Lifting congruence closure with free variables to λ-free higher-order logic via SAT encoding

Sophie Tourret (1) , Pascal Fontaine (2, 3, 1) , Daniel El Ouraoui (1) , Haniel Barbosa (4)

1 VERIDIS - Modeling and Verification of Distributed Algorithms and Systems
2 Université de Liège
3 MOSEL - Proof-oriented development of computer-based systems
4 UFMG - Universidade Federal de Minas Gerais = Federal University of Minas Gerais [Belo Horizonte, Brazil]

Recent work in extending SMT solvers to higher-order logic (HOL) has not explored lifting quantifier instantiation algorithms to perform higher-order unification. As a consequence, widely used instantiation techniques, such as trigger-and particularly conflictbased, can only be applied in a limited manner. Congruence closure with free variables (CCFV) is a decision procedure for the E-ground (dis-)unification problem, which is at the heart of these instantiation techniques. Here, as a first step towards fully supporting trigger-and conflict-based instantiation in HOL, we define the E-ground (dis-)unification problem in λ-free higher-order logic (λfHOL), an extension of first-order logic where function symbols may be partially applied and functional variables may occur, and extend CCFV to solve it. To improve scalability in the context of handling higher-order variables, we rely on an encoding of the CCFV search as a propositional formula. We present a solution reconstruction procedure so that models for the propositional formula lead to solutions for the E-ground (dis-)unification problem. This is instrumental to port triggerand conflict-based instantiation to be fully applied in λfHOL. * The order of authors is inverse alphabetic.

Domaines: Logique en informatique [cs.LO]

Sophie Tourret, Pascal Fontaine, Daniel El Ouraoui, Haniel Barbosa. Lifting congruence closure with free variables to λ-free higher-order logic via SAT encoding. SMT 2020 - 18th International Workshop on Satisfiability Modulo Theories, Jul 2020, Online COVID-19, France. ⟨hal-03049088⟩