An Approach to Fuzzy Modal Logic of Time Intervals

Temporal reasoning based on intervals is nowadays ubiquitous in artificial intelligence, and the most representative interval temporal logic, called HS, was introduced by Halpern and Shoham in the eighties. There has been a great effort in the past in studying the expressive power and computational properties of the satisfiability problem for HS and its fragments, but only recently HS has been proposed as a suitable formalism for artificial intelligence applications. Such applications highlighted some of the intrinsic limits of HS: Sometimes, when dealing with real-life data one is not able to express temporal relations and propositional labels in a definite, crisp way. In this paper, following the seminal ideas of Fitting and Zadeh, among others, we present a fuzzy generalization of HS that partially solves such problems of expressive power, and we prove that, as in the crisp case, its satisfiability problem is generally undecidable

Data Structures for Symbolic Multi-Valued Model-Checking

Multi-valued logics provide an interesting alternative to classical boolean logic for modeling and reasoning about systems. Such logics can be used for reasoning about partially-specified systems, effectively encode vacuity detection and query-checking problems, help in detecting inconsistencies, and many others. In our earlier work, we identified a useful family of multi-valued logics: those specified over finite distributive lattices where negation preserves involution, i.e., �¦������ � for every element � of the logic. Such structures are called quasi-boolean algebras, and model-checking over these not only extends the domain of applicability of automated reasoning to new problems, but can also speed up solutions to some classical verification problems. Symbolic model-checking over quasi-boolean algebras can be cast in terms of operations over multi-valued sets: sets whose membership functions are multi-valued. In this paper, we propose and empirically evaluate several choices for implementing multi-valued sets with decision diagrams. In particular, we describe two major approaches: (1) representing the multi-valued membership function canonically, using MDDs or ADDs; (2) representing multi-valued sets as a collection of classical sets, using a vector of either MBTDDs or BDDs. The naive implementation of (2) includes having a classical set for each value of the algebra. We exploit a result of lattice theory to reduce the number of such sets that need to be represented. The major contribution of this paper is the evaluation of the different implementations of multivalued sets, done via a series of experiments and using several case studies.

Data Structures for Symbolic Multi-Valued Model-Checking

Multi-valued logics can be effectively used to reason about incomplete and/or inconsistent systems, e.g. during early software requirements or as the systems evolve. In our earlier work we identified a useful family of multi-valued logics: those specified over finite distributive lattices where negation preserves involution, i.e., �¦������ � for every element � of the logic. Model-checking over this family of logics allows not only to extend the domain of applicability of automated reasoning to new problems, but also to speed up some classical verification problems. Symbolic model-checking over multi-valued logics can be cast in terms of operations over multivalued sets: sets whose membership functions are multi-valued. In this paper we propose and empirically evaluate several choices for implementing multi-valued sets with decision diagrams. In particular, we describe two major approaches: (1) representing the multi-valued membership function canonically, using MDDs or ADDs; (2) representing multi-valued sets as a collection of classical sets, using a vector of either MBTDDs or BDDs. The naive implementation of (2) includes having a classical set for each value of the logic. We exploit a result of lattice theory to reduce the number of such sets that need to be represented. The major contribution of this paper is the evaluation of the different implementations of multivalued sets, done via a series of experiments and using several case studies.