12 research outputs found
Team logic : axioms, expressiveness, complexity
Team semantics is an extension of classical logic where statements do not refer to single states of a system, but instead to sets of such states, called teams. This kind of semantics has applications for example in mathematical logic, verification of dynamic systems as well as in database theory.
In this thesis, we focus on the propositional, modal and first-order variant of team logic.
We study the classical questions of formal logic: Expressiveness (can we formalize sufficiently interesting properties of models?), axiomatizability (can all true statements be deduced in some formal system?) and complexity (can problems such as satisfiability and model checking be solved algorithmically?). Finally, we classify existing team logics and show approaches how team semantics can be defined for arbitrary other logics.Team-Semantik ist eine Erweiterung klassischer Logik, bei der Aussagen nicht über einzelne Zustände eines Systems getroffen werden, sondern über Mengen solcher Zustände, genannt Teams. Diese Art von Semantik besitzt unter anderem Anwendungen in der mathematischen Logik, in der Verifikation dynamischer Systeme sowie in der Datenbanktheorie. In dieser Arbeit liegt der Fokus auf der aussagenlogischen, der modallogischen und der prädikatenlogischen Variante der Team-Logik. Es werden die klassischen Fragestellungen formaler Logik untersucht: Ausdruckskraft (können hinreichend interessante Eigenschaften von Modellen formalisiert werden?), Axiomatisierbarkeit (lassen sich alle wahren Aussagen in einem Kalkül ableiten?) und Komplexität (können Probleme wie Erfüllbarkeit und Modellprüfung algorithmisch gelöst werden?). Schlussendlich werden existierende Team-Logiken klassifiziert und es werden Ansätze aufgezeigt, wie Team-Semantik für beliebige weitere Logiken definiert werden kann
Topics in elliptic problems: from semilinear equations to shape optimization
In this paper, which corresponds to an updated version of the author's
Habilitation lecture in Mathematics, we do an overview of several topics in
elliptic problems. We review some old and new results regarding the Lane-Emden
equation, both under Dirichlet and Neumann boundary conditions, then focus on
sign-changing solutions for Lane-Emden systems. We also survey some results
regarding fully nontrivial solutions to gradient elliptic systems with mixed
cooperative and competitive interactions. We conclude by exhibiting results on
optimal partition problems, with cost functions either related to Dirichlet
eigenvalues or to the Yamabe equation. Several open problems are referred along
the text.Comment: Review article focused on the author's own work(expanded version of
his Habilitation lecture).Draws heavily from:
arXiv:2305.02870,arXiv:2211.04839,arXiv:2209.02113,
arXiv:2109.14753,arXiv:2106.03661,arXiv:2106.00579,arXiv:1908.11090,arXiv:1807.03082,
arXiv:1706.08391, arXiv:1701.05005,
arXiv:1508.01783,arXiv:1412.4336,arXiv:1409.5693,arXiv:1405.5549,arXiv:1403.6313,arXiv:1307.3981,arXiv:1201.520
Efficient local search for Pseudo Boolean Optimization
Algorithms and the Foundations of Software technolog
Compound Logics for Modification Problems
We introduce a novel model-theoretic framework inspired from graph
modification and based on the interplay between model theory and algorithmic
graph minors. The core of our framework is a new compound logic operating with
two types of sentences, expressing graph modification: the modulator sentence,
defining some property of the modified part of the graph, and the target
sentence, defining some property of the resulting graph. In our framework,
modulator sentences are in counting monadic second-order logic (CMSOL) and have
models of bounded treewidth, while target sentences express first-order logic
(FOL) properties along with minor-exclusion. Our logic captures problems that
are not definable in first-order logic and, moreover, may have instances of
unbounded treewidth. Also, it permits the modeling of wide families of problems
involving vertex/edge removals, alternative modulator measures (such as
elimination distance or -treewidth), multistage modifications, and
various cut problems. Our main result is that, for this compound logic,
model-checking can be done in quadratic time. All derived algorithms are
constructive and this, as a byproduct, extends the constructibility horizon of
the algorithmic applications of the Graph Minors theorem of Robertson and
Seymour. The proposed logic can be seen as a general framework to capitalize on
the potential of the irrelevant vertex technique. It gives a way to deal with
problem instances of unbounded treewidth, for which Courcelle's theorem does
not apply. The proof of our meta-theorem combines novel combinatorial results
related to the Flat Wall theorem along with elements of the proof of
Courcelle's theorem and Gaifman's theorem. We finally prove extensions where
the target property is expressible in FOL+DP, i.e., the enhancement of FOL with
disjoint-paths predicates
Using Model Theory to Find Decidable and Tractable Description Logics with Concrete Domains
Concrete domains have been introduced in the area of Description Logic (DL) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability.
