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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
Formalizing and operationalizing industrial standards
 Fundamental Approaches to Software Engineering
, 2011
"... Abstract. Industrial standards establish technical criteria for various engineering artifacts, materials, or services, with a view to ensuring their functionality, safety, and reliability. We develop a methodology and tools to systematically formalize such standards, in particular their domain speci ..."
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Abstract. Industrial standards establish technical criteria for various engineering artifacts, materials, or services, with a view to ensuring their functionality, safety, and reliability. We develop a methodology and tools to systematically formalize such standards, in particular their domain specific calculation methods, in order to support the automatic verification of functional properties for concrete physical artifacts. We approach this problem in the setting of the Bremen heterogeneous tool set HETS, which allows for the integrated use of a wide range of generic and custommade logics. Specifically, we (i) design a domain specific language for the formalization of industrial standards; (ii) formulate a semantics of this language in terms of a translation into the higherorder specification language HASCASL, and (iii) integrate computer algebra systems (CAS) with the HETS framework via a generic CASInterface in order to execute explicit and implicit calculations specified in the standard. This enables a wide variety of addedvalue services based on formal reasoning, including verification of parameterized designs and simplification of standards for particular configurations. We illustrate our approach using the European standard EN 1591, which concerns calculation methods for gasketed flange connections that assure the impermeability and mechanical strength of the flangeboltgasket system. 1
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
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.
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.
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(
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