144 research outputs found

    The Diophantine problem in Chevalley groups

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    In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R)G_\pi (\Phi,R), where Φ\Phi is an indecomposable root system of rank >1> 1, RR is an arbitrary commutative ring with 11. We establish a variant of double centralizer theorem for elementary unipotents xα(1)x_\alpha(1). This theorem is valid for arbitrary commutative rings with 11. The result is principle to show that any one-parametric subgroup XαX_\alpha, αΦ\alpha \in \Phi, is Diophantine in GG. Then we prove that the Diophantine problem in Gπ(Φ,R)G_\pi (\Phi,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in RR. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.Comment: 44 page

    On some modifications and applications of the post correspondence problem

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    The Post Correspondence Problem was introduced by Emil Post in 1946. The problem considers pairs of lists of sequences of symbols, or words, where each word has its place on the list determined by its index. The Post Correspondence Problem asks does there exist a sequence of indices so that, when we write the words in the order of the sequence as single words from both lists, the two resulting words are equal. Post proved the problem to be undecidable, that is, no algorithm deciding it can exist. A variety of restrictions and modifications have been introduced to the original formulation of the problem, that have then been shown to be either decidable or undecidable. Both the original Post Correspondence Problem and its modifications have been widely used in proving other decision problems undecidable. In this thesis we consider some modifications of the Post Correspondence Problem as well as some applications of it in undecidability proofs. We consider a modification for sequences of indices that are infinite to two directions. We also consider a modification to the original Post Correspondence Problem where instead of the words being equal for a sequence of indices, we take two sequences that are conjugates of each other. Two words are conjugates if we can write one word by taking the other and moving some part of that word from the end to the beginning. Both modifications are shown to be undecidable. We also use the Post Correspondence Problem and its modification for injective morphisms in proving two problems from formal language theory to be undecidable; the first problem is on special shuffling of words and the second problem on fixed points of rational functions

    Mechanised metamathematics : an investigation of first-order logic and set theory in constructive type theory

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    In this thesis, we investigate several key results in the canon of metamathematics, applying the contemporary perspective of formalisation in constructive type theory and mechanisation in the Coq proof assistant. Concretely, we consider the central completeness, undecidability, and incompleteness theorems of first-order logic as well as properties of the axiom of choice and the continuum hypothesis in axiomatic set theory. Due to their fundamental role in the foundations of mathematics and their technical intricacies, these results have a long tradition in the codification as standard literature and, in more recent investigations, increasingly serve as a benchmark for computer mechanisation. With the present thesis, we continue this tradition by uniformly analysing the aforementioned cornerstones of metamathematics in the formal framework of constructive type theory. This programme offers novel insights into the constructive content of completeness, a synthetic approach to undecidability and incompleteness that largely eliminates the notorious tedium obscuring the essence of their proofs, as well as natural representations of set theory in the form of a second-order axiomatisation and of a fully type-theoretic account. The mechanisation concerning first-order logic is organised as a comprehensive Coq library open to usage and contribution by external users.In dieser Doktorarbeit werden einige Schlüsselergebnisse aus dem Kanon der Metamathematik untersucht, unter Verwendung der zeitgenössischen Perspektive von Formalisierung in konstruktiver Typtheorie und Mechanisierung mit Hilfe des Beweisassistenten Coq. Konkret werden die zentralen Vollständigkeits-, Unentscheidbarkeits- und Unvollständigkeitsergebnisse der Logik erster Ordnung sowie Eigenschaften des Auswahlaxioms und der Kontinuumshypothese in axiomatischer Mengenlehre betrachtet. Aufgrund ihrer fundamentalen Rolle in der Fundierung der Mathematik und ihrer technischen Schwierigkeiten, besitzen diese Ergebnisse eine lange Tradition der Kodifizierung als Standardliteratur und, besonders in jüngeren Untersuchungen, eine zunehmende Bedeutung als Maßstab für Mechanisierung mit Computern. Mit der vorliegenden Doktorarbeit wird diese Tradition fortgeführt, indem die zuvorgenannten Grundpfeiler der Methamatematik uniform im formalen Rahmen der konstruktiven Typtheorie analysiert werden. Dieses Programm ermöglicht neue Einsichten in den konstruktiven Gehalt von Vollständigkeit, einen synthetischen Ansatz für Unentscheidbarkeit und Unvollständigkeit, der großteils den berüchtigten, die Essenz der Beweise verdeckenden, technischen Aufwand eliminiert, sowie natürliche Repräsentationen von Mengentheorie in Form einer Axiomatisierung zweiter Ordnung und einer vollkommen typtheoretischen Darstellung. Die Mechanisierung zur Logik erster Ordnung ist als eine umfassende Coq-Bibliothek organisiert, die offen für Nutzung und Beiträge externer Anwender ist

    Undecidable problems in quantum field theory

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    In the hope of providing some intellectual entertainment on April Fools' day of this dark year, we point out that some questions in quantum field theory are undecidable in a precise mathematical sense. More concretely, it will be demonstrated that there is no algorithm answering whether a given 2d supersymmetric Lagrangian theory breaks supersymmetry or not. It will also be shown that there is a specific 2d supersymmetric Lagrangian theory which breaks supersymmetry if and only if the standard Zermelo-Fraenkel set theory with the axiom of choice is consistent, which can never be proved or disproved as the consequence of G\"odel's second incompleteness theorem. The article includes a brief and informal introduction to the phenomenon of undecidability and its previous appearances in theoretical physics.Comment: 13 pages; a 15-minute video presentation of the content is available at https://www.youtube.com/watch?v=H548i3dnsW

    Light On String Solving: Approaches to Efficiently and Correctly Solving String Constraints

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    Widespread use of string solvers in formal analysis of string-heavy programs has led to a growing demand for more efficient and reliable techniques which can be applied in this context, especially for real-world cases. Designing an algorithm for the (generally undecidable) satisfiability problem for systems of string constraints requires a thorough understanding of the structure of constraints present in the targeted cases. We target the aforementioned case in different perspectives: We present an algorithm which works by reformulating the satisfiability of bounded word equations as a reachability problem for non-deterministic finite automata. Secondly, we present a transformation-system-based technique to solving string constraints. Thirdly, we investigate benchmarks presented in the literature containing regular expression membership predicates and design a decission procedure for a PSPACE-complete sub-theory. Additionally, we introduce a new benchmarking framework for string solvers and use it to showcase the power of our algorithms via an extensive empirical evaluation over a diverse set of benchmarks

    Mathematical Logic and Its Applications 2020

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    The issue "Mathematical Logic and Its Applications 2020" contains articles related to the following three directions: Descriptive Set Theory (3 articles). Solutions for long-standing problems, including those of A. Tarski and H. Friedman, are presented. Exact combinatorial optimization algorithms, in which the complexity relative to the source data is characterized by a low, or even first degree, polynomial (1 article). III. Applications of mathematical logic and the theory of algorithms (2 articles). The first article deals with the Jacobian and M. Kontsevich’s conjectures, and algorithmic undecidability; for these purposes, non-standard analysis is used. The second article provides a quantitative description of the balance and adaptive resource of a human. Submissions are invited for the next issue "Mathematical Logic and Its Applications 2021
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