type theory In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of ...

, a branch of mathematics, the identity type represents the concept of

equality Equality generally refers to the fact of being equal, of having the same value. In specific contexts, equality may refer to: Society * Egalitarianism, a trend of thought that favors equality for all people ** Political egalitarianism, in which ...

. It is also known as propositional equality to differentiate it from "judgemental equality". Equality in type theory is a complex topic and has been the subject of research, such as the field of

homotopy type theory In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory ap ...

Comparison with Judgemental Equality

The identity type is one of 2 different notions of equality in type theory. The more fundamental notion is "judgemental equality", which is a

judgement Judgement (or judgment) is the evaluation of given circumstances to make a decision. Judgement is also the ability to make considered decisions. In an informal context, a judgement is opinion expressed as fact. In the context of a legal tria ...

Beyond Judgemental Equality

The identity type can do more than what judgemental equality can do. It can be used to show "for all

x, x+1=1+x

", which is impossible to show with judgemental equality. This is accomplished by using the eliminator (or "recursor") of the natural numbers, known as "R". The "R" function lets us define a new function on the natural numbers. That new function "P" is defined to be "(λ x:nat . x+1 = 1+x)". The other arguments act like the parts of an induction proof. The argument "PZ : P 0" becomes the base case "0+1 = 1+0", which is the term "refl nat 1". The argument "PS : P n → P (S n)" becomes the inductive case. Essentially, this says that when "x+1 = 1+x" has "x" replaced with a canonical value, the expression will be the same as "refl nat (x+1)".

Versions of the Identity Type

The identity type is complex and is the subject of research in type theory. While every version agrees on the constructor, "refl". Their properties and eliminator functions differ dramatically. For "extensional" versions, any identity type can be converted into a judgemental equality. A computational version is known as "Axiom K" due to Thomas Streicher. These are not very popular lately.

Complexity of Identity Type

Martin Hofmann and Thomas Streicher refuted that idea type theory required all terms of the identity type to be the same. A popular branches of research into the identity type are

and its

Cubical type theory Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...

References

{{Formal semantics Type theory