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Peter Clote [10]P. Clote [8]Peter G. Clote [5]
  1.  22
    Members of Countable Π10 Classes.Douglas Cenzer, Peter Clote, Rick L. Smith, Robert I. Soare & Stanley S. Wainer - 1986 - Annals of Pure and Applied Logic 31:145-163.
  2.  17
    Bounded Arithmetic for NC, ALogTIME, L and NL.P. Clote & G. Takeuti - 1992 - Annals of Pure and Applied Logic 56 (1-3):73-117.
    We define theories of bounded arithmetic, whose definable functions and relations are exactly those in certain complexity classes. Based on a recursion-theoretic characterization of NC in Clote , the first-order theory TNC, whose principal axiom scheme is a form of short induction on notation for nondeterministic polynomial-time computable relations, has the property that those functions having nondeterministic polynomial-time graph Θ such that TNC x y Θ are exactly the functions in NC, computable on a parallel random-access machine in polylogarithmic parallel (...)
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  3.  13
    The Metamathematics of Scattered Linear Orderings.P. Clote - 1989 - Archive for Mathematical Logic 29 (1):9-20.
    Pursuing the proof-theoretic program of Friedman and Simpson, we begin the study of the metamathematics of countable linear orderings by proving two main results. Over the weak base system consisting of arithmetic comprehension, II 1 1 -CA0 is equivalent to Hausdorff's theorem concerning the canonical decomposition of countable linear orderings into a sum over a dense or singleton set of scattered linear orderings. Over the same base system, ATR0 is equivalent to a version of the Continuum Hypothesis for linear orderings, (...)
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  4.  10
    On Some Formalized Conservation Results in Arithmetic.P. Clote, P. Hájek & J. Paris - 1990 - Archive for Mathematical Logic 30 (4):201-218.
    IΣ n andBΣ n are well known fragments of first-order arithmetic with induction and collection forΣ n formulas respectively;IΣ n 0 andBΣ n 0 are their second-order counterparts. RCA0 is the well known fragment of second-order arithmetic with recursive comprehension;WKL 0 isRCA 0 plus weak König's lemma. We first strengthen Harrington's conservation result by showing thatWKL 0 +BΣ n 0 is Π 1 1 -conservative overRCA 0 +BΣ n 0 . Then we develop some model theory inWKL 0 and illustrate (...)
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  5.  48
    Cutting Planes, Connectivity, and Threshold Logic.Samuel R. Buss & Peter Clote - 1996 - Archive for Mathematical Logic 35 (1):33-62.
    Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCP q ofCP, forq≥2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem forCP (...)
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  6.  26
    Two Further Combinatorial Theorems Equivalent to the 1-Consistency of Peano Arithmetic.Peter Clote & Kenneth Mcaloon - 1983 - Journal of Symbolic Logic 48 (4):1090-1104.
  7. Ash, CJ, Stability of Recursive Structures in Arithmetical Degrees Ash, CJ, [email protected] in Hyperarithmetical Degrees.D. Cenzer, P. Clote, R. L. Smith, S. S. Wainer, K. J. Compton, C. W. Henson & S. Shelah - 1988 - Annals of Pure and Applied Logic 40:307-310.
  8. Review: Harvey M. Friedman, Stephen G. Simpson, Rick L. Smith, Countable Algebra and Set Existence Axioms; Harvey M. Friedman, Stephen G. Simpson, Rick L. Smith, Addendum to "Countable Algebra and Set Existence Axioms.". [REVIEW]Peter G. Clote - 1987 - Journal of Symbolic Logic 52 (1):276-278.
  9. Review: Stephen Simpson, Logic and Combinatorics. [REVIEW]P. Clote - 1992 - Journal of Symbolic Logic 57 (4):1491-1497.
  10.  14
    Wilfried Sieg. Fragments of Arithmetic. Annals of Pure and Applied Logic, Vol. 28 , Pp. 33–71.Peter G. Clote - 1987 - Journal of Symbolic Logic 52 (4):1054-1055.
