<p> Propositional proof complexity is a field of theoretical computer science which concerns itself with the lengths of formal proofs in various propositional proof systems. Frege systems are an important class of propositional proof systems. Extended Frege augments them by allowing the introduction of new variables to abbreviate formulas. Perhaps the largest open question in propositional proof complexity is whether or not Extended Frege is significantly more powerful that Frege. Several proof systems, each introducing new rules or syntax to Frege, have been developed in an attempt to shed some light on this problem.</p> <p> We introduce one such system, which we call H, which allows for the quantification of transpositions of propositional variables. We show that H is sound and complete, and that H's transposition quantifiers efficiently represent any permutation.</p> <p> The most important contribution is showing that a fragment of this proof system, H*1, is equivalent in power to Extended Frege. This is a complicated and rather technical result, and is achieved by showing that H*1 can efficiently prove translations of the first-order logical theory ∀PLA, a logical theory well suited for reasoning about linear algebra and properties of graphs.</p> / Thesis / Master of Science (MSc)

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....

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第23317号 / 情博第753号 / 新制||情||129(附属図書館) / 京都大学大学院情報学研究科社会情報学専攻 / (主査)教授 神田 崇行, 教授 吉川 正俊, 教授 岡部 寿男 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Automated proof system for logic of correlated knowledge is presented in the dissertation. The system consists of the sequent calculus GS-LCK and the proof search procedure GS-LCK-PROC. Sequent calculus is sound, complete and satisfy the properties of invertibility of rules, admissibility of weakening, contraction and cut. The procedure GS-LCK-PROC is terminating and allows to check if the sequent is provable. Also decidability of logic of correlated knowledge has been proved. Using the terminating procedure GS-LCK-PROC the validity of all formulas of logic of correlated knowledge can be checked. / Automatinė įrodymų sistema koreliatyvių žinių logikai yra pristatoma disertacijoje. Sistemą sudaro sekvencinis skaičiavimas GS-LCK ir įrodymo paieškos procedūra GS-LCK-PROC. Sekvencinis skaičiavimas yra pagrįstas, pilnas ir tenkina taisyklių apverčiamumo, silpninimo, prastinimo ir pjūvio leistinumo savybes. Procedūra GS-LCK-PROC yra baigtinė ir leidžia patikrinti, ar sekvencija yra išvedama. Taip pat buvo įrodytas koreliatyvių žinių logikos išsprendžiamumas. Naudojant baigtinę procedūra GS-LCK-PROC, visų koreliatyvių žinių logikos formulių tapatus teisingumas gali būti patikrintas.

Automatinė įrodymų sistema koreliatyvių žinių logikai yra pristatoma disertacijoje. Sistemą sudaro sekvencinis skaičiavimas GS-LCK ir įrodymo paieškos procedūra GS-LCK-PROC. Sekvencinis skaičiavimas yra pagrįstas, pilnas ir tenkina taisyklių apverčiamumo, silpninimo, prastinimo ir pjūvio leistinumo savybes. Procedūra GS-LCK-PROC yra baigtinė ir leidžia patikrinti, ar sekvencija yra išvedama. Taip pat buvo įrodytas koreliatyvių žinių logikos išsprendžiamumas. Naudojant baigtinę procedūra GS-LCK-PROC, visų koreliatyvių žinių logikos formulių tapatus teisingumas gali būti patikrintas. / Automated proof system for logic of correlated knowledge is presented in the dissertation. The system consists of the sequent calculus GS-LCK and the proof search procedure GS-LCK-PROC. Sequent calculus is sound, complete and satisfy the properties of invertibility of rules, admissibility of weakening, contraction and cut. The procedure GS-LCK-PROC is terminating and allows to check if the sequent is provable. Also decidability of logic of correlated knowledge has been proved. Using the terminating procedure GS-LCK-PROC the validity of all formulas of logic of correlated knowledge can be checked.

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Les smooth (ou universal) projective hash functions ont été introduites par Cramer et Shoup, à Eurocrypt'02, comme un outil pour construire des schémas de chiffrement efficaces et sûrs contre les attaques à chiffrés choisis. Depuis, elles ont trouvé de nombreuses applications, notamment pour la construction de schémas d'authentification par mot de passe, d'oblivious transfer, de signatures en blanc, et de preuves à divulgation nulle de connaissance. Elles peuvent êtres vues comme des preuves implicites d'appartenance à certains langages. Un problème important est de caractériser pour quels langages de telles fonctions existent.Dans cette thèse, nous avançons dans la résolution de ce problème en proposant la notion de diverse modules. Un diverse module est une représentation d'un langage, comme un sous-module d'un module plus grand, un module étant un espace vectoriel sur un anneau. À n'importe quel diverse module est associée une smooth projective hash function pour le même langage. Par ailleurs, presque toutes les smooth projective hash functions actuelles sont construites de cette manière.Mais les diverse modules sont aussi intéressants en eux-mêmes. Grâce à leur structure algébrique, nous montrons qu'ils peuvent facilement être combinés pour permettre de nouvelles applications, comme les preuves implicites à divulgation nulle de connaissance (une alternative légère aux preuves non-interactives à divulgation nulle de connaissance), ainsi que des preuves non-interactives à divulgation nulle de connaissance et one-time simulation-sound très efficaces pour les langages linéaires sur les groupes cycliques. / Smooth (or universal) projective hash functions were first introduced by Cramer and Shoup, at Eurocrypt'02, as a tool to construct efficient encryption schemes, indistinguishable under chosen-ciphertext attacks. Since then, they have found many other applications, including password-authenticated key exchange, oblivious transfer, blind signatures, and zero-knowledge arguments. They can be seen as implicit proofs of membership for certain languages. An important question is to characterize which languages they can handle.In this thesis, we make a step forward towards this goal, by introducing diverse modules. A diverse module is a representation of a language, as a submodule of a larger module, where a module is essentially a vector space over a ring. Any diverse module directly yields a smooth projective hash function for the corresponding language, and almost all the known smooth projective hash functions are constructed this way.Diverse modules are also valuable in their own right. Thanks to their algebraic structural properties, we show that they can be easily combined to provide new applications related to zero-knowledge notions, such as implicit zero-knowledge arguments (a lightweight alternative to non-interactive zero-knowledge arguments), and very efficient one-time simulation-sound (quasi-adaptive) non-interactive zero-knowledge arguments for linear languages over cyclic groups.