## Biform theories in Chiron (2007)

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Venue: | Towards Mechanized Mathematical Assistants, volume 4573 of Lecture Notes in Computer Science |

Citations: | 8 - 5 self |

### BibTeX

@INPROCEEDINGS{Farmer07biformtheories,

author = {William M. Farmer},

title = {Biform theories in Chiron},

booktitle = {Towards Mechanized Mathematical Assistants, volume 4573 of Lecture Notes in Computer Science},

year = {2007},

pages = {66--79},

publisher = {Springer-Verlag}

}

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### Abstract

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 von-Neumann-Bernays-Gödel (nbg) set theory that is intended to be a practical, general-purpose 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 well-suited 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

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Citation Context ...that provide a means to reason about syntax. Many approaches for formalizing the syntax of expressions have been proposed starting with K. Gödel’s famous arithmetization of syntax via Gödel numbering =-=[12]-=-. Two good surveys of this research area are [14] and the extended version of [17]. A great deal of research has been directed to the problem of how to integrate computer theorem proving and computer ... |

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Citation Context ...e most important. M is derived from a structure, consisting of a nonempty domain Dc of classes and a membership relation ∈ on Dc, that satisfies the axioms of nbg set theory as given, for example, in =-=[13]-=- or [16]. The values of M include sets, classes, superclasses, truth values, the undefined value, and operations. A class of M is a member of Dc. A set of M is a member x of Dc such that x ∈ y for som... |

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Citation Context ...oject [3] or has been presented at the Calculemus symposia that began in 1996. Two research initiatives that are closely related to biform theories and the MathScheme project are the Theorema project =-=[2]-=- at the RISC Research Institute for Symbolic Computation [18] and the work by H. Barendregt and F. Wiedijk on the foundations of computerized mathematics [1]. The development and application of Chiron... |

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Citation Context ... approaches for formalizing the syntax of expressions have been proposed starting with K. Gödel’s famous arithmetization of syntax via Gödel numbering [12]. Two good surveys of this research area are =-=[14]-=- and the extended version of [17]. A great deal of research has been directed to the problem of how to integrate computer theorem proving and computer algebra. Much of this research has been done in c... |

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Citation Context ... ˆπ transforms the formula A into the formula A[x ↦→ t]. Again the expression π(E1, E2, E3) cannot be an expression in EF. ✷ Example 3. Let STT be a general logic representation of simple type theory =-=[8]-=-. Suppose T = (L, Γ) is an axiomatic theory of a complete ordered field in STT and that we have defined in T a type real of real numbers and the basic concepts of calculus such as limits, continuity, ... |

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Citation Context ...athScheme project are the Theorema project [2] at the RISC Research Institute for Symbolic Computation [18] and the work by H. Barendregt and F. Wiedijk on the foundations of computerized mathematics =-=[1]-=-. The development and application of Chiron is a long-range research project composed of the following four tasks: 1. The design of Chiron. 2. The design of a proof system for Chiron. 3. The developme... |

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Citation Context ...nd there is a clear definition of what a derived formula or rule is in a biform theory. The notion of a biform theory was first introduced as part of ffmm, a Formal Framework for Managing Mathematics =-=[11]-=- developed as part of the MathSchemesChiron 3 project [15] at McMaster University. One of the principal goals of ffmm is to integrate and generalize computer theorem proving and computer algebra. Bifo... |

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Citation Context ...yntax of expressions have been proposed starting with K. Gödel’s famous arithmetization of syntax via Gödel numbering [12]. Two good surveys of this research area are [14] and the extended version of =-=[17]-=-. A great deal of research has been directed to the problem of how to integrate computer theorem proving and computer algebra. Much of this research has been done in connection with the Calculemus Pro... |

7 | Chiron: A multi-paradigm logic
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Citation Context ...theory. Hence an algorithm in the form of a program in a high-level programming language can be made into a perfectly legitimate rule if a meaning formula for it can be expressed in the logic. Chiron =-=[8,9]-=- 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 includes elements of type theory, a scheme for ... |

6 | Chiron: A set theory with types, undefinedness, quotation, and evaluation
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(Show Context)
Citation Context ...theory. Hence an algorithm in the form of a program in a high-level programming language can be made into a perfectly legitimate rule if a meaning formula for it can be expressed in the logic. Chiron =-=[7,9]-=- 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 includes elements of type theory, a scheme for ... |

5 | Transformers for symbolic computation and formal deduction - Farmer, Mohrenschildt - 2000 |

4 | Trustable communication between mathematical systems - Carette, Farmer, et al. - 2003 |

4 |
Introduction to Mathematical Logic. Chapman & Hall/CRC, fourth edition
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Citation Context ...mportant. M is derived from a structure, consisting of a nonempty domain Dc of classes and a membership relation ∈ on Dc, that satisfies the axioms of nbg set theory as given, for example, in [13] or =-=[16]-=-. The values of M include sets, classes, superclasses, truth values, the undefined value, and operations. A class of M is a member of Dc. A set of M is a member x of Dc such that x ∈ y for some member... |

3 |
Chiron: A multi-paradigm logic. Studies in Logic, Grammar and Rhetoric. Special issue
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Citation Context ...theory. Hence an algorithm in the form of a program in a high-level programming language can be made into a perfectly legitimate rule if a meaning formula for it can be expressed in the logic. Chiron =-=[7,9]-=- 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 includes elements of type theory, a scheme for ... |

2 | A rational reconstruction of a system for experimental mathematics - Carette, Farmer, et al. - 2007 |