Complexity of Non-Monotonic Logics

Over the past few decades, non-monotonic reasoning has developed to be one of the most important topics in computational logic and artificial intelligence. Different ways to introduce non-monotonic aspects to classical logic have been considered, e.g., extension with default rules, extension with modal belief operators, or modification of the semantics. In this survey we consider a logical formalism from each of the above possibilities, namely Reiter's default logic, Moore's autoepistemic logic and McCarthy's circumscription. Additionally, we consider abduction, where one is not interested in inferences from a given knowledge base but in computing possible explanations for an observation with respect to a given knowledge base. Complexity results for different reasoning tasks for propositional variants of these logics have been studied already in the nineties. In recent years, however, a renewed interest in complexity issues can be observed. One current focal approach is to consider parameterized problems and identify reasonable parameters that allow for FPT algorithms. In another approach, the emphasis lies on identifying fragments, i.e., restriction of the logical language, that allow more efficient algorithms for the most important reasoning tasks. In this survey we focus on this second aspect. We describe complexity results for fragments of logical languages obtained by either restricting the allowed set of operators (e.g., forbidding negations one might consider only monotone formulae) or by considering only formulae in conjunctive normal form but with generalized clause types. The algorithmic problems we consider are suitable variants of satisfiability and implication in each of the logics, but also counting problems, where one is not only interested in the existence of certain objects (e.g., models of a formula) but asks for their number.Comment: To appear in Bulletin of the EATC

The complexity of counting locally maximal satisfying assignments of Boolean CSPs

We investigate the computational complexity of the problem of counting the maximal satisfying assignments of a Constraint Satisfaction Problem (CSP) over the Boolean domain {0,1}. A satisfying assignment is maximal if any new assignment which is obtained from it by changing a 0 to a 1 is unsatisfying. For each constraint language Gamma, #MaximalCSP(Gamma) denotes the problem of counting the maximal satisfying assignments, given an input CSP with constraints in Gamma. We give a complexity dichotomy for the problem of exactly counting the maximal satisfying assignments and a complexity trichotomy for the problem of approximately counting them. Relative to the problem #CSP(Gamma), which is the problem of counting all satisfying assignments, the maximal version can sometimes be easier but never harder. This finding contrasts with the recent discovery that approximately counting maximal independent sets in a bipartite graph is harder (under the usual complexity-theoretic assumptions) than counting all independent sets.Comment: V2 adds contextual material relating the results obtained here to earlier work in a different but related setting. The technical content is unchanged. V3 (this version) incorporates minor revisions. The title has been changed to better reflect what is novel in this work. This version has been accepted for publication in Theoretical Computer Science. 19 page

Minimization for Generalized Boolean Formulas

The minimization problem for propositional formulas is an important optimization problem in the second level of the polynomial hierarchy. In general, the problem is Sigma-2-complete under Turing reductions, but restricted versions are tractable. We study the complexity of minimization for formulas in two established frameworks for restricted propositional logic: The Post framework allowing arbitrarily nested formulas over a set of Boolean connectors, and the constraint setting, allowing generalizations of CNF formulas. In the Post case, we obtain a dichotomy result: Minimization is solvable in polynomial time or coNP-hard. This result also applies to Boolean circuits. For CNF formulas, we obtain new minimization algorithms for a large class of formulas, and give strong evidence that we have covered all polynomial-time cases

Can there ever be a theory of utterance interpretation?

In this paper, I tackle what appears to be a rather simple question: can there ever be a theory of utterance interpretation? It will be contended that a theory of utterance interpretation is not beyond the intellectual grasp of present-day pragmatists so much as it is a construct which lacks sense and is unintelligible. Although many of our most successful theories exhibit desiderata such as simplicity, completeness and explanatory power, it will be argued that these same desiderata are problematic when it is utterance interpretation that is the focus of theoretical efforts. The case in support of this claim sets out from a detailed analysis of the rational, intentional, holistic character of utterance interpretation and draws on the insights of the American philosopher Hilary Putnam. To the extent that a theory of utterance interpretation is not a difficult empirical possibility to realize so much as it is an endeavour which leads to an unintelligible outcome, we consider where this situation leaves pragmatists who have a substantial appetite for theory construction

Open biomedical pluralism : formalising knowledge about breast cancer phenotypes

We demonstrate a heterogeneity of representation types for breast cancer phenotypes and stress that the characterisation of a tumour phenotype often includes parameters that go beyond the representation of a corresponding empirically observed tumour, thus reflecting significant functional features of the phenotypes as well as epistemic interests that drive the modes of representation. Accordingly, the represented features of cancer phenotypes function as epistemic vehicles aiding various classifications, explanations, and predictions. In order to clarify how the plurality of epistemic motivations can be integrated on a formal level, we give a distinction between six categories of human agents as individuals and groups focused around particular epistemic interests. We analyse the corresponding impact of these groups and individuals on representation types, mapping and reasoning scenarios. Respecting the plurality of representations, related formalisms, expressivities and aims, as they are found across diverse scientific communities, we argue for a pluralistic ontology integration. Moreover, we discuss and illustrate to what extent such a pluralistic integration is supported by the distributed ontology language DOL, a meta-language for heterogeneous ontology representation that is currently under standardisation as ISO WD 17347 within the OntoIOp (Ontology Integration and Interoperability) activity of ISO/TC 37/SC 3. We particularly illustrate how DOL supports representations of parthood on various levels of logical expressivity, mapping of terms, merging of ontologies, as well as non-monotonic extensions based on circumscription allowing a transparent formal modelling of the normal/abnormal distinction in phenotypes

Counting Constraint Satisfaction Problems

This chapter surveys counting Constraint Satisfaction Problems (counting CSPs, or #CSPs) and their computational complexity. It aims to provide an introduction to the main concepts and techniques, and present a representative selection of results and open problems. It does not cover holants, which are the subject of a separate chapter