natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
Systems of formal logic, such as type theory, try to transform expressions into a canonical form which then serves as the end result of the given computation or deduction. A formal system is said to enjoy canonicity if every expression reduced to canonical form.
More precisely, in type theory, a term belonging to some type is said to be of canonical form if it is explicitly built up using the constructors of that type. A canonical form is in particular a normal form? (one not admitting any reductions), but the converse need not hold.
For example, the terms of type $\mathbb{N}$ (a natural numbers type) of canonical form are the numerals
A type theory is said to enjoy canonicity if every term computes to a canonical form. This is held to be an important meta-theoretic property of type theory, especially considered as a programming language or as a computational foundation for mathematics.
Adding axioms to type theory, such as the principle of excluded middle or the usual version of the univalence axiom, can destroy canonicity. The axioms result in “stuck terms” which are not of canonical form, yet neither can they be “computed” any further.
For instance, if we assume the law of excluded middle, then we can build a term $case(LEM(Goldbach),0,1) \colon \mathbb{N}$ which is $0$ or $1$ according as the Goldbach conjecture is true or false. Clearly this term doesn’t “compute”, but neither is it of canonical form (a numeral).
Similarly, using the univalence axiom, we can obtain a term $p : (\mathbb{N}=\mathbb{N})$ corresponding to the automorphism of $\mathbb{N}$ which adds one to every even number and subtracts one from every odd number. Then the term $transport(p,0)$ also has type $\mathbb{N}$, but doesn’t “compute” because the computer gets “stuck” on the univalence term.
It is conjectured that univalence, unlike excluded middle, can be given a “computational” interpretation while preserving canonicity. Some partial progress towards this can be found here.
Discussion of canonicity in homotopy type theory with univalence is discussed in
Simon Huber (with Thierry Coquand), Towards a computational justification of the Axiom of Univalence , talk at TYPES 2011 (pdf)
Dan Licata, Robert Harper, Computing with Univalence (2012) (pdf)
Dan Licata, Canonicity for 2-Dimensional Type theory (2011) (web)