Abstract
Decades of work have gone into developing efficient proof calculi, data structures, algorithms, and heuristics for first-order automatic theorem proving. Higher-order provers lag behind in terms of efficiency. Instead of developing a new higher-order prover from the ground up, we propose to start with the state-of-the-art superposition prover E and gradually enrich it with higher-order features. We explain how to extend the prover’s data structures, algorithms, and heuristics to lambda -free higher-order logic, a formalism that supports partial application and applied variables. Our extension outperforms the traditional encoding and appears promising as a stepping stone toward full higher-order logic.
Highlights
Superposition provers such as E [45], SPASS [57], and Vampire [27] are among the most successful first-order reasoning systems
We introduce an intermediate transition system, −→, that focuses on a single pair of terms but that solves the constraints in a depth-first, left-to-right fashion and builds the substitution incrementally
How useful are Ehoh’s new heuristics? And how does Ehoh perform compared with E, used directly or in tandem with the applicative encoding, and compared with other provers? To answer the first question, we evaluated each new parameter independently
Summary
Superposition provers such as E [45], SPASS [57], and Vampire [27] are among the most successful first-order reasoning systems. Research on higher-order automatic provers has resulted in systems such as LEO [11], Leo-II [13], and Leo-III [46], based on resolution and paramodulation, and Satallax [18], based on tableaux and SAT solving. They feature a “cooperative” architecture, pioneered by LEO: They are full-fledged higher-order provers that regularly invoke an external firstorder prover with a low time limit as a terminal procedure, in an attempt to finish the proof quickly using only first-order reasoning. At the 2017 installment of the CADE ATP System Competition (CASC) [50], Leo-III, which uses E as a backend, proved 652 out of 2000 first-order problems in the Sledgehammer division, compared with 1185 for E on its own and 1433 for Vampire
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