eta-conversion (changes) in nLab

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η\eta-conversion

Idea

In type theory, η\eta-conversion is a conversion rule for “computation” which is “dual” to beta-conversion.

Whereas β\beta-reduction tells us how to simplify a term that involves an eliminator applied to a constructor, η\eta-reduction tells us how to simplify a term that involves a constructor applied to an eliminator.

In contrast to β\beta-reduction, whose “directionality” is universally agreed upon, the directionality of η\eta-conversion is not always the same. Sometimes one may talk about η\eta-reduction, which (usually) simplifies a constructor–eliminator pair by removing it, but we may also talk about η\eta-expansion, which (usually) makes a term more complicated by introducing a constructor–eliminator pair. Although one might expect that of course we always want to use reduction to simplify, it is possible to put bounds on η\eta-expansion, and η\eta-reduction is ill-defined for the unit type (the exception prompting ‘usually’ above).

However, from a category-theoretic POV, it is more natural to consider η\eta to be an expansion, which paired with β\beta as a reduction can be used as a syntax for a lax 2-adjunction (Seely).

The equivalence relation generated by η\eta-reduction/expansion is called η\eta-equivalence, and the whole collection of processes is called η\eta-conversion.

Examples

For function types

The most common example is for a function type A→BA \to B.

In this case, the constructor of A→BA \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 A→BA \to B.

The eliminator of A→BA \to B says that given a term ff of type A→BA \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.

An etaeta-redex? (a term that can be reduced by η\eta-reduction) is then of the form λx.f(x)\lambda x.\, f(x) – the constructor (lambda expression) applied to the eliminator (application). Eta reduction reduces such a redex to the term ff.

Conversely, η\eta-expansion expands any bare function term ff to the form λx.f(x)\lambda x.\, f(x). If η\eta-expansion is applied again to this, we get λx.(λy.f(y))(x)\lambda x.\, (\lambda y.\, f(y))(x), but β\beta-reduction returns this to λx.f(x)\lambda x.\, f(x); therefore, this last form is considered to be fully η\eta-expanded. In general, the rule when applying η\eta-expansion is to use it only when the result is not a β\beta-redex.

For product types

Although function types are the most publicized notion of η\eta-reduction, basically all types in type theory can 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 η\eta-expansion rule then says that for a term p:A×Bp\colon A\times B, the term (π 1p,π 2p)(\pi_1 p, \pi_2 p) — the constructor applied to the eliminators — is equivalent to pp itself. (Again, we do not repeat the η\eta-expansion, as this would produce a β\beta-redex.) If we use η\eta-reduction instead, then we simplify any subterm of the form (π 1p,π 2p)(\pi_1 p, \pi_2 p) to pp (and leave anything not of that form alone).

For unit types

Above we did a product type with two factors, although it's easy to generalise to any natural number of factors. The case with zero factors is known as the unit type, and η\eta-conversion behaves a bit oddly there; let us examine it.

In the negative presentation of the unit type 11, the constructor is an empty list ():1()\colon 1, while there are no eliminators. The η\eta-expansion rule then says that any term p:1p\colon 1 is equivalent to the term ()() — the constructor applied to no eliminators. In this case, if we repeat the η\eta-expansion, this does not produce a β\beta-redex (indeed, there is no β\beta-reduction for the unit type), but simply makes no change. If we try to apply η\eta-reduction to ()(), then this is ill-defined; we could ‘simplify’ this to any term p:1p\colon 1 that we might be able to construct.

The positive presentation of the unit type does have a well-defined η\eta-reduction, however; see unit type.

Implementation

Eta-reduction/expansion is not as well-behaved formally as beta-reduction, and its introduction can make computational equality undecidable. For this reason and others, it is not always implemented in computer proof assistants.

Coq versions 8.3 and prior do not implement η\eta-equivalence (definitionally), but versions 8.4 and higher do implement it for dependent product types (which include function types). Even in Coq v8.4, η\eta-equivalence is not implemented for other types, such as inductive and coinductive types. This is a good thing for homotopy type theory, since η\eta-equivalence for identity types forces us into set-level type theory.

When η\eta-equivalence is not an implemented as a direct identity, it may be derived for a defined (coarser than identity) equality. For example, if f= A→Bgf =_{A \to B} g is defined to mean ∀x.f(x)= Bg(x)\forall x.\, f(x) =_B g(x) (where = B=_B is assumed to have been previously defined) and (λx.b)(a)(\lambda x.\, b)(a) is taken to be identical to b[a/x]b[a/x] (implementing β\beta-reduction), then ff and λx.f(x)\lambda x.\, f(x) are provably equal even if not identical. Thus, eta-equivalence for function types follows from function extensionality (relative to any appropriate notion of equality).

Similarly, if “equality” refers to a Martin-Löf identity type in dependent type theory, then a suitable form of η\eta-equivalence is provable for inductively defined types (with β\beta-reduction and a dependent eliminator). This includes the identity types themselves, but this form of η\eta-equivalence does not imply the identity types are valued in propositions because the identity type itself must be incorporated in stating the equivalence. See the next section.

Propositional η\eta-conversion

In dependent type theory, an important role is played by propositional η\eta-conversions which “compute to identities” along constructors. For example, consider binary products with β\beta-reduction, but not (definitional) η\eta-conversion. We say that η\eta-conversion holds propositionally if

1. For any p:A×Bp\colon A\times B we have a term η p:Id A×B(p,(π 1p,π 2p))\eta_p \colon Id_{A\times B}(p, (\pi_1 p, \pi_2 p)), and

2. For a:Aa\colon A and b:Bb\colon B we have a definitional equality η (a,b)=1 (a,b)\eta_{(a,b)} = 1_{(a,b)} (where 1 (a,b)1_{(a,b)} denotes the reflexivity constructor of the identity type).

Similar definitions apply for any other type.

The reason this notion is important is that it is “equivalent” to the ability to extend the eliminator of non-dependent type theory to a dependent eliminator, where the type being eliminated into is dependent on the type under consideration.

For instance, in the case of the binary product, suppose that η\eta-conversion holds propositionally as above, and that we have a dependent type z:A×B⊢C(z):Typez\colon A\times B \vdash C(z)\colon Type along with a term x:A,y:B⊢c(x,y):C((x,y))x\colon A, y\colon B \vdash c(x,y) \colon C((x,y)) defined over the constructor. Then for any p:A×Bp\colon A\times B we can “transport” along η p\eta_p to obtain a term defined over pp, yielding the dependent eliminator. The rule η (a,b)=1 (a,b)\eta_{(a,b)} = 1_{(a,b)} ensures that this dependent eliminator satisfies the appropriate β\beta-reduction rule.

Conversely, if we have a dependent eliminator, then η p\eta_p can be defined by eliminating into the dependent type z:A×B⊢id A×B(z,(π 1z,π 2z))z\colon A\times B \vdash id_{A\times B}(z,(\pi_1 z, \pi_2 z)), since when zz is (x,y)(x,y) we have 1 (x,y)1_{(x,y)} inhabiting this type.

Note that this “equivalence” is itself only “propositional”, however; if we go back and forth, we should not expect to get literally the same dependent eliminator or propositional η\eta term, only a propositionally-equal one.

The same principle applies to other types, particularly dependent sum types and dependent product types/function types (although the latter are a bit trickier).

• lambda-calculus

• beta-reduction

References

• Seely, Modelling computations: a 2-categorical framework, pdf