## Journal of Symbolic Logic

### Completeness and Herbrand theorems for nominal logic

James Cheney

#### Abstract

Nominal logic is a variant of first-order logic in which abstract syntax with names and binding is formalized in terms of two basic operations: name-swapping and freshness. It relies on two important principles: equivariance (validity is preserved by name-swapping), and fresh name generation (“new” or fresh names can always be chosen). It is inspired by a particular class of models for abstract syntax trees involving names and binding, drawing on ideas from Fraenkel-Mostowski set theory: finite-support models in which each value can depend on only finitely many names.

Although nominal logic is sound with respect to such models, it is not complete. In this paper we review nominal logic and show why finite-support models are insufficient both in theory and practice. We then identify (up to isomorphism) the class of models with respect to which nominal logic is complete: ideal-supported models in which the supports of values are elements of a proper ideal on the set of names.

We also investigate an appropriate generalization of Herbrand models to nominal logic. After adjusting the syntax of nominal logic to include constants denoting names, we generalize universal theories to nominal-universal theories and prove that each such theory has an Herbrand model.

#### Article information

Source
J. Symbolic Logic, Volume 71, Issue 1 (2006), 299-320.

Dates
First available in Project Euclid: 22 February 2006

https://projecteuclid.org/euclid.jsl/1140641176

Digital Object Identifier
doi:10.2178/jsl/1140641176

Mathematical Reviews number (MathSciNet)
MR2210069

Zentralblatt MATH identifier
1100.03016

#### Citation

Cheney, James. Completeness and Herbrand theorems for nominal logic. J. Symbolic Logic 71 (2006), no. 1, 299--320. doi:10.2178/jsl/1140641176. https://projecteuclid.org/euclid.jsl/1140641176

