nLab alpha-equivalence

 Idea

In logic and type theory, $\alpha$-equivalence is the principle that two syntactic expressions (types, terms, propositions, contexts, whatever) are equivalent for all purposes if their only difference is the renaming of bound variables.

Depending on the technicalities of how variables are managed, $\alpha$-equivalence may be a necessary axiom, a provable theorem, or entirely trivial. In any case, it is often (usually? always?) seen as a technicality devoid of conceptual interest. However, it can be a technically nontrivial task to implement $\alpha$-conversion (which is necessary to avoid capture of free variables upon substitution) when programming logic or type theory into a computer.

Definition

We work in a dependent type theory which acts as the metatheory for our object type theory.

The type $\mathbb{A}$ is a countable set (and thus a type with decidable equality) whose terms are called names or atoms. The type $\mathrm{Exp}$ of syntactic expressions is an h-set which is inductively defined by binary functions $P:\mathrm{Exp} \times \mathrm{Exp} \to \mathrm{Exp}$ and $B([(-)](-)):\mathbb{A} \times \mathrm{Exp} \to \mathrm{Exp}$. There are dependent types $E:\mathrm{Exp} \vdash \mathrm{occ} \; \mathrm{type}$ representing the occuring atoms, $E:\mathrm{Exp} \vdash \mathrm{free} \; \mathrm{type}$ representing the free atoms, and $E:\mathrm{Exp} \vdash \mathrm{bnd} \; \mathrm{type}$ representing the bound atoms, such that for each expression $E:\mathrm{Exp}$, there are embeddings

$$i_\mathrm{occ}(E):\mathrm{occ}(E) \hookrightarrow \mathbb{A}$$

$$i_\mathrm{free}(E):\mathrm{free}(E) \hookrightarrow \mathbb{A}$$

$$i_\mathrm{bnd}(E):\mathrm{bnd}(E) \hookrightarrow \mathbb{A}$$

which may be defined by recursion on $\mathrm{Exp}$.

We define the following dependent types $a:\mathbb{A}, E:\mathrm{Exp} \vdash a \lhd E \; \mathrm{type}$ as

$$a \lhd E \coloneqq \sum_{x:\mathrm{occ}(E)} i_\mathrm{occ}(E)(x) =_\mathbb{A} a$$

representing that $a$ occurs in $E$ at least once,

$$a \lhd_\mathrm{free} E \coloneqq \sum_{x:\mathrm{free}(E)} i_\mathrm{free}(E)(x) =_\mathbb{A} a$$

representing that $a$ freely occurs in $E$ at least once, and

$$a \# E \coloneqq (a \lhd_\mathrm{free} E) \to \emptyset$$

representing that $a$ is fresh (does not freely occurs in $E$ at least once)

…

See also

• equality, syntactic equality, definitional equality
• substitution
• rewriting

References

• Roy L. Crole, Alpha equivalence equalities, Theoretical Computer Science, Volume 433, 18 May 2012, Pages 1-19, (doi:10.1016/j.tcs.2012.01.030)

The distinction between $\alpha$-equivalence and syntactic equality of expressions is briefly discussed in:

• Benno van den Berg, Martijn den Besten, Quadratic type checking for objective type theory (arXiv:2102.00905)