Overlap (term rewriting) explained

In mathematics, computer science and logic, overlap, as a property of the reduction rules in term rewriting system, describes a situation where a number of different reduction rules specify potentially contradictory ways of reducing a reducible expression, also known as a redex, within a term.^[1]

More precisely, if a number of different reduction rules share function symbols on the left-hand side, overlap can occur. Often we do not consider trivial overlap with a redex and itself.

Examples

Consider the term rewriting system defined by the following reduction rules:

\rho _{1 : f(g(x),} y) → y \rho _{2 : g(x)} → f(x, x)

The term

f(g(x), y) can be reduced via ρ₁ to yield, but it can also be reduced via ρ₂ to yield f(f(x, x), y). . Note how the redex g(x) is contained in the redex f(g(x), y) . The result of reducing different redexes is described in a what is known as a critical pair; the critical pair arising out of this term rewriting system is (f(f(x, x), y), y) .

Overlap may occur with fewer than two reduction rules.

Consider the term rewriting system defined by the following reduction rule:

\rho _{1 : g(g(x))} → x The term g(g(g(x))) has overlapping redexes, which can be either applied to the innermost occurrence or to the outermost occurrence of the g(g(x)) term.

Notes and References

1. Book: Term Rewriting Systems . Marc Bezem . Jan Willem Klop . Roel de Vrijer . Cambridge Tracts in Theoretical Computer Science . . 2003 . 0-521-39115-6 . 48. Cambridge, UK .