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A Quantitative Version of Simple Types

Authors Daniele Pautasso, Simona Ronchi Della Rocca

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Daniele Pautasso
  • Dipartimento di Informatica, University of Torino, Italy
Simona Ronchi Della Rocca
  • Dipartimento di Informatica, University of Torino, Italy


We would like to thank Delia Kesner and Antonio Bucciarelli for many useful discussions about the topic of this paper. We are also grateful to the anonymous referees for their careful reading and suggestions.

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Daniele Pautasso and Simona Ronchi Della Rocca. A Quantitative Version of Simple Types. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 29:1-29:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


This work introduces a quantitative version of the simple type assignment system, starting from a suitable restriction of non-idempotent intersection types. The resulting system is decidable and has the same typability power as the simple type system; thus, assigning types to terms supplies the very same qualitative information given by simple types, but at the same time can provide some interesting quantitative information. It is well known that typability for simple types is equivalent to unification; we prove a similar result for the newly introduced system. More precisely, we show that typability is equivalent to a unification problem which is a non-trivial extension of the classical one: in addition to unification rules, our typing algorithm makes use of an expansion operation that increases the cardinality of multisets whenever needed.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • λ-calculus
  • intersection types
  • unification


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