Building on the Diamonds between Theories: Theory Presentation Combinators

Carette, Jacques; O'Connor, Russell; Sharoda, Yasmine

doi:10.48550/arxiv.1812.08079

Cited by 3 publications

(18 citation statements)

References 41 publications

(63 reference statements)

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“…We suggest automating some of the definitions of concepts derivable via known techniques. We have tested our implementation on a library of 227 theories, including Ring and BoundedDistributedLattice, built using the tiny theories approach [5] and the combinators of [9]. A theory defined declaratively using the combinators elaborate into a Tog record, which is then manipulated to generate the constructions presented in Section 3.…”

Section: Discussionmentioning

confidence: 99%

“…This is the problem we continue [1,6,8,9] to tackle here, and that others [11] have started to look at as well. It is worthwhile noting that some programming languages already provide interesting features in this direction.…”

Section: Introductionmentioning

confidence: 93%

“…[11] addresses the important problem of library maintainability, especially when dealing with changes to the hierarchy. We have proposed an alternate solution in [9], based on the categorical structures already present in dependent type theories. The algebraic library of Lean [12] is of particular interest, as its developers are quite concerned with automation.…”

Section: Related Workmentioning

confidence: 99%

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Leveraging the Information Contained in Theory Presentations

Carette¹,

Farmer²,

Sharoda³

2020

Preprint

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A theorem prover without an extensive library is much less useful to its potential users. Algebra, the study of algebraic structures, is a core component of such libraries. Algebraic theories also are themselves structured, the study of which was started as Universal Algebra. Various constructions (homomorphism, term algebras, products, etc) and their properties are both universal and constructive. Thus they are ripe for being automated. Unfortunately, current practice still requires library builders to write these by hand. We first highlight specific redundancies in libraries of existing systems. Then we describe a framework for generating these derived concepts from theory definitions. We demonstrate the usefulness of this framework on a test library of 227 theories.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 93%

Section: Related Workmentioning

confidence: 99%

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Leveraging the Information Contained in Theory Presentations

Carette¹,

Farmer²,

Sharoda³

2020

Preprint

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“…• Build a library of over 200 theories describing the algebraic hierarchy, implemented using the combinators in [Carette et al, 2019] (Chapter 7).…”

Section: Contributionsmentioning

confidence: 99%

“…• [Carette et al, 2019] Contributed to an extended paper discussing the MathScheme combinators that were initially published in [Carette and O'Connor, 2012]. The • [Rabe and Sharoda, 2019] Used the diagram infrastructure developed in MMT [Rabe and Kohlhase, 2013a] and described in the paper to implement the MathScheme combinators described in [Carette et al, 2019]. Since the combinators compute a theory and some arrows, we considered treating their inputs and outputs as diagrams.…”

Section: Publicationsmentioning

confidence: 99%

Leveraging the Information Contained in Theory Presentations

Carette

Farmer

Sharoda

2020

Lecture Notes in Computer Science

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Building a large library of mathematical knowledge is a complex and labour-intensive task. By examining current libraries of mathematics, we see that the human effort put in building them is not entirely directed towards tasks that need human creativity.Instead, a non-trivial amount of work is spent on providing definitions that could have been mechanically derived.In this work, we propose a generative approach to library building, so definitions that can be automatically derived are computed by meta-programs. We focus our attention on libraries of algebraic structures, like monoids, groups, and rings. These structures are highly inter-related and their commonalities have been well-studied in universal algebra. We use theory presentation combinators to build a library of algebraic structures. Definitions from universal algebra and programming languages meta-theory are used to derive library definitions of constructions, like homomorphisms and term languages, from algebraic theory presentations. The result is an interpreter that, given 227 theory expressions, builds a library of over 5000 definitions. This library is, then, exported to Agda and Lean.iii To my family, You are my greatest blessing iv I would like to express my sincere thanks to my supervisors Dr. Jacques Carette and Dr. William Farmer for their continuous support to my learning journey. Your expertise and feedback were invaluable in shaping this research direction and throughout my studies. I learned from you a lot about how to do research and communicate it. I

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Revisiting Language Support for Generic Programming: When Genericity Is a Core Design Goal

2022

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Context Generic programming, as defined by Stepanov, is a methodology for writing efficient and reusable algorithms by considering only the required properties of their underlying data types and operations. Generic programming has proven to be an effective means of constructing libraries of reusable software components in languages that support it. Generics-related language design choices play a major role in how conducive generic programming is in practice. Inquiry Several mainstream programming languages (e.g. Java and C ++ ) were first created without generics; features to support generic programming were added later, gradually. Much of the existing literature on supporting generic programming focuses thus on retrofitting generic programming into existing languages and identifying related implementation challenges. Is the programming experience significantly better, or different when programming with a language designed for generic programming without limitations from prior language design choices? Approach We examine Magnolia, a language designed to embody generic programming. Magnolia is representative of an approach to language design rooted in algebraic specifications. We repeat a well-known experiment, where we put Magnolia's generic programming facilities under scrutiny by implementing a subset of the Boost Graph Library, and reflect on our development experience. Knowledge We discover that the idioms identified as key features for supporting Stepanov-style generic programming in the previous studies and work on the topic do not tell a full story. We clarify which of them are more of a means to an end, rather than fundamental features for supporting generic programming. Based on the development experience with Magnolia, we identify variadics as an additional key feature for generic programming and point out limitations and challenges of genericity by property. Grounding Our work uses a well-known framework for evaluating the generic programming facilities of a language from the literature to evaluate the algebraic approach through Magnolia, and we draw comparisons with well-known programming languages. Importance This work gives a fresh perspective on generic programming, and clarifies what are fundamental language properties and their trade-offs when considering supporting Stepanov-style generic programming. The understanding of how to set the ground for generic programming will inform future language design. ACM CCS 2012Software and its engineering → Very high level languages;

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Building on the Diamonds between Theories: Theory Presentation Combinators

Cited by 3 publications

References 41 publications

Leveraging the Information Contained in Theory Presentations

Leveraging the Information Contained in Theory Presentations

Leveraging the Information Contained in Theory Presentations

Revisiting Language Support for Generic Programming: When Genericity Is a Core Design Goal

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