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Biform theories in Chiron
 Towards Mechanized Mathematical Assistants, volume 4573 of Lecture Notes in Computer Science
, 2007
"... Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical k ..."
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Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical knowledge both declaratively and procedurally. Since the algorithms of algorithmic theories manipulate the syntax of expressions, biform theories—as well as algorithmic theories—are difficult to formalize in a traditional logic without the means to reason about syntax. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It includes elements of type theory, a scheme for handling undefinedness, and a facility for reasoning about the syntax of expressions. It is an exceptionally wellsuited logic for formalizing biform theories. This paper defines the notion of a biform theory, gives an overview of Chiron, and illustrates how biform theories can be formalized in Chiron. 1
Chiron: A set theory with types, undefinedness, quotation, and evaluation
, 2007
"... Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. Unlike traditional set theories such as ZermeloFraenkel (zf) and nbg, Chiron is equipped with a type system, lambda notation, and definite and ..."
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Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. Unlike traditional set theories such as ZermeloFraenkel (zf) and nbg, Chiron is equipped with a type system, lambda notation, and definite and indefinite description. The type system includes a universal type, dependent types, dependent function types, subtypes, and possibly empty types. Unlike traditional logics such as firstorder logic and simple type theory, Chiron admits undefined terms that result, for example, from a function applied to an argument outside its domain or from an improper definite or indefinite description. The most noteworthy part of Chiron is its facility for reasoning about the syntax of expressions. Quotation is used to refer to a set called a construction that represents the syntactic structure of an expression, and evaluation is used to refer to the value of the expression that a construction
The MathScheme library: Some preliminary experiments. Manuscript arXiv:1106.1862v1
, 2011
"... Abstract. We present some of the experiments we have performed to besttestourdesignforalibraryforMathScheme,themechanizedmathematics software system we are building. We wish for our library design to use and reflect, as much as possible, the mathematical structure present in the objects which popula ..."
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Abstract. We present some of the experiments we have performed to besttestourdesignforalibraryforMathScheme,themechanizedmathematics software system we are building. We wish for our library design to use and reflect, as much as possible, the mathematical structure present in the objects which populate the library. 1
Mathscheme: Project description
 Intelligent Computer Mathematics, volume 6824 of Lecture Notes in Computer Science
, 2011
"... 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) the ..."
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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 longterm 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 shortterm, we are developing tools and techniques to support this approach, with the longterm 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 metaprogramming techniques
Frameworks for reasoning about syntax that utilize quotation and evaluation
, 2013
"... It is often useful, if not necessary, to reason about the syntactic structure of an expression in an interpreted language (i.e., a language with a semantics). This paper introduces a mathematical structure called a syntax framework that is intended to be an abstract model of a system for reasoning a ..."
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It is often useful, if not necessary, to reason about the syntactic structure of an expression in an interpreted language (i.e., a language with a semantics). This paper introduces a mathematical structure called a syntax framework that is intended to be an abstract model of a system for reasoning about the syntax of an interpreted language. Like many concrete systems for reasoning about syntax, a syntax framework contains a mapping of expressions in the interpreted language to syntactic values that represent the syntactic structures of the expressions; a language for reasoning about the syntactic values; a mechanism called quotation to refer to the syntactic value of an expression; and a mechanism called evaluation to refer to the value of the expression represented by a syntactic value. A syntax framework provides a basis for integrating reasoning about the syntax of the expressions with reasoning about what the expressions mean. The notion of a syntax framework is used to discuss how quotation and evaluation can be built into a language and to define what quasiquotation is. Several examples of syntax frameworks are presented.
Chiron: Mechanizing Mathematics in OCaml By
"... Computer algebra systems such as Maple [2] and Mathematica [12] are good at symbolic computation, while theorem proving systems such as Coq [11] and pvs [9] are welldeveloped for creating formal proofs. However, people are searching for a mechanized mathematics system which can provide highly integ ..."
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Computer algebra systems such as Maple [2] and Mathematica [12] are good at symbolic computation, while theorem proving systems such as Coq [11] and pvs [9] are welldeveloped for creating formal proofs. However, people are searching for a mechanized mathematics system which can provide highly integrated symbolic computation and formal deduction capabilities at the same time. My work is to design and implement the basis for a mechanized mathematics system based on a formal framework, which was previously developed as part of the MathScheme project at McMaster University. The core idea of the framework consists of the notion of a biform theory, which is simultaneously an axiomatic theory and an algorithmic theory, providing a formal context for both deduction and computation. A mechanized mathematics system which utilizes biform theories to represent mathematics requires a logic in which biform theories can be expressed. Chiron, as a derivative of vonNeumannBernaysGödel set theory, is the logic we choose for our MMS development. It is intended to be a practical, generalpurpose logic
iv Table of Contents
"... Many issues stand in the way of the development of contemporary mechanized mathematics systems (MMS) and the following are two major obstacles: • Dedicated languages with mathematical specifications for MMS. • A wellendowed theory library which serves as a database of mathematics. We implement a Ma ..."
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Many issues stand in the way of the development of contemporary mechanized mathematics systems (MMS) and the following are two major obstacles: • Dedicated languages with mathematical specifications for MMS. • A wellendowed theory library which serves as a database of mathematics. We implement a MathScheme Language (MSL), which represents theory types useful for covering basic algebraic structures and improving the expressive power of mathematical modeling systems. The development of MSL primarily focuses on language syntax and its logic independence. More importantly, we present a library of theory types developed based on module systems of typed programming languages and algebraic specification languages. The modularity mechanism used in our library aims for the interface manipulation and high level expressivity of MMSs. The theories are organized according to the little theories method [10]. Our module system extensively supports several building operations to construct new theories from existing theories. iii Acknowledgements I would first and foremost like to express my deepfelt gratitude to my supervisor, Dr. Jacques Carette, who shared with me a lot of his expertise and research insight through my studies. This thesis was made possible by his advice, assistance and guidance. My special thanks and appreciation goes to the members of my examination committee, Dr. Jacques Carette, Dr. William M. Farmer, and Dr. Spencer