Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-18T11:28:54.367Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free

Published online by Cambridge University Press:  12 March 2014

Murdoch J. Gabbay
Murdoch J. Gabbay*
Affiliation:
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, Scotland, URL: http://www.gabbay.org.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

By operations on models we show how to relate completeness with respect to permissive-nominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atoms-abstraction.

The results are of interest in their own right, but also, we factor the mathematics so as to maximise the chances that it could be used off-the-shelf for other nominal reasoning systems too. Models with infinite support can be easier to work with, so it is useful to have a semi-automatic theorem to transfer results from classes of infinitely-supported nominal models to the more restricted class of models with finite support.

In conclusion, we consider different permissive-nominal syntaxes and nominal models and discuss how they relate to the results proved here.

Keywords

permissive-nominal techniquesinfinite supportfinite supportnominal-algebrapermissive-nominal logiccompletenessinfinite atoms-abstraction
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 3 , September 2012 , pp. 828 - 852
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Cheney, James, Completeness and Herbrand theorems for nominal logic, this Journal, vol. 71 (2006), pp. 299–320.Google Scholar
[2] de Bruijn, Nicolaas G., Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem, Indagationes Mathematicae, vol. 34 (1972), no. 5, pp. 381–392.CrossRefGoogle Scholar
[3] Dowek, Gilles and Gabbay, Murdoch J., Permissive nominal logic, Proceedings of the 12th international ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming (PPDP 2010), 2010, pp. 165–176.Google Scholar
[4] Dowek, Gilles and Gabbay, Murdoch J., Permissive nominal logic, Transactions on Computational Logic, (2012), in press (journal version).Google Scholar
[5] Dowek, Gilles, Gabbay, Murdoch J., and Mulligan, Dominic P., Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques, Logic Journal of the IGPL, vol. 18 (2010), no. 6, pp. 769–822, (journal version).CrossRefGoogle Scholar
[6] Gabbay, Murdoch J., FM-HOL a higher-order theory of names, 35 years of Automath (Kamareddine, F., editor), 2002.Google Scholar
[7] Gabbay, Murdoch J., A general mathematics of names, Information and Computation, vol. 205 (2007), no. 7, pp. 982–1011.Google Scholar
[8] Gabbay, Murdoch J., Nominal algebra and the HSP theorem, Journal of Logic and Computation, vol. 19 (2009), no. 2, pp. 341–367.Google Scholar
[9] Gabbay, Murdoch J., Foundations of nominal techniques: logic and semantics of variables in abstract syntax, The Bulletin of Symbolic Logic, vol. 17 (2011), no. 2, pp. 161–229.CrossRefGoogle Scholar
[10] Gabbay, Murdoch J., Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables, Mathematical Structures in Computer Science, vol. 21 (2011), pp. 997–1033.Google Scholar
[11] Gabbay, Murdoch J., Meta-variables as infinite lists in nominal terms unification and rewriting, Logic Journal of the IGPL, (2012), in press.Google Scholar
[12] Gabbay, Murdoch J., Nominal terms and nominal logics: from foundations to meta-mathematics, Handbook of philosophical logic, vol. 17, Kluwer, 2012.Google Scholar
[13] Gabbay, Murdoch J. and Mathijssen, Aad, A formal calculus for informal equality with binding, WoLLIC'07: 14th Workshop on Logic, Language, Information and Computation, Lecture Notes in Computer Science, vol. 4576, Springer, 2007, pp. 162–176.Google Scholar
[14] Gabbay, Murdoch J., Nominal universal algebra: equational logic with names and binding, Journal of Logic and Computation, vol. 19 (2009), no. 6, pp. 1455–1508.Google Scholar
[15] Gabbay, Murdoch J. and Pitts, Andrew M., A new approach to abstract syntax with variable binding, Formal Aspects of Computing, vol. 13 (2001), no. 3–5, pp. 341–363.Google Scholar
[16] Hodges, Wilfrid, Model theory, Cambridge University Press, 1993.Google Scholar
[17] Mulligan, Dominic P., Online nominal bibliography, 2010, http://www.citeulike.org/group/11951/.Google Scholar
[18] Newman, M. H. A., On theories with a combinatorial definition of equivalence. Annals of Mathematics, vol. 43 (1942), no. 2, pp. 223–243.Google Scholar
[19] Pitts, Andrew M., Nominal logic, a first order theory of names and binding, Information and Computation, vol. 186 (2003), no. 2, pp. 165–193.Google Scholar
[20] Urban, Christian, Pitts, Andrew M., and Gabbay, Murdoch J., Nominal unification, Theoretical Computer Science, vol. 323 (2004), no. 1–3, pp. 473–497.Google Scholar