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. Schroer in 1965. ==Pure untyped lambda calculus==

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 ( \lambda x . t) s is contracted by the substitution t [ x := s], where t, s are two lambda expressions. If β-reduction is denoted by \rightarrow_\beta and its reflexive, transitive closure by \twoheadrightarrow_\beta then the Church–Rosser theorem is that: :\forall M, N_1, N_2 \in \Lambda: \text{if}\ M\twoheadrightarrow_\beta N_1 \ \text{and}\ M\twoheadrightarrow_\beta N_2 \ \text{then}\ \exists X\in \Lambda: N_1\twoheadrightarrow_\beta X \ \text{and}\ N_2\twoheadrightarrow_\beta X A consequence of this property is that two terms equal in \lambda\beta must reduce to a common term: :\forall M, N\in \Lambda: \text{if}\ \lambda\beta \vdash M=N \ \text{then}\ \exists X: M \twoheadrightarrow_\beta X \ \text{and}\ N\twoheadrightarrow_\beta X The theorem also applies to η-reduction, in which a subterm \lambda x.Sx is replaced by S. It also applies to βη-reduction, the union of the two reduction rules. Proof For β-reduction, one proof method originates from William W. Tait and Per Martin-Löf. Say that a binary relation \rightarrow satisfies the diamond property if: :\forall M, N_1, N_2 \in \Lambda: \text{if}\ M\rightarrow N_1 \ \text{and}\ M\rightarrow N_2 \ \text{then}\ \exists X\in \Lambda: N_1\rightarrow X \ \text{and}\ N_2\rightarrow X Then the Church–Rosser property is the statement that \twoheadrightarrow_\beta satisfies the diamond property. We introduce a new reduction \rightarrow_{\|} whose reflexive transitive closure is \twoheadrightarrow_\beta and which satisfies the diamond property. By induction on the number of steps in the reduction, it thus follows that \twoheadrightarrow_\beta satisfies the diamond property. The relation \rightarrow_{\|} has the formation rules: • M \rightarrow_{\|} M • If M \rightarrow_{\|} M' and N \rightarrow_{\|} N' then \lambda x.M \rightarrow_{\|} \lambda x.M' and MN \rightarrow_{\|} M'N' and (\lambda x. M)N \rightarrow_{\|} M'[x:=N'] The η-reduction rule can be proved to be Church–Rosser directly. Then, it can be proved that β-reduction and η-reduction commute in the sense that: :If M \rightarrow_\beta N_1 and M \rightarrow_\eta N_2 then there exists a term X such that N_1 \rightarrow_\eta X and N_2\rightarrow_\beta X. Hence we can conclude that βη-reduction is Church–Rosser. ==Variants==

The Church–Rosser theorem also holds for many variants of the lambda calculus, such as the simply typed lambda calculus, many calculi with advanced type systems, and Gordon Plotkin's beta-value calculus. Plotkin also used a Church–Rosser theorem to prove that the evaluation of functional programs (for both lazy evaluation and eager evaluation) is a function from programs to values (a subset of the lambda terms). In older research papers, a rewriting system is said to be Church–Rosser, or to have the Church–Rosser property, when it is confluent. == Notes ==