13 research outputs found

    Unified Foundations of Team Semantics via Semirings

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    Semiring semantics for first-order logic provides a way to trace how facts represented by a model are used to deduce satisfaction of a formula. Team semantics is a framework for studying logics of dependence and independence in diverse contexts such as databases, quantum mechanics, and statistics by extending first-order logic with atoms that describe dependencies between variables. Combining these two, we propose a unifying approach for analysing the concepts of dependence and independence via a novel semiring team semantics, which subsumes all the previously considered variants for first-order team semantics. In particular, we study the preservation of satisfaction of dependencies and formulae between different semirings. In addition we create links to reasoning tasks such as provenance, counting, and repairs

    Parameterized aspects of team-based formalisms and logical inference

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    Parameterized complexity is an interesting subfield of complexity theory that has received a lot of attention in recent years. Such an analysis characterizes the complexity of (classically) intractable problems by pinpointing the computational hardness to some structural aspects of the input. In this thesis, we study the parameterized complexity of various problems from the area of team-based formalisms as well as logical inference. In the context of team-based formalism, we consider propositional dependence logic (PDL). The problems of interest are model checking (MC) and satisfiability (SAT). Peter Lohmann studied the classical complexity of these problems as a part of his Ph.D. thesis proving that both MC and SAT are NP-complete for PDL. This thesis addresses the parameterized complexity of these problems with respect to a wealth of different parameterizations. Interestingly, SAT for PDL boils down to the satisfiability of propositional logic as implied by the downwards closure of PDL-formulas. We propose an interesting satisfiability variant (mSAT) asking for a satisfiable team of size m. The problem mSAT restores the ‘team semantic’ nature of satisfiability for PDL-formulas. We propose another problem (MaxSubTeam) asking for a maximal satisfiable team if a given team does not satisfy the input formula. From the area of logical inference, we consider (logic-based) abduction and argumentation. The problem of interest in abduction (ABD) is to determine whether there is an explanation for a manifestation in a knowledge base (KB). Following Pfandler et al., we also consider two of its variants by imposing additional restrictions over the size of an explanation (ABD and ABD=). In argumentation, our focus is on the argument existence (ARG), relevance (ARG-Rel) and verification (ARG-Check) problems. The complexity of these problems have been explored already in the classical setting, and each of them is known to be complete for the second level of the polynomial hierarchy (except for ARG-Check which is DP-complete) for propositional logic. Moreover, the work by Nord and Zanuttini (resp., Creignou et al.) explores the complexity of these problems with respect to various restrictions over allowed KBs for ABD (ARG). In this thesis, we explore a two-dimensional complexity analysis for these problems. The first dimension is the restrictions over KB in Schaefer’s framework (the same direction as Nord and Zanuttini and Creignou et al.). What differentiates the work in this thesis from an existing research on these problems is that we add another dimension, the parameterization. The results obtained in this thesis are interesting for two reasons. First (from a theoretical point of view), ideas used in our reductions can help in developing further reductions and prove (in)tractability results for related problems. Second (from a practical point of view), the obtained tractability results might help an agent designing an instance of a problem come up with the one for which the problem is tractable

    On elementary logics for quantitative dependencies

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    We define and study logics in the framework of probabilistic team semantics and over metafinite structures. Our work is paralleled by the recent development of novel axiomatizable and tractable logics in team semantics that are closed under the Boolean negation. Our logics employ new probabilistic atoms that resemble so-called extended atoms from the team semantics literature. We also define counterparts of our logics over metafinite structures and show that all of our logics can be translated into functional fixed point logic implying a polynomial time upper bound for data complexity with respect to BSS-computations.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Peer reviewe

