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An Overview of A Formal Framework For Managing Mathematics
 Annals of Mathematics and Artificial Intelligence
, 2003
"... Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform t ..."
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Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform theory which is simultaneously an axiomatic theory and an algorithmic theory. Representing a collection of mathematical models, a biform theory provides a formal context for both deduction and computation. The framework includes facilities for deriving theorems via a mixture of deduction and computation, constructing sound deduction and computation rules, and developing networks of biform theories linked by interpretations. The framework is not tied to a specific underlying logic; indeed, it is intended to be used with several background logics simultaneously. Many of the ideas and mechanisms used in the framework are inspired by the imps Interactive Mathematical Proof System and the Axiom computer algebra system.
A Set Theory with Support for Partial Functions
 STUDIA LOGICA
, 2000
"... Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its dom ..."
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Cited by 10 (2 self)
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Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambdanotation and sorts or types as in type theory. This paper introduces a version of vonNeumannBernaysGödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial firstorder logic, so classvalued terms may be nondenoting. Functions can be specified using lambdanotation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the imps Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.
Chiron: A multiparadigm logic
 University of Bialystok
, 2007
"... Abstract. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It supports several reasoning paradigms by integrating nbg set theory with elements of type theory, a scheme for handling undefinednes ..."
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Cited by 8 (6 self)
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Abstract. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It supports several reasoning paradigms by integrating nbg set theory with elements of type theory, a scheme for handling undefinedness, and a facility for reasoning about the syntax of expressions. This paper gives a quick, informal presentation of the syntax and semantics of Chiron and then discusses some of the benefits Chiron provides as a multiparadigm logic. 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
Discovering Theorems using GOEDEL: A Case Study
"... Combining an interactive symbolic manipulation program with a theorem prover allows one to discover theorems as well to prove them. The specific focus in this paper is on illustrating how to use the GOEDEL program, a Mathematica implementation of Gödel's algorithm for class formation, to ..."
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Cited by 2 (1 self)
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Combining an interactive symbolic manipulation program with a theorem prover allows one to discover theorems as well to prove them. The specific focus in this paper is on illustrating how to use the GOEDEL program, a Mathematica implementation of Gödel's algorithm for class formation, to help discover theorems about sets satisfying some property hereditarily. Similar techniques are applicable to other topics in set theory. Formal proofs of many of these theorems have been obtained using McCune's first order automated reasoning program Otter.
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
A Foundation for a Semantic Web
"... A semantic web is a web of data designed to be processed by machines. It enables processing based on the meaning of the data. To be useful, semantic data will be combined from several sources. This paper focuses on the relation of the combined data to its sources. Using the method of interpretations ..."
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A semantic web is a web of data designed to be processed by machines. It enables processing based on the meaning of the data. To be useful, semantic data will be combined from several sources. This paper focuses on the relation of the combined data to its sources. Using the method of interpretations between theories in a logic with undefined terms, it establishes criteria for combining information in a fashion that preserves the inferences available in the original information. The formalism can be used to evaluate existing languages for semantic data on the web, such as the Simple html Ontology Extensions (shoe). 1 Logic and the Web In the traditional web, information is structured and shared in forms that facilitate its display for human consumption. For example, an html document is divided into sections, paragraphs, and lists. The document's structure guides a browser's rendering of the document on a computer screen.
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(
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 10 (23) 2007 Chiron: A MultiParadigm Logic
"... Abstract. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It supports several reasoning paradigms by integrating nbg set theory with elements of type theory, a scheme for handling undefinednes ..."
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Abstract. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It supports several reasoning paradigms by integrating nbg set theory with elements of type theory, a scheme for handling undefinedness, and a facility for reasoning about the syntax of expressions. This paper gives a quick, informal presentation of the syntax and semantics of Chiron and then discusses some of the benefits Chiron provides as a multiparadigm logic. 1
Decidability Results for Sets with Atoms ∗
"... Formal Set Theory is traditionally concerned with pure sets; consequently, the satisfiability problem for fragments of set theory was most often addressed (and in many cases positively solved) in the pure framework. In practical applications, however, it is common to assume the existence of a number ..."
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Formal Set Theory is traditionally concerned with pure sets; consequently, the satisfiability problem for fragments of set theory was most often addressed (and in many cases positively solved) in the pure framework. In practical applications, however, it is common to assume the existence of a number of primitive objects (sometimes called atoms) that can be members of sets but behave differently from them. If these entities are assumed to be devoid of members, the standard extensionality axiom must be revised; then decidability results can sometimes be achieved via reduction to the pure case and sometimes can be based on direct goaldriven algorithms. An alternative approach to modeling atoms, that allows one to retain the original formulation of extensionality, was proposed by Quine: atoms are selfsingletons. In this paper we adopt this approach in coping with the satisfiability problem: we show the decidability of this problem relativized to ∃ ∗ ∀sentences, and develop a goaldriven unification algorithm.