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

Automated Conjecturing in QuickSpec

A key component of mathematical reasoning is the ability to formulate interesting conjectures about a problem domain at hand. This task has not yet been widely studied by the automated reasoning and AI communities, but we believe interest is growing. In this paper, we give a brief overview of a theory exploration system called QuickSpec, able to automatically discover interesting conjectures about a given set of functions. QuickSpec works by interleaving term generation with random testing to form candidate equational conjectures. This is made tractable by starting from small sizes and ensuring that only terms that are irreducible with respect to already discovered equalities 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 also give a small survey of different approaches to conjecture discovery, and speculate about future directions combining symbolic methods and machine learning

Conditional Lemma Discovery and Recursion Induction in Hipster

Hipster is a theory exploration tool for the proof assistant Isabelle/HOL. It automatically discovers lemmas about given recursive functions and datatypes and proves them by induction. Previously, only equational properties could be discovered. Conditional lemmas, for example required when reasoning about sorting, has been beyond the scope of theory exploration. In this paper we describe an extension to Hipster to also support discovery and proof of conditional lemmas. We also present a new automated tactic, which uses recursion induction. Recursion induction follows the recursive structure of a function definition through its termina- tion order, as opposed to structural induction, which follows that of the datatype. We find that the addition of recursion induction increases the number of proofs completed automatically, both for conditional and equational statements.

Best-First Rippling

Rippling is a form of rewriting that guides search by only performing steps that reduce the syntactic differences between formulae. Termination is normally ensured by a measure that is decreases with each rewrite step. Because of this restriction, rippling will fail to prove theorems about, for example, mutual recursion as steps that temporarily increase the differences are necessary. Best-first rippling is an extension to rippling where the restrictions have been recast as heuristic scores for use in best-first search. If nothing better is available, previously illegal steps can be considered, making best-first rippling more flexible than ordinary rippling. We have implemented best-first rippling in the IsaPlanner system together with a mechanism for caching proof-states that helps remove symmetries in the search space, and machinery to ensure termination based on term embeddings. Our experiments show that the implementation of best-first rippling is faster on average than IsaPlanner’s version of traditional depth-first rippling, and solves a range of problems where ordinary rippling fails

Pacing Patterns of Half-Marathon Runners: An analysis of ten years of results from Gothenburg Half Marathon

ObjectivesEvery year over 40 000 runners complete Gothenburg Half Marathon, one of the world’s largest half-marathons. As participation in recreational races become more common among e.g., older people and those without extensive training experience, providing advice on how to plan the pacing during race is valuable and provide a safer, more positive experience, lessening the risk of over-straining, injury, or collapse.MethodsWe conduct a large-scale data analysis of 10 years (2011 – 2019) of publicly available results data (n=423 496). We calculate how many runners experience slowdowns >25% somewhere during the race, and how many avoid losing time on the second half. We investigate differences between runners depending on age, sex, and ability. Furthermore, we calculate the relationship between temperature on the race day with the average finishing times and proportion of runners who hit the wall each year.ResultsAmong recreational runners, men are about twice as likely to hit the wall compared to women, across all age groups and ability levels. Younger runners more likely to hit the wall than the middle-aged. In warmer years especially, more runners hit the wall, with a steeper increase among the men.ConclusionUsing only easily accessible publicly available results- and weather data, we see that most runners loose time on the second half and would have benefited from pacing advice, especially in warmer years. Our results can be used by race organisers to provide advice to participants based on e.g., the weather prognosis on the race day, as well as estimating need for medical assistance

Machine Learning of Pacing Patterns for Half Marathon

Every year over 40 000 runners participate in Gothenburg Half Marathon, one of the world’s largest half-marathons. Based on publicly available results data (423 496 entries) for ten years (2010 – 2019)1, we investigate machine learning models for two tasks: prediction of finishing times and identification of runners risking hitting the wall. Our models improve results over the current baseline on finish time prediction and manage to correctly identify many of the runners who later hit the wall, although it also misclassifies many who do not

The Effect of Scaling, Retrieval Augmentation and Form on the Factual Consistency of Language Models

