Learning-assisted Theorem Proving with Millions of Lemmas

Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and re-used in later proofs by formal mathematicians. In this work, we suggest and implement criteria defining the estimated usefulness of the HOL Light lemmas for proving further theorems. We use these criteria to mine the large inference graph of the lemmas in the HOL Light and Flyspeck libraries, adding up to millions of the best lemmas to the pool of statements that can be re-used in later proofs. We show that in combination with learning-based relevance filtering, such methods significantly strengthen automated theorem proving of new conjectures over large formal mathematical libraries such as Flyspeck.Comment: journal version of arXiv:1310.2797 (which was submitted to LPAR conference

Lemmas: Generation, Selection, Application

Noting that lemmas are a key feature of mathematics, we engage in an investigation of the role of lemmas in automated theorem proving. The paper describes experiments with a combined system involving learning technology that generates useful lemmas for automated theorem provers, demonstrating improvement for several representative systems and solving a hard problem not solved by any system for twenty years. By focusing on condensed detachment problems we simplify the setting considerably, allowing us to get at the essence of lemmas and their role in proof search

A Framework for the Flexible Integration of a Class of Decision Procedures into Theorem Provers

The role of decision procedures is often essential in theorem proving. Decision procedures can reduce the search space of heuristic components of a prover and increase its abilities. However, in some applications only a small number of conjectures fall within the scope of the available decision procedures. Some of these conjectures could in an informal sense fall ‘just outside’ that scope. In these situations a problem arises because lemmas have to be invoked or the decision procedure has to communicate with the heuristic component of a theorem prover. This problem is also related to the general problem of how to exibly integrate decision procedures into heuristic theorem provers. In this paper we address such problems and describe a framework for the exible integration of decision procedures into other proof methods. The proposed framework can be used in different theorem provers, for different theories and for different decision procedures. New decision procedures can be simply ‘plugged-in’ to the system. As an illustration, we describe an instantiation of this framework within the Clam proof-planning system, to which it is well suited. We report on some results using this implementation

Cooperation between Top-Down and Bottom-Up Theorem Provers

Top-down and bottom-up theorem proving approaches each have specific advantages and disadvantages. Bottom-up provers profit from strong redundancy control but suffer from the lack of goal-orientation, whereas top-down provers are goal-oriented but often have weak calculi when their proof lengths are considered. In order to integrate both approaches, we try to achieve cooperation between a top-down and a bottom-up prover in two different ways: The first technique aims at supporting a bottom-up with a top-down prover. A top-down prover generates subgoal clauses, they are then processed by a bottom-up prover. The second technique deals with the use of bottom-up generated lemmas in a top-down prover. We apply our concept to the areas of model elimination and superposition. We discuss the ability of our techniques to shorten proofs as well as to reorder the search space in an appropriate manner. Furthermore, in order to identify subgoal clauses and lemmas which are actually relevant for the proof task, we develop methods for a relevancy-based filtering. Experiments with the provers SETHEO and SPASS performed in the problem library TPTP reveal the high potential of our cooperation approaches

Synthesizing Lemmas for Inductive Reasoning

Recursively defined structures and properties about them are naturally expressed in first-order logic with least fixpoint definitions (FO+lfp), but automated reasoning for such logics has not seen much progress. Such logics, unlike pure FOL, do not even admit complete procedures, let alone decidable ones. In this paper, we undertake a foundational study of finding proofs that use induction to reason with these logics. By treating proofs as purely FO proofs punctuated by declarations of induction lemmas, we separate proofs into deductively reasoned components that can be automated and statements of lemmas that need to be divined, respectively. While humans divine such lemmas with intuition, we propose a counterexample driven technique that guides the synthesis of such lemmas, where counterexamples are finite models that witness inability of proving the theorem as well as other proposed lemmas. We develop relatively complete procedures for synthesizing such lemmas using techniques and tools from program/expression synthesis, for powerful FO+lfp logics that have background sorts constrained by natural theories such as arithmetic and set theory. We evaluate our procedures and show that over a class of theorems that require finding inductive proofs, our automatic synthesis procedure is effective in proving them

Automating and simplifying agreement and secrecy verification using PVS

In this thesis we present a system for assisting with theorem proving of security protocols. The desirability of theorem proving is examined and a method of automating the encoding, and some sections of the proof, are demonstrated. We also discuss various aspects of two different classes of security properties: secrecy and agreement. We demonstrate how our system can be used via two case study protocols, NetBill and SET. The proof can be decomposed into various sub-lemmas, most of which can be proven automatically, and then used to simplify the proofs of the final theorems of interest

Conjectures, tests and proofs: An overview of theory exploration

A key component of mathematical reasoning is the ability to formulate interesting conjectures about a problem domain at hand. In this paper, we give a brief overview of a theory exploration system called QuickSpec, which is able to automatically discover interesting conjectures about a given set of functions. QuickSpec works by interleaving term generation with random testing to form candidate conjectures. This is made tractable by starting from small sizes and ensuring that only terms that are irreducible with respect to already discovered conjectures are considered. QuickSpec has been successfully applied to generate lemmas for automated inductive theorem proving as well as to generate specifications of functional programs. We give an overview of typical use-cases of QuickSpec, as well as demonstrating how to easily connect it to a theorem prover of the user’s choice