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Tennenbaum phenomena in models of arithmetic
Kriško, Lukáš ; Thapen, Neil (advisor) ; Bulín, Jakub (referee)
The main aim of this text is to present and investigate some basic arithmetical func- tions and relations with regard to being recursive in a countable non-standard model of Peano arithmetic, PA for short, or some weaker fragment, like I∆0 or IΣ1, of PA. In PART I, we present a known result called Tennenbaum's theorem. It states that every non-standard model M of PA with domain N can have neither +M nor ×M recursive. Moreover, we present the case for + in a strengthened version for I∆0, which is due to K. McAloon. To show that not everything is lost, we also present a well know result stating that < and the successor function can be simultaneously recursive in some non-standard model of PA with domain N. In PART II, we make our own investigation into the questions related to whether there can be a non-standard model of PA s.t. x div y, the quotient function, and x mod y, the remainder function, are recursive in it. Furthermore, we often restrict y to some standard number n. To give a non-exhaustive list of problems we have solved, we showed that there can be no non-standard model of PA with both x div n and x mod n recursive. Furthermore, there can be no non-standard model of IΣ1 with x div y recursive. On the other hand, x div n and x mod n can be separately recursive in a non-standard model of PA. 1
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Kombinatorika matematických struktur
Paták, Pavel ; Krajíček, Jan (advisor) ; Thapen, Neil (referee)
The combinatorics of a first order mathematical structure is the class of all formulas valid in all in it definable structures. This notion was first introduced by Kraj'ček in [6]. In the present work we try to characterize and compare the combinatorics of several different prominent structures (reals, complex number, dense linear order, . . . ). We also study the question of algorithmical complexity, i.e. the question how hard it is to check whether a given formula lies in the combinatorics of a given structure. We prove that this question is corecursively enumeratively complete and therefore algorithmicaly undecidable in the case of models of complete theories without strict order property (SOP) and in the case of pseudofinite structures.
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Model constructions for bounded arithmetic
Garlík, Michal ; Krajíček, Jan (advisor) ; Buss, Samuel (referee) ; Thapen, Neil (referee)
Title: Model constructions for bounded arithmetic Author: Michal Garlík Abstract: We study constructions of models of bounded arithmetic theories. Us- ing basic techniques of model theory we give a new proof of Ajtai's completeness theorem for nonstandard finite structures. Working in the framework of restricted reduced powers (a generalization of the ultrapower construction) we devise two methods of constructing models of bounded arithmetic. The first one gives a new proof of Buss's witnessing theorem. Using the second method we show that the theory R1 2 is stronger than its variant strictR1 2 under a plausible computational assumption (the existence of a strong enough one-way permutation), and that the theory PV1 + Σb 1(PV ) − LLIND is stronger than PV1 + strictΣb 1(PV ) − LLIND under the same assumption. Considering relativized theories, we show that R1 2(α) is stronger than strictR1 2(α) (unconditionally). 1
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On the Power of Weak Extensions of V0
Müller, Sebastian Peter ; Krajíček, Jan (advisor) ; Thapen, Neil (referee) ; Kolodziejczyk, Leszek (referee)
Název práce: O síle slabých rozšírení teorie V0 Autor: Sebastian Müller Katedra: Katedra Algebry Vedoucí disertační práce: Prof. RNDr. Jan Krajíček, DrSc., Katedra Algebry. Abstrakt: V predložené disertacní práci zkoumáme sílu slabých fragmentu arit- metiky. Činíme tak jak z modelově-teoretického pohledu, tak z pohledu důkazové složitosti. Pohled skrze teorii modelu naznačuje, že malý iniciální segment libo- volného modelu omezené aritmetiky bude modelem silnější teorie. Jako příklad ukážeme, že každý polylogaritmický řez modelu V0 je modelem VNC. Užitím známé souvislosti mezi fragmenty omezené aritmetiky a dokazatelností v ro- zličných důkazových systémech dokážeme separaci mezi rezolucí a TC0 -Frege systémem na náhodných 3CNF-formulích s jistým poměrem počtu klauzulí vůci počtu proměnných. Zkombinováním obou výsledků dostaneme slabší separační výsledek pro rezoluci a Fregeho důkazové systémy omezené hloubky. Klíčová slova: omezená aritmetika, důkazová složitost, Fregeho důkazový systém, Fregeho důkazový systém omezené hloubky, rezoluce Title: On the Power of Weak Extensions of V0 Author: Sebastian Müller Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajíček, DrSc., Department of Algebra....
