The Role of Entropy in Guiding a Connection Prover

In this work we study how to learn good algorithms for selecting reasoning steps in theorem proving. We explore this in the connection tableau calculus implemented by leanCoP where the partial tableau provides a clean and compact notion of a state to which a limited number of inferences can be applied. We start by incorporating a state-of-the-art learning algorithm -- a graph neural network (GNN) -- into the plCoP theorem prover. Then we use it to observe the system's behaviour in a reinforcement learning setting, i.e., when learning inference guidance from successful Monte-Carlo tree searches on many problems. Despite its better pattern matching capability, the GNN initially performs worse than a simpler previously used learning algorithm. We observe that the simpler algorithm is less confident, i.e., its recommendations have higher entropy. This leads us to explore how the entropy of the inference selection implemented via the neural network influences the proof search. This is related to research in human decision-making under uncertainty, and in particular the probability matching theory. Our main result shows that a proper entropy regularisation, i.e., training the GNN not to be overconfident, greatly improves plCoP's performance on a large mathematical corpus

The formal verification of the ctm approach to forcing

We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model $M$ of $\mathit{ZFC}$, of generic extensions satisfying $\mathit{ZFC}+\neg\mathit{CH}$ and $\mathit{ZFC}+\mathit{CH}$. Moreover, let $\mathcal{R}$ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset $\Omega\subseteq\mathcal{R}$ and defined $\mathcal{F}:\mathcal{R}\to\mathcal{R}$ such that for every $\Phi\subseteq\mathcal{R}$ and $M$-generic $G$, $M\models \mathit{ZC} \cup \mathcal{F}\text{``}\Phi \cup \Omega$ implies $M[G]\models \mathit{ZC} \cup \Phi \cup \{ \neg \mathit{CH} \}$, where $\mathit{ZC}$ is Zermelo set theory with Choice. To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.Comment: 20pp + 14pp in bibliography & appendices, 2 table

Derivational Complexity and Context-Sensitive Rewriting

[EN] Context-sensitive rewriting is a restriction of rewriting where reduction steps are allowed on specific arguments mu(f) subset of {1, ..., k} of k-ary function symbols f only. Terms which cannot be further rewritten in this way are called mu-normal forms. For left-linear term rewriting systems (TRSs), the so-called normalization via mu-normalization procedure provides a systematic way to obtain normal forms by the stepwise computation and combination of intermediate mu-normal forms. In this paper, we show how to obtain bounds on the derivational complexity of computations using this procedure by using bounds on the derivational complexity of context-sensitive rewriting. Two main applications are envisaged: Normalization via mu-normalization can be used with non-terminating TRSs where the procedure still terminates; on the other hand, it can be used to improve on bounds of derivational complexity of terminating TRSs as it discards many rewritings.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32 and PROMETEO/2019/098.Lucas Alba, S. (2021). Derivational Complexity and Context-Sensitive Rewriting. Journal of Automated Reasoning. 65(8):1191-1229. https://doi.org/10.1007/s10817-021-09603-11191122965

Automatización para el entorno Isabelle / ZF

Tesis (Lic. en Cs. de la Computación)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2021.Al formalizar en Isabelle/ZF las definiciones asociadas a Forcing para demostrar la independencia de la Hipótesis del Continuo, se presenta una cantidad significativa de tareas sistemáticas y repetitivas, entre las que se destacan la relativización de términos y predicados, por un lado, y la síntesis de fórmulas internalizadas, por el otro. Por lo tanto, se desea evitar el trabajo manual todo lo posible. Este trabajo consiste en brindar herramientas automáticas que se encarguen de dichas tareas y minimicen la cantidad de intervenciones manuales requeridas. Más aún, se justificará con cierto grado de formalidad la corrección de los métodos implementados, y también se detallará la intuición detrás de las partes más complejas. Finalmente, se mostrará cuál es la disciplina a seguir a la hora de utilizar los comandos implementados.When the definitions regarding Forcing are being formalised in Isabelle/ZF, in order to prove the independence of the Continuum Hypothesis, a lot of systematic and repetitive tasks are required. Among them, relativization of terms and predicates, on the one hand, and synthesis of internalized formulas, on the other hand, are the most important ones. Thus, it is desired to reduce manual intervention as much as possible. In this thesis, some automatic tools will be provided to take care of those tasks, and will reduce the amount of manual interventions required. Furthermore, the soundness of the implemented methods will be formally justified, and the intuition behind the most complex parts will also be detailed. Finally, the whole discipline to use the commands will be shown.publishedVersionFil: Steinberg, Matías Uriel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina

Applications and extensions of context-sensitive rewriting

[EN] Context-sensitive rewriting is a restriction of term rewriting which is obtained by imposing replacement restrictions on the arguments of function symbols. It has proven useful to analyze computational properties of programs written in sophisticated rewriting-based programming languages such asCafeOBJ, Haskell, Maude, OBJ*, etc. Also, a number of extensions(e.g., to conditional rewritingor constrained equational systems) and generalizations(e.g., controlled rewritingor forbidden patterns) of context-sensitive rewriting have been proposed. In this paper, we provide an overview of these applications and related issues. (C) 2021 Elsevier Inc. All rights reserved.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32 and PROMETEO/2019/098.Lucas Alba, S. (2021). Applications and extensions of context-sensitive rewriting. Journal of Logical and Algebraic Methods in Programming. 121:1-33. https://doi.org/10.1016/j.jlamp.2021.10068013312