## Mathematical Society of Japan Memoirs

### Syntax and Semantics of Type Assignment Systems

Hirofumi Yokouchi

#### Abstract

This paper gives a survey on syntax and semantics for type assignment systems, with a special attention to semantic completeness of the systems. Starting with the most basic system with function types only, it introduces polymorphic types, intersection types, union types, and existential type quantifier in a step-by-step manner. It also provides several sequent-style formulations of type assignment systems. With the sequent calculi, it shows the properties of type assignment systems concerning the completeness and the conservativity of various systems.

#### Chapter information

Source
Masako Takahashi, Mitsuhiro Okada and Mariangiola Dezani-Ciancaglini, Eds. Theories of Types and Proofs (Tokyo: The Mathematical Society of Japan, 1998), 99-141

Dates
First available in Project Euclid: 17 January 2014

https://projecteuclid.org/euclid.msjm/1389985701

Digital Object Identifier
doi:10.2969/msjmemoirs/00201C030

Mathematical Reviews number (MathSciNet)
MR1728760

Zentralblatt MATH identifier
0935.03025

Rights

#### Citation

Yokouchi, Hirofumi. Syntax and Semantics of Type Assignment Systems. Theories of Types and Proofs, 99--141, The Mathematical Society of Japan, Tokyo, Japan, 1998. doi:10.2969/msjmemoirs/00201C030. https://projecteuclid.org/euclid.msjm/1389985701

