Higher-Order Positive Set Constraints

We introduce a natural notion of positive set constraints on simplytyped -terms. We show that satisfiability of these so-called positive higher-order set constraints is decidable in 2-NEXPTIME. We explore a number of subcases solvable in 2-DEXPTIME, among which higher-order definite set constraints, a.k.a., emptiness of higher-order pushdown processes. This uses a first-order clause format on so-called shallow higher-order patterns, and automated deduction techniques based on ordered resolution with splitting. This technique is then applied to the task of approximating success sets for a restricted subset of - Prolog, a la Fr uhwirth et al