Undecidability of the Emptiness Problem of Deterministic Propositional While Programs with Graph Loop: Hypothesis Elimination Using Loops

We show that the emptiness (unsatisfiability) problem is undecidable and $\mathrm{\Pi}^{0}_{1}$-complete for deterministic propositional while programs with (graph) loop. To this end, we introduce a hypothesis elimination using loops. Using this, we give reductions from the complement of the periodic domino problem. Moreover, as a corollary via hypothesis eliminations, we also show that the equational theory is $\mathrm{\Pi}^{0}_{1}$-complete for the positive calculus of relations with transitive closure and difference. Additionally, we show that the emptiness problem is PSPACE-complete for the existential calculus of relations with transitive closure.