Deleting string rewriting systems preserve regularity

A string rewriting system 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 induced by a deleting system 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 the length of the right-hand side of any rule does not exceed one. 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 (J. ACM 21(3) (1974) 446-453).

[1]  Dieter Hofbauer,et al.  Deleting String Rewriting Systems Preserve Regularity , 2003, Developments in Language Theory.

[2]  Dirk Siefkes,et al.  Finite Automata, Their Algebras and Grammars , 1990 .

[3]  J. Richard Büchi,et al.  Canonical systems which produce periodic sets , 2005, Mathematical systems theory.

[4]  Zohar Manna,et al.  Proving termination with multiset orderings , 1979, CACM.

[5]  Thomas N. Hibbard.  Context-Limited Grammars , 1974, JACM.

[6]  Seymour Ginsburg,et al.  Mappings which Preserve Context Sensitive Languages , 1966, Inf. Control..

[7]  Julius Richard Büchi,et al.  Finite automata, their algebras and grammars - towards a theory of formal expressions , 1989 .

[8]  Dominique Perrin,et al.  Finite Automata , 1958, Philosophy.

[9]  Bala Ravikumar.  Peg-Solitaire, String Rewriting Systems and Finite Automata , 1997, ISAAC.

[10]  Friedrich Otto,et al.  String-Rewriting Systems , 1993, Text and Monographs in Computer Science.

[11]  Roland Fraïssé.  Theory of relations , 1986 .

[12]  Ronald V. Book,et al.  Monadic Thue Systems , 1982, Theor. Comput. Sci..

[13]  Sophie Tison.  Tree Automata and Term Rewrite Systems , 2000, RTA.

[14]  Dieter Hofbauer,et al.  Match-Bounded String Rewriting Systems , 2003, Applicable Algebra in Engineering, Communication and Computing.

[15]  Jean Berstel,et al.  Transductions and context-free languages , 1979, Teubner Studienbücher : Informatik.

[16]  Friedrich Otto.  Some Undecidability Results Concerning the Property of Preserving Regularity , 1998, Theor. Comput. Sci..

[17]  Emil L. Post.  Formal Reductions of the General Combinatorial Decision Problem , 1943 .

[18]  J. R. Büchi.  Regular Canonical Systems , 1964 .