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. 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: The term can be reduced via ρ1 to yield , but it can also be reduced via ρ2 to yield . Note how the redex is contained in the redex . 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 . Overlap may occur with fewer than two reduction rules. Consider the term rewriting system defined by the following reduction rule: The term has overlapping redexes, which can be either applied to the innermost occurrence or to the outermost occurrence of the term. References Rewriting systems