Superposition with First-class Booleans and Inprocessing Clausification

Abstract

We present a complete superposition calculus for first-order logic with an interpreted Boolean type. Our motivation is to lay the foundation for refutationally complete calculi in more expressive logics with Booleans, such as higher-order logic, and to make superposition work efficiently on problems that would be obfuscated when using clausification as preprocessing. Working directly on formulas, our calculus avoids the costly axiomatic encoding of the theory of Booleans into first-order logic and offers various ways to interleave clausification with other derivation steps. We evaluate our calculus using the Zipperposition theorem prover, and observe that, with no tuning of parameters, our approach is on a par with the state-of-the-art approach.

Lattice Structure of Some Closed Classes for Three-Valued Logic and Its Applications

“…However, for the case of multivalued logic (in particular, for three-valued logic), this is still an open problem. Different techniques and tools have been applied to this problem, but the current results mainly include either the description of general approaches or attempts to apply the mathematical tool of linear algebra (nonlogical tools) to synthesize nonbinary digital current circuits [53], or the more complicated pure theoretical tools that are impossible to realize [54], or the realization for some special operators is only given [55], or a complete superposition calculus is only provided for first-order-type logic [56]. I proved the result for the three-valued case and showed that any logic function (digital circuit) can be synthesized from a finite number of different microcircuits.…”

Superposition for Higher-Order Logic

“…Standard superposition Bachmair and Ganzinger [9] Superposition with ← → and delayed CNF Ganzinger and Stuber [61] Superposition with Booleans Nummelin et al [109] λ-free superposition First milestone Superposition with combinators Bhayat and Reger [28] Boolean-free λ-superposition Second milestone…”

“…In the second experiment, we explore the clausification methods introduced at the end of Section 7.3: inner delayed clausification, which relies on the core calculus to reason about logical symbols; outer delayed clausification, which clausifies step-by-step guided by the outermost logical symbols; and immediate clausification, which eagerly applies a monolithic clausification algorithm when encountering top-level logical symbols. The modes inner and outer employ the RENAME rule developed in my work with Nummelin et al [109] headed by logical symbols using a Tseitin-like transformation if they occur at least four times in the proof state. Vukmirović and Nummelin [138] observed that outer clausification can greatly help prove higher-order problems and we expected it perform well for our calculus, too.…”

“…In this chapter, we introduce first-order logic with interpreted Booleans (Section 6.2), present a ground calculus for the logic (Section 6.3), and prove it to be refutationally complete (Section 6.4). The nonground version of this calculus, simplification rules, and an empirical evaluation are in progress [109].…”

A Comprehensive Framework for Saturation Theorem Proving

“…When users of the framework instantiate these prover architectures with a concrete saturation calculus, they obtain the dynamic refutational completeness of the combination from the properties of the prover architecture and the static refutational completeness proof for the calculus. The framework is applicable to a wide range of calculi, including ordered resolution [6], unfailing completion [2], standard superposition [5], constraint superposition [29], theory superposition [44], and hierarchic superposition [8], It is already used in several published and ongoing works on combinatory superposition [15], λ-free superposition [12], λ-superposition [13,14], superposition with interpreted Booleans [33], AVATAR-style splitting [21], and superposition with SAT-inspired inprocessing [43].…”