HoCHC: A Refutationally Complete and Semantically Invariant System of Higher-order Logic Modulo Theories

Abstract

We present a simple resolution proof system for higher-order constrained Horn clauses (HoCHC)-a system of higher-order logic modulo theories-and prove its soundness and refutational completeness w.r.t. both standard and Henkin semantics. As corollaries, we obtain the compactness theorem and semi-decidability of HoCHC for semi-decidable background theories, and we prove that HoCHC satisfies a canonical model property. Moreover a variant of the well-known translation from higher-order to 1st-order logic is shown to be sound and complete for HoCHC in both semantics. We illustrate how to transfer decidability results for (fragments of) 1st-order logic modulo theories to our higher-order setting, using as example the Bernays-Schönfinkel-Ramsey fragment of HoCHC modulo a restricted form of Linear Integer Arithmetic.

Initial Limit Datalog: a New Extensible Class of Decidable Constrained Horn Clauses

“…The setting for our language is the fragment of higherorder logic known as higher-order constrained Horn clauses (HoCHC) [3,7]. Higher-order constrained Horn clauses allow for the description of predicates of higher-types (i.e., whose subjects may themselves be predicates).…”

“…Such predicates can be described by clauses built from terms of the simply typed λcalculus when equipped with the appropriate logical constants. As in [3,7], we forgo the use of explicit abstraction to simplify the Horn clause format.…”

“…The former semi-decision procedure is an immediate consequence of (P1), (P3) and (P4). The latter is an application of the semi-decidability of HoCHC unsatisfiability, via a refutationally complete resolution proof system (Γ is unsatisfiable if, and only if, there is a resolution proof of ⊥ from Γ) [7].…”

“…Remark II.2. Unlike [7], we do not distinguish pre-frame and frame. Because λ-abstractions are not part of the HoCHC syntax here, the (weak) comprehension axiom in [7, p. 3] does not apply.…”

“…(iii) If Σ bg contains a predicate interpreted by A as equality (as with Σ LIA and A LIA ), then we may write terms of integer type inside foreground atoms. For example X (x + y + 5) is equivalent to X z ∧ (z = x + y + 5) (iv) Every satisfiable set of clauses Γ has, in each frame, a canonical model B which arises by saturating under all immediate consequences [7,Thm. 23].…”