Grid structures and undecidable constraint theories

Franck Seynhaeve, Sophie Tison, Marc Tommasi, Ralf Treinen

Abstract: We prove three new undecidability results for computational mechanisms over finite trees: There is a linear, ultra-shallow, noetherian and strongly confluent rewrite system R such that the ∃ ∗ ∀ ∗ -fragment of the first-order theory of one-step-rewriting by R is undecidable; the emptiness problem for tree automata with equality tests between cousins is undecidable; and the ∃ ∗ ∀ ∗ -fragment of the first-order theory of set constraints with the union operator is undecidable. The common feature of these three computational mechanisms is that they allow us to describe the set of first-order terms that represent grids. We extend our representation of grids by terms to a representation of linear two-dimensional patterns by linear terms, which allows us to transfer classical techniques on the grid to terms and thus to obtain our undecidability results.