A framework for reasoning under uncertainty based on non-deterministic distance semantics

Ofer Arieli, Anna Zamansky
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/sy2zvsxl4vdejh3zsp3utmplry" style="color: black;">International Journal of Approximate Reasoning</a> </i> &nbsp;
In this paper, we introduce a general and modular framework for formalizing reasoning with incomplete and inconsistent information. Our framework is composed of non-deterministic semantic structures and distance-based considerations. This combination leads to a variety of entailment relations that can be used for reasoning about non-deterministic phenomena and are inconsistency-tolerant. We investigate the basic properties of these entailments, as well as some of their computational aspects,
more &raquo; ... demonstrate their usefulness in the context of model-based diagnostic systems. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j a r Matrices, Nmatrices and their families We start with the simplest semantic structures used for defining logics (see, for instance, [28, 52, 55] ). Definition 3 (Deterministic matrices). A (deterministic) matrix for L is a triple M ¼ hV; D; Oi, where V is a non-empty set of truth values, D is a non-empty proper subset of V, consisting of the designated elements of V, and for every n-ary connective } of L; O includes an n-ary function e } M : V n ! V. A matrix M ¼ hV; D; Oi consists, then, of a set V of the truth-values, a subset D of the values representing 'true assertions', and a set O with an interpretation (a 'truth table') for each connective in the language L. We say that M is finite if so is V. In case that V ¼ ft; fg and D ¼ ftg we say that the matrix is two-valued (or a 2matrix). Definition 4 (Models and satisfiability). Let M be a matrix for L. -An M-valuation for L is a function m : W L ! V such that for every n-ary connective } of L and every w 1 ; . . . ; w n 2 W L ; mð}ðw 1 ; . . . ; w n ÞÞ ¼ e }ðmðw 1 Þ; . . . ; mðw n ÞÞ. We denote by K s M the set of all the M-valuations of L. 4 -A valuation m 2 K s M M-satisfies a formula w (alternatively, m is an M-model of w), if mðwÞ 2 D. We denote this by m M w. Example 2. The most common matrix-based logic is, of-course, classical logic, which is induced, e.g., by the two-valued matrix M cl ¼ hft; fg; ftg; f e _; e :gi, interpreting the conjunction _ and the negation : in the standard way. Deterministic matrices do not always faithfully represent incompleteness, or situations in which the truth-value of a formula cannot be strictly determined. This brings us to the second type of structures, defined by non-deterministic matrices (Nmatrices for short). These are a natural generalization of the standard many-valued matrices, in which the truth-value assigned to a complex formula is chosen non-deterministically out of a set of options. This idea allows to express uncertainty by the semantic structures themselves (as opposed to some other multi-valued logics, such as annotated logic [33, 34] , where uncertainty is reflected by the syntax of the underlying language). Definition 6 (Non-deterministic matrices). [11] A non-deterministic matrix for L (henceforth, an Nmatrix) is a triple N ¼ hV; D; Oi, 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 e } N : V n ! 2 V n f;g. Again, we say that an Nmatrix N is finite if so is V. When V ¼ ft; fg and D ¼ ftg, N is called two-valued Nmatrix (alternatively, 2Nmatrix). Example 3. Consider an AND-gate that operates correctly when its input lines have the same value and is unpredictable otherwise. The behaviour of such faulty gate may be described by the following non-deterministic truth-table: 3 Clearly, there are other ways of introducing non-determinism into the semantics, such as probabilistic or stochastic logics, but these methods are outside the scope of this paper. 4 The letter 's' stands here for 'static' semantics. This notation will be useful in the context of non-deterministic matrices, to distinguish between static and dynamic semantics. We use it already for deterministic matrices to keep the notations uniform.
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