    Parametrised enumeration

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    In this thesis, we develop a framework of parametrised enumeration complexity. At first, we provide the reader with preliminary notions such as machine models and complexity classes besides proving them to be well-chosen. Then, we study the interplay and the landscape of these classes and present connections to classical enumeration classes. Afterwards, we translate the fundamental methods of kernelisation and self-reducibility into equivalent techniques in the setting of parametrised enumeration. Subsequently, we illustrate the introduced classes by investigating the parametrised enumeration complexity of Max-Ones-SAT and strong backdoor sets as well as sharpen the first result by presenting a dichotomy theorem for Max-Ones-SAT. After this, we extend the definitions of parametrised enumeration algorithms by allowing orders on the solution space. In this context, we study the relations ``order by size'' and ``lexicographic order'' for graph modification problems and observe a trade-off between enumeration delay and space requirements of enumeration algorithms. These results then yield an enumeration technique for generalised modification problems that is illustrated by applying this method to the problems closest string, weak and strong backdoor sets, and weighted satisfiability. Eventually, we consider the enumeration of satisfying teams of formulas of poor man's propositional dependence logic. There, we present an enumeration algorithm with FPT delay and exponential space which is one of the first enumeration complexity results of a problem in a team logic. Finally, we show how this algorithm can be modified such that only polynomial space is required, however, by increasing the delay to incremental FPT time.In diesem Werk begrĂŒnden wir die Theorie der parametrisierten Enumeration, prĂ€sentieren die grundlegenden Definitionen und prĂŒfen ihre Sinnhaftigkeit. Im nĂ€chsten Schritt, untersuchen wir das Zusammenspiel der eingefĂŒhrten KomplexitĂ€tsklassen und zeigen Verbindungen zur klassischen EnumerationskomplexitĂ€t auf. Anschließend ĂŒbertragen wir die zwei fundamentalen Techniken der Kernelisierung und Selbstreduzierbarkeit in Entsprechungen in dem Gebiet der parametrisierten Enumeration. Schließlich untersuchen wir das Problem Max-Ones-SAT und das Problem der AufzĂ€hlung starker Backdoor-Mengen als typische Probleme in diesen Klassen. Die vorherigen Resultate zu Max-Ones-SAT werden anschließend in einem Dichotomie-Satz vervollstĂ€ndigt. Im nĂ€chsten Abschnitt erweitern wir die neuen Definitionen auf Ordnungen (auf dem Lösungsraum) und erforschen insbesondere die zwei Relationen \glqq GrĂ¶ĂŸenordnung\grqq\ und \glqq lexikographische Reihenfolge\grqq\ im Kontext von Graphen-Modifikationsproblemen. Hierbei scheint es, als mĂŒsste man zwischen Delay und Speicheranforderungen von AufzĂ€hlungsalgorithmen abwĂ€gen, wobei dies jedoch nicht abschließend gelöst werden kann. Aus den vorherigen Überlegungen wird schließlich ein generisches Enumerationsverfahren fĂŒr allgemeine Modifikationsprobleme entwickelt und anhand der Probleme Closest String, schwacher und starker Backdoor-Mengen sowie gewichteter ErfĂŒllbarkeit veranschaulicht. Im letzten Abschnitt betrachten wir die parametrisierte EnumerationskomplexitĂ€t von ErfĂŒllbarkeitsproblemen im Bereich der Poor Man's Propositional Dependence Logic und stellen einen AufzĂ€hlungsalgorithmus mit FPT Delay vor, der mit exponentiellem Platz arbeitet. Dies ist einer der ersten AufzĂ€hlungsalgorithmen im Bereich der Teamlogiken. Abschließend zeigen wir, wie dieser Algorithmus so modifiziert werden kann, dass nur polynomieller Speicherplatz benötigt wird, bezahlen jedoch diese Einsparung mit einem Anstieg des Delays auf inkrementelle FPT Zeit (IncFPT)

    Parametrised Complexity of Model Checking and Satisfiability in Propositional Dependence Logic

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    In this paper, we initiate a systematic study of the parametrised complexity in the field of Dependence Logics which finds its origin in the Dependence Logic of V\"a\"an\"anen from 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parametrisations with respect to the central decision problems. The model checking problem (MC) of PDL is NP-complete. The subject of this research is to identify a list of parametrisations (formula-size, treewidth, treedepth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) showing a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is in FPT. Finally, we introduce a variant of the satisfiability problem, asking for teams of a given size, and show for this problem an almost complete picture.Comment: Update includes refined result

    Workshop on Logics of Dependence and Independence (LoDE 2020V)

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