Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T05:34:50.441Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear lambda terms as invariants of rooted trivalent maps

Published online by Cambridge University Press:  04 November 2016

NOAM ZEILBERGER
NOAM ZEILBERGER*
Affiliation:
Inria, Campus de l'École Polytechnique, 91120 Palaiseau, France (e-mail: noam.zeilberger@gmail.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main aim of the paper is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between α-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact-oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the “easy” direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.

Type
Theoretical Pearls
Information
Journal of Functional Programming , Volume 26 , 2016 , e21
Copyright
Copyright © Cambridge University Press 2016 

References

Barendregt, H. P. (1984) The Lambda Calculus: Its Syntax and Semantics, Studies in Logic, vol. 103, 2nd revised ed., Amsterdam, North-Holland.Google Scholar
Bar-Natan, D. (1997) Lie algebras and the four color theorem. Combinatorica 17, 43–52.CrossRefGoogle Scholar
Bodini, O., Gardy, D. & Jacquot, A. (2013) Asymptotics and random sampling for BCI and BCK lambda terms. Theor. Comput. Sci. 502, 227–238.CrossRefGoogle Scholar
Curry, H. B., Hindley, J. R. & Seldin, J. P. (1972) Combinatory Logic, vol. II, North-Holland.Google Scholar
Hyland, M. (2013) Classical lambda calculus in modern dress. Mathematical Structures in Computer Science. In special issue dedicated to Corrado Böhm for his 90th birthday, pp. 1–20.Google Scholar
Jacobs, B. (1993) Semantics of lambda-I and of other substructural lambda calculi. In Typed Lambda Calculus and Applications, LNCS, vol. 664. Springer, pp. 195–208.CrossRefGoogle Scholar
Jones, G. A. & Singerman, D. (1978) Theory of maps on orientable surfaces. Proc. Lond. Math. Soc. 37, 273–307.CrossRefGoogle Scholar
Jones, G. A. & Singerman, D. (1994) Maps, hypermaps, and triangle groups. In The Grothendieck Theory of Dessins d'Enfants, Schneps, L. (ed), London Mathematical Society Lecture Note Series, vol. 200. Cambridge University Press, pp. 115–145.CrossRefGoogle Scholar
Joyal, A. & Street, R. (1991) The geometry of tensor calculus I. Adv. Math. 102, 20–78.Google Scholar
Kauffman, L. H. (1990) Map coloring and the vector cross product. J. Comb. Theory B 48, 145–154.CrossRefGoogle Scholar
Knuth, D. E. (1970) Examples of formal semantics. In Symposium on Semantics of Algorithmic Languages, Engeler, E. (ed), Lecture Notes in Mathematics, vol. 188. Springer, pp. 212–235.Google Scholar
Lando, S. K. & Zvonkin, A. K. (2004) Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences, vol. 141. Springer-Verlag.CrossRefGoogle Scholar
Mairson, H. G. (2002) From Hilbert spaces to Dilbert spaces: Context semantics made simple. In Proceedings of the 22nd Conference on Foundations of Software Technology and Theoretical Computer Science. Kanpur, India, Springer, pp. 2–17.Google Scholar
Mairson, H. G. (2004) Linear lambda calculus and PTIME-completeness. J. Funct. Program. 14, 6.CrossRefGoogle Scholar
OEIS Foundation Inc. (2016) The On-Line Encyclopedia of Integer Sequences.Google Scholar
Penrose, R. (1971) Applications of negative dimensional tensors. In Combinatorial Mathematics and its Application, Welsh, D. (ed). Academic Press, pp. 221–243.Google Scholar
Scott, D. S. (1980) Relating theories of the λ-calculus. In To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, Hindley & Seldin (eds). Academic Press, pp. 403–450.Google Scholar
Seely, R. A. G. (1987) Modelling computations: A 2-categorical framework. In Proceedings of the 2nd Annual IEEE Symposium on Logic in Computer Science, Ithaca, NY, USA, IEEE Computer Society, pp. 65–71.Google Scholar
Selinger, P. (2011) A survey of graphical languages for monoidal categories. In New Structures for Physics, Coecke, B. (ed), Springer Lecture Notes in Physics, vol. 813. Springer, pp. 289–355.Google Scholar
Stay, M. (2013) Compact closed bicategories. Theory and Applications of Categories 31 (2016), pp. 755–798.Google Scholar
Thomas, R. (1998) An update on the four-color theorem. Not. Am. Math. Soc. 45 (7), 848–859.Google Scholar
Tutte, W. T. (1962) A census of Hamiltonian polygons. Can. J. Math. 14, 402–417.CrossRefGoogle Scholar
Tutte, W. T. (1968) On the enumeration of planar maps. Bull. Am. Math. Soc. 74, 64–74.CrossRefGoogle Scholar
Vidal, S. (2010) Groupe Modulaire et Cartes Combinatoires: Génération et Comptage. PhD Thesis, Université Lille I, France (July).Google Scholar
Zeilberger, N. (2015a) Balanced polymorphism and linear lambda calculus. Talk at TYPES 2015, Tallinn, Estonia (18 May).Google Scholar
Zeilberger, N. (2015b) Counting isomorphism classes of β-normal linear lambda terms. arXiv:1509.07596 (25 September)Google Scholar
Zeilberger, N. & Giorgetti, A. (2015) A correspondence between rooted planar maps and normal planar lambda terms. Log. Methods Comput. Sci. 11 (3:22), 1–39.Google Scholar
Submit a response

Discussions

No Discussions have been published for this article.