Large Language Models (LLMs) make natural interfaces to factual knowledge, but their usefulness is limited by their tendency to deliver inconsistent answers to semantically equivalent questions. For example, a model might predict both "Anne Redpath passed away in Edinburgh." and "Anne Redpath's life ended in London." In this work, we identify potential causes of inconsistency and evaluate the effectiveness of two mitigation strategies: up-scaling and augmenting the LM with a retrieval corpus. Our results on the LLaMA and Atlas models show that both strategies reduce inconsistency while retrieval augmentation is considerably more efficient. We further consider and disentangle the consistency contributions of different components of Atlas. For all LMs evaluated we find that syntactical form and other evaluation task artifacts impact consistency. Taken together, our results provide a better understanding of the factors affecting the factual consistency of language models.Comment: Accepted at EMNLP 202

Automated discovery of inductive lemmas

The discovery of unknown lemmas, case-splits and other so called eureka steps are challenging problems for automated theorem proving and have generally been assumed to require user intervention. This thesis is mainly concerned with the automated discovery of inductive lemmas. We have explored two approaches based on failure recovery and theory formation, with the aim of improving automation of firstand higher-order inductive proofs in the IsaPlanner system. We have implemented a lemma speculation critic which attempts to find a missing lemma using information from a failed proof-attempt. However, we found few proofs for which this critic was applicable and successful. We have also developed a program for inductive theory formation, which we call IsaCoSy. IsaCoSy was evaluated on different inductive theories about natural numbers, lists and binary trees, and found to successfully produce many relevant theorems and lemmas. Using a background theory produced by IsaCoSy, it was possible for IsaPlanner to automatically prove more new theorems than with lemma speculation. In addition to the lemma discovery techniques, we also implemented an automated technique for case-analysis. This allows IsaPlanner to deal with proofs involving conditionals, expressed as if- or case-statements. ii

Den ”gröna” ambitionen

Denna uppsats utforskar hur greenwashing yttrar sig inom den fysiska planeringen, med syfte att synliggöra problemet och på detta sätt förebygga att framtida misstag sker. Till en början undersöks ämnet övergripande och senare analyseras planeringsprocessen och resultatet av två olika stadsbyggnadsprojekt, Hammarby Sjöstad i Stockholm och Västra Hamnen i Malmö. Resultatet av studien tyder på att det finns en problematik kopplat till miljömässiga ambitioner inom den fysiska planeringen och att en del av denna ligger mycket nära greenwashing. Tendenser till greenwashing visade sig kunna finnas i olika skeden av planeringen, exempelvis inom översiktsplaneringen, i projektspecifika plandokument samt visionsbilder. Att vagt underbyggda, ”gröna” begrepp används frekvent inom branschen är en del av problemet, lika så att plandokument samt visionsbilder lovar mer miljömässiga framgångar än vad resultatet visar. Resultatet visar även att miljöfördelaktigt arbete gärna får huvudfokus i projekt medan nackdelarna aktivt skyms undan, vilket kan ge en falsk bild av projektet. I projekten som används som exempel i uppsatsen finnes tendenser till greenwashing framför allt i jämförelsen av dess miljöprogram (Hammarby Sjöstad) respektive kvalitetsprogram (Västra Hamnen) och resultatet från verkligheten, där det bland annat visade sig att miljömål ej nåddes. Resultatet visar också på att det funnits en problematik i att projekten arbetat med miljökrav som inte inneburit några sanktioner om dessa ej följdes.This essay explores how greenwashing is used in spatial planning, with the aim to highlight the problem and through this prevent future mistakes. Initially the topic is investigated comprehensively and later the planning process and results are being analyzed in two urban construction projects, Hammarby Sjöstad (Hammarby Waterfront City) in Stockholm and Västra Hamnen (Western Harbour) in Malmö. The result of the study indicates that there are problems regarding environmental work in spatial planning, and that some of these are close to greenwashing. Tendencies towards greenwashing were found in various stages of the planning, for example within the comprehensive planning, in project- specific planning documents and visualizations. That vaguely substantiated, “green” concepts are being used frequently in the industry is part of the problem, as are planning documents (and visualizations) that promise more environmental success than the result shows. The result also shows that environmentally beneficial work gets a lot of focus while the disadvantages are hidden, which could give a false picture of the project. In the projects used as examples in the essay there are tendencies towards greenwashing mainly in the comparison of its environmental program (Hammarby Sjöstad) and quality program (Västra Hamnen) and the result from reality, where it turned out, among other things, that environmental targets were not reached. The result also shows that there was a problem in that the projects worked with environmental requirements that did not entail any sanctions if these were not followed