The Approximate Well-Founded Semantics for Logic Programs with Uncertainty

Abstract

Abstract. The management of uncertain information in logic programs becomes to be important whenever the real world information to be represented is of imperfect nature and the classical crisp true, false approximation is not adequate. A general framework, called Parametric Deductive Databases with Uncertainty (PDDU) framework [10], was proposed as a unifying umbrella for many existing approaches towards the manipulation of uncertainty in logic programs. We extend PDDU with (non-monotonic) negation, a well-known and important feature of logic programs. We show that, dealing with uncertain and incomplete knowledge, atoms should be assigned only approximations of uncertainty values, unless some assumption is used to complete the knowledge. We rely on the closed world assumption to infer as much default "false" knowledge as possible. Our approach leads also to a novel characterizations, both epistemic and operational, of the well-founded semantics in PDDU, and preserves the continuity of the immediate consequence operator, a major feature of the classical PDDU framework.

Query Answering in Normal Logic Programs Under Uncertainty

“…Negation can be defined as ¬ x, y = ¬y, ¬x , where ¬ is a negation operator on L. This approach has been used, e.g. in [14], where L is the unit interval Logic Programs. Fix a bilattice.…”

“…For an interpretation I, for any ground atom A with (unique) A ← ϕ ∈ P * , Φ P (I)(A) = I(ϕ). We can show that (based on [9,14]) the function Φ P is k -continuous over I(B), the set of fixed-points of Φ P is a complete lattice under k and, thus, Φ P has a k -least fixed-point and I is a model of a program P iff I is a fixed-point of Φ P . Therefore, the Kripke-Kleene model of P coincides with the least fixed-point of Φ P under k , which can be computed in the usual way by iterating Φ P over I ⊥ and is attained after at most ω iterations.…”

“…Essentially, they differ in the underlying notion of uncertainty 1 (e.g. probability theory [15,18], fuzzy set theory [22], multi-valued logic [4,12,13,14], possibilistic logic [8]) and how uncertainty values, associated to rules and facts, are managed. Roughly, these frameworks can be classified into annotation based (AB) and implication based (IB).…”

“…Exception to this limitation are e.g. [16,17] in which the stable semantics has been considered, but limited to the case where the underlying uncertainty formalism is probability theory; in [7] the underlying truth-space are lattices, but its formulations goes over bilattices [11] (a slightly more general structure than lattices); while [14] uses normal logic programs over bilattices under the IB framework.…”

Reducing fuzzy answer set programming to model finding in fuzzy logics

“…Most notably are the probabilistic (Baral et al 2007;Damásio and Pereira 2000;Fuhr 2000;Lukasiewicz 1998;Lukasiewicz 1999;Ng and Subrahmanian 1993;Ng and Subrahmanian 1994;Straccia 2008) and possibilistic (Alsinet et al 2002;Bauters et al 2010;Nicolas et al 2005;Nicolas et al 2006) extensions to handle uncertainty, the fuzzy extensions (Cao 2000;Ishizuka and Kanai 1985;Lukasiewicz 2006;Lukasiewicz and Straccia 2007a;Lukasiewicz and Straccia 2007b;Madrid and Ojeda-Aciego 2008;Madrid and Ojeda-Aciego 2009;Saad 2009a;Straccia 2008;Van Nieuwenborgh et al 2007b;Vojtás 2001;Wagner 1998) which allow to encode the intensity to which the predicates are satisfied, and, more generally, many-valued extensions Damásio et al 2007;Damásio and Pereira 2001a;Damásio and Pereira 2001b;Damásio and Pereira 2004;Emden 1986;Fitting 1991;Kifer and Li 1988;Kifer and Subrahmanian 1992;Lakshmanan 1994;Lakshmanan and Sadri 1994;Lakshmanan and Sadri 1997;Lakshmanan and Shiri 2001;Lakshmanan 1997;Loyer and Straccia 2002;Loyer and Straccia 2003;…”