Reasoning with Uncertainty by Nmatrix–Metric Semantics [chapter]

Ofer Arieli, Anna Zamansky
2008 Lecture Notes in Computer Science  
Non-deterministic matrices, a natural generalization of manyvalued matrices, are semantic structures in which the value assigned to a complex formula may be chosen non-deterministically from a given set of options. We show that by combining Nmatrices and preferential metric-based considerations, one obtains a family of logics that are useful for reasoning with uncertainty. We investigate the basic properties of these logics and demonstrate their usefulness in handling incomplete and
more » ... information. requires a retraction of old assertions. To cope with this, Shoham [22] introduced the notion of preferential semantics (see also [20] ), according to which an order relation, reflecting some condition or preference criteria, is defined on a set of valuations, and only the valuations that are minimal with respect to this order are relevant for making inferences from a given theory. Following this idea, we use metric-like considerations as our primary preference criteria. Such distance minimization considerations are a cornerstone behind many paradigms of handling incomplete or inconsistent information, such as belief revision [9, 14, 18, 23] database integration systems [1, 5, 10, 19] , and formalisms for commonsense reasoning in the context of social choice theory [16, 21] . In [2, 3, 7] this approach is described in terms of entailment relations, based on a standard truth-functional semantics. As argued above, this cannot capture non-deterministic behavior, so instead, in this paper, we use logics based on Nmatrices as the underlying formalism for a preferential metric-based approach. We also consider some of the properties of the entailment relations that are obtained, demonstrate their applicability for reasoning under uncertainty by some case studies, and show the relation between reasoning in these cases and some well-known SAT problems. Distance-Based Non-Deterministic Semantics Non-Deterministic Matrices In what follows, L denotes a propositional language with a set Atoms of atomic formulas. A theory Γ is a finite multiset of L-formulas, for which Atoms(Γ ) and SF(Γ ) denote, respectively, the atomic formulas of Γ and the subformulas of Γ . Below, we shortly reproduce the main definitions from [8]. Definition 1. A non-deterministic matrix (henceforth, Nmatrix ) for L is a tuple M = V, D, O , where V is a non-empty set of truth values, D is a non-empty proper subset of V, and for every n-ary connective of L, O includes an n-ary function from V n to 2 V − {∅}.
doi:10.1007/978-3-540-69937-8_8 fatcat:b4tuebrg7nayho4hdz6r7xgfgu