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Church–Rosser theorem

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Dec 15, 2021

In lambda calculus, the Church–Rosser theorem states that, when applying reduction rules to terms, the ordering in which the reductions are chosen does not make a difference to the eventual result.

More precisely, if there are two distinct reductions or sequences of reductions that can be applied to the same term, then there exists a term that is reachable from both results, by applying (possibly empty) sequences of additional reductions.[1] The theorem was proved in 1936 by Alonzo Church and J. Barkley Rosser, after whom it is named.

The theorem is symbolized by the adjacent diagram: If term a can be reduced to both b and c, then there must be a further term d (possibly equal to either b or c) to which both b and c can be reduced. Viewing the lambda calculus as an abstract rewriting system, the Church–Rosser theorem states that the reduction rules of the lambda calculus are confluent. As a consequence of the theorem, a term in the lambda calculus has at most one normal form, justifying reference to “the normal form” of a given normalizable term.

. . . Church–Rosser theorem . . .

In 1936, Alonzo Church and J. Barkley Rosser proved that the theorem holds for β-reduction in the λI-calculus (in which every abstracted variable must appear in the term’s body). The proof method is known as “finiteness of developments”, and it has additional consequences such as the Standardization Theorem, which relates to a method in which reductions can be performed from left to right to reach a normal form (if one exists). The result for the pure untyped lambda calculus was proved by D. E. Shroer in 1965.[2]

One type of reduction in the pure untyped lambda calculus for which the Church–Rosser theorem applies is β-reduction, in which a subterm of the form

(λx.t)s{displaystyle (lambda x.t)s}

is contracted by the substitution

t[x:=s]{displaystyle t[x:=s]}

. If β-reduction is denoted by

β{displaystyle rightarrow _{beta }}

and its reflexive, transitive closure by

β{displaystyle twoheadrightarrow _{beta }}

then the Church–Rosser theorem is that:[3]

M,N1,N2Λ:if MβN1 and MβN2 then XΛ:N1βX and N2βX{displaystyle forall M,N_{1},N_{2}in Lambda

 :{text{if}} Mtwoheadrightarrow _{beta }N_{1} {text{and}} Mtwoheadrightarrow _{beta }N_{2} {text{then}} exists Xin Lambda :N_{1}twoheadrightarrow _{beta }X {text{and}} N_{2}twoheadrightarrow _{beta }X}

A consequence of this property is that two terms equal in

λβ{displaystyle lambda beta }

must reduce to a common term:[4]

M,NΛ:if λβM=N then X:MβX and NβX{displaystyle forall M,Nin Lambda

 :{text{if}} lambda beta vdash M=N {text{then}} exists X:Mtwoheadrightarrow _{beta }X {text{and}} Ntwoheadrightarrow _{beta }X}

The theorem also applies to η-reduction, in which a subterm

λx.Sx{displaystyle lambda x.Sx}

is replaced by

S{displaystyle S}

. It also applies to βη-reduction, the union of the two reduction rules.

. . . Church–Rosser theorem . . .

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. . . Church–Rosser theorem . . .