Possibilistic reasoning with partially ordered beliefs

International audienceThis paper presents the extension of results on reasoning with totally ordered belief bases to the partially ordered case. The idea is to reason from logical bases equipped with a partial order expressing relative certainty and to construct a partially ordered deductive closure. The difficult point lies in the fact that equivalent definitions in the totally ordered case are no longer equivalent in the partially ordered one. At the syntactic level we can either use a language expressing pairs of related formulas and axioms describing the properties of the ordering, or use formulas with partially ordered symbolic weights attached to them in the spirit of possibilistic logic. A possible semantics consists in assuming the partial order on formulas stems from a partial order on interpretations. It requires the capability of inducing a partial order on subsets of a set from a partial order on its elements so as to extend possibility theory functions. Among different possible definitions of induced partial order relations, we select the one generalizing necessity orderings (closely related to epistemic entrenchments). We study such a semantic approach inspired from possibilistic logic, and show its limitations when relying on a unique partial order on interpretations. We propose a more general sound and complete approach to relative certainty, inspired by conditional modal logics, in order to get a partial order on the whole propositional language. Some links between several inference systems, namely conditional logic, modal epistemic logic and non-monotonic preferential inference are established. Possibilistic logic with partially ordered symbolic weights is also revisited and a comparison with the relative certainty approach is made via mutual translations

On the Semantics of Partially Ordered Bases

International audienceThis paper presents first results toward the extension of possibilistic logic when the total order on formulas is replaced by a partial preorder. Few works have dealt with this matter in the past but they include some by Halpern, and Benferhat et al. Here we focus on semantic aspects, namely the construction of a partial order on interpretations from a partial order on formulas and conversely. It requires the capability of inducing a partial order on subsets of a set from a partial order on its elements. The difficult point lies in the fact that equivalent definitions in the totally ordered case are no longer equivalent in the partially ordered one. We give arguments for selecting one approach extending comparative possibility and its preadditive refinement, pursuing some previous works by Halpern. It comes close to non-monotonic inference relations in the style of Kraus Lehmann and Magidor. We define an intuitively appealing notion of closure of a partially ordered belief base from a semantic standpoint, and show its limitations in terms of expressiveness, due to the fact that a partial ordering on subsets of a set cannot be expressed by means of a single partial order on the sets of elements. We also discuss several existing languages and syntactic inference techniques devised for reasoning from partially ordered belief bases in the light of this difficulty. The long term purpose is to find a proof method adapted to partially ordered formulas, liable of capturing a suitable notion of semantic closure

Generic filters in partially ordered sets

The concept of partial-order valued models and that of D-generic filters play a central role in the present day development of Set Theory;In this dissertation, we consider questions related to both of the above concepts;In Section 2, we prove the existence of a model (M,(ELEM)) for the unrestricted Comprehension Scheme;((FOR ALL)x)((x (epsilon) M) (---\u3e) ((THERE EXISTS)s)((s(ELEM)M) (WEDGE) ((VBAR)(VBAR)x (ELEM) s(VBAR)(VBAR) = (VBAR)(VBAR)F(x)(VBAR)(VBAR))));in a certain n-valued logic (L(,n), N, D) with connectives N and D defined by appropriate truth tables;In Section 3, we construct partial-order valued models where (VBAR)(VBAR)x (epsilon) y(VBAR)(VBAR) is a subset of a partially ordered set P with (VBAR)(VBAR)(IL-PERP)F(VBAR)(VBAR) defined as z (VBAR)z (epsilon) P and z is incompatible with every element of (VBAR)(VBAR)F(VBAR)(VBAR) and with the other connectives defined in their usual set-theoretical interpretations. These models are reduced to two valued models via the notion of a generic filter of P. In Section 4, we introduce the notion of a molecule m of P by requiring that every two elements compatible with m be themselves compatible. Then, we prove the equivalence of the existence of a generic filter to that of a molecule in a partially ordered set. In Section 5, we introduce some P-lattice algebras and prove the existence of a D-complete ultrafilter in a Boolean algebra for the denumerable case;In Section 6, we introduce the notion of k-inducive partially ordered sets as partially ordered sets in which every inversely well-ordered subset of cardinality less then k of nonzero elements has a nonzero lower bound. Based on k-inducive partially ordered set, we prove the existence of some E-complete filters of partially ordered sets with the condition E(\u27 )\u3c(\u27 )k imposed on cardinality of E;In Sections 7 to 10, we prove some equivalent forms of Martin\u27s Axiom in connection with Boolean algebras and also we give some consequences of Martin\u27s Axiom pertaining to the cardinal exponentiation;Finally, in Section 11 we introduce the notion of a receding sequence (S(,i))(,

Security Theorems via Model Theory

A model-theoretic approach can establish security theorems for cryptographic protocols. Formulas expressing authentication and non-disclosure properties of protocols have a special form. They are quantified implications for all xs . (phi implies for some ys . psi). Models (interpretations) for these formulas are *skeletons*, partially ordered structures consisting of a number of local protocol behaviors. Realized skeletons contain enough local sessions to explain all the behavior, when combined with some possible adversary behaviors. We show two results. (1) If phi is the antecedent of a security goal, then there is a skeleton A_phi such that, for every skeleton B, phi is satisfied in B iff there is a homomorphism from A_phi to B. (2) A protocol enforces for all xs . (phi implies for some ys . psi) iff every realized homomorphic image of A_phi satisfies psi. Hence, to verify a security goal, one can use the Cryptographic Protocol Shapes Analyzer CPSA (TACAS, 2007) to identify minimal realized skeletons, or "shapes," that are homomorphic images of A_phi. If psi holds in each of these shapes, then the goal holds