Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes

Set based failure diagnosis for concurrent constraint programming

Oz is a recent high-level programming language, based on an extension of the concurrent constraint model by higher-order procedures and state. Oz is a dynamically typed language like Prolog, Scheme, or Smalltalk. We investigate two approaches of making static type analysis available for Oz: Set-based failure diagnosis and strong typing. We define a new system of set constraints over feature trees that is appropriate for the analysis of record structures, and we investigate its satisfiability, emptiness, and entailment problem. We present a set-based diagnosis for constraint logic programming and concurrent constraint programming as first-order fragments of Oz, and we prove that it correctly detects inevitable run-time errors. We also propose an analysis for a larger sublanguage of Oz. Complementarily, we define an Oz-style language called Plain that allows an expressive strong type system. We present such a type system and prove its soundness.Oz ist eine anwendungsnahe Programmiersprache, deren Grundlage eine Erweiterung des Modells nebenläufiger Constraintprogrammierung um Prozeduren höherer Stufe und Zustand ist. Oz ist eine Sprache mit dynamischer Typüberprüfung wie Prolog, Scheme oder Smalltalk. Wir untersuchen zwei Ansätze, statische Typüberprüfung für Oz zu ermöglichen: Mengenbasierte Fehlerdiagnose und Starke Typisierung. Wir definieren ein neues System von Mengenconstraints über Featurebäumen, das für die Analyse von Recordstrukturen geeignet ist, und wir untersuchen das Erfüllbarkeits-, das Leerheits- und das Subsumtionsproblem für dieses Constraintsystem. Wir präsentieren eine mengenbasierte Diagnose für Constraint-Logikprogrammierung und für nebenläufige Constraintprogrammierung als Teilsprachen von Oz, und wir beweisen, daß diese unvermeidliche Laufzeitfehler erkennt. Wir schlagen auch eine mengenbasierte Analyse für eine grössere Teilsprache von Oz vor. Komplementär dazu definieren wir eine Oz-artige Sprache genannt Plain, die ein expressives starkes Typsystem erlaubt. Wir stellen ein solches Typsystem vor und beweisen seine Korrektheit

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis

Description Logic (abbrv. DL) belongs to the field of knowledge representation and reasoning. DL researchers have developed a large family of logic-based languages, so-called description logics (abbrv. DLs). These logics allow their users to explicitly represent knowledge as ontologies, which are finite sets of (human- and machine-readable) axioms, and provide them with automated inference services to derive implicit knowledge. The landscape of decidability and computational complexity of common reasoning tasks for various description logics has been explored in large parts: there is always a trade-off between expressibility and reasoning costs. It is therefore not surprising that DLs are nowadays applied in a large variety of domains: agriculture, astronomy, biology, defense, education, energy management, geography, geoscience, medicine, oceanography, and oil and gas. Furthermore, the most notable success of DLs is that these constitute the logical underpinning of the Web Ontology Language (abbrv. OWL) in the Semantic Web. Formal Concept Analysis (abbrv. FCA) is a subfield of lattice theory that allows to analyze data-sets that can be represented as formal contexts. Put simply, such a formal context binds a set of objects to a set of attributes by specifying which objects have which attributes. There are two major techniques that can be applied in various ways for purposes of conceptual clustering, data mining, machine learning, knowledge management, knowledge visualization, etc. On the one hand, it is possible to describe the hierarchical structure of such a data-set in form of a formal concept lattice. On the other hand, the theory of implications (dependencies between attributes) valid in a given formal context can be axiomatized in a sound and complete manner by the so-called canonical base, which furthermore contains a minimal number of implications w.r.t. the properties of soundness and completeness. In spite of the different notions used in FCA and in DLs, there has been a very fruitful interaction between these two research areas. My thesis continues this line of research and, more specifically, I will describe how methods from FCA can be used to support the automatic construction and extension of DL ontologies from data

Set based failure diagnosis for concurrent constraint programming

Oz is a recent high-level programming language, based on an extension of the concurrent constraint model by higher-order procedures and state. Oz is a dynamically typed language like Prolog, Scheme, or Smalltalk. We investigate two approaches of making static type analysis available for Oz: Set-based failure diagnosis and strong typing. We define a new system of set constraints over feature trees that is appropriate for the analysis of record structures, and we investigate its satisfiability, emptiness, and entailment problem. We present a set-based diagnosis for constraint logic programming and concurrent constraint programming as first-order fragments of Oz, and we prove that it correctly detects inevitable run-time errors. We also propose an analysis for a larger sublanguage of Oz. Complementarily, we define an Oz-style language called Plain that allows an expressive strong type system. We present such a type system and prove its soundness.Oz ist eine anwendungsnahe Programmiersprache, deren Grundlage eine Erweiterung des Modells nebenläufiger Constraintprogrammierung um Prozeduren höherer Stufe und Zustand ist. Oz ist eine Sprache mit dynamischer Typüberprüfung wie Prolog, Scheme oder Smalltalk. Wir untersuchen zwei Ansätze, statische Typüberprüfung für Oz zu ermöglichen: Mengenbasierte Fehlerdiagnose und Starke Typisierung. Wir definieren ein neues System von Mengenconstraints über Featurebäumen, das für die Analyse von Recordstrukturen geeignet ist, und wir untersuchen das Erfüllbarkeits-, das Leerheits- und das Subsumtionsproblem für dieses Constraintsystem. Wir präsentieren eine mengenbasierte Diagnose für Constraint-Logikprogrammierung und für nebenläufige Constraintprogrammierung als Teilsprachen von Oz, und wir beweisen, daß diese unvermeidliche Laufzeitfehler erkennt. Wir schlagen auch eine mengenbasierte Analyse für eine grössere Teilsprache von Oz vor. Komplementär dazu definieren wir eine Oz-artige Sprache genannt Plain, die ein expressives starkes Typsystem erlaubt. Wir stellen ein solches Typsystem vor und beweisen seine Korrektheit

Approximate equivalence relations

Generalizing results for approximate subgroups, we study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover’s normal form is valid here; if one begins with a discrete logic structure, it reduces arbitrary formulas of probability logic to correlations between quantifier-free formulas. We completely classify binary correlations in terms of the Kim–Pillay space, leading to strong results on the interpretative power of pure probability logic over a binary language. Assuming higher amalgamation of independent types, we prove a higher stationarity statement for such correlations. We also give a short model-theoretic proof of a categoricity theorem for continuous logic structures with a measure of full support, generalizing theorems of Gromov–Vershik and Keisler, and often providing a canonical model for a complete pure probability logic theory. These results also apply to local probability logic, providing in particular a canonical model for a local pure probability logic theory with a unique 1-type and geodesic metric. For sequences of approximate equivalence relations with an “approximately unique” probability logic 1-type, we obtain a structure theorem generalizing the “Lie model” theorem for approximate subgroups (Theorem 5.5). The models here are Riemannian homogeneous spaces, fibered over a locally finite graph. Specializing to definable graphs over finite fields, we show that after a finite partition, a definable binary relation converges in finitely many self-compositions to an equivalence relation of geometric origin. This generalizes the main lemma for strong approximation of groups. For NIP theories, pursuing a question of Pillay’s, we prove an archimedean finite-dimensionality statement for the automorphism groups of definable measures, acting on a given type of definable sets. This can be seen as an archimedean analogue of results of Macpherson and Tent on NIP profinite groups