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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:

${\displaystyle \rho _{1}\ :\ f(g(x),y)\rightarrow y}$

${\displaystyle \rho _{2}\ :\ g(x)\rightarrow f(x,x)}$

The term ${\displaystyle f(g(x),y)}$ can be reduced via ρ₁ to yield y, but it can also be reduced via ρ₂ to yield ${\displaystyle f(f(x,x),y).}$. Note how the redex ${\displaystyle g(x)}$ is contained in the redex ${\displaystyle f(g(x),y)}$. 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 ${\displaystyle (f(f(x,x),y),y)}$.

Overlap may occur with fewer than two reduction rules.

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

${\displaystyle \rho _{1}\ :\ g(g(x))\rightarrow x}$

The term ${\displaystyle g(g(g(x)))}$ has overlapping redexes, which can be either applied to the innermost occurrence or to the outermost occurrence of the ${\displaystyle g(g(x))}$ term.

References

1. Marc Bezem; Jan Willem Klop; Roel de Vrijer (2003). Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science. Cambridge, UK: Cambridge University Press. p. 48. ISBN 0-521-39115-6.

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