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Relating the Provable Collapse of P to NC¹ and the Power of Logical Theories
"... We show that the following three statements are equivalent: QPV is conservative over QALV, QALV proves its open induction formulas, and QALV proves P=NC¹. Here QPV and QALV are first order theories whose function symbols range over polynomialtime and NC¹ functions, respectively. ..."
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We show that the following three statements are equivalent: QPV is conservative over QALV, QALV proves its open induction formulas, and QALV proves P=NC¹. Here QPV and QALV are first order theories whose function symbols range over polynomialtime and NC¹ functions, respectively.
Sharply Bounded Alternation within P
, 1996
"... We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilineartime computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. T ..."
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Cited by 5 (3 self)
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We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilineartime computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We define both SBH (QL) and its corresponding hierarchy of function classes, FSBH(QL),and present a variety of problems in these classes, including ql m complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that certain simple structural conditions on it would imply P 6= PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on firstorder definability, as well as recursiontheoretic characterizations of function classes corresponding to SBH (QL).
Sharply bounded alternation and quasilinear time
 Theory of Computing Systems
, 1998
"... We de ne the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P, using quasilineartime computation and quanti cation over strings of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The n ..."
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Cited by 4 (0 self)
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We de ne the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P, using quasilineartime computation and quanti cation over strings of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy hasseveral alternative characterizations. We de ne both SBH (QL) and its corresponding hierarchy of function classes, ql and present a variety of problems in these classes, including mcomplete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that determining its precise relationship to deterministic time classes can imply P 6 = PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on rstorder de nability, aswell as recursiontheoretic characterizations of function classes corresponding to SBH (QL).
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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) 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...
A Note on the Relation between Polynomial Time Functionals and Constable's Class K
 IN KLEINEBUNING, EDITOR, COMPUTER SCIENCE LOGIC. SPRINGER LECTURE NOTES IN COMPUTER SCIENCE
, 1996
"... . A result claimed without proof by R. Constable in a STOC73 paper is here corrected: a strictly increasing function f is presented for which Constable's class K(f) is properly contained in FP (f ), the collection of functions polynomial time computable in f . ..."
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. A result claimed without proof by R. Constable in a STOC73 paper is here corrected: a strictly increasing function f is presented for which Constable's class K(f) is properly contained in FP (f ), the collection of functions polynomial time computable in f .
Circuit Complexity and the Expressive Power of Generalized FirstOrder Formulas
"... . The circuit complexity classes AC 0 ; ACC; and CC (also called pureACC) can be characterized as the classes of languages definable in certain extensions of firstorder logic. All of the known and conjectured inclusions among these classes have been shown to be equivalent to a single conjecture ..."
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. The circuit complexity classes AC 0 ; ACC; and CC (also called pureACC) can be characterized as the classes of languages definable in certain extensions of firstorder logic. All of the known and conjectured inclusions among these classes have been shown to be equivalent to a single conjecture concerning the form of the formulas required to define the regular languages they contain. (The conjecture states, roughly, that when a formula defines a regular language, predicates representing numerical relations on the positions in a string can be replaced by predicates computed by finite state automata.) Here this conjecture is established in a special case: It is shown that the conjecture holds for the subclasses of AC 0 ; ACC; and CC defined by restricting all the numerical predicates occurring in the logical formulas to be either unary relations, or the order relation ! : 1 Introduction Certain formulas of predicate logic can be interpreted in a natural way in strings over a fini...
Invariant Definability and P/poly
, 1999
"... . We look at various uniform and nonuniform complexity classes within P=poly and its variations L=poly, NL=poly, NP=poly and PSpace=poly, and look for analogues of the AjtaiImmerman theorem which characterizes AC0 as the nonuniformly First Order Definable classes of finite structures. We have pr ..."
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. We look at various uniform and nonuniform complexity classes within P=poly and its variations L=poly, NL=poly, NP=poly and PSpace=poly, and look for analogues of the AjtaiImmerman theorem which characterizes AC0 as the nonuniformly First Order Definable classes of finite structures. We have previously observed that the AjtaiImmerman theorem can be rephrased in terms of invariant definability: A class of finite structures is FOL invariantly definable iff it is in AC0 . Invariant definability is a notion closely related to but different from implicit definability and \Deltadefinability. Its exact relationship to these other notions of definability has been determined in [Mak97]. Our first results are a slight generalization of similar results due to Molzan and can be stated as follows: let C be one of L; NL;P, NP, PSpace and L be a logic which captures C on ordered structures. Then the nonuniform Linvariantly definable classes of (not necessarily ordered) finite structures are...
Nondeterministic Stack Register Machines
, 1996
"... For integer k 0, let srm(n O(1) ; k) denote the collection of relations computable by a stack register machine with stack registers bounded by a polynomial p(n) in the input n, and work registers bounded by k. Let nsrm(n O(1) ; k) denote the analogous class accepted by nondeterministic stack r ..."
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For integer k 0, let srm(n O(1) ; k) denote the collection of relations computable by a stack register machine with stack registers bounded by a polynomial p(n) in the input n, and work registers bounded by k. Let nsrm(n O(1) ; k) denote the analogous class accepted by nondeterministic stack register machines. In this paper, nondeterminism is shown to provide no additional power. Specifically, nsrm(n O(1) ; 0) = srm(n O(1) ; 0) nsrm(n O(1) ; 1) = srm(n O(1) ; 1) nsrm(n O(1) ; k) = srm(n O(1) ; k); for k 4 srm(n O(1) ; k) = alintime ; for k 4: