Scheme-based theorem discovery and concept invention

Abstract

We describe an approach to automatically invent/explore new mathematical theories, with the goal of producing results comparable to those produced by humans, as represented, for example, in the libraries of the Isabelle proof assistant. Our approach is based on 'schemes', which are terms in higher-order logic. We show that it is possible to automate the instantiation process of schemes to generate conjectures and definitions. We also show how the new definitions and the lemmata discovered during the exploration of a theory can be used, not only to help with the proof obligations during the exploration, but also to reduce redundancies inherent in most theory formation systems.We implemented our ideas in an automated tool, called IsaScheme, which employs Knuth-Bendix completion and recent automatic inductive proof tools.We have evaluated our system in a theory of natural numbers and a theory of lists.

Hipster: Integrating Theory Exploration in a Proof Assistant

“…There are two other theory exploration systems available for Isabelle/HOL, IsaCoSy [13] and IsaScheme [16]. They differ in the way they generate conjectures, and both discover similar lemmas as HipSpec/Hipster.…”

“…However, they leave the generation of new conjectures relating different concepts largely to the user. Recently, a number of different systems have been implemented to address the conjecture synthesis aspect of theory exploration [15,13,16,5]. Our work goes one step further by integrating the discovery and proof of new conjectures in the workflow of the interactive theorem prover Isabelle/HOL.…”

European collaboration on automated reasoning

“…Termination: of recursive programs was a closely related problem [30,63]. [10,20,44,54,57]. Verification: of computer systems was the main application of inductive proof [18,35,38,41,52,58,64], with a continuing interest in the synthesis of programs from their specifications [2,7,24,49] and a new interest in security applications [1,59].…”

ACL2(ml): Machine-Learning for ACL2

“…Lemma Analogy in ACL2(ml). Symbolic tools for automatic discovery of lemmas, like IsaCoSy [17] and IsaScheme [22], synthesise new terms using different kinds of heuristics. These tools generate candidate conjectures which are then filtered through a counterexample checker.…”