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Algorithm Invention and Verification by Lazy Thinking
 Proceedings of SYNASC 2003 (Symbolic and Numeric Algorithms for Scientific Computing
, 2003
"... the frame of the Theorema system starting from my earlier versions of the induction prover, the conjecture generation algorithm, and the cascade. In this paper, we study algorithm invention and verification as a specific variant of systematic theory exploration and propose the "lazy thinking paradig ..."
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Cited by 13 (7 self)
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the frame of the Theorema system starting from my earlier versions of the induction prover, the conjecture generation algorithm, and the cascade. In this paper, we study algorithm invention and verification as a specific variant of systematic theory exploration and propose the "lazy thinking paradigm " for inventing and verifying algorithms automatically; i.e., for a given predicate logic specification of the problem in terms of a set of operations (functions and predicates), the method produces an algorithm that solves the problem together with a correctness proof for the algorithm. In the ideal case, the only information that has to be provided by the user consists of the formal problem specification and a complete knowledge base for the operations that occur in the problem specification. The "lazy thinking paradigm " is characterized ∘ by using a library of algorithm schemes ∘ and by using the information contained in failing attempts to prove the correctness theorem for an algorithm scheme in order to invent sufficient requirements on the auxiliary functions in the algorithm scheme.
Understanding expression simplification
 Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (ISSAC 2004
, 2004
"... We give the first formal definition of the concept of simplification for general expressions in the context of Computer Algebra Systems. The main mathematical tool is an adaptation of the theory of Minimum Description Length, which is closely related to various theories of complexity, such as Kolmog ..."
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Cited by 13 (2 self)
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We give the first formal definition of the concept of simplification for general expressions in the context of Computer Algebra Systems. The main mathematical tool is an adaptation of the theory of Minimum Description Length, which is closely related to various theories of complexity, such as Kolmogorov Complexity and Algorithmic Information Theory. In particular, we show how this theory can justify the use of various “magic constants ” for deciding between some equivalent representations of an expression, as found in implementations of simplification routines.
7 A joint meeting with the SGML�Holland group is scheduled in spring
 J. of Automated Reasoning
, 1990
"... Abstract. The vision of a computerised assistant to mathematicians has existed since the inception of theorem proving systems. The Alcor system has been designed to investigate and explore how a mathematician might interact with such an assistant by providing an interface to Mizar and the Mizar Math ..."
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Cited by 9 (2 self)
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Abstract. The vision of a computerised assistant to mathematicians has existed since the inception of theorem proving systems. The Alcor system has been designed to investigate and explore how a mathematician might interact with such an assistant by providing an interface to Mizar and the Mizar Mathematical Library. Our current research focuses on the integration of searching and authoring while proving. In this paper we use a scenario to elaborate the nature of the interaction. We abstract from this two distinct types of searching and describe how the Alcor interface implements these with keyword and LSIbased search. Though Alcor is still in its early stages of development, there are clear implications for the general problem of integrating searching and authoring, as well as technical issues with Mizar.
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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Cited by 8 (5 self)
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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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Cited by 6 (5 self)
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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
Trustable Communication Between Mathematics Systems
 IN PROC. OF CALCULEMUS 2003
, 2003
"... This paper presents a rigorous, unified framework for facilitating communication between mathematics systems. A mathematics system is given one or more interfaces which oer deductive and computational services to other mathematics systems. To achieve communication between systems, a client inter ..."
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Cited by 4 (3 self)
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This paper presents a rigorous, unified framework for facilitating communication between mathematics systems. A mathematics system is given one or more interfaces which oer deductive and computational services to other mathematics systems. To achieve communication between systems, a client interface is linked to a server interface by an asymmetric connection consisting of a pair of translations. Answers to requests are trustable in the sense that they are correct provided a small set of prescribed conditions are satis ed. The framework is robust with respect to interface extension and can process requests for abstract services, where the server interface is not fully specified.
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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Cited by 2 (2 self)
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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
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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Cited by 1 (1 self)
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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