Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecompositionsubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem

homotopy levels




In type theory, β\beta-reduction is a process of “computation”, which generally replaces more complicated terms with simpler ones. It was originally identified in the lambda-calculus, where it contrasts with α\alpha-equivalence and η\eta-expansion; this is the version described below for function types. The analogous reduction for inductive types may also be known as ι\iota-reduction.


In its most general form, β\beta-reduction consists of rules which specify, for any given type TT, if we apply an “eliminator” for TT to the result of a “constructor” for TT, how to “evaluate” the result. We write

s βts \to_\beta t

if the term ss beta-reduces to the term tt. Sometimes we write s β *ts \to_\beta^* t if this reduction takes nn steps (leaving off the ** to denote n=1n=1). The relation “reduces to” generates an equivalence relation on the set of terms called beta equivalence and often denoted s= βts =_\beta t or s βts \equiv_\beta t.

Function types

The most common (and original) example is when TT is a function type ABA \to B.

In this case, the constructor of ABA \to B is a λ\lambda-expression: given a term bb of type BB containing a free variable xx of type AA, then λx.b\lambda x.\, b is a term of type ABA \to B.

The eliminator of ABA \to B says that given a term ff of type ABA \to B and a term aa of type AA, we can apply ff to aa to obtain a term f(a)f(a) of type BB.

Now if we first construct a term λx.b:AB\lambda x.\, b\colon A \to B, and then apply this term to a:Aa\colon A, we obtain a term (λx.b)(a):B(\lambda x.\, b)(a)\colon B. The rule of β\beta-reduction then tells us that this term evaluates or computes or reduces to b[a/x]b[a/x], the result of substituting the term aa for the variable xx in the term bb.

See lambda calculus for more.

Product types

Although function types are the most publicized notion of β\beta-reduction, basically all types in type theory have a form of it. For instance, in the negative presentation of a product type A×BA \times B, the constructor is an ordered pair (a,b):A×B(a,b)\colon A \times B, while the eliminators are projections π 1\pi_1 and π 2\pi_2 which yield elements of AA or BB.

The beta reduction rules then say that if we first apply a constructor (a,b)(a,b), then apply an eliminator to this, the resulting terms π 1(a,b)\pi_1(a,b) and π 2(a,b)\pi_2(a,b) compute to aa and bb respectively.

Revised on October 14, 2013 12:45:47 by Toby Bartels (