To regain decidability of the DL ALC in the presence of GCIs, quite strong restrictions, called ω-admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of ω-admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate ω-admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield ω-admissible concrete domains. This allows us to show ω-admissibility of concrete domains using existing results from model theory.
When integrating concrete domains into lightweight DLs of the EL family, achieving decidability of reasoning is not enough. One wants the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Although ω-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into ALC yields
decidable DLs.
DL systems that can handle concrete domains allow their users to employ a fixed set of predicates of one or more fixed concrete domains when modelling concepts.
They do not provide their users with means for defining new predicates, let alone new concrete domains. The good news is that finitely bounded homogeneous structures offer precisely that. We show that integrating concrete domains based on finitely bounded homogeneous structures into ALC yields decidable DLs even if we allow predicates specified by first-order formulas. This class of structures also provides effective means for defining new ω-admissible concrete domains with at most binary predicates. The bad news is that defining ω-admissible concrete domains with predicates of higher arities is computationally hard. We obtain two new lower bounds for this meta-problem, but leave its decidability open. In contrast, we prove that there is no algorithm that would facilitate defining p-admissible concrete domains already for binary signatures.:1. Introduction . . . 1
2. Preliminaries . . . 5
3. Description Logics with Concrete Domains . . . 9
3.1. Basic definitions and undecidability results . . . 9
3.2. Decidable and tractable DLs with concrete domains . . . 16
4. A Model-Theoretic Analysis of ω-Admissibility . . . 23
4.1. Homomorphism ω-compactness via ω-categoricity . . . 23
4.2. Patchworks via homogeneity . . . 24
4.3. JDJEPD via decomposition into orbits . . . 27
4.4. Upper bounds via finite boundedness . . . 28
4.5. ω-admissible finitely bounded homogeneous structures . . . 32
4.6. ω-admissible homogeneous cores with a decidable CSP . . . 34
4.7. Coverage of the developed sufficient conditions . . . 36
4.8. Closure properties: homogeneity & finite boundedness . . . 39
5. A Model-Theoretic Analysis of p-Admissibility . . . 47
5.1. Convexity via square embeddings . . . 47
5.2. Convex ω-categorical structures . . . 50
5.3. Convex numerical structures . . . 52
5.4. Ages defined by forbidden substructures . . . 54
5.5. Ages defined by forbidden homomorphic images . . . 56
5.6. (Non-)closure properties of convexity . . . 59
6. Towards user-definable concrete domains . . . 61
6.1. A proof-theoretic perspective . . . 65
6.2. Universal Horn sentences and the JEP . . . 66
6.3. Universal sentences and the AP: the Horn case . . . 77
6.4. Universal sentences and the AP: the general case . . . 90
7. Conclusion . . . 99
7.1. Contributions and future outlook . . . 99
A. Concrete Domains without Equality . . . 103
Bibliography . . . 107
List of figures . . . 115
Alphabetical Index . . . 11
Beyond Logic. Proceedings of the Conference held in Cerisy-la-Salle, 22-27 May 2017
The project "Beyond Logic" is devoted to what hypothetical reasoning is all about when we go beyond the realm of "pure" logic into the world where logic is applied. As such extralogical areas we have chosen philosophy of science as an application within philosophy, informatics as an application within the formal sciences, and law as an application within the field of social interaction. The aim of the conference was to allow philosophers, logicians and computer scientists to present their work in connection with these three areas. The conference took place 22-27 May, 2017 in Cerisy-la-Salle at the Centre Culturel International de Cerisy. The proceedings collect abstracts, slides and papers of the presentations given, as well as a contribution from a speaker who was unable to attend