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Automorphisms of types in certain type theories and representation of finite groups

  • SERGEI SOLOVIEV (a1) (a2)


The automorphism groups of types in several systems of type theory are studied. It is shown that in simply typed λ-calculus λ1βη and in its extension with surjective pairing and terminal object these groups correspond exactly to the groups of automorphisms of finite trees. In second-order λ-calculus and in Luo's framework (LF) with dependent products, any finite group may be represented.



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In memoriam Kosta Doen



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Asperti, A. and Longo, G. (1991). Categories, Types and Structures. Category Theory for the Working Computer Scientist, M.I.T. Press.
Babai, L. (1995). Automorphism groups, isomorphism, reconstruction. In: Graham, R. L. et al., eds. Handbook of Combinatorics, Elsevier, vol. 2, 14471541.
Balat, V. and Di Cosmo, R. (1999). A Linear Logical View of Linear Type Isomorphisms. Lecture Notes in Computer Science, Springer-Verlag, vol. 1683, 250265.
Barendregt, H. (1984). The Lambda Calculus: Its Syntax and Semantics (revised edition), North-Holland Plc.
Barthe, G. (2005). A computational view of implicit coercions in type theory. Mathematical Structures in Computer Science 15 (5) 839874.
Brown, R., Higgins, Ph. G., and Sivera, R. (2011). Nonabelian Algebraic Topology, European Math. Society.
Bruce, K. and Longo, G. (1985). Provable isomorphisms and domain equations in models of typed languages. In: Proceedings of the ACM Symposium on Theory of Computing (STOC 85) 263–272.
Bruce, K., Di Cosmo, R., and Longo, G. (1992). Provable isomorphisms of types. Mathematical Structures in Computer Science (Special Issue) 2 (2) 231247.
Clairambault, P. (2012). Isomorphisms of types in the presence of higher-order references. Logical Methods in Computer Science 8 128.
Coppo, M., Dezani-Ciancaglini, M., Margaria, I., Zacchi, M. (2017). Isomorphism of intersection and union types. Mathematical Structures in Computer Science 27 (5) 603625.
Coppo, M., Dezani-Ciancaglini, M., Díaz-Caro, A., Margaria, I., Zacchi, M. (2016). Retractions in intersection types. In: Kobayashi, N. (ed.) ITRS 2016, EPTCS 242, 31–47.
Coquand, T. and Huet, G. (1988). The calculus of constructions. Information and Computation 76 (23) 95120.
Delahaye, D. (1999). Information retrieval in a coq proof library using type isomorphisms. In: Altenkirch, T. et al., (eds.) TYPES 1999, Lecture Notes in Computer Science, Springer-Verlag, vol. 1956, 131147.
Dezani-Ciancaglini, M. (1976). Characterization of normal forms possessing inverse in the λ-β-η-calculus. Theoretical Computer Science 2 323337.
Di Cosmo, R. (1995). Isomorphisms of Types: From Lambda-Calculus to Information Retrieval and Language Design, Birkhauser.
Dosen, K. and Petric, Z. (1997). Isomorphic objects in symmetric monoidal closed categories. Mathematical Structures in Computer Science 7 639662.
Fiore, M. (2004). Isomorphisms of generic recursive polynomial types. In: POPL '04, New York, NY, USA, ACM, 7788.
Fiore, M., Di Cosmo, R., and Balat, V. (2006). Remarks on isomorphisms in typed lambda calculi with empty and sum types. Annals of Pure and Applied Logic 141 (1) 3550.
Frucht, R. (1938). Herstellung von Graphen mit vorgegebener abstrakten Gruppe. Kompositio Math. 6 3950.
Frucht, R. (1949). Graphs of degree three with a given abstract group. Canadian Journal of Mathematics 1 365378.
Goguen, H. (1994). A Typed Operational Semantics for Type Theory. PhD thesis, University of Edinburgh, UK.
Hankin, Chr. (1994). Lambda Calculi: A Guide for Computer Scientists, Clarendon Press, Oxford.
Harari, F. (1969). Graph Theory, Addison-Wesley PLC.
Hall, M. (Jr.) (1959). The Theory of Groups, The Macmillan Company, N.-Y.
Lambek, J. and Scott, P.J. (1988). Introduction to Higher-Order Categorical Logic, Cambridge Studies in Advanced Mathematics, Cambridge University Press, UK.
Longo, G., Milsted, K. and Soloviev, S. (1993). The genericity theorem and effective parametricity in polymorphic lambda-calculus. Theoretical Computer Science 121 323349.
Luo, Z. (1994). Computation and Reasoning. A Type Theory for Computer Science. International Series of Monographs on Computer Science, vol. 11, Oxford Science Publications, Clarendon Press, Oxford, UK.
Luo, Z., Soloviev, S. and Xue, T. (2013). Coercive subtyping: Theory and implementation. Information and Computation 223 1842.
Mac Lane, S. (1976). Topology and logic as a source of algebra. Bulletin of the American Mathematical Society 82 (1) 140.
Marie-Magdeleine, L. (2009). Sous-typage coercitif en présence de réductions non-standards dans un système aux types dépendants. Thèse de doctorat. Université Toulouse-3 Paul Sabatier.
Pólya, G. (1937). Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Mathematica 68 145254.
Soloviev, S. and Luo, Z. (2002). Coercion completion and conservativity in coercive subtyping. Annals of Pure and Applied Logic 113 (1–3) 297322.
Solov'ev, S. V. (1983). The category of finite sets and cartesian closed categories. Journal of Soviet Mathematics 22 (3) 13871400.
Soloviev, S. V. (1993). A complete axiom system for isomorphism of types in closed categories. In: Voronkov, A., (ed.) LPAR'93, Lecture Notes in Artificial Intelligence, Springer-Verlag, vol. 698, 360371.
Soloviev, S. (2015). On isomorphism of dependent products in a typed logical framework. In: Post-proceedings of 20th International Conference on Types for Proofs and Programs, TYPES 2014, May 12–15, 2014, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Paris, France, LIPICS, vol. 39, 275288.
Stirling, C. (2013). Proof systems for retracts in simply typed lambda calculus. In: Fomin, F. V. et al., (eds.) ICALP (2) Lecture Notes in Computer Science 7966, Springer-Verlag, 398409.
The Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Available at, Institute for Advanced Study, Princeton.
White, A. T. (1984). Graphs, Groups and Surfaces, North-Holland, Amsterdam.

Automorphisms of types in certain type theories and representation of finite groups

  • SERGEI SOLOVIEV (a1) (a2)


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