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  11.  13
    Harvey M. Friedman, Stephen G. Simpson, and Rick L. Smith. Countable Algebra and Set Existence Axioms. Annals of Pure and Applied Logic, Vol. 25 , Pp. 141–181. - Harvey M. Friedman, Stephen G. Simpson, and Rick L. Smith. Addendum to “Countable Algebra and Set Existence Axioms.” Annals of Pure and Applied Logic, Vol. 28 , Pp. 319–320. [REVIEW]Peter G. Clote - 1987 - Journal of Symbolic Logic 52 (1):276-278.
  12.  10
    Wilfried Sieg. Reductions of Theories for Analysis. Foundations of Logic and Linguistics, Problems and Their Solutions, Edited by Georg Dorn and P. Weingartner, Plenum Press, New York and London1985, Pp. 199– 231. [REVIEW]Peter Clote - 1990 - Journal of Symbolic Logic 55 (1):354-354.
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  13.  16
    Krajíček Jan. Bounded Arithmetic, Propositional Logic, and Complexity Theory. Encyclopedia of Mathematics and its Applications, Vol. 60. Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1995, Xiv + 343 Pp. [REVIEW]P. Clote - 1999 - Journal of Symbolic Logic 64 (3):1357-1362.
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  14.  13
    Logic and Combinatorics, Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference Held August 4–10, 1985, Edited by Simpson Stephen, Contemporary Mathematics, Vol. 65, American Mathematical Society, Providence 1987, Xi + 394 Pp. [REVIEW]P. Clote - 1992 - Journal of Symbolic Logic 57 (4):1491-1497.
  15.  12
    Review: Jan Krajicek, Bounded Arithmetic, Propositional Logic, and Complexity Theory. [REVIEW]P. Clote - 1999 - Journal of Symbolic Logic 64 (3):1357-1362.
  16.  28
    A Generalization of the Limit Lemma and Clopen Games.Peter Clote - 1986 - Journal of Symbolic Logic 51 (2):273-291.
    We give a new characterization of the hyperarithmetic sets: a set X of integers is recursive in e α if and only if there is a Turing machine which computes X and "halts" in less than or equal to the ordinal number ω α of steps. This result represents a generalization of the well-known "limit lemma" due to J. R. Shoenfield [Sho-1] and later independently by H. Putnam [Pu] and independently by E. M. Gold [Go]. As an application of this (...)
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  17.  5
    Countable Algebra and Set Existence Axioms.Peter G. Clote - 1987 - Journal of Symbolic Logic 52 (1):276-278.
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  18.  7
    Editor's Introduction.Peter Clote - 1995 - Notre Dame Journal of Formal Logic 36 (4):499-501.
    This collection of articles on Models of Arithmetic is dedicated to the memory of Zygmunt Ratajczyk, who contributed a number of important results to the field, and who unexpectedly died in February 1994.
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  19. Review: Wilfried Sieg, Georg Dorn, P. Weingartner, Reductions of Theories for Analysis. [REVIEW]Peter Clote - 1990 - Journal of Symbolic Logic 55 (1):354-354.
  20.  13
    A Recursion Theoretic Analysis of the Clopen Ramsey Theorem.Peter Clote - 1984 - Journal of Symbolic Logic 49 (2):376-400.
    Solovay has shown that if F: [ω] ω → 2 is a clopen partition with recursive code, then there is an infinite homogeneous hyperarithmetic set for the partition (a basis result). Simpson has shown that for every 0 α , where α is a recursive ordinal, there is a clopen partition F: [ω] ω → 2 such that every infinite homogeneous set is Turing above 0 α (an anti-basis result). Here we refine these results, by associating the "order type" of (...)
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  21.  1
    Review: Wilfried Sieg, Fragments of Arithmetic. [REVIEW]Peter G. Clote - 1987 - Journal of Symbolic Logic 52 (4):1054-1055.