Expressive power of digraph solvability

Abstract

A kernel of a directed graph is a set of vertices without edges between them, such that every other vertex has a directed edge to a vertex in the kernel. A digraph possessing a kernel is called solvable. We show that solvability of digraphs is equivalent to satisfiability of theories of propositional logic. Based on a new normal form for such theories, this equivalence relates finitely branching digraphs to propositional logic, and arbitrary digraphs to infinitary propositional logic. Furthermore, we show that the existence of a kernel for a digraph is equivalent to the existence of a kernel for its lifting to an infinitely-branching dag. While the computational complexity of solvability differs between finite dags (trivial, since always solvable) and finite digraphs (NP-complete), this difference disappears in the infinite case: we prove that solvability for recursive dags and digraphs is Σ11-complete. This implies that satisfiability for recursive theories of infinitary propositional logic is also Σ11 -complete. Finally, using solvability of dags we formulate a new equivalent of the axiom of choice.