Deleting String Rewriting Systems Preserve Regularity
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A string rewriting system R is called deleting if there exists a partial ordering on its alphabet such that each letter in the right hand side of a rule is less than some letter in the corresponding left hand side. We show that the rewrite relation R* induced by R can be represented as the composition of a finite substitution (into an extended alphabet), a rewrite relation of an inverse context-free system (over the extended alphabet), and a restriction (to the original alphabet). Here, a system is called inverse context-free if |r| ≤ 1 for each rule ℓ → r. The decomposition result directly implies that deleting systems preserve regularity, and that inverse deleting systems preserve context-freeness. The latter result was already obtained by Hibbard [Hib74].
KeywordsRegular Language Canonical System Tree Automaton Pivot Rule Bubble Sort
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