Uncertainty and Partial Non-uniform Assumptions in Parametric Deductive Databases

Abstract

Abstract. Different many-valued logic programming frameworks have been proposed to manage uncertain information in deductive databases and logic programming. A feature of these frameworks is that they rely on a predefined assumption or hypothesis, i.e. an interpretation that assigns the same default truth value to all the atoms of a program, e.g. in the open world assumption, by default all atoms have unknown truth value. In this paper we extend these frameworks along three directions: (i) we will introduce non-monotonic modes of negation; (ii) the default truth values of atoms need not necessarily to be all equal each other; and (iii) a hypothesis can be a partial interpretation. We will show that our approach extends the usual ones: if we restrict our attention to classical logic programs and consider total uniform hypotheses, then our semantics reduces to the usual semantics of logic programs. In particular, under the everything false assumption, our semantics captures and extends the well-founded semantics to these frameworks.

Default Knowledge in Logic Programs with Uncertainty

“…Applying a usual approach, like the well-founded semantics [31] or the Kripke-Kleene semantics [6], would lead us to choose the conservative solution 1. This was also the approach taken in [19]. Such a semantics seems to be too weak, in the sense that it loses some knowledge (e.g.…”

“…A first attempt to give a semantics to PDDU with negation under non-uniform assumptions is described in [19]. It is based on an extension of the alternating fixpoint semantics of van Gelder [30], i.e.…”

“…The approach proposed here is based on a more intuitive management of negation where no distinction is made between positive and negative literals. Moreover, our approach is more general: in fact (i) in [19], some knowledge is lost whenever the value of an atom oscillates between two different uncertainty values, while in our approach, an approximation (interval) of the uncertainty value is assigned. For instance, with respect to Example 8 under the assumption I ⊥ , [19] assigns ⊥ = [0, 1] to both A and B, while our approach assigns [0.2, 0.7] to A and [0.3, 0.8] to B; and (ii) our approach does not impose any restriction on the kind of interpretations that can be used as assumptions (any value, even an interval, can be considered as a default value).…”

“…Well-known are (i) the Closed World Assumption (CWA), which asserts that any atom whose truth-value cannot be inferred from the facts and rules is supposed to be false, and (ii) the Open World Assumption (OWA), which asserts that every such atom is supposed to be unknown or undefined. In the first case, fall the approaches, which end up with a stable model [9,10], or well-founded [31] semantics, like [5,19,23,24], while in the latter case fall the others, including [17] as well. Exceptions are a previous attempt [19] and those works related to Extended and Disjunctive Logic Programs [10], where some non-uniformity is allowed (see, e.g.…”

Epistemic Foundation of the Well-Founded Semantics over Bilattices

“…Bilattices, where FOUR is the simplest non-trivial one, play an important role in logic programming, and in knowledge representation in general, allowing in particular reasoning with inconsistency and uncertainty (see e.g. [1,4,6,14,15,16,17,18]). Fitting proposed an extension of the stable models family by extending the Gelfond-Lifschitz transform from logic programming over two-or three-valued logics to logic programming over bilattices [8].…”