function application


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




A function ff is defined by its association to each input value xx (belonging to some allowable domain of values) of an output value, usually denoted f(x)f(x) or fxf x. The process of passing from ff and xx to f(x)f(x) is called function application, and one speaks of applying ff to xx to produce f(x)f(x).

The determination of the allowable domain for xx, given ff, depends a bit on foundational choices. In type theory and structural set theory, all functions have a type (a function type, naturally) which specifies their domain and codomain. In material set theory, a function is sometimes defined to be simply a particular sort of set of ordered pairs, with its domain specified implicitly as the set of elements occurring as first components of some such pair. (However, even in material set theory it is sometimes important for a function to come with a specified domain and/or codomain, in which case it can be defined to be an ordered triple.)

Syntax versus semantics

In formalized logic and type theory, ff, xx, and f(x)f(x) are terms (or more precisely, metavariables? standing for terms), and the process of function application is a rule of term formation. This is something which belongs to the realm of syntax. On propositions ((-1)-truncated types) this is the modus ponens deduction rule.

Under a denotational semantics, each of these terms denotes a particular object, and we also refer to the object denoted by f(x)f(x) as the result of applying the object denoted by ff to the object denoted by xx. For instance, in a material set-theoretic semantics, ff would denote a set of ordered pairs such that for any aa, there is at most one bb such that (a,b)f(a,b)\in f, and xx would denote some aa such that there does exist such a bb, and f(x)f(x) would denote that uniquely specified bb. The distinction between the terms ff, xx, and f(x)f(x) and what they denote is usually (and harmlessly) blurred in ordinary mathematical practice, but when studying logic and type theory formally it becomes important.

Under an operational semantics, by contrast, the “meaning” of the term f(x)f(x) lies in how it is “evaluated”. Usually this proceeds by beta-reduction and related rules. For instance, if ff is the term λx.x*x\lambda x. x*x and xx is the term s(s(0))s(s(0)) (the numeral two?), then f(x)f(x) is (λx.x*x)(s(s(0)))(\lambda x.x*x)(s(s(0))) which beta-reduces to s(s(0))*s(s(0))s(s(0))*s(s(0)). The definition of ** can then be invoked to cause futher beta-reductions resulting in s(s(s(s(0))))s(s(s(s(0)))) (the numeral four).

Revised on January 26, 2014 07:12:56 by Urs Schreiber (