Leivant and Bellantoni&Cook. Key words. Circuit complexity � subrecursion. Subject classi cations. 68Q15, 03D20, 94C99. 1.
Divide and Conquer in Parallel Complexity and Proof Theory
, 1992
"... Copyright Stephen Austin Bloch, 1992 All rights reserved. The dissertation of Stephen Bloch is approved, and it is acceptable in quality and form for publication on micro ..."
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Cited by 4 (2 self)
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Copyright Stephen Austin Bloch, 1992 All rights reserved. The dissertation of Stephen Bloch is approved, and it is acceptable in quality and form for publication on micro
Some applications of logic to feasibility in higher types
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2002
"... While it is commonly accepted that computability on a Turing machine in polynomial time represents a correct formalisation of the notion of a feasibly computable function, there is no similar agreement on how to extend this notion on functionals, i.e., what functionals should be considered feasible. ..."
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Cited by 3 (0 self)
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While it is commonly accepted that computability on a Turing machine in polynomial time represents a correct formalisation of the notion of a feasibly computable function, there is no similar agreement on how to extend this notion on functionals, i.e., what functionals should be considered feasible. One possible paradigm was introduced by Mehlhorn, who extended Cobham’s definition of feasible functions to type 2 functionals. Subsequently, this class of functionals (with inessential changes of the definition) was studied by Townsend who calls this class POLY, and by Kapron and Cook who call the same class basic feasible functionals. Kapron and Cook gave an oracle Turing machine model characterisation of this class. In this paper we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalise the corresponding properties of the class of feasible functions, thus giving further evidence that the notion of feasibility of functionals mentioned above is correctly chosen. We also improve the Kapron and Cook result on machine representation. Our proofs are based on essential applications of logic. We introduce a weak fragment of second order arithmetic with second order variables ranging over functions from N N which suitably characterises basic feasible functionals, and show that it is a useful tool for investigating the
On parallel hierarchies and R i k
 Submitted Annals of Pure and Applied Logic
, 1996
"... This paper de nes natural hierarchies of function and relation classes, constructed from parallel complexity classes in a manner analogous to the polynomialtime hierarchy. A number of structural results about these classes are proven: relationships between them and the levels of PH � relationships ..."
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This paper de nes natural hierarchies of function and relation classes, constructed from parallel complexity classes in a manner analogous to the polynomialtime hierarchy. A number of structural results about these classes are proven: relationships between them and the levels of PH � relationships between these classes and de nability in the bounded arithmetic theories Ri k � and several results relating conservation among theories of bounded arithmetic to the collapse of complexity classes. 1
Recursion Schemata For Slow Growing Depth Circuit Classes
"... . In this note we characterize iterated log depth circuit classes LD i and ND i by Cobhamlike bounded recursion schemata. We also give alternative characterizations which utilizes the safe recursion method developed by Bellantoni and Cook. 1. Introduction The search for recursion theoretic ..."
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. In this note we characterize iterated log depth circuit classes LD i and ND i by Cobhamlike bounded recursion schemata. We also give alternative characterizations which utilizes the safe recursion method developed by Bellantoni and Cook. 1. Introduction The search for recursion theoretic characterizations of various complexity classes was began by A. Cobham [Cob], who characterized the class of polynomial time computable functions by a scheme now called bounded recursion on notation. (See also [Ro] for the proof.) The essence of this recursion scheme is two fold: firstly, on input x the recursive call is made for jxj O(1) times where jxj is the length of x, and . secondly, the growth rate is bounded by a previously defined polynomial time function. The second condition is crucial for the characterization of resource bounded computations since the computation on each recursive call takes the value of the function as an argument, so the number of steps that each recursive ...
A Note on Induction Schemas in Bounded Arithmetic
, 2008
"... As is well known, Buss ’ theory of bounded arithmetic S 1 2 proves Σ b 0(Σ b 1) − LIND; however, we show that Allen’s D 1 2 does not prove Σ b 0(Σ b 1) − LLIND unless P = NC. We also give some interesting alternative axiomatisations of S 1 2. We assume familiarity with the theory of bounded arithm ..."
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As is well known, Buss ’ theory of bounded arithmetic S 1 2 proves Σ b 0(Σ b 1) − LIND; however, we show that Allen’s D 1 2 does not prove Σ b 0(Σ b 1) − LLIND unless P = NC. We also give some interesting alternative axiomatisations of S 1 2. We assume familiarity with the theory of bounded arithmetic S 1 2 as introduced in Buss ’ [2], as well as with the theory D 1 2 formulated by Allen in [1]. In particular, we use the general notation as introduced in [2] and in [1]. We denote the language of the theory S 1 2 by Lb, and the language of the theory D 1 2 by Ld. Thus, Lb = {0, S, +, ·, x, ⌊ 1 2 x⌋, #, ≤}, and Ld = Lb ∪ {. −, Bit(x, y), Msp(x, y),Lsp(x,y)}. The most basic theory for bounded arithmetic (which corresponds to Robinson’s Q in case of PA) for the language Lb is BASIC, introduced by Buss (see [2]), and for the language Ld is BASIC + introduced by Allen (see [1]) which extends BASIC by a few additional axioms for the extra symbols. Following [1], we abbreviate Msp(x, ⌊ 1 2 x⌋) by Fh(x), and Lsp(x, ⌊1 2 x⌋) by Bh(x). We use the usual hierarchies of formulas to measure the (bounded) quantifier complexity of formulas in our first order theories: Σb i, Πbi and Σb0 (Σbi). Here 1 Σ b 0 (Σb i) denotes the class of formulas obtained as the least closure of Σb i formulas for Boolean connectives and sharply bounded quantifiers. Theory D 1 2 is defined as BASIC + together with the schema of Σ b 1DCI: A(0) ∧ A(1) ∧ (∀x)(A(Fh(x)) ∧ A(Bh(x)) → A(x)) → (∀x)A(x). It is shown in [1] that D 1 2 is a subtheory of (an extension by definitions) of S 1 2. In the same paper Allen proves that the following schemas are provable in