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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barendregt, H., Wiedijk, F.: The challenge of computer mathematics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 363, 2351–2375 (2005)
Buchberger, B., Craciun, A., Jebelean, T., Kovacs, L., Kutsia, T., Nakagawa, K., Piroi, F., Popov, N., Robu, J., Rosenkranz, M., Windsteiger, W.: Theorema: Towards computer-aided mathematical theory exploration. Journal of Applied Logic 4, 470–504 (2006)
Calculemus Project: Systems for Integrated Computation and Deduction. Web site at http://www.calculemus.net/
Carette, J.: Understanding expression simplification. In: Gutierrez, J. (ed.) ISSAC 2004. Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, pp. 72–79. ACM Press, New York (2004)
Carette, J., Farmer, W.M., Sorge, V.: A rational reconstruction of a system for experimental mathematics. In: Kauers, M., Windsteiger, W. (eds.) Towards Mechanized Mathematical Assistants. LNCS (LNAI), vol. 4573, Springer, Heidelberg (2007)
Carette, J., Farmer, W.M., Wajs, J.: Trustable communication between mathematics systems. In: Hardin, T., Rioboo, R. (eds.) Calculemus 2003, pp. 58–68, Rome, Italy, Aracne (2003)
Farmer, W.M.: Chiron: A multi-paradigm logic. Studies in Logic, Grammar and Rhetoric. Special issue: Matuszewski, R., Rudnicki, P., Zalewska, A. (eds.) From Insight to Proof, forthcoming
Farmer, W.M.: The seven virtues of simple type theory. SQRL Report No. 18, McMaster University, 2003. Revised (2006)
Farmer, W.M.: Chiron: A set theory with types, undefinedness, quotation, and evaluation. SQRL Report No. 38, McMaster University (2007)
Farmer, W.M., von Mohrenschildt, M.: Transformers for symbolic computation and formal deduction. In: Colton, S., Martin, U., Sorge, V. (eds.) Automated Deduction - CADE-17. LNCS, vol. 1831, pp. 36–45. Springer, Heidelberg (2000)
Farmer, W.M., von Mohrenschildt, M.: An overview of a formal framework for managing mathematics. Annals of Mathematics and Artificial Intelligence 38, 165–191 (2003)
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mthematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198 (1931)
Gödel, K.: The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with the Axioms of Set Theory. In: Annals of Mathematical Studies, vol. 3, Princeton University Press, Princeton (1940)
Harrison, J.: Metatheory and reflection in theorem proving: A survey and critique. Technical Report CRC-053, SRI Cambridge, Millers Yard, Cambridge, UK (1995), available at http://www.cl.cam.ac.uk/jrh13/papers/reflect.ps.gz
MathScheme: An Integrated Framework for Computer Algebra and Computer Theorem Proving, Web site at http://www.imps.mcmaster.ca/mathscheme/
Mendelson, E.: Introduction to Mathematical Logic, vol. 4. Chapman & Hall/CRC, Sydney (1997)
Nogin, A., Kopylov, A., Yu, X., Hickey, J.: A computational approach to reflective meta-reasoning about languages with bindings. In: Momigliano, A., Pollack, R. (eds.) MERLIN 2005. Proceedings of the Third ACM SIGPLAN Workshop on Mechanized Reasoning about Languages with Variable Binding. An extended version is available as a California Institute of Technology technical report, CaltechCSTR:2005.003, pp. 2–12. ACM Press, New York (2005)
RISC Research Institute for Symbolic Computation, Web site at http://www.risc.uni-linz.ac.at//
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Farmer, W.M. (2007). Biform Theories in Chiron. In: Kauers, M., Kerber, M., Miner, R., Windsteiger, W. (eds) Towards Mechanized Mathematical Assistants. MKM Calculemus 2007 2007. Lecture Notes in Computer Science(), vol 4573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73086-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-73086-6_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73083-5
Online ISBN: 978-3-540-73086-6
eBook Packages: Computer ScienceComputer Science (R0)