Buchi automata augmented with spatial constraints: simulating an alternating with a nondeterministic and deciding the emptiness problem for the latter

Abstract: The aim of this work is to thoroughly investigate Buchi automata augmented with spatial constraints. The input trees of such an automaton are infinite k-ary Sigma-trees, with the nodes standing for time points, and Sigma including, additionally to its uses in classical k-ary Sigma-trees, the description of the snapshot of an n-object spatial scene of interest. The constraints, from an RCC8-like spatial Relation Algebra (RA) x, are used to impose spatial constraints on objects of the spatial scene, eventually at different nodes of the input trees. We show that a Buchi alternating automaton augmented with spatial constraints can be simulated with a classical Buchi nondeterministic automaton of the same type, augmented with spatial constraints. We then provide a nondeterministic doubly depth-first polynomial space algorithm for the emptiness problem of the latter automaton. Our main motivation came from another work, also submitted to this conference, which defines a spatio-temporalisation of the well-known family ALC(D) of description logics with a concrete domain: together, the two works provide an effective solution to the satisfiability problem of a concept of the spatio-temporalisation with respect to a weakly cyclic TBox.