An Undecidable Fragment of the Theory of Set Constraints

Set constraints are inclusions between expressions denoting sets of trees. In atomic set constraints, the syntax of set expressions is restricted not to contain any Boolean set operators. Using a reduction from the Hilbert's Tenth Problem we prove the undecidability of the 9 8-fragment of the rst-order theory of atomic set constraints. This is the minimal undecidable fragment of the rst-order theory of set constraints if all Boolean connectives (_; :) are admitted.