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.

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 lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel 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 first-order logic, so classvalued terms may be nondenoting. Functions can be specified using lambda-notation, 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.

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 plug-and-play 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

Abstract. Chiron is a derivative of von-Neumann-Bernays-Gödel (nbg) set theory that is intended to be a practical, general-purpose 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 multi-paradigm logic. 1

Chiron is a derivative of von-Neumann-Bernays-Gödel (nbg) set theory that is intended to be a practical, general-purpose logic for mechanizing mathematics. Unlike traditional set theories such as Zermelo-Fraenkel (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 first-order 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 represents. Using quotation and evaluation, syntactic side conditions, schemas, syntactic transformations used in deduction and computation rules, and other such things can be directly expressed in Chiron. This paper presents the syntax and semantics of Chiron and illustrates its use with some simple examples.

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.

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