Nominal (universal) algebra: Equational logic with names and binding

Journal of Logic and Computation

Volumn 19, Issue 6, 2009, Pages 1455-1508

  Gabbay, Murdoch J ^a   Mathijssen, Aad ^b  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Binding; Equational logic; Names; Nominal techniques; Universal algebra

Indexed keywords

DESIGN DECISIONS; EQUATIONAL LOGIC; LOGIC OF EQUALITIES; LOGICAL FRAMEWORKS; NOMINAL SETS; NOMINAL TECHNIQUES; THEORETICAL COMPUTER SCIENCE; UNIVERSAL ALGEBRA;

ALGEBRA;

EID: 72649092775     PISSN: 0955792X     EISSN: 1465363X     Source Type: Journal    
DOI: 10.1093/logcom/exp033     Document Type: Article

References (63)

1. M.Abadi andA. D. Gordon.Acalculus for cryptographic protocols: the spificalculus. In CCS '97: Proceedings of the 4th ACMconference on Computer and Communications Security, pp. 36-47. ACM Press, 1997.
2. H. Andréka, I. Németi, and I. Sain. Algebraic logic. In Handbook of Philosophical Logic, 2nd edn, vol.2, D. M. Gabbay and F. Guenthner, eds, pp. 133-249. Kluwer, 2001.
3. J. Barwise. Admissible Sets and Structures: an Approach to Definability Theory. Perspectives in mathematical logic. Springer, 1975.
4. H. P. Barendregt. The Lambda Calculus: its Syntax and Semantics (revised edn.). North-Holland, 1984.
5. H. P. Barendregt. Lambda calculi with types. In Handbook of Logic in Computer Science, vol.2, S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, eds, pp. 117-309. OUP, 2000.
6. J. C. M. Baeten andW. P.Weijland. Process Algebra,Vol. 18 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1990.
7. N. Brunner. 75 years of independence proofs by Fraenkel-Mostowski permutation models. Mathematica Japonica, 43, 177-199, 1996.
8. S. Burris and H. Sankappanavar.ACourse in Universal Algebra. GraduateTexts in Mathematics. Springer, 1981.
9. J. Cheney. The complexity of equivariant unification. In Proceedings of 31st International Colloquium on Automata, Languages and Programming (ICALP 2004), Vol.3142 of Lecture Notes in Computer Science, pp. 332-344. Springer, 2004.
10. J. Cheney. Completeness and Herbrand theorems for nominal logic. Journal of Symbolic Logic, 71, 299-320, 2006.
11. R. A. Clouston and A. M. Pitts. Nominal equational logic. Electronic Notes in Theoretical Computer Science, 172, 223-257, 2007.
12. H. B. Curry and R. Feys. Combinatory Logic, vol.1. North Holland, 1958.
13. N. G. de Bruijn. Checking mathematics with computer assistance. Notices of the American Mathematical Society (AMS), 38, 8-15, 1991.
14. M. Fernández and Murdoch J. Gabbay. Nominal rewriting. Information and Computation, 205, 917-965, 2007.
15. M. Fernández, Murdoch J. Gabbay and I. Mackie. Nominal rewriting systems. In Proceedings of the 6th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP'2004), pp. 108-119. ACM Press, 2004.
16. M. Fiore and C.-K. Hur. Term equational systems and logics. Electronic Notes in Theoretical Computer Science, 218, 171-192, 2008.
17. M. P. Fiore, G. D. Plotkin, and D. Turi. Abstract syntax and variable binding. In LICS '99: 14th Annual Symposium on Logic in Computer Science, pp. 193-202. IEEE, 1999.
18. Murdoch J. Gabbay. A Theory of Inductive Definitions with alpha-Equivalence. PhD Thesis, Cambridge University, 2000.
19. Murdoch J. Gabbay. A NEW calculus of contexts. In PPDP '05: Proceedings of the 7th ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming, pp. 94-105. ACM, 2005.
20. Murdoch J. Gabbay. Fresh logic. Journal of Applied Logic, 5, 356-387, 2007.
21. Murdoch J. Gabbay. A general mathematics of names. Information and Computation, 205, 982-1011, 2007.
22. Murdoch J. Gabbay. Hierarchical nominal terms and their theory of rewriting. Electronic Notes in Theoretical Computer Science, 174, 37-52, 2007.
23. Murdoch J. Gabbay. Nominal algebra and the HSPtheorem. Journal of Logic and Computation, 19, 341-367, 2009.
24. Murdoch J. Gabbay and S. Lengrand. The lambda-context calculus. Electronic Notes in Theoretical Computer Science, 196, 19-35, 2008.
25. Murdoch J. Gabbay and A. Mathijssen. Capture-avoiding substitution as a nominal algebra. In ICTAC 2006: Theoretical Aspects of Computing, Vol.4281 of Lecture Notes in Computer Science, pp. 198-212, Springer, 2006.
26. Murdoch J. Gabbay and A. Mathijssen. Nominal algebra. In 18th Nordic Workshop on Programming Theory, 2006.
27. Murdoch J. Gabbay and A. Mathijssen. Nominal algebra. Technical Report HW-MACS-TR- 0045. Heriott-Watt, 2006.
28. Murdoch J. Gabbay and A. Mathijssen. One-and-a-halfth-order logic. In PPDP '06: Proceedings of the 8th ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming, pp. 189-200. ACM, 2006.
29. Murdoch J. Gabbay and A. Mathijssen. A formal calculus for informal equality with binding. In WoLLIC'07: 14th Workshop on Logic, Language, Information and Computation, Vol.4576 of Lecture Notes in Computer Science, pp. 162-176. Springer, 2007.
