This technical note describes a monotone and continuous fixpoint operator to compute the answer sets of programs with aggregates. The fixpoint operator relies on the notion of aggregate solution. Under certain conditions, this operator behaves identically to the three-valued immediate consequence operator Φ P for aggregate programs, independently proposed in (Pelov 2004; Pelov et al. 2004). Thi...

We give in this paper a sufficient condition under which the least fixpoint of the equation X = a + f(X)X equals the least fixpoint of the equation X = a+ f(a)X. We then apply that condition to recursive logic programs containing chain rules: we translate it into a sufficient condition under which a recursive logic program containing n ≥ 2 recursive calls in the bodies of the rules is equivalen...

Cyclic definitions in description logics have until now been investigated only for description logics allowing for value restrictions. Even for the most basic language which allows for conjunction and value restrictions only, deciding subsumption in the presence of terminological cycles is a PSPACE-complete problem. This paper investigates subsumption in the presence of terminological cycles fo...

Abstract interpretation is used in program static analysis and model checking to cope with infinite state spaces and/or with com­ puter resource limitations. One common problem is to check abstract fixpoints for specifications. The abstraction is partially complete when the checking algorithm is exact in that, if the algorithm ever terminates, its answer is always affirmative for correct specif...

This paper introduces a new higher-order typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a strong version of Martin-Lof’s ‘iteration type’ [ll]. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial...

The modal μ-calculus, introduced by Dexter Kozen, is an extension of modal logic with fixpoint operators. Its axiomatization, Koz, was introduced at the same time and is an extension of the minimal modal logic K with the so-called Park fixpoint induction principle. It took more than a decade for the completeness of Koz to be proven, finally achieved by Igor Walukiewicz. However, his proof is fa...

The classical method for program analysis by abstract interpretation consists in computing a increasing sequence with widening, which converges towards a correct solution, then computing a decreasing sequence of correct solutions without widening. It is generally admitted that, when the decreasing sequence reaches a fixpoint, it cannot be improved further. As a consequence, all efforts for impr...

A crucial component of an intelligent system is its knowledge base that contains knowledge about a problem domain. Knowledge base development involves domain analysis, context space definition, ontological specification, and knowledge acquisition, codification and verification. Knowledge base anomalies can affect the correctness and performance of an intelligent system. In this chapter, we desc...