Complete Restrictions of the Intersection Type Discipline

Abstract In this paper the intersection type discipline as defined in Barendregt (1983) is studied. We will present two different and independent complete restrictions of the intersection type discipline. The first restricted system, the strict type assignment system, is presented in Section 2. Its major feature is the absence of the derivation rule (⩽) and it is based on a set of strict types. We will show that these together give rise to a strict filter lambda model that is essentially different from the one presented in Barendregt. We will show that the strict type assignment system is the nucleus of the full system, i.e. for every derivation in the intersection type discipline there is a derivation in which (⩽) is used only at the very end. Finally we will prove that strict type assignment is complete for inference semantics. The second restricted system is presented in Section 3. Its major feature is the absence of the type ω. We will show that this system gives rise to a filter λ I-model and that type assignment without ω is complete for the λ I-calculus. Finally we will prove that a lambda term is typeable in this system if and only if it is strongly normalizable.

[1]  M. Coppo.  Type theories, normal forms, and D?-lambda-models*1 , 1987 .

[2]  Haskell B. Curry,et al.  Combinatory Logic, Volume I , 1959 .

[3]  Simona Ronchi Della Rocca,et al.  Principal Type Schemes for an Extended Type Theory , 1984, Theor. Comput. Sci..

[4]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[5]  M. Dezani-Ciancaglini,et al.  Extended Type Structures and Filter Lambda Models , 1984 .

[6]  John C. Mitchell,et al.  Polymorphic Type Inference and Containment , 1988, Inf. Comput..

[7]  G. Pottinger,et al.  A type assignment for the strongly normalizable |?-terms , 1980 .

[8]  D. A. Turner,et al.  Miranda: A Non-Strict Functional language with Polymorphic Types , 1985, FPCA.

[9]  Gordon D. Plotkin,et al.  The category-theoretic solution of recursive domain equations , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[10]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[11]  R. Hindley.  The Principal Type-Scheme of an Object in Combinatory Logic , 1969 .

[12]  P. Sallé.  Une extension de la theorie des types en λ-calcul , 1978 .

[13]  Mariangiola Dezani-Ciancaglini,et al.  An extension of the basic functionality theory for the λ-calculus , 1980, Notre Dame J. Formal Log..

[14]  Mariangiola Dezani-Ciancaglini,et al.  Functional Characters of Solvable Terms , 1981, Math. Log. Q..

[15]  Mariangiola Dezani-Ciancaglini,et al.  A filter lambda model and the completeness of type assignment , 1983, Journal of Symbolic Logic.

[16]  Simona Ronchi Della Rocca,et al.  Principal Type Scheme and Unification for Intersection Type Discipline , 1988, Theor. Comput. Sci..

[17]  Daniel Leivant,et al.  Polymorphic type inference , 1983, POPL '83.

[18]  Mariangiola Dezani-Ciancaglini,et al.  A Characterization of F-Complete Type Assignments , 1986, Theor. Comput. Sci..

[19]  Mariangiola Dezani-Ciancaglini,et al.  F-Semantics for Intersection Type Discipline , 1984, Semantics of Data Types.

[20]  H B Curry,et al.  Functionality in Combinatory Logic. , 1934, Proceedings of the National Academy of Sciences of the United States of America.

[21]  J. Roger Hindley,et al.  The Completeness Theorem for Typing lambda-Terms , 1983, Theor. Comput. Sci..

[22]  Robin Milner,et al.  A Theory of Type Polymorphism in Programming , 1978, J. Comput. Syst. Sci..