20 found
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  1.  22
    [Image] -Determinacy, Comprehension and Induction.Medyahya Ould Medsalem & Kazuyuki Tanaka - 2007 - Journal of Symbolic Logic 72 (2):452 - 462.
    We show that each of $\Delta _{3}^{1}-{\rm CA}_{0}+\Sigma _{3}^{1}-{\rm IND}$ and $\Pi _{2}^{1}-{\rm CA}_{0}+\Pi _{3}^{1}-{\rm TI}$ proves $\Delta _{3}^{0}-{\rm Det}$ and that neither $\Sigma _{3}^{1}-{\rm IND}$ nor $\Pi _{3}^{1}-{\rm TI}$ can be dropped. We also show that neither $\Delta _{3}^{1}-{\rm CA}_{0}+\Sigma _{\infty}^{1}-{\rm IND}$ nor $\Pi _{2}^{1}-{\rm CA}_{0}+\Pi _{\infty}^{1}-{\rm TI}$ proves $\Sigma _{3}^{0}-{\rm Det}$. Moreover, we prove that none of $\Delta _{2}^{1}-{\rm CA}_{0}$, $\Sigma _{3}^{1}-{\rm IND}$ and $\Pi _{2}^{1}-{\rm TI}$ is provable in $\Delta _{1}^{1}-{\rm Det}_{0}={\rm ACA}_{0}+\Delta _{1}^{1}-{\rm Det}$.
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  2.  17
    Δ 0 3 -Determinacy, Comprehension and Induction.MedYahya Ould MedSalem & Kazuyuki Tanaka - 2007 - Journal of Symbolic Logic 72 (2):452-462.
    We show that each of Δ13-CA0 + Σ13-IND and Π12-CA0 + Π13-TI proves Δ03-Det and that neither Σ31-IND nor Π13-TI can be dropped. We also show that neither Δ13-CA0 + Σ1∞-IND nor Π12-CA0 + Π1∞-TI proves Σ03-Det. Moreover, we prove that none of Δ21-CA0, Σ31-IND and Π21-TI is provable in Δ11-Det0 = ACA0 + Δ11-Det.
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  3.  14
    Weak Axioms of Determinacy and Subsystems of Analysis II.Kazuyuki Tanaka - 1991 - Annals of Pure and Applied Logic 52 (1-2):181-193.
    In [10], we have shown that the statement that all ∑ 1 1 partitions are Ramsey is deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition,but the reversal needs П 1 1 - CA 0 rather than ATR 0 . By contrast, we show in this paper that the statement that all ∑ 0 2 games are determinate is also deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition, but the (...)
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  4.  10
    Fixed Point Theory in Weak Second-Order Arithmetic.Naoki Shioji & Kazuyuki Tanaka - 1990 - Annals of Pure and Applied Logic 47 (2):167-188.
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  5.  26
    Weak Axioms of Determinacy and Subsystems of Analysis I: Δ20 Games.Kazuyuki Tanaka - 1990 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (6):481-491.
  6.  18
    On Formalization of Model-Theoretic Proofs of Gödel's Theorems.Makoto Kikuchi & Kazuyuki Tanaka - 1994 - Notre Dame Journal of Formal Logic 35 (3):403-412.
    Within a weak subsystem of second-order arithmetic , that is -conservative over , we reformulate Kreisel's proof of the Second Incompleteness Theorem and Boolos' proof of the First Incompleteness Theorem.
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  7.  28
    Non‐Standard Analysis in WKL0.Kazuyuki Tanaka - 1997 - Mathematical Logic Quarterly 43 (3):396-400.
    Within a weak subsystem of second-order arithmetic WKL0, we develop basic part of non-standard analysis up to the Peano existence theorem.
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  8.  36
    Some Conservation Results on Weak König's Lemma.Stephen G. Simpson, Kazuyuki Tanaka & Takeshi Yamazaki - 2002 - Annals of Pure and Applied Logic 118 (1-2):87-114.
