Skip to main content
×
Home
    • Aa
    • Aa

An Isbell duality theorem for type refinement systems

  • PAUL-ANDRÉ MELLIÈS (a1) and NOAM ZEILBERGER (a2)
Abstract

Any refinement system (= functor) has a fully faithful representation in the refinement system of presheaves, by interpreting types as relative slice categories, and refinement types as presheaves over those categories. Motivated by an analogy between side effects in programming and context effects in linear logic, we study logical aspects of this ‘positive’ (covariant) representation, as well as of an associated ‘negative’ (contravariant) representation. We establish several preservation properties for these representations, including a generalization of Day's embedding theorem for monoidal closed categories. Then, we establish that the positive and negative representations satisfy an Isbell-style duality. As corollaries, we derive two different formulas for the positive representation of a pushforward (inspired by the classical negative translations of proof theory), which express it either as the dual of a pullback of a dual or as the double dual of a pushforward. Besides explaining how these constructions on refinement systems generalize familiar category-theoretic ones (by viewing categories as special refinement systems), our main running examples involve representations of Hoare logic and linear sequent calculus.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

R. Atkey , P. Johann and N. Ghani (2012). Refining inductive types. Logical Methods in Computer Science 8 (2:9).

D. Čubrić , P. Dybjer and P. Scott (1998). Normalization and the Yoneda Embedding. Mathematical Structures in Computer Science 8 (2) 153192.

N. Ghani , P. Johann and C. Fumex (2013). Indexed induction and coinduction, fibrationally. Logical Methods in Computer Science 9 (3:6) 131.

J.-Y. Girard (1987). Linear logic. Theoretical Computer Science 50 (1) 1102.

C. Hermida and B. Jacobs (1998). Structural induction and coinduction in a fibrational setting. Information and Computation 145 (2) 107152.

C.A.R. Hoare (1969). An axiomatic basis for computer programming. Communications of the ACM 12 (10).

S. Mac Lane (1971). Categories for the Working Mathematician, Springer.

A. Nanevski , G. Morrisett and L. Birkedal (2008). Hoare type theory, polymorphism and separation. Journal of Functional Programming 18 (5–6) 865911.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 48 *
Loading metrics...

* Views captured on Cambridge Core between 20th March 2017 - 25th May 2017. This data will be updated every 24 hours.