Theorem Provers For Every Normal Modal Logic

Abstract

We present a procedure for algorithmically embedding problems formulated in higherorder modal logic into classical higher-order logic. The procedure was implemented as a stand-alone tool and can be used as a preprocessor for turning TPTP THF-compliant theorem provers into provers for various modal logics. The choice of the concrete modal logic is thereby specified within the problem as a meta-logical statement. This specification format as well as the underlying semantics parameters are discussed, and the implementation and the operation of the tool are outlined.By combining our tool with one or more THF-compliant theorem provers we accomplish the most widely applicable modal logic theorem prover available to date, i.e. no other available prover covers more variants of propositional and quantified modal logics. Despite this generality, our approach remains competitive, at least for quantified modal logics, as our experiments demonstrate.

Extensional Paramodulation for Higher-Order Logic and Its Effective Implementation Leo-III

“…The practical evidence that quantified modal logics can effectively be modeled as a fragment of classical higher-order logic furthermore suggests that higherorder logic can serve as an universal meta-logic [Ben17b]. This claim is further substantiated by more recent evaluations [BOR12,GSB17,SB18] and the observation that there exist analogous reductions for numerous further non-classical logics [BS16,SB16,Ben17a,Ben17c,BFP18], for many of which there exist none or only few specialized reasoning systems. A strong foundation for the automation of higher-order logic and an effective implementation of a corresponding deduction system thus enables computer-assisted reasoning in even more, practically relevant, logical systems and application areas.…”

“…Recently, the expressivity of higher-order logic has been exploited for encoding various expressive non-classical logics within HOL. Semantical embeddings of, among others, higher-order modal logics [BP13a,GSB17], conditional logics [Ben17a], many-valued logics [SB16], deontic logic [BFP18], free logics [BS16], access control logics [Ben09] and combinations of such logics [Ben11] can be used to automate reasoning within the respective logic using ATP systems for classical HOL. A prominent result from the applications of automated reasoning in non-classical logics, here in quantified modal logics, was the detection of a major flaw in Gödel's Ontological Argument [FB17,BWP17] as well as the verification of Scott's variant of that argument [BWP15] using the LEO-II theorem prover [BPST15] and the interactive proof assistant Isabelle/HOL [NWP02].…”

The Higher-Order Prover Leo-III

“…This is due to the encoding of the S5 accessibility relation in Leo-III 1.2 as the universal relation between possible worlds as opposed to its prior encoding (cf. [16,13]) as an equivalence relation. Leo-III contributes 199 solutions to previously unsolved problems.…”

“…In addition to its HOL reasoning capabilities, it is the first ATP that natively supports a very wide range of normal HOMLs. To achieve this, Leo-III internally implements a shallow semantical embeddings approach [12,13]. The key idea in this approach is to provide and exploit faithful mappings for HOML input problems to HOL that encode its Kripke-style semantics.…”

Leo-III Version 1.1 (System description)

“…Such logics are of strong interest in many different fields of research, for example in mathematics, artificial intelligence, and philosophy. In its current state, our system is already capable of reasoning for a range of embedded logics using an external pre-processor [16]. The next version of Leo-III will natively integrate such a pre-processor in order to offer out-of-thebox automation for non-classical logics.…”