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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
A rational reconstruction of a system for experimental mathematics
 Towards Mechanized Mathematical Assistants, Lecture Notes in Computer Science
, 2007
"... Over the last decade several environments and formalisms for the combination and integration of mathematical software systems have been proposed. Many of these systems aim at a traditional automated theorem proving approach, in which a given conjecture is to be proved or refuted by the cooperation o ..."
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Over the last decade several environments and formalisms for the combination and integration of mathematical software systems have been proposed. Many of these systems aim at a traditional automated theorem proving approach, in which a given conjecture is to be proved or refuted by the cooperation of different reasoning engines. However, they offer little support for experimental mathematics in which new conjectures are constructed by an interleaved process of model computation, model inspection, property conjecture and verification. In particular, despite some previous research in that direction, there are currently no systems available that provide, in an easy to use environment, the flexible combination of diverse reasoning system in a plugandplay fashion via a high level specification of experiments. [2, 3] presents an integration of more than a dozen different reasoning systems — first order theorem provers, SAT solvers, SMT solvers, model generators, computer algebra, and machine learning systems — in a general bootstrapping algorithm to generate novel theorems in the specialised algebraic domain of
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
Chiron: A Set Theory with Types, Undefinedness, Quotation, and Evaluation*
, 2007
"... Abstract Chiron is a derivative of vonNeumannBernaysG"odel (nbg) set theorythat is intended to be a practical, generalpurpose logic for mechanizing mathematics. Unlike traditional set theories such as ZermeloFraenkel( ..."
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Abstract Chiron is a derivative of vonNeumannBernaysG&quot;odel (nbg) set theorythat is intended to be a practical, generalpurpose logic for mechanizing mathematics. Unlike traditional set theories such as ZermeloFraenkel(
The Formalization of SyntaxBased Mathematical Algorithms Using Quotation and Evaluation ⋆
, 2013
"... Abstract. Algorithms like those for differentiating functional expressions manipulate the syntactic structure of mathematical expressions in a mathematically meaningful way. A formalization of such an algorithm should include a specification of its computational behavior, a specification of its math ..."
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Abstract. Algorithms like those for differentiating functional expressions manipulate the syntactic structure of mathematical expressions in a mathematically meaningful way. A formalization of such an algorithm should include a specification of its computational behavior, a specification of its mathematical meaning, and a mechanism for applying the algorithm to actual expressions. Achieving these goals requires the ability to integrate reasoning about the syntax of the expressions with reasoning about what the expressions mean. A syntax framework is a mathematical structure that is an abstract model for a syntax reasoning system. It contains a mapping of expressions to syntactic values that represent the syntactic structures of the expressions; a language for reasoning about syntactic values; a quotation mechanism to refer to the syntactic value of an expression; and an evaluation mechanism to refer to the value of the expression represented by a syntactic value. We present and compare two approaches, based on instances of a syntax framework, to formalize a syntaxbased mathematical algorithm in a formal theory T. In the first approach the syntactic values for the expressions manipulated by the algorithm are members of an inductive type in T, but quotation and evaluation are functions defined in the metatheory of T. In the second approach every expression in T is represented by a syntactic value, and quotation and evaluation are operators in T itself. 1
CTPbased programming languages? Considerations about an experimental design
"... This paper discusses plans for joint work in order to gain early feedback from the community. Three lines of work pursued independently so far shall be joined: (1) narrowing the gap between declarative program specification and program generation already working in Isabelle, (2) reusing work, which ..."
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This paper discusses plans for joint work in order to gain early feedback from the community. Three lines of work pursued independently so far shall be joined: (1) narrowing the gap between declarative program specification and program generation already working in Isabelle, (2) reusing work, which embedded an inputresponseloop resembling Computer Algebra Systems (CAS) into HOL Light, and (3) reconstructing an experimental language for applied mathematics by exploiting established as well as emerging features of Isabelle/Isar. These plans have to be seen as part of a variety of highly active research areas — on “integration of the deduction and the computational power ” of Computer Theorem Proving (CTP) and CAS respectively (Calculemus), on “innovative theoretical and technological solutions for contentbased systems ” (MKM), on “Programming Languages for Mechanized Mathematics Systems” (PLMMS), just to cite from some related interest groups. Facing the abundant variety of approaches, of intermediate results and of ongoing developments, and taking under consideration the many difficulties in integrating such approaches, we pursue pragmatic goals: Design a component indispensable for working engineers, a programming language for engineering applications. Use Isabelle for an experimental embedding of the language, which is useful at least in engineering education as soon as possible.