MathScheme: Project Description

Abstract

The mission of mechanized mathematics is to develop software systems that support the process people use to create, explore, connect, and apply mathematics. Working mathematicians routinely leverage a powerful synergy between deduction and computation. The artificial division between (axiomatic) theorem proving systems and (algorithmic) computer algebra systems has broken this synergy. To significantly advance mechanized mathematics, this synergy needs to be recaptured within a single framework. MathScheme [6] is a long-term project being pursued at McMaster University with the aim of producing such a framework in which formal deduction and symbolic computation are tightly integrated. In the short-term, we are developing tools and techniques to support this approach, with the long-term objective to produce a new system.Towards this aim, we have already developed several techniques, with some laying the theoretical foundations of our framework, while others are implementation techniques. In particular, we rely on biform theories and an expressive logic (Chiron) for grounding. We rely on various meta-programming techniques as well as the increased safety offered by a modern statically typed programming language (Objective Caml [8]) to greatly simplify our implementation burden.A biform theory [1, 3] is a combination of an axiomatic theory and an algorithmic theory. It is the basic unit of mathematical knowledge that consists of a set of concepts, transformers, and facts. The concepts are symbols that denote mathematical values and, together with the transformers, form a language L for the theory. The transformers are programs whose input and output are expressions of L. Transformers represent syntax-manipulating operations such as inference and computation rules. The facts are statements expressed in L about the concepts and transformers. In a typical biform theory, the concepts are classified as primitive or defined, the transformers as primitive or derived, and the facts as axioms, definitions, or theorems. A pure axiomatic theory is a biform theory with no transformers, and a pure algorithmic theory is a biform theory with no facts or only facts about the transformers.Since transformers manipulate the syntax of expressions, biform theories are difficult to formalize in a traditional logic without the means to reason about ⋆

Incorporating quotation and evaluation into Church's type theory

“…Since transformers manipulate the syntax of expressions, biform theories are difficult to formalize in a traditional logic. The notion of a biform theory is a key component of a framework for integrating axiomatic and algorithmic mathematics that is being developed under the MathScheme project [28] at McMaster University, led by Jacques Carette and the author. One of the main goals of the MathScheme is to see if a logic like ctt qe can be used to develop a library of biform theories connected by meaning preserving theory morphisms.…”

“…(24) is an instance of part 5 of Theorem 9.3.1;(25) follows from (24) by Weakening;(26) and(27) follow from the hypothesis E 2 o and Axioms B5.1-6 by Universal Generalization, Universal Instantiation, the Equality Rules and propositional logic;(28) and(29)follow from (26), (27), and the hypothesis D o by Universal Instantiation, parts 5 and 7 of Lemma 9.3.2, the Equality Rules, and propositional logic; (30) and (31) follow from (28), (29), and the hypothesis E 2 ′ and propositional logic; (40)-(43) follow from the hypothesis E 2 o and the definition of IS-EFFECTIVE by Beta-Reduction by Substitution, part 4 of Lemma 6.6.2 and Lemma 9.3.2; and (44) follows from (41)-(43) by Lemma 6.3.7.…”

Realms: A Structure for Consolidating Knowledge about Mathematical Theories

“…The next step in our investigation will be to realize and test such support in the OMDoc/MMT [MMT;KRZ10] and the MathScheme [CFO11] systems, fully develop the examples sketched in this paper, and test the interactions on developers, students, and practitioners (see section 3).…”

“…Our work on OMDoc/MMT [MMT;KRZ10] and the MathScheme [CFO11] systems have given us a decent intuition (or so we feel) regarding the services that a theory-graph based system should provide. We now extent this to realms.…”

Meaning Formulas for Syntax-Based Mathematical Algorithms

“…This research is part of the MathScheme project [2], a long-term initiative at McMaster University led by Dr. Jacques Carette and the author with the aim of producing a framework in which formal deduction and symbolic computation are tightly integrated.…”