#### References

• [1] F. Alessi and F. Barbanera. Strong conjunction and intersection types. In A. Tarlecki, editor, Mathematical Foundations of Computer Science 1991, volume 520 of Lecture Notes in Computer Science, pages 64-73, Kazimierz Dolny, Poland, 9-13 Sept. 1991. Springer.
• [2] F. Barbanera, M. Dezani-Ciancaglini, and U. de'Liguoro. Intersection and union types: Syntax and semantics. Inform. Comp., 119(2):202-230, June 1995.
• [3] H. Barendregt. Lambda calculi with types. In S. Abramsky, D. M. Gabbai, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume II, pages 117-309. Oxford University Press, 1992.
• [4] H. Barendregt, M. Coppo, and M. Dezani-Ciancaglini. A filter lambda model and the completeness of type assignment. J. Symbolic Logic, 48:931-940, 1983.
• [5] H. P. Barendregt. The Lambda Calculus: Its Syntax and Semantics. North- Holland, Amsterdam, 1984.
• [6] F. Cardone and M. Coppo. Two extensions of Curry's type inference system. In P. Odifreddi, editor, Logic and Computer Science, pages 19-75. Academic Press, London, 1990.
• [7] F. Cardone, M. Dezani-Ciancaglini, and U. de'Liguoro. Combining type disciplines. Ann. Pure Appl. Logic, 66(3):197-230,5 Apr. 1994.
• [8] A. Church. A formulation of the simple theory of types. J. Symbolic Logic, 5, 1940.
• [9] M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Z. Math. Log. Grund Math., 27:45-58, 1981.
• [10] H. B. Curry. Functionality in combinatory logic. In Proc. Nat. Acad. Science USA 20, pages 584-590, 1934.
• [11] H. B. Curry and R. Feys. Combinatory Logic, Vol. I. North-Holland, Amsterdam, 1958.
• [12] H. B. Curry, J. R. Hindley, and J. P. Seldin. Combinatory Logic, Vol. II. North-Holland, Amsterdam, 1972.
• [13] N. G. de Bruijn. A survey of the project automath. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 579-606. Academic Press, London, 1980.
• [14] M. Dezani-Ciancaglini and M. Coppo. An extension of basic functionality theory for lambda-calculus. Notre Dame J. Formal Log., 21:685-693, 1980.
• [15] M. Dezani-Ciancaglini, S. Ghilezan, and B. Venneri. The relevance'' of intersection and union types. Notre Dame J. Formal Logic, 38(2):246-269, Spring 1997.
• [16] M. Dezani-Ciancaglini and I. Margaria. A characterization of $F$-complete assignments. Theoret. Comput. Sci., 45(2):121-157, 1986. Fundamental study.
• [17] J. H. Gallier. On Girard's candidats de reductibilité''. In P. Odifreddi, editor, Logic and Computer Science, pages 123-203. Academic Press, London, 1990.
• [18] G. Gentzen. Untersuchungen über das logische Schilie\ssen, I, II. Math. Zeitschr., 39:176-210, 405-431, 1934.
• [19] J. Y. Girard. Interprétation fonctionnelle et élimination des coupures de l'arithmétique d'ordre supérieur. PhD thesis, University of Paris VII, 1972.
• [20] R. Hindley. The completeness theorem for typing $\lambda$-terms. Theoret. Comput. Sci., 22(1-2):1-17, Jan. 1983.
• [21] R. Hindley. Curry's type-rules are complete with respect to the $F$ -semantics too. Theoret. Comput. Sci., 22(1-2):127-133, Jan. 1983.
• [22] W. Howard. The formulae-as-types notation of construction. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 479-501. Academic Press, London, 1980.
• [23] D. Leivant. Typing and computational properties of lambda expressions. Theoret. Comput. Sci., 44(1):51-68, 1986.
• [24] D. MacQueen, G. Plotkin, and R. Sethi. An ideal model for recursive polymorphic types. Inform. Comp., 71 (1/2):95-130, Oct. /Nov. 1986.
• [25] D. MacQueen and R. Sethi. A semantic model of types for applicative languages. In ACM Symposium on Lisp and Functional Programming, pages 243-252, 1982.
• [26] A. R. Meyer, J. C. Mitchell, E. Moggi, and R. Statman. Empty types in polymorphic lambda calculus. In Conference Record of the Fourteenth Annual ACM Symposium on Principles of Programming Languages, pages 253-262, Munich, Germany, Jan. 1987.
• [27] J. C. Mitchell. Polymorphic type inference and containment. Inform. Comp., 76 (2/3):211-249, Feb. /Mar. 1988.
• [28] J. C. Mitchell and E. Moggi. Kripke-style models for typed lambda calculus. Ann. Pure Appl. Logic, 51 (1-2):99-124, 1991.
• [29] J. C. Mitchell and G. D. Plotkin. Abstract types have existential type. ACM Trans. Prog. Lang. Sys., 10(3):470-502, July 1988.
• [30] J. Palsberg and C. Pavlopoulou. From polyvariant flow information to intersection and union types. In Conference Record of POPL '98: The 25th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 197-208, San Diego, California, 19-21 Jan. 1998.
• [31] B. C. Pierce. Programming with intersection types, union types, and polymorphism. Technical report, CMU-CS-91-106, Carnegie Mellon University, February 1991.
• [32] B. C. Pierce. Intersection types and bounded polymorphism. Mathematical Structures in Computer Science, 7(2):129-193, Apr. 1997.
• [33] G. Plotkin. A semantics for static type inference. Inform. Comp., 109(1/2):256-299,15 Feb. /Mar. 1994.
• [34] J. Reynolds. Preliminary design of the programming language Forsythe. Technical report, CMU-CS-88-159, Carnegie Mellon University, June 1988.
• [35] J. C. Reynolds. Towards a theory of type structure. In Mathematical Foundations of Software Development, volume 146 of Lecture Notes in Computer Science, pages 408-426, Berlin, 1972. Springer-Verlag.
• [36] H. Yokouchi. F-semantics for type assignment systems. Theoret. Comput. Sci., 129(1):39-77, 20 June 1994. Fundamental Study.
• [37] H. Yokouchi. Embedding a second order type system into an intersection type system. Inform. Comp., 117(2):206-220, Mar. 1995.
• [38] H. Yokouchi. Completeness of type assignment systems with intersection, union, and type quantifiers (full paper). Available at http: //www. keim.cs.gunma-u.ac.jp/\tilde yokouchi, 1998.
• [39] H. Yokouchi. Completeness of type assignment systems with intersection, union, and type quantifiers. In Proceedings, Thirteenth Annual IEEE Symposium on Logic in Computer Science, pages 368-379, Indianapolis, Indiana, 1998. IEEE Computer Society Press.
• [40] H. Yokouchi and R. Kashima. Sequent calculi for type assignment and their completeness. Available at http: //www.keim.cs.gunma-u.ac.jp/\tilde yokouchi, 1997.