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Logika a kryptografie
Wagner, Vojtěch ; Krajíček, Jan (advisor) ; Thapen, Neil (referee)
Title: Logic and cryptography Author: Bc.Vojtěch Wagner Department: Department of Algebra Supervisor: prof. RNDr. Jan Krajíček, DrSc. Abstract: This work is devoted to a study of a formal method of formalization of cryptographic constructions. It is based on defining a multi-sorted formal logic theory T composed of strings, integers and objects of sort k - k-ary functions. We allow some operations on them, formulate axioms, terms and formulas. We also have a special type of integers called the counting integers. It denotes the number of x from a given interval satisfying formula ϕ(x). It allows us to talk about probabilities and use terms of probability theory. The work first describes this theory and then it brings a formalization of the Goldreich-Levin theorem. The goal of this work is to adapt all needed cryptographic terms into the language of T and then prove the theorem using objects, rules and axioms of T. Presented definitions and principles are ilustrated on examples. The purpose of this work is to show that such theory is sufficiently strong to prove such cryptographic constructions and verify its correctness and security. Keywords: cryptography, protocol verifying, Soundness theorem, formal logic theory, the Goldreich-Levin theorem 1
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Model constructions for bounded arithmetic
Garlík, Michal ; Krajíček, Jan (advisor) ; Buss, Samuel (referee) ; Thapen, Neil (referee)
Title: Model constructions for bounded arithmetic Author: Michal Garlík Abstract: We study constructions of models of bounded arithmetic theories. Us- ing basic techniques of model theory we give a new proof of Ajtai's completeness theorem for nonstandard finite structures. Working in the framework of restricted reduced powers (a generalization of the ultrapower construction) we devise two methods of constructing models of bounded arithmetic. The first one gives a new proof of Buss's witnessing theorem. Using the second method we show that the theory R1 2 is stronger than its variant strictR1 2 under a plausible computational assumption (the existence of a strong enough one-way permutation), and that the theory PV1 + Σb 1(PV ) − LLIND is stronger than PV1 + strictΣb 1(PV ) − LLIND under the same assumption. Considering relativized theories, we show that R1 2(α) is stronger than strictR1 2(α) (unconditionally). 1
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The BSS model and cryptography
Hostáková, Kristina ; Krajíček, Jan (advisor) ; Thapen, Neil (referee)
Real numbers are usually represented by various discrete objects such as floating points or partial decimal expansions. This is mainly because the clas- sical computability theory relates to computers which work with discrete data. Nevertheless, for theoretical purposes it is interesting to look at models of com- putation that deal with real numbers as with objects of unit size. A very natural such model was suggested by Blum, Shub and Smale in 1989. In 2012 Grigoriev and Nikolenko studied various cryptographic tasks involv- ing real numbers (for example, biometric authentication) and they considered the BSS machine model. In this work we focus on hard to invert functions in this model of computation. Our main theme is to analyse whether there are real functions of one variable that are easier to compute than to invert by a BSS machine. 1
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On the Power of Weak Extensions of V0
Müller, Sebastian Peter ; Krajíček, Jan (advisor) ; Thapen, Neil (referee) ; Kolodziejczyk, Leszek (referee)
Název práce: O síle slabých rozšírení teorie V0 Autor: Sebastian Müller Katedra: Katedra Algebry Vedoucí disertační práce: Prof. RNDr. Jan Krajíček, DrSc., Katedra Algebry. Abstrakt: V predložené disertacní práci zkoumáme sílu slabých fragmentu arit- metiky. Činíme tak jak z modelově-teoretického pohledu, tak z pohledu důkazové složitosti. Pohled skrze teorii modelu naznačuje, že malý iniciální segment libo- volného modelu omezené aritmetiky bude modelem silnější teorie. Jako příklad ukážeme, že každý polylogaritmický řez modelu V0 je modelem VNC. Užitím známé souvislosti mezi fragmenty omezené aritmetiky a dokazatelností v ro- zličných důkazových systémech dokážeme separaci mezi rezolucí a TC0 -Frege systémem na náhodných 3CNF-formulích s jistým poměrem počtu klauzulí vůci počtu proměnných. Zkombinováním obou výsledků dostaneme slabší separační výsledek pro rezoluci a Fregeho důkazové systémy omezené hloubky. Klíčová slova: omezená aritmetika, důkazová složitost, Fregeho důkazový systém, Fregeho důkazový systém omezené hloubky, rezoluce Title: On the Power of Weak Extensions of V0 Author: Sebastian Müller Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajíček, DrSc., Department of Algebra....
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Kombinatorika matematických struktur
Paták, Pavel ; Krajíček, Jan (advisor) ; Thapen, Neil (referee)
The combinatorics of a first order mathematical structure is the class of all formulas valid in all in it definable structures. This notion was first introduced by Kraj'ček in [6]. In the present work we try to characterize and compare the combinatorics of several different prominent structures (reals, complex number, dense linear order, . . . ). We also study the question of algorithmical complexity, i.e. the question how hard it is to check whether a given formula lies in the combinatorics of a given structure. We prove that this question is corecursively enumeratively complete and therefore algorithmicaly undecidable in the case of models of complete theories without strict order property (SOP) and in the case of pseudofinite structures.
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