## Automata and zigzag codes. (Automates et codes zigzag.)(French)Zbl 0735.68050

Let $$X$$ be an alphabet. $$X^*$$ is the free monoid generated by $$X$$ with 1 as identity element. Let $$L$$ be a language over $$X$$. On $$X^*\times X^*$$ one defines re-writing steps $$(u,v)\to(u',v')$$ as follows. This step is possible if $$u=u'x$$ and $$v'=xv$$ or if $$v=xv'$$ and $$u'=ux$$. For a word $$w$$, a sequence of re-writing steps without a loop that leads from $$(1,w)$$ to $$(w,1)$$ is called a zigzag factorization of $$w$$ with respect to $$L$$. A language $$L$$ is called a zigzag code if every word in $$X^*$$ has at most one zigzag factorization. Every zigzag code is indeed a code.
These definitions are motivated by Isbell’s theory of dominions in semigroups and the “zigzag theorem” [J. R. Isbell, Epimorphisms and dominions, Proc. Conf. Categorical Algebra, La Jolla 1965, 232–246 (1966; Zbl 0194.01601); see also J. M. Howie, An introduction to semigroup theory. London etc.: Academic Press (1976; Zbl 0355.20056)].
The zigzag re-writing steps correspond to elementary moves of 2-way automata (automates boustrophédons). This connection is explored in detail and used to prove that the property of being a zigzag code is decidable for finite $$L$$. For infinite rational $$L$$ this decidability problem is open in general.
Reviewer: H. Jürgensen

### MSC:

 68Q45 Formal languages and automata

### Keywords:

zigzag factorization; zigzag code

### Citations:

Zbl 0355.20056; Zbl 0194.01601
Full Text:

### References:

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