30. Murdoch J. Gabbay and A. Mathijssen. Capture-avoiding substitution as a nominal algebra. Formal Aspects of Computing, 20, 451-479, 2008.
31. Murdoch J. Gabbay and A. Mathijssen. A nominal axiomatisation of the lambda calculus. Technical Report 08-18. Technische Universiteit Eindhoven, 2008.
32. Murdoch J. Gabbay and A. Mathijssen. One-and-a-halfth-order Logic. Journal of Logic and Computation, 18, 521-562, 2008.
33. Murdoch J. Gabbay andA. Mathijssen.The lambda-calculus is nominal algebraic. In Reasoning in Simple Type Theory: Festschrift in Honour of Peter B. Andrews on his 70th Birthday, Studies in Logic and the Foundations of Mathematics. C. Benzmueller, C. E. Brown, J. Siekman, R. Statman, eds, IFCoLog, College Publications, 2008.
34. Murdoch J. Gabbay and A. Mathijssen. A nominal axiomatisation of the lambda-calculus. Journal of Logic and Computation, 2009, To appear.
35. Murdoch J. Gabbay and A. M. Pitts. A new approach to abstract syntax with variable binding. Formal Aspects of Computing, 13, 341-363, 2001.
36. H. Geuvers and G. I. Jojgov. Open proofs and open terms: a basis for interactive logic. In Computer Science Logic: 16th International Workshop, Vol.2471 of Lecture Notes in Computer Science, Springer, pp. 537-552, 2002.
37. J. F. Groote. The syntax and semantics of timed ?CRL. Technical Report SEN-R9709. CWI, Amsterdam, 1997.
38. J. F. Groote, A. Mathijssen, M. A. Reniers, Y. S. Usenko and M. vanWeerdenburg. Analysis of distributed systems with mCRL2. In Process Algebra for Parallel and Distributed Processing, M. Alexander andW. Gardner, eds, pp. 99-128. Chapman and Hall, 2008.
39. D. Hendriks andV. van Oostrom.Adbmal. InCADE2003: Conference onAutomated Deduction, Vol.2471 of Lecture Notes in Computer Science, Springer, pp. 136-150, 2003.
40. L. Henkin, J. D. Monk, and A. Tarski. Cylindric Algebras. North Holland, 1971 and 1985. Parts I and II.
41. M. Hyland and J. Power. The category theoretic understanding of universal algebra: Lawvere theories and monads. Electronic Notes in Theoretical Computer Science, 172, 437-458, 2007.
42. P. T. Johnstone. Notes on Logic and Set Theory. Cambridge University Press, 1987.
43. G. I. Jojgov. Incomplete Proofs and Terms and Their Use in Interactive Theorem Proving. PhD Thesis. Technische Universiteit Eindhoven, 2004.
44. J.-P. Katoen and P. R. D'Argenio. General distributions in process algebra. In Lectures onFormal Methods and Performance Analysis: First EEF/Euro Summer School on Trends in Computer Science, pp. 375-429. Springer, 2002.
45. J. R. Kennaway, J. W. Klop, M. R. Sleep, and F. J. de Vries. Infinitary lambda calculus. Theoretical Computer Science, 175, 93-125, 1997.
46. F.W. Lawvere. Functorial Semantics of Algebraic Theories. PhD Thesis, Columbia University, 1963. Available with commentary as TAC Reprint 5.
47. P. Lescanne. From lambda-sigma to lambda-upsilon: a journey through calculi of explicit substitutions. In Proceedings of the 21st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL'94), pp. 60-69. ACM Press, 1994.
48. S. Lusin and A. Salibra. The lattice of lambda theories. Journal of Logic and Computation, 14, 373-394, 2004.
49. B. Luttik. Choice Quantification in Process Algebra. PhD Thesis, University of Amsterdam, 2002.
50. A. Mathijssen. Logical Calculi for Reasoning with Binding. PhDThesis,Technische Universiteit Eindhoven, 2007.
51. K. Meinke. Universal algebra in higher types. Theoretical Computer Science, 100, 385-417, 1992.
52. M. Miculan. Developing (meta)theory of lambda-calculus in the theory of contexts. Electronic Notes in Theoretical Computer Science, 58, 37-58, 2001.
53. P. W. O'Hearn and D. J. Pym. The logic of bunched implications. Bulletin of Symbolic Logic, 2, 215-244, 1999.
54. J. Parrow. An introduction to the pi-calculus. In Handbook of Process Algebra, J. Bergstra, A. Ponse, and S. Smolka, eds, pp. 479-543. Elsevier Science, 2001.
55. A. M. Pitts. Nominal logic, a first order theory of names and binding. Information and Computation, 186, 165-193, 2003.
56. D. Prawitz. Natural Deduction: A Proof Theoretical Study. Almqvist andWiksell, 1965.
57. G. Restall. An Introduction to Substructural Logics. Routledge, 1999.
58. J. C. Reynolds. Separation logic: a logic for shared mutable data structures. In LICS '02: Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science, pp. 55-74. IEEE Computer Society, 2002.
59. A. Salibra. On the algebraic models of lambda calculus. Theoretical Computer Science, 249, 197-240, 2000.
60. Y. Sun.An algebraic generalization of Frege structures-binding algebras. Theoretical Computer Science, 211, 189-232, 1999.
61. C. Urban, A. M. Pitts, and Murdoch J. Gabbay. Nominal unification. Theoretical Computer Science, 323, 473-497, 2004.
62. C. Urban. Nominal reasoning techniques in Isabelle/HOL. Journal of Automatic Reasoning, 40, 327-356, 2008.
63. J. van Benthem. Higher-order logic. In Handbook of Philosophical Logic, 2nd edn, D. Gabbay and F. Guenther (eds), vol.1, pp. 189-244. Kluwer, 2001.