Flatness Is Not a Weakness

Abstract

We propose an extension, called $\mathcal{L}_p^ + , o$, of the temporal logic LTL, which enables talking about finitely many register values: the models are infinite words over tuples of integers (resp. real numbers). The formulas of $\mathcal{L}_p^ + , o$ are flat: on the left of an until, only atomic formulas or LTL formulas are allowed. We prove, in the spirit of the correspondence between automata and temporal logics, that the models of a $\mathcal{L}_p^ + , o$ formula are recognized by a piecewise flat counter machine; for each state q, at most one loop of the machine on q may modify the register values.

Emptiness of (piecewise). at counter machines is decidable (this follows from a result in [9]). It follows that satisfiability and model-checking the negation of a formula are decidable for $\mathcal{L}_p^ + , o$. On the other hand, we show that inclusion is undecidable for such languages. This shows that validity and model-checking positive formulas are undecidable.