    By , we denote the system of second-order arithmetic based on recursive comprehension axioms and Σ10 induction. is defined to be plus weak König's lemma: every infinite tree of sequences of 0's and 1's has an infinite path. In this paper, we first show that for any countable model M of , there exists a countable model M′ of whose first-order part is the same as that of M, and whose second-order part consists of the M-recursive sets and sets not (...)
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  9.  4
    The Galvin-Prikry Theorem and Set Existen Axioms.Kazuyuki Tanaka - 1989 - Annals of Pure and Applied Logic 42 (1):81-104.
  10.  45
    A Non-Standard Construction of Haar Measure and Weak König's Lemma.Kazuyuki Tanaka & Takeshi Yamazaki - 2000 - Journal of Symbolic Logic 65 (1):173-186.
    In this paper, we show within RCA 0 that weak Konig's lemma is necessary and sufficient to prove that any (separable) compact group has a Haar measure. Within WKL 0 , a Haar measure is constructed by a non-standard method based on a fact that every countable non-standard model of WKL 0 has a proper initial part isomorphic to itself [10].
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  11.  39
    The Strong Soundness Theorem for Real Closed Fields and Hilbert’s Nullstellensatz in Second Order Arithmetic.Nobuyuki Sakamoto & Kazuyuki Tanaka - 2004 - Archive for Mathematical Logic 43 (3):337-349.
    By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA 0.
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  12.  24
    A Game-Theoretic Proof of Analytic Ramsey Theorem.Kazuyuki Tanaka - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):301-304.
  13.  36
    Infinite Games in the Cantor Space and Subsystems of Second Order Arithmetic.Takako Nemoto, MedYahya Ould MedSalem & Kazuyuki Tanaka - 2007 - Mathematical Logic Quarterly 53 (3):226-236.
    In this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems:1. RCA0 ⊢ equation image-Det* ↔ equation image-Det* ↔ WKL0.2. RCA0 ⊢ 2-Det* ↔ ACA0.3. RCA0 ⊢ equation image-Det* ↔ equation image-Det* ↔ equation image-Det ↔ equation image-Det ↔ ATR0.4. For 1 < k < ω, RCA0 ⊢ k-Det* ↔ k –1-Det.5. RCA0 ⊢ equation image-Det* ↔ equation image-Det.Here, Det* stands for (...)
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  14.  9
    A Game‐Theoretic Proof of Analytic Ramsey Theorem.Kazuyuki Tanaka - 1992 - Mathematical Logic Quarterly 38 (1):301-304.
    We give a simple game-theoretic proof of Silver's theorem that every analytic set is Ramsey. A set P of subsets of ω is called Ramsey if there exists an infinite set H such that either all infinite subsets of H are in P or all out of P. Our proof clarifies a strong connection between the Ramsey property of partitions and the determinacy of infinite games.
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  15.  9
    Weak Axioms of Determinacy and Subsystems of Analysis I: Δmath Image Games.Kazuyuki Tanaka - 1990 - Mathematical Logic Quarterly 36 (6):481-491.
  16.  10
    Statistical Analysis of the Expectation-Maximization Algorithm with Loopy Belief Propagation in Bayesian Image Modeling.Shun Kataoka, Muneki Yasuda, Kazuyuki Tanaka & D. M. Titterington - 2012 - Philosophical Magazine 92 (1-3):50-63.
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  17.  18
    Maximum Marginal Likelihood Estimation and Constrained Optimization in Image Restoration.Kazuyuki Tanaka - 2001 - 人工知能学会論文誌: Transactions of the Japanese Society for Artificial Intelligence 16:246-258.
  18.  4
    Maximum Marginal Likelihood Estimation and Constrained Optimization in Image Restoration.Kazuyuki Tanaka - 2001 - Transactions of the Japanese Society for Artificial Intelligence 16:246-258.
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  19.  7
    TAP Equation for Non-Negative Boltzmann Machine.Muneki Yasuda & Kazuyuki Tanaka - 2012 - Philosophical Magazine 92 (1-3):192-209.
  20.  14
    Infinite Games and Transfinite Recursion of Multiple Inductive Definitions.Keisuke Yoshii & Kazuyuki Tanaka - 2012 - In S. Barry Cooper (ed.), How the World Computes. pp. 374--383.
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