Guarded Cubical Type Theory: Path Equality for Guarded Recursion

TY - GEN

T1 - Guarded Cubical Type Theory

T2 - 25th EACSL Annual Conference on Computer Science Logic

AU - Birkedal, Lars

AU - Bizjak, Aleš

AU - Clouston, Ranald

AU - Grathwohl, Hans Bugge

AU - Spitters, Bas

AU - Vezzosi, Andrea

N1 - 17 pages, to be published in proceedings of CSL 2016

PY - 2016

Y1 - 2016

N2 - This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-L\"of type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.

AB - This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-L\"of type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.

KW - cs.LO

KW - cs.PL

KW - F.3.3; F.3.2

M3 - Article in proceedings

T3 - Leibniz International Proceedings in Informatics

SP - 1

EP - 17

BT - CSL 2016

A2 - Talbot, Jean-Marc

A2 - Regnier, Laurent

Y2 - 29 August 2016 through 1 September 2016

ER -