Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
194 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A Complete Finite Axiomatisation of the Equational Theory of Common Meadows (2307.04270v3)

Published 9 Jul 2023 in cs.LO

Abstract: We analyse abstract data types that model numerical structures with a concept of error. Specifically, we focus on arithmetic data types that contain an error value $\bot$ whose main purpose is to always return a value for division. To rings and fields, we add a division operator $x/y$ and study a class of algebras called common meadows wherein $x/0 = \bot$. The set of equations true in all common meadows is named the equational theory of common meadows. We give a finite equational axiomatisation of the equational theory of common meadows and prove that it is complete and that the equational theory is decidable.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (58)
  1. Perspecx Machine VIII, axioms of transreal arithmetic. In J. Latecki, D. M. Mount and A. Y. Wu (eds), Proc. SPIE 6499. Vision Geometry XV, 649902, 2007.
  2. James A. Anderson and Jan A. Bergstra. 2021. Review of Suppes 1957 proposals for division by zero. Transmathematica, (2021). https://doi.org/10.36285/tm.53.
  3. Franz Baader and Tobias Nipkow. 1998. Term Rewriting and All That. Cambridge University Press, 1998. https://doi:10.1017/CBO9781139172752
  4. Jan A. Bergstra. 2019. Division by zero, a survey of options. Transmathematica, (2019). https://doi.org/10.36285/tm.v0i0.17.
  5. Jan A. Bergstra. 2020. Arithmetical data types, fracterms, and the fraction definition problem. Transmathematica, (2020). https://doi.org/10.36285/tm.33.
  6. Jan A. Bergstra, Inge Bethke and Alban Ponse. 2013. Cancellation meadows: a generic basis theorem and some applications. The Computer Journal, 56 (1) (2013), 3–14. Also arxiv.org/abs/0803.3969.
  7. Equations for formally real meadows. Journal of Applied Logic, 13 (2) (2015), 1–23.
  8. Meadows and the equational specification of division. Theoretical Computer Science, 410 (12) (2009), 1261–1271.
  9. Division by zero in non-involutive meadows. Journal of Applied Logic, 13(1): 1–12 (2015). https://doi.org/10.1016/j.jal.2014.10.001
  10. Jan A. Bergstra and Cornelis A. Middelburg. 2015. Transformation of fractions into simple fractions in divisive meadows. Journal of Applied Logic, 16 (2015), 92–110. Also https://arxiv.org/abs/1510.06233.
  11. Jan A. Bergstra and Alban Ponse. 2015. Division by zero in common meadows. In R. de Nicola and R. Hennicker (eds), Software, Services, and Systems: Wirsing Festschrift, Lecture Notes in Computer Science 8950, Springer, 2015, 46–61. For an improved version (2021), see: arXiv:1406.6878v4.
  12. Jan A. Bergstra and Alban Ponse. 2016. Fracpairs and fractions over a reduced commutative ring. Indigationes Mathematicae, 27, (2016), 727–748. Also https://arxiv.org/abs/1411.4410.
  13. Equational specifications, complete term rewriting systems, and computable and semicomputable algebras. Journal of the ACM, Vol. 42 (6), 1194-1230 (1995).
  14. Jan A. Bergstra and John V. Tucker. 2007. The rational numbers as an abstract data type. Journal of the ACM, 54 (2) (2007), Article 7.
  15. Jan A. Bergstra and John V. Tucker. 2020. The transrational numbers as an abstract data type. Transmathematica, (2020). https://doi.org/10.36285/tm.47.
  16. Jan A. Bergstra and John V. Tucker. 2021. The wheel of rational numbers as an abstract data type. In Roggenbach M. (editor), Recent Trends in Algebraic Development Techniques. WADT 2020. Lecture Notes in Computer Science 12669, Springer, 2021, 13–30.
  17. Jan A. Bergstra and John V. Tucker. 2022. On the axioms of common meadows: Fracterm calculus, flattening and incompleteness. The Computer Journal. Online first, 8pp. https://doi.org/10.1093/comjnl/bxac026
  18. Jan A. Bergstra and John V. Tucker. 2022. Totalising partial algebras: Teams and splinters. Transmathematica, https://doi.org/10.36285/tm.57
  19. Jan A. Bergstra and John V. Tucker. 2022. Symmetric transrationals: The data type and the algorithmic degree of its equational theory, in N. Jansen et al. (eds.) A Journey From Process Algebra via Timed Automata to Model Learning - A Festschrift Dedicated to Frits Vaandrager on the Occasion of His 60th Birthday, Lecture Notes in Computer Science 13560, 63-80. Springer, 2022.
  20. Jan A. Bergstra and John V. Tucker. 2023. Eager term rewriting for the fracterm calculus of common meadows, The Computer Journal, 2023. bxad106. https://doi.org/10.1093/comjnl/bxad106
  21. Garrett Birkhoff. 1935. On the Structure of Abstract Algebras. Mathematical Proceedings of the Cambridge Philosophical Society, 31 (4), 433-454.
  22. Garrett Birkhoff and Saunders MacLane. 1965. Survey of Modern Algebra. Macmillan, 1965.
  23. Michel Bidoit and Peter D Mosses. 2004. Casl User Manual - Introduction to Using the Common Algebraic Specification Language, Lecture Notes in Computer Science 2900, Springer, 2004. https://doi.org/10.1007/b11968
  24. Jesper Carlström. 2004. Wheels – on division by zero, Mathematical Structures in Computer Science, 14 (1), (2004), 143-184.
