Temporal Logics over Transitive States [chapter]

Boris Konev, Frank Wolter, Michael Zakharyaschev
2005 Lecture Notes in Computer Science  
We investigate the computational behaviour of 'two-dimensional' propositional temporal logics over (N, <) (with and without the next-time operator Ä) that are capable of reasoning about states with transitive relations. Such logics are known to be undecidable (even Π 1 1complete) if the domains of states with those relations are assumed to be constant. Motivated by applications in the areas of temporal description logic and specification & verification of hybrid systems, in this paper we
more » ... the computational impact of allowing the domains of states to expand. We show that over finite expanding domains (with an arbitrary, tree-like, quasi-order, or linear transitive relation) the logics are recursively enumerable, but undecidable. If these finite domains eventually become constant then the resulting Ä-free logics are decidable (but not in primitive recursive time); on the other hand, when equipped with Ä they are not even recursively enumerable. Finally, we show that temporal logics over infinite expanding domains as above are undecidable even for the language with the sole temporal operator 'eventually.' The proofs are based on Kruskal's tree theorem and reductions of reachability problems for lossy channel systems. The monodic fragment of first-order LTL allows applications of temporal operators to formulas with at most one free variable [14] . Thus, in the framework of this fragment we can only control the temporal change of properties-i.e., unary predicates-of states, while binary, ternary, etc. relations can change arbitrarily. The full monodic fragment turns out to be semi-decidable [32] , and if we restrict the first-order part to a decidable fragment (for example, to the two-variable or guarded fragments), then the resulting monodic fragment is usually decidable as well. The simplest interesting fragment of this sort is the one-variable firstorder LTL (Sistla and German [29] considered it in the context of verification). Various spatio-temporal logics based on spatial formalisms like RCC-8, etc. can be encoded in the one-variable first-order temporal logic [8, 9] and therefore inherit its good computational properties. Monodic fragments of this kind are usually decidable in elementary time [8] , and both tableau-and resolutionbased provers have been developed and implemented for monodic temporal logics [19, 17, 16] . The idea of monodicity is based on two conditions: the 'positive' Mono + : temporal constraints can be imposed on unary predicates and the 'negative' Dya − : no temporal constraint can be imposed on n-nary predicates for n ≥ 2. Having in mind possible applications of temporal logic mentioned above, condition Dya − appears too restrictive. In fact, in temporal knowledge bases (say, temporal description logics), more sophisticated spatio-temporal formalisms, in particular, dynamic topological logics (designed for reasoning about safety and
doi:10.1007/11532231_14 fatcat:bjvs534zkzc2ff2nyu2zj4k5cq