Certified Symbolic Manipulation: Bivariate Simplicial Polynomials

Certified symbolic manipulation is an emerging new field where programs are accompanied by certificates that, suitably interpreted, ensure the correctness of the algorithms. In this paper, we focus on algebraic algorithms implemented in the proof assistant ACL2, which allows us to verify correctness in the same programming environment. The case study is that of bivariate simplicial polynomials, a data structure used to help the proof of properties in Simplicial Topology. Simplicial polynomials can be computationally interpreted in two ways. As symbolic expressions, they can be handled algorithmically, increasing the automation in ACL2 proofs. As representations of functional operators, they help proving properties of categorical morphisms. As an application of this second view, we present the definition in ACL2 of some morphisms involved in the Eilenberg-Zilber reduction, a central part of the Kenzo computer algebra system. We have proved the ACL2 implementations are correct and tested that they get the same results as Kenzo does.Ministerio de Ciencia e Innovación MTM2009-13842Unión Europea nr. 243847 (ForMath

A visual approach for solving problems with fractions

This article discusses the importance of visual models in problem solving, in the scope of rational numbers. We seek to highlight the potential of this approach, as a structuring theme in the mathematical development of students in elementary education and the connections it allows to establish. In order for students to be mathematically competent and creative, they must be able not only to solve traditional computational problems but also to use models/visual representations when solving all types of mathematical problems, including those in which the visual component is not evident. We developed a qualitative study based on a didactical experience involving 14 future teachers who were attending a Didactics of Mathematics unit course that included a module about problem solving with emphasizes in visual approaches. The main purpose of the study was to identify the strategies used by the future teachers when solving problems with multiple solutions, before and after that module. Data was collected through observation and the written productions of the participants. It was possible to conclude that they tended to privilege analytical approaches before the intervention and, after the teaching experience, they started to value visual approaches, which generated an increase of the productions involving this type of solutions.0D1E-4824-1244 | Ana Cristina Coelho BarbosaN/

Functional declarative language design and predicate calculus: A practical approach

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Dynamic IFC Theorems for Free!

We show that noninterference and transparency, the key soundness theorems for dynamic IFC libraries, can be obtained "for free", as direct consequences of the more general parametricity theorem of type abstraction. This allows us to give very short soundness proofs for dynamic IFC libraries such as faceted values and LIO. Our proofs stay short even when fully mechanized for Agda implementations of the libraries in terms of type abstraction.Comment: CSF 2021 final versio

The use of data-mining for the automatic formation of tactics

This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques

Unifying heterogeneous state-spaces with lenses

Most verification approaches embed a model of program state into their semantic treatment. Though a variety of heterogeneous state-space models exists,they all possess common theoretical properties one would like to capture abstractly,such as the common algebraic laws of programming. In this paper,we propose lenses as a universal state-space modelling solution. Lenses provide an abstract interface for manipulating data types through spatially-separated views. We define a lens algebra that enables their composition and comparison,and apply it to formally model variables and alphabets in Hoare and He’s Unifying Theories of Programming (UTP). The combination of lenses and relational algebra gives rise to a model for UTP in which its fundamental laws can be verified. Moreover,we illustrate how lenses can be used to model more complex state notions such as memory stores and parallel states. We provide a mechanisation in Isabelle/HOL that validates our theory,and facilitates its use in program verification

Robust Computer Algebra, Theorem Proving, and Oracle AI

In the context of superintelligent AI systems, the term "oracle" has two meanings. One refers to modular systems queried for domain-specific tasks. Another usage, referring to a class of systems which may be useful for addressing the value alignment and AI control problems, is a superintelligent AI system that only answers questions. The aim of this manuscript is to survey contemporary research problems related to oracles which align with long-term research goals of AI safety. We examine existing question answering systems and argue that their high degree of architectural heterogeneity makes them poor candidates for rigorous analysis as oracles. On the other hand, we identify computer algebra systems (CASs) as being primitive examples of domain-specific oracles for mathematics and argue that efforts to integrate computer algebra systems with theorem provers, systems which have largely been developed independent of one another, provide a concrete set of problems related to the notion of provable safety that has emerged in the AI safety community. We review approaches to interfacing CASs with theorem provers, describe well-defined architectural deficiencies that have been identified with CASs, and suggest possible lines of research and practical software projects for scientists interested in AI safety.Comment: 15 pages, 3 figure