Interval-based Synthesis

In this paper, we introduce the synthesis problem for Halpern and Shoham’s interval temporal logic [5] extended with an equivalence relation ∼ over time points (HS ∼ for short). In analogy to the case of monadic second-order logic of one successor [2], given an HS ∼ formula φ and a finite set ΣT of proposition letters and temporal requests, the problem consists of establishing whether or not, for all possible evaluations of elements in ΣT in every interval structure, there is an evaluation of the remaining proposition letters and temporal requests such that the resulting structure is a model for φ . We focus our attention on the decidability of the synthesis problem for some meaningful fragments of HS ∼, whose modalities are drawn from {A (meets), Ā (met by), B (begun by), B̄ (begins)}, interpreted over finite linear orders and natural numbers. We prove that the synthesis problem for ABB∼ over finite linear orders is decidable (non-primitive recursive hard), while AĀBB̄ turns out to be undecidable. In addition, we show that if we replace finite linear orders by natural numbers, then the problem becomes undecidable even for ABB̄.