arXiv subjects

Nicholas Smallbone

Publications and source records attributed to Nicholas Smallbone.

7 recordsLinked to original sources

arXiv2026

Twitch: Learning Abstractions for Equational Theorem Proving

Several successful strategies in automated reasoning rely on human-supplied guidance about which term or clause shapes are interesting. In this paper we aim to discover interesting term shapes automatically. Specifically, we discover abstractions : term patterns that occur over and over again in relevant proofs. We present our tool Twitch which discovers abstractions with the help of Stitch, a tool originally developed for discovering reusable library functions in program synthesis tasks. Twitch can produce abstractions in two ways: (1) from a partial, failed proof of a conjecture; (2) from successful proofs of other theorems in the same domain. We have also extended Twee, an equational theorem prover, to use these abstractions. We evaluate Twitch on a set of unit equality (UEQ) problems from TPTP, and show that it can prove 12 rating-1 problems as well as yielding significant speed-ups on many other problems.

cs.LO↗

arXiv2025

Lemmanaid: Neuro-Symbolic Lemma Conjecturing

Mathematicians and computer scientists are increasingly leveraging proof assistants to formalize and check complex proofs, a task that demands substantial expertise. Can we lower the bar by automating the conjecturing of helpful, interesting and novel lemmas? We present the first neuro-symbolic lemma conjecturing tool, LEMMANAID, designed to discover conjectures by drawing analogies between mathematical theories. LEMMANAID uses a fine-tuned LLM to generate lemma templates that describe the shape of a lemma, and symbolic methods to fill in the details. We compare LEMMANAID against the same LLM fine-tuned to generate lemmas directly, as well as a fully symbolic conjecturing method. On test sets from Isabelle's HOL library and Archive of Formal Proofs (AFP), LEMMANAID consistently outperforms both neural and symbolic methods. Using DeepSeek-coder-6.7B as a backend, LEMMANAID discovers 50% (HOL) and 29% (AFP) of the gold standard lemmas, increasing to 55% and 35% when ensembling prompting strategies. In a case study on Octonions, LEMMANAID discovers 79% of the gold standard lemmas, compared to 62% for neural-only and 23% for the state of the art symbolic tool. Furthermore, in a targeted comparison, LEMMANAID discovers more gold standard lemmas than both Claude Opus 4.5 and GPT-5.2. Our results show that LEMMANAID can conjecture a significant number of interesting lemmas across complex formalizations in mathematics and computer science.

cs.AI↗

arXiv2024

QuickerCheck: Implementing and Evaluating a Parallel Run-Time for QuickCheck

This paper introduces a new parallel run-time for QuickCheck, a Haskell library and EDSL for specifying and randomly testing properties of programs. The new run-time can run multiple tests for a single property in parallel, using the available cores. Moreover, if a counterexample is found, the run-time can also shrink the test case in parallel, implementing a parallel search for a locally minimal counterexample. Our experimental results show a 3--9$\times$ speed-up for testing QuickCheck properties on a variety of heavy-weight benchmark problems. We also evaluate two different shrinking strategies; deterministic shrinking, which guarantees to produce the same minimal test case as standard sequential shrinking, and greedy shrinking, which does not have this guarantee but still produces a locally minimal test case, and is faster in practice.

cs.PL↗

arXiv2021

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.

cs.LO↗

arXiv2019

Enhancing Temporal Logic Falsification with Specification Transformation and Valued Booleans

Cyber-Physical Systems (CPSs) are systems with both physical and software components, for example cars and industrial robots. Since these systems exhibit both discrete and continuous dynamics, they are complex and it is thus difficult to verify that they behave as expected. Falsification of temporal logic properties is an approach to find counterexamples to CPSs by means of simulation. In this paper, we propose two additions to enhance the capability of falsification and make it more viable in a large-scale industrial setting. The first addition is a framework for transforming specifications from a signal-based model into Signal Temporal Logic. The second addition is the use of Valued Booleans and an additive robust semantics in the falsification process. We evaluate the performance of the additive robust semantics on a set of benchmark models, and we can see that which semantics are preferable depend both on the model and on the specification.

eess.SY↗

arXiv2016

Encoding Monomorphic and Polymorphic Types

Many automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter when translating monomorphic to untyped first-order logic. Here we pursue this approach systematically, analysing formally a variety of encodings that further improve on efficiency while retaining soundness and completeness. We extend the approach to rank-1 polymorphism and present alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of "cover". The new encodings are implemented in Isabelle/HOL as part of the Sledgehammer tool. We include informal proofs of soundness and correctness, and have formalised the monomorphic part of this work in Isabelle/HOL. Our evaluation finds the new encodings vastly superior to previous schemes.

cs.LO↗

arXiv2014

Hipster: Integrating Theory Exploration in a Proof Assistant

This paper describes Hipster, a system integrating theory exploration with the proof assistant Isabelle/HOL. Theory exploration is a technique for automatically discovering new interesting lemmas in a given theory development. Hipster can be used in two main modes. The first is exploratory mode, used for automatically generating basic lemmas about a given set of datatypes and functions in a new theory development. The second is proof mode, used in a particular proof attempt, trying to discover the missing lemmas which would allow the current goal to be proved. Hipster's proof mode complements and boosts existing proof automation techniques that rely on automatically selecting existing lemmas, by inventing new lemmas that need induction to be proved. We show example uses of both modes.

cs.LO↗