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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 (or ''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.
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| For example, consider the term rewriting system defined by the reduction rules
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| : <math>\rho_1\ :\ f(g(x), y) \rightarrow y</math>
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| : <math>\rho_2\ :\ g(x) \rightarrow f(x, x)</math> | |
| The term ''f''(''g''(''x''), ''y'') can be reduced via ρ<sub>1</sub> to yield ''y'', but it can also be reduced via ρ<sub>2</sub> 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 ''[[Critical pair (logic)|critical pair]]'' -- the critical pair arising out of this term rewriting system is (''f''(''f''(''x'', ''x''), ''y''), ''y'').
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| Overlap may occur with fewer than two reduction rules. Consider the term rewriting system defined by the reduction rule
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| : <math> \rho_1\ :\ g(g(x)) \rightarrow x</math> | |
| 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.
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| ==References==
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| * {{cite book | title=Term Rewriting Systems | author="Terese" | coauthors=Marc Bezem, J. W. Klop, Roel de Vrijer | series=Cambridge Tracts in Theoretical Computer Science | publisher=[[Cambridge University Press]] | year=2003 | isbn=0-521-39115-6 | page=48 }}
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| {{math-stub}}
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| [[Category:Rewriting systems]]
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