The complexity of normal form rewrite sequences for Associativity

The complexity of normal form rewrite sequences for Associativity. Abstract The complexity of a particular term-rewrite system is considered: the rule of associativity (x * y) * z ⊲ x * (y * z). Algorithms and exact calculations are given for the longest and shortest sequences of applications of ⊲ that result in normal form (NF). The shortest NF sequence for a term x is always n − d rm (x), where n is the number of occurrences of * in x and d rm (x) is the depth of the rightmost leaf of x. The longest NF sequence for any term is of length n(n − 1)/2. (Klop 1992) provides an overview of the theory of term rewrite systems. There is relatively little known about the complexity of various term rewrite systems. Here, I consider a particular rewrite system with one binary connective * and one rewrite rule (x * y) * z ⊲ x * (y * z). A rewrite system − → is Strongly Normalizing (SN) iff every sequence of applications of − → is finite. A rewrite system is Church-Rosser (CR) just in case ∀x, y. A rewrite system is Weakly Church-Rosser (WCR) just in case ∀x, y, w. Let −→ be the relation between two terms such that x−→ y just in case x contains a subterm, the redex, which matches the left hand side of the rule ⊲ , and replacing the redex by the corresponding right hand side, the contractum, yields the new term y. A term is in normal form (NF) if it contains no redex. Given a term x, define λ(x) (resp. ρ(x)) refers to its the left (right) child of x.