#### References

• Serge Abiteboul, Richard Hull, and Victor Vianu Foundations of Databases, Addison-Wesley,1995.
• H. P. Barendregt The Lambda Calculus, North-Holland,1984.
• N. G. de Bruijn Lambda-calculus notation with nameless dummies, a tool for automatic formula manipulation, Indagationes Mathematicae, vol. 34 (1972), no. 5, pp. 381--392.
• Norbert Brunner The Fraenkel-Mostowski method, revisited, Notre Dame Journal of Formal Logic, vol. 31 (1990), no. 1, pp. 64--75.
• James Cheney The complexity of equivariant unification, Proceedings of the 31$^st$ International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 3142, Springer-Verlag,2004, pp. 332--344.
• -------- Nominal logic programming, Ph.D. thesis, Cornell University, Ithaca, NY, August 2004.
• -------- Equivariant unification, Proceedings of the Conference on Rewriting Techniques and Applications (RTA), Lecture Notes in Computer Science, vol. 3467, Springer-Verlag,2005, pp. 74--89.
• -------- A simpler proof theory for nominal logic, Proceedings of the Conference on Foundations of Software Science and Computation Structures (FOSSACS), Lecture Notes in Computer Science, vol. 3441, Springer-Verlag,2005, pp. 379--394.
• James Cheney and Christian Urban Alpha-Prolog: A logic programming language with names, binding and alpha-equivalence, Proceedings of the 20$^th$ International Conference on Logic Programming (ICLP), Lecture Notes in Computer Science, vol. 3132, Springer-Verlag,2004, pp. 269--283.
• Alonzo Church A formulation of the simple theory of types, Journal of Symbolic Logic, vol. 5 (1940), pp. 56--68.
• H. B. Curry and R. Feys Combinatory Logic, North-Holland,1958.
• Ulrich Felgner Models of ZF-Set Theory, Lecture Notes in Mathematics, vol. 223, Springer-Verlag,1970.
• A. Fraenkel The concept definite'' and the indepedence of the Auswahlsaxiom, From Frege to Gödel (J. van Heijenoort, editor), Harvard University Press,1967, pp. 284--289.
• G. Frege Begriffsschrift: A formula language, modeled upon that of arithmetic, for pure thought, From Frege to Gödel (J. van Heijenoort, editor), Harvard University Press,1967, pp. 1--82.
• Murdoch Gabbay A theory of inductive definitions with alpha-equivalence, Ph.D. thesis, University of Cambridge,2001.
• -------- FM-HOL, a higher-order theory of names, Workshop on Thirty Five Years of Automath (Heriot Watt, UK) (F. Kamareddine, editor), April 2002.
• -------- Fresh logic: A logic of FM,2003, preprint.
• -------- A general mathematics of names in syntax, preprint, March 2004.
• Murdoch Gabbay and James Cheney A sequent calculus for nominal logic, Proceedings of the Annual IEEE Symposium on Logic in Computer Science (LICS) (Turku, Finland),2004, pp. 139--148.
• Murdoch Gabbay and Andrew Pitts A new approach to abstract syntax with variable binding, Formal Aspects of Computing, vol. 13 (2002), pp. 341--363.
• C. A. Gunter and J. C. Mitchell (editors) Theoretical Aspects of Object-Oriented Programming: Types, Semantics, and Language Design, The MIT Press,1994.
• Robert Harper, Furio Honsell, and Gordon Plotkin A framework for defining logics, Journal of the ACM, vol. 40 (1993), no. 1, pp. 143--184.
• Leon Henkin The completeness of the first-order functional calculus, Journal of Symbolic Logic, vol. 14 (1949), no. 3, pp. 159--166.
• John E. Hopcroft and Jeffrey D. Ullmann Introduction to Automata Theory, Languages, and Computation, Addison-Wesley,1979.
• Thomas J. Jech About the axiom of choice, Handbook of Mathematical Logic (J. Barwise, editor), North-Holland,1977, pp. 345--370.
• Fairouz Kamarredine, Twan Laan, and Rob Nederpelt Types in logic and mathematics before 1940, Bulletin of Symbolic Logic, vol. 8 (2002), no. 2, pp. 185--245.
• J. R. Kennaway, J. W. Klop, M. R. Sleep, and F. J. de Vries Infinitary lambda calculus, Theoretical Computer Science, vol. 175 (1997), no. 1, pp. 93--125.
• Gavin Lowe An attack on the Needham-Schroeder public-key authentication protocol, Information Processing Letters, vol. 56 (1995), no. 3, pp. 131--133.
• Saunders Mac Lane and Ieke Moerdijk Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag,1992.
• Dale Miller and Alwen Tiu A proof theory for generic judgments (extended abstract), Proceedings of the 18$^th$ Symposium on Logic in Computer Science (LICS), IEEE Press,2003, pp. 118--127.
• Robin Milner, Joachim Parrow, and David Walker A calculus of mobile processes, I-II, Information and Computation, vol. 100 (1992), no. 1, pp. 1--77.
• Robin Milner, Mads Tofte, Robert Harper, and David MacQueen The Definition of Standard ML - Revised, MIT Press,1997.
• G. Morrisett and R. Harper Semantics of memory management for polymorphic languages, Higher Order Operational Techniques in Semantics, Publications of the Newton Institute, Cambridge University Press,1997.
• Roger M. Needham and Michael D. Schroeder Using encryption for authentication in large networks of computers, Communications of the ACM, vol. 21 (1978), no. 12, pp. 993--999.
• Flemming Nielson, Hanne Riis Nielson, and Chris Hankin Principles of Program Analysis, 2nd ed., Springer,2005.
• P. O'Hearn and D. J. Pym The logic of bunched implications, Bulletin of Symbolic Logic, vol. 5 (1999), no. 2, pp. 215--244.
• Frank Pfenning and Conal Elliott Higher-order abstract syntax, Proceedings of the ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), ACM Press,1989, pp. 199--208.
• A. M. Pitts Nominal logic, A first order theory of names and binding, Information and Computation, vol. 183 (2003), pp. 165--193.
• Dag Prawitz Natural Deduction: A Proof-Theoretical Study, Almquist and Wiksell,1965.
• Graham Priest An Introduction to Non-Classical Logic, Cambridge University Press,2001.
• M. Schönfinkel On the building blocks of mathematical logic, From Frege to Gödel (J. van Heijenoort, editor), Harvard University Press,1967, pp. 355--366.
• Ulrich Schöpp and Ian Stark A dependent type theory with names and binding, Proceedings of the Computer Science Logic Conference (Karpacz, Poland), Lecture Notes in Computer Science, vol. 3210, Springer-Verlag,2004, pp. 235--249.
• Joseph E. Stoy Denotational Semantics: The Scott-Strachey Aapproach to Programming Language Ttheory, Series in Computer Systems, vol. 1, MIT Press,1981.
• J. K. Truss Permutations and the axiom of choice, Automorphisms of First-Order Structures (Richard Kaye and Dugald Macpherson, editors), Oxford,1994, pp. 131--152.
• René Vestergaard and James Brotherston A formalised first-order confluence proof for the $\lambda$-calculus using one-sorted variable names, Information and Computation, vol. 183 (2003), pp. 212--244.
• Luca Viganó Labelled Non-Classical Logics, Kluwer Academic Publishers,2000.