Duality between Unprovability and Provability in Forward Refutation-search for Intuitionistic Propositional Logic

Abstract

The inverse method is a saturation based theorem proving technique; it relies on a forward proof-search strategy and can be applied to cut-free calculi enjoying the subformula property. Here we apply this method to derive the unprovability of a goal formula G in Intuitionistic Propositional Logic. To this aim we design a forward calculus FRJ(G) for Intuitionistic unprovability which is prone to constructively ascertain the unprovability of a formula G by providing a concise countermodel for it; in particular we prove that the generated countermodels have minimal height. Moreover, we clarify the role of the saturated database obtained as result of a failed proof-search in FRJ(G) by showing how to extract from such a database a derivation witnessing the Intuitionistic validity of the goal.

Efficient SAT-based Proof Search in Intuitionistic Propositional Logic

“…Instead, intuitR requires 45 calls to the SAT-solvers, 8 restarts and yields a countermodel consisting of 4 worlds; the set W contains 26 worlds before the first restart, one world before the remaining ones. With all the benchmarks the models generated during the computation are small (typically, big models occur before the first restart); however, differently from [7,8,9], we cannot guarantee that countermodels have minimum depth or minimum number of worlds. To complete the picture, the scatter plot in Fig.…”

Refutation Systems: An Overview and Some Applications to Philosophical Logics

“…• Likewise, in [Fiorentini and Ferrari, 2018], Fiorentini and Ferrari explore the duality between unprovability and provability in forward proof-search for intuitionistic propositional logic, following on their previous, already mentioned work [Fiorentini and Ferrari, 2017].…”

Hybrid Deduction–Refutation Systems

“…Even though with different motivation and agenda, and with no technical results of the type pursued here, this work appears to be the closest in spirit to the idea of hybrid deduction-refutation systems studied in the present work. • Likewise, in [15], Fiorentini and Ferrari explore the duality between unprovability and provability in forward proof-search for intuitionistic propositional logic and develop a refutation-complete sequent-based forward refutation calculus for it, following on their previous work [16].…”