Guarded Teams: The Horizontally Guarded Case

Team semantics admits reasoning about large sets of data, modelled by sets of assignments (called teams), with first-order syntax. This leads to high expressive power and complexity, particularly in the presence of atomic dependency properties for such data sets. It is therefore interesting to explore fragments and variants of logic with team semantics that permit model-theoretic tools and algorithmic methods to control this explosion in expressive power and complexity. We combine here the study of team semantics with the notion of guarded logics, which are well-understood in the case of classical Tarski semantics, and known to strike a good balance between expressive power and algorithmic manageability. In fact there are two strains of guardedness for teams. Horizontal guardedness requires the individual assignments of the team to be guarded in the usual sense of guarded logics. Vertical guardedness, on the other hand, posits an additional (or definable) hypergraph structure on relational structures in order to interpret a constraint on the component-wise variability of assignments within teams. In this paper we investigate the horizontally guarded case. We study horizontally guarded logics for teams and appropriate notions of guarded team bisimulation. In particular, we establish characterisation theorems that relate invariance under guarded team bisimulation with guarded team logics, but also with logics under classical Tarski semantics

A Simple Logic of Functional Dependence

This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games.Comment: 56 pages. Journal of Philosophical Logic (2021

SyTeCi: Automating Contextual Equivalence for Higher-Order Programs with References

International audienc

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

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

Characterising Fixed Parameter Tractability for Query Evaluation over Guarded TGDs

We consider the parameterized complexity of evaluating Ontology Mediated Queries (OMQ) based on Guarded TGDs (GTGD) and Unions of Conjunctive Queries, in the case where relational symbols have unrestricted arity and where the parameter is the size of the OMQ. We establish exact criteria for fixed-parameter tractable (fpt) evaluation of recursively enumerable (r.e.) classes of such OMQs (under the widely held Exponential Time Hypothesis). One of the main technical tools introduced in the paper is an fpt-reduction from deciding parameterized uniform CSPs to parameterized OMQ evaluation. The reduction preserves measures known to be essential for classifying r.e. classes of parameterized uniform CSPs: submodular width (according to the well known result of Marx for unrestricted-arity schemas) and treewidth (according to the well known result of Grohe for bounded-arity schemas). As such, it can be employed to obtain hardness results for evaluation of r.e. classes of parameterized OMQs based on GTGD both in the unrestricted and in the bounded arity case. Previously, for bounded arity schemas, this has been tackled using a technique requiring full introspection into the construction employed by Grohe

A Generic Polynomial Time Approach to Separation by First-Order Logic Without Quantifier Alternation

We look at classes of languages associated to fragments of first-order logic B??, in which quantifier alternations are disallowed. Each such fragment is fully determined by choosing the set of predicates on positions that may be used. Equipping first-order logic with the linear ordering and possibly the successor relation as predicates yields two natural fragments, which were investigated by Simon and Knast, who proved that these two variants have decidable membership: "does an input regular language belong to the class ?". We extend their results in two orthogonal directions. - First, instead of membership, we explore the more general separation problem: decide if two regular languages can be separated by a language from the class under study. - Second, we use more general inputs: classes ? of group languages (i.e., recognized by a DFA in which each letter induces a permutation of the states) and extensions thereof, written ?^+. We rely on a characterization of B?? by the operator BPol: given an input class ?, it outputs a class BPol(?) that corresponds to a variant of B?? equipped with special predicates associated to ?. The classes BPol(?) and BPol(?^+) capture many natural variants of B?? which use predicates such as the linear ordering, the successor, the modular predicates or the alphabetic modular predicates. We show that separation is decidable for BPol(?) and BPol(?^+) when this is the case for ?. This was already known for BPol(?) and for two particular classes of the form BPol(?^+). Yet, the algorithms were indirect and relied on involved frameworks, yielding poor upper complexity bounds. In contrast, our approach is direct. We work only with elementary concepts (mainly, finite automata). Our main contribution consists in polynomial time Turing reductions from both BPol(?)- and BPol(?^+)-separation to ?-separation. This yields polynomial time algorithms for several key variants of B??, including those equipped with the linear ordering and possibly the successor and/or the modular predicates