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Beyond Fuzzy: Parameterized Approximations of Heyting Algebras for Uncertain Knowledge

We propose a parameterized framework based on a Heyting algebra and Lukasiewicz negation for modeling uncertainty for belief. We adopt a probability theory as mathematical formalism for manipulating uncertainty. An agent can express the uncertainty in her knowledge about a piece of information in the form of belief types: as a single probability, as an interval (lower and upper boundary for a probability) or as a confidence level. The probabilistic logic programs can be parameterized by different kinds of conjunctive/disjunctive ”probabilistic strategies” for their rules based on residuum-implication. The underlying algebra for belief computation is a parameterized approximation of strict (without negation) Heyting (or briefly ’parameterized Heyting’) algebra with a unique epistemic negation: it is a set of Lukasiewicz-style residuated lattices and extension of fuzzy logic technique to wide family of probabilistic logic programming and deductive databases. Such framework offers a clear semantics for the satisfaction relation of different probabilistic formalisms used for handling uncertainty, and is open toward the extension of logic languages for formulae with residuum-based implications also in the body of rules.