Notre Dame Journal of Formal Logic

The "Relevance" of Intersection and Union Types

Mariangiola Dezani-Ciancaglini, Silvia Ghilezan, and Betti Venneri


The aim of this paper is to investigate a Curry-Howard interpretation of the intersection and union type inference system for Combinatory Logic. Types are interpreted as formulas of a Hilbert-style logic L, which turns out to be an extension of the intuitionistic logic with respect to provable disjunctive formulas (because of new equivalence relations on formulas), while the implicational-conjunctive fragment of L is still a fragment of intuitionistic logic. Moreover, typable terms are translated in a typed version, so that $\vee$-$\wedge$-typed combinatory logic terms are proved to completely codify the associated logical proofs.

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Notre Dame J. Formal Logic Volume 38, Number 2 (1997), 246-269.

First available in Project Euclid: 12 December 2002

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Zentralblatt MATH identifier

Primary: 03B40: Combinatory logic and lambda-calculus [See also 68N18]
Secondary: 03B46 03B55: Intermediate logics 68N15: Programming languages


Dezani-Ciancaglini, Mariangiola; Ghilezan, Silvia; Venneri, Betti. The "Relevance" of Intersection and Union Types. Notre Dame J. Formal Logic 38 (1997), no. 2, 246--269. doi:10.1305/ndjfl/1039724889.

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  • [1] Abramsky, S., ``Domain theory in logical form,'' Annals of Pure and Applied Logic, vol. 51 (1991), pp. 1--77.
  • [2] Alessi, F., and F. Barbanera, ``Strong conjunction and intersection types,'' LNCS, vol. 520 (1991), Springer-Verlag, Berlin, pp. 64--73.
  • [3] Barbanera, F., and M. Dezani-Ciancaglini, ``Intersection and union types,'' LNCS, vol. 526 (1991), Springer-Verlag, Berlin, pp. 651--74.
  • [4] Barbanera, F., M. Dezani-Ciancaglini, and U. de'Liguoro, ``Intersection and union types: syntax and semantics,'' Information and Computation, vol. 119 (1995), pp. 202--30.
  • [5] Barbanera, F., and S. Martini, ``Proof functional connectives and realizability,'' Archive for Mathematical Logic, vol. 33 (1994), pp. 189--211.
  • [6] Coppo, M., M. Dezani-Ciancaglini, and B. Venneri, ``Functional characters of solvable terms,'' Zeitschrift für mathematiche Logik und Grundlagen der Mathematik, vol. 27 (1981), pp. 45--58.
  • [7] Coppo, M., and A. Ferrari, ``Type inference, abstract interpretation and strictness analysis,'' Theoretical Computer Science, vol. 121 (1993), pp. 113--43.
  • [8] Curry, H. B., and K. Feys, Combinatory Logic, North-Holland, Amsterdam, 1958.
  • [9] Dezani-Ciancaglini, M., and J. R. Hindley, ``Intersection types for combinatory logic,'' Theoretical Computer Science, vol. 100 (1992), pp. 303--24.
  • [10] Dummet, M., ``A propositional calculus with denumerable matrics,'' The Journal Symbolic Logic, vol. 24 (1959), pp. 97--106.
  • [11] Harrop, R., ``Concerning formulas of the types $A\rightarrow B\vee C$, $A\rightarrow (Ex)B(x)$ in intuitionistic formal systems,'' The Journal of Symbolic Logic, vol. 25 (1960), pp. 27--32.
  • [12] Hindley, J. R., ``Coppo-Dezani types do not correspond to propositional logic,'' Theoretical Computer Science, vol. 28 (1984), pp. 235--36.
  • [13] Hindley, J. R., and J. P. Seldin, Introduction to Combinators and $\lambda $-Calculus, Cambridge University Press, Cambridge, 1986.
  • [14] Jensen, T., ``Strictness analysis in logical form,'' LNCS, vol. 532 (1991), Springer-Verlag, Berlin, pp. 352--66.
  • [15] Lopez-Escobar, E. G. K., ``Proof functional connectives,'' LNM, vol. 1130 (1985), Springer-Verlag, Berlin, pp. 208--21.
  • [16] MacQueen, D., G. Plotkin, and R. Sethi, ``An ideal model for recursive polymorphic types,'' Information and Control, vol. 71 (1986), pp. 95--130.
  • [17] Meyer, R. K., and R. Routley, ``Algebraic analysis of entailment I,'' Logique et Analyse, vol. 15 (1972), pp. 407--28.
  • [18] Meyer, R. K., and R. Routley, ``The semantics of entailment III,'' Journal of Philosophical Logic, vol. 1 (1972), pp. 192--208.
  • [19] Mints, G. E., ``The completeness of provable realizability,'' Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 420--41.
  • [20] Venneri, B., ``Intersection types as logical formulae,'' Journal of Logic and Computation, vol. 4 (1994), pp. 109--124.