Complexity of Nested Circumscription and Nested Abnormality Theories

The need for a circumscriptive formalism that allows for simple yet elegant modular problem representation has led Lifschitz (AIJ, 1995) to introduce nested abnormality theories (NATs) as a tool for modular knowledge representation, tailored for applying circumscription to minimize exceptional circumstances. Abstracting from this particular objective, we propose L_{CIRC}, which is an extension of generic propositional circumscription by allowing propositional combinations and nesting of circumscriptive theories. As shown, NATs are naturally embedded into this language, and are in fact of equal expressive capability. We then analyze the complexity of L_{CIRC} and NATs, and in particular the effect of nesting. The latter is found to be a source of complexity, which climbs the Polynomial Hierarchy as the nesting depth increases and reaches PSPACE-completeness in the general case. We also identify meaningful syntactic fragments of NATs which have lower complexity. In particular, we show that the generalization of Horn circumscription in the NAT framework remains CONP-complete, and that Horn NATs without fixed letters can be efficiently transformed into an equivalent Horn CNF, which implies polynomial solvability of principal reasoning tasks. Finally, we also study extensions of NATs and briefly address the complexity in the first-order case. Our results give insight into the ``cost'' of using L_{CIRC} (resp. NATs) as a host language for expressing other formalisms such as action theories, narratives, or spatial theories.Comment: A preliminary abstract of this paper appeared in Proc. Seventeenth International Joint Conference on Artificial Intelligence (IJCAI-01), pages 169--174. Morgan Kaufmann, 200

Modularity in answer set programs

Answer set programming (ASP) is an approach to rule-based constraint programming allowing flexible knowledge representation in variety of application areas. The declarative nature of ASP is reflected in problem solving. First, a programmer writes down a logic program the answer sets of which correspond to the solutions of the problem. The answer sets of the program are then computed using a special purpose search engine, an ASP solver. The development of efficient ASP solvers has enabled the use of answer set programming in various application domains such as planning, product configuration, computer aided verification, and bioinformatics. The topic of this thesis is modularity in answer set programming. While modern programming languages typically provide means to exploit modularity in a number of ways to govern the complexity of programs and their development process, relatively little attention has been paid to modularity in ASP. When designing a module architecture for ASP, it is essential to establish full compositionality of the semantics with respect to the module system. A balance is sought between introducing restrictions that guarantee the compositionality of the semantics and enforce a good programming style in ASP, and avoiding restrictions on the module hierarchy for the sake of flexibility of knowledge representation. To justify a replacement of a module with another, that is, to be able to guarantee that changes made on the level of modules do not alter the semantics of the program when seen as an entity, a notion of equivalence for modules is provided. In close connection with the development of the compositional module architecture, a transformation from verification of equivalence to search for answer sets is developed. The translation-based approach makes it unnecessary to develop a dedicated tool for the equivalence verification task by allowing the direct use of existing ASP solvers. Translations and transformations between different problems, program classes, and formalisms are another central theme in the thesis. To guarantee efficiency and soundness of the translation-based approach, certain syntactical and semantical properties of transformations are desirable, in terms of translation time, solution correspondence between the original and the transformed problem, and locality/globality of a particular transformation. In certain cases a more refined notion of minimality than that inherent in ASP can make program encodings more intuitive. Lifschitz' parallel and prioritized circumscription offer a solution in which certain atoms are allowed to vary or to have fixed values while others are falsified as far as possible according to priority classes. In this thesis a linear and faithful transformation embedding parallel and prioritized circumscription into ASP is provided. This enhances the knowledge representation capabilities of answer set programming by allowing the use of existing ASP solvers for computing parallel and prioritized circumscription

Strong Equivalence of Qualitative Optimization Problems

We introduce the framework of qualitative optimization problems (or, simply, optimization problems) to represent preference theories. The formalism uses separate modules to describe the space of outcomes to be compared (the generator) and the preferences on outcomes (the selector). We consider two types of optimization problems. They differ in the way the generator, which we model by a propositional theory, is interpreted: by the standard propositional logic semantics, and by the equilibrium-model (answer-set) semantics. Under the latter interpretation of generators, optimization problems directly generalize answer-set optimization programs proposed previously. We study strong equivalence of optimization problems, which guarantees their interchangeability within any larger context. We characterize several versions of strong equivalence obtained by restricting the class of optimization problems that can be used as extensions and establish the complexity of associated reasoning tasks. Understanding strong equivalence is essential for modular representation of optimization problems and rewriting techniques to simplify them without changing their inherent properties

Circumscribing datalog: Expressive power and complexity

AbstractIn this paper we study a generalization of datalog, the language of function-free definite clauses. It is known that standard datalog semantics (i.e., least Herbrand model semantics) can be obtained by regarding programs as theories to be circumscribed with all predicates to be minimized. The extension proposed here, called datalogcirc, consists in considering the general form of circumscription, where some predicates are minimized, some predicates are fixed, and some vary. We study the complexity and the expressive power of the language thus obtained. We show that this language (and, actually, its non-recursive fragment) is capable of expressing all the queries in DB-co-NP and, as such, is much more powerful than standard datalog, whose expressive power is limited to a strict subset of PTIME queries. Both data and combined complexities of answering datalogcirc queries are studied. Data complexity is proved to be co-NP-complete. Combined complexity is shown to be in general hard for co-NE and complete for co-NE in the case of Herbrand bases containing k distinct constant symbols, where k is bounded

DRUM-II : efficient model based diagnosis of technical systems

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Controlled Natural Languages for Knowledge Representation and Reasoning

Controlled natural languages (CNLs) are effective languages for knowledge representation and reasoning. They are designed based on certain natural languages with restricted lexicon and grammar. CNLs are unambiguous and simple as opposed to their base languages. They preserve the expressiveness and coherence of natural languages. In this paper, it mainly focuses on a class of CNLs, called machine-oriented CNLs, which have well-defined semantics that can be deterministically translated into formal languages to do logical reasoning. Although a number of machine-oriented CNLs emerged and have been used in many application domains for problem solving and question answering, there are still many limitations: First, CNLs cannot handle inconsistencies in the knowledge base. Second, CNLs are not powerful enough to identify different variations of a sentence and therefore might not return the expected inference results. Third, CNLs do not have a good mechanism for defeasible reasoning. This paper addresses these three problems and proposes a research plan for solving these problems. It also shows the current state of research: a paraconsistent logical framework from which six principles that guide the user to encode CNL sentences were created. Experiment results show this paraconsistent logical framework and these six principles can consistently and effectively solve word puzzles with injections of inconsistencies

Proceedings of the Workshop on Change of Representation and Problem Reformulation

The proceedings of the third Workshop on Change of representation and Problem Reformulation is presented. In contrast to the first two workshops, this workshop was focused on analytic or knowledge-based approaches, as opposed to statistical or empirical approaches called 'constructive induction'. The organizing committee believes that there is a potential for combining analytic and inductive approaches at a future date. However, it became apparent at the previous two workshops that the communities pursuing these different approaches are currently interested in largely non-overlapping issues. The constructive induction community has been holding its own workshops, principally in conjunction with the machine learning conference. While this workshop is more focused on analytic approaches, the organizing committee has made an effort to include more application domains. We have greatly expanded from the origins in the machine learning community. Participants in this workshop come from the full spectrum of AI application domains including planning, qualitative physics, software engineering, knowledge representation, and machine learning