  25. Maude: specification and programming in rewriting logic, Theoretical Computer Science, 285 (2), 2002,187-243, https://doi.org/10.1016/S0304-3975(01)00359-0.
  26. All About Maude - A High-Performance Logical Framework, How to Specify, Program and Verify Systems in Rewriting Logic, Lecture Notes in Computer Science 4350, Springer, 2007. https://doi.org/10.1007/978-3-540-71999-1_18.
  27. Claude Chevalley. 1956. Fundamental Concepts of Algebra. Academic Press, 1956.
  28. Construction of the transreal numbers and algebraic transfields. IAENG International Journal of Applied Mathematics, 46 (1) (2016), 11–23. http://www.iaeng.org/IJAM/issues_v46/issue_1/IJAM_46_1_03.pdf
  29. Specification of Abstract Data Types. Vieweg Teubner, 1997.
  30. H. Ehrig and B. Mahr. 1985. Fundamentals of Algebraic Specification 1: Equations und Initial Semantics, EATCS Monographs on Theoretical Computer Science, Vol. 6, Springer, 1985.
  31. Technical Report RC 6487, IBM T. J. Watson Research Center, October, 1976. Reprinted in: Raymond Yeh (editor), Current Trends in Programming Methodology, IV. Prentice-Hall, 1978, 80-149.
  32. Joseph A. Goguen. 1989. Memories of ADJ. 1989. Bulletin of the EATCS no. 36, October 1989. (Available at https://cseweb.ucsd.edu/~goguen/pps/beatcs-adj.ps).
  33. Wilfrid Hodges. 1993. Model Theory. Cambridge University Press, 1993.
  34. William Kahan. 2011. Desperately Needed Remedies for the Undebuggability of Large Floating-Point Computations in Science and Engineering https://people.eecs.berkeley.edu/~wkahan/Boulder.pdf
  35. John von Neumann and Hermann Goldstine. 1947. Numerical inverting of matrices of high order. 1947. Bulletin American Mathematical Society, 53 (11), 1021-1099.
  36. Serge Lang. 1965. Algebra. Addison Wesley, 1965.
  37. Serge Lang. 2002. Algebra. Graduate Texts in Mathematics, Vol. 211, Third revised edition, 2002. Springer.
  38. A.I. Mal’tsev. 1973. Algebraic systems. Springer-Verlag, 1973.
  39. The Maude System. http://maude.cs.illinois.edu/w/index.php/The_Maude_System
  40. K. Meinke and J. V. Tucker. 1992. Universal Algebra. In S Abramsky and D Gabbay and T Maibaum, Handbook of Logic for Computer Science, Oxford University Press, 1992, 189–411.
  41. Jose Meseguer and Joseph Goguen. 1985. Initiality, induction and computability. In Maurice Nivat and John C. Reynolds (editors), Algebraic Methods in Semantics Cambridge University Press, 1985, 459-541.
  42. Ramon E Moore. 1966. Interval Analysis. Prentice-Hall, 1966.
  43. Hiroshi Okumura, Saburou Saitoh and Tsutomu Matsuura. 2017. Relations of zero and ∞\infty∞. Journal of Technology and Social Science, (2017) 1 (1).
  44. Hiroshi Okumura. 2018. Is it really impossible to divide by zero? Biostatistics and Biometrics Open Acc. J. 7 (1) 555703. DOI: 10.19080/BBOJ.2018.07.555703, (2018)
  45. Hiroakira Ono. 1983. Equational theories and universal theories of fields. Journal of the Mathematical Society of Japan, 35 (2) (1983), 289-306.
  46. M. Overton. 2001. Numerical Computing with IEEE Floating Point Arithmetic. SIAM, 2001.
  47. S Rump. 2010. Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica,19, 287-449. doi:10.1017/S096249291000005X
  48. Anton Setzer. 1997. Wheels (Draft), Unpublished. 1997.
  49. Albrecht Fröhlich and John C. Shepherdson Effective procedures in field theory, 1956. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 248 (1956), 407-432 http://doi.org/10.1098/rsta.1956.0003
  50. Ian Stewart. 1972. Galois Theory. Chapman Hall, 1972.
  51. Viggo Stoltenberg-Hansen and John V. Tucker. 1999. Computable rings and fields, in Edward Griffor (ed), Handbook of Computability Theory, Elsevier, 1999, 363-447.
  52. Viggo Stoltenberg-Hansen and John V. Tucker. Concrete models of computation for topological algebras, Theoretical Computer Science, 219 (1999) 347-378 https://doi.org/10.1016/S0304-3975(98)00296-5
  53. Patrick Suppes. 1957. Introduction to Logic. Van Nostrand Reinhold, 1957.
  54. Terese, Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science 55. Cambridge University Press, 2003.
  55. Warwick Tucker. 2011. Validated Numerics: A Short Introduction to Rigorous Computations’. Princeton University Press, 2011.
  56. John V Tucker. 2022. Unfinished Business: Abstract data types and computer arithmetic. BCS FACS FACTS, The Newsletter of the Formal Aspects of Computing Science BCS Specialist Group, Issue 2022-1, February 2022, 60-68. https://www.bcs.org/media/8289/facs-jan22.pdf
  57. B L van der Waerden. Modern Algebra. Volume 1. Frederick Ungar Publishing Company, 1970.
  58. Wolfgang Wechler. Universal Algebra for Computer Scientists. Springer-Verlag, 1992.
Citations (3)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now