Theory Presentation Combinators

Carette, Jacques; O'Connor, Russell

doi:10.1007/978-3-642-31374-5_14

Cited by 14 publications

(15 citation statements)

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“…Interesting categorical motivation of such an approach is presented in [7]. Modules can be treated globally for all proof assistants, and a kind of interface allowing for information interchange is proposed as MMT -a module system for mathematical theories with scalable formalism [34].…”

Section: Discussionmentioning

confidence: 99%

On algebraic hierarchies in mathematical repository of Mizar

Grabowski

Korniłowicz

Schwarzweller

2016

Annals of Computer Science and Information Systems

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Abstract-Mathematics, especially algebra, uses plenty of structures: groups, rings, integral domains, fields, vector spaces to name a few of the most basic ones. Classes of structures are closely connected -usually by inclusion -naturally leading to hierarchies that has been reproduced in different forms in different mathematical repositories. In this paper we give a brief overview of some existing algebraic hierarchies and report on the latest developments in the Mizar computerized proof assistant system. In particular we present a detailed algebraic hierarchy that has been defined in Mizar and discuss extensions of the hierarchy towards more involved domains. Taking fully formal approach into account we meet new difficulties comparing with its informal mathematical framework.

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Section: Discussionmentioning

confidence: 99%

On algebraic hierarchies in mathematical repository of Mizar

Grabowski

Korniłowicz

Schwarzweller

2016

Annals of Computer Science and Information Systems

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“…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%

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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“…A complex body of mathematical knowledge can be represented in accordance with the little theories method [30] (or even the tiny theories method [18]) as a theory graph [36] consisting of axiomatic theories as nodes and theory morphisms as directed edges. A theory morphism is a meaning-preserving mapping from the formulas of one axiomatic theory to the formulas of another.…”

Section: Background Ideasmentioning

confidence: 99%

“…In [18], we developed combinators for combining theory presentations. There is no significant difference between axiomatic and biform theories with respect to the semantics of these combinators, and we expect that these will continue to work as well as they did in [8].…”

Section: Biform Theory Graphsmentioning

confidence: 99%

Biform Theories: Project Description

Carette¹,

Farmer²,

Sharoda³

2018

Preprint

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A biform theory is a combination of an axiomatic theory and an algorithmic theory that supports the integration of reasoning and computation. These are ideal for specifying and reasoning about algorithms that manipulate mathematical expressions. However, formalizing biform theories is challenging as it requires the means to express statements about the interplay of what these algorithms do and what their actions mean mathematically. This paper describes a project to develop a methodology for expressing, manipulating, managing, and generating mathematical knowledge as a network of biform theories. It is a subproject of MathScheme, a long-term project at McMaster University to produce a framework for integrating formal deduction and symbolic computation.We present the Biform Theories project, a subproject of MathScheme [11] (a long-term project to produce a framework integrating formal deduction and symbolic computation). MotivationType 2 * 3 into your favourite computer algebra system, press enter, and you will receive (unsurprisingly) 6. But what if you want to go in the opposite direction? Easy: you ask ifactors(6) in Maple or FactorInteger [6] in Mathematica. 1 The Maple command ifactors returns a 2-element list, with the first element the unit (1 or -1) and the second element a list of pairs (encoded as two-element lists) with (distinct) primes in the first component and the prime's multiplicity in the second. Mathematica's FactorInteger is similar, except that it omits the unit (and thus does not document what happens for negative integers).This simple example illustrates the difference between a simple computation 2 * 3 and a more complex symbolic query, factoring. The reason for using lists-of-lists in both systems is that multiplication and powering are both functions that evaluate immediately in these systems. So that factoring 6 cannot This research is supported by NSERC. 1 Other computer algebra systems have similar commands.

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Theory Presentation Combinators

Abstract: Abstract. We motivate and give semantics to theory presentation combinators as the foundational building blocks for a scalable library of theories. The key observation is that the category of contexts and fibered categories are the ideal theoretical tools for this purpose.

Cited by 14 publications

References 15 publications

On algebraic hierarchies in mathematical repository of Mizar

On algebraic hierarchies in mathematical repository of Mizar

Leveraging the Information Contained in Theory Presentations

Biform Theories: Project Description

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