Ultimate approximations in nonmonotonic knowledge representation systems

We study fixpoints of operators on lattices. To this end we introduce the notion of an approximation of an operator. We order approximations by means of a precision ordering. We show that each lattice operator O has a unique most precise or ultimate approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We apply our theory to logic programming and introduce the ultimate Kripke-Kleene, well-founded and stable semantics. We show that the ultimate Kripke-Kleene and well-founded semantics are more precise then their standard counterparts We argue that ultimate semantics for logic programming have attractive epistemological properties and that, while in general they are computationally more complex than the standard semantics, for many classes of theories, their complexity is no worse.Comment: This paper was published in Principles of Knowledge Representation and Reasoning, Proceedings of the Eighth International Conference (KR2002

Clafer: Lightweight Modeling of Structure, Behaviour, and Variability

Embedded software is growing fast in size and complexity, leading to intimate mixture of complex architectures and complex control. Consequently, software specification requires modeling both structures and behaviour of systems. Unfortunately, existing languages do not integrate these aspects well, usually prioritizing one of them. It is common to develop a separate language for each of these facets. In this paper, we contribute Clafer: a small language that attempts to tackle this challenge. It combines rich structural modeling with state of the art behavioural formalisms. We are not aware of any other modeling language that seamlessly combines these facets common to system and software modeling. We show how Clafer, in a single unified syntax and semantics, allows capturing feature models (variability), component models, discrete control models (automata) and variability encompassing all these aspects. The language is built on top of first order logic with quantifiers over basic entities (for modeling structures) combined with linear temporal logic (for modeling behaviour). On top of this semantic foundation we build a simple but expressive syntax, enriched with carefully selected syntactic expansions that cover hierarchical modeling, associations, automata, scenarios, and Dwyer's property patterns. We evaluate Clafer using a power window case study, and comparing it against other notations that substantially overlap with its scope (SysML, AADL, Temporal OCL and Live Sequence Charts), discussing benefits and perils of using a single notation for the purpose

Heuristic Ranking in Tightly Coupled Probabilistic Description Logics

The Semantic Web effort has steadily been gaining traction in the recent years. In particular,Web search companies are recently realizing that their products need to evolve towards having richer semantic search capabilities. Description logics (DLs) have been adopted as the formal underpinnings for Semantic Web languages used in describing ontologies. Reasoning under uncertainty has recently taken a leading role in this arena, given the nature of data found on theWeb. In this paper, we present a probabilistic extension of the DL EL++ (which underlies the OWL2 EL profile) using Markov logic networks (MLNs) as probabilistic semantics. This extension is tightly coupled, meaning that probabilistic annotations in formulas can refer to objects in the ontology. We show that, even though the tightly coupled nature of our language means that many basic operations are data-intractable, we can leverage a sublanguage of MLNs that allows to rank the atomic consequences of an ontology relative to their probability values (called ranking queries) even when these values are not fully computed. We present an anytime algorithm to answer ranking queries, and provide an upper bound on the error that it incurs, as well as a criterion to decide when results are guaranteed to be correct.Comment: Appears in Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (UAI2012

The Logic of Counting Query Answers

We consider the problem of counting the number of answers to a first-order formula on a finite structure. We present and study an extension of first-order logic in which algorithms for this counting problem can be naturally and conveniently expressed, in senses that are made precise and that are motivated by the wish to understand tractable cases of the counting problem