Analytic Tableaux for Higher-Order Logic with Choice

Abstract

While many higher-order interactive theorem provers include a choice operator, higher-order automated theorem provers so far have not. In order to support automated reasoning in the presence of a choice operator, we present a cut-free ground tableau calculus for Church's simple type theory with choice. The tableau calculus is designed with automated search in mind. In particular, the rules only operate on the top level structure of formulas. Additionally, we restrict the instantiation terms for quantifiers to a universe that depends on the current branch. At base types the universe of instantiations is finite. Both of these restrictions are intended to minimize the number of rules a corresponding search procedure is obligated to consider. We prove completeness of the tableau calculus relative to Henkin models.

Extending Sledgehammer with SMT Solvers

“…To evaluate the SMT integration, we ran their benchmark suite with the latest versions of Sledgehammer on the same seven formalizations. 3 We also added two formalizations (QE and S2S) that rely heavily on arithmetic to exercise the SMT decision procedures. The formalizations are listed below.…”

“…The refactoring seemed worthwhile, especially since it also benefits other provers that we might want to interface with Sledgehammer, such as higher-order ATPs [3,6]. The Sledgehammer-SMT integration is, to our knowledge, the first of its kind, and we had no clear idea of how successful it would be as we started the implementation work.…”

Higher-order aspects and context in SUMO

“…Satallax [58] is a higher-order automated theorem prover with additional model finding capabilities. The system is based on a complete ground tableau calculus for HOL with a choice operator [59]. An initial tableau branch is formed from the axioms of the problem and negation of the conjecture (if any is given).…”

Progress in Automating Higher-Order Ontology Reasoning

“…Available HO-ATPs include LEO-II [10], TPS [2], IsabelleP, IsabelleM/N 2 and Satallax [3]. These systems are available online via the SystemOnTPTP tool [25], they support the TPTP THF infrastructure, and they employ THF0 [11], the simple type theory fragment of the THF language, as input language.…”