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On infinitary finite length codes. (English) Zbl 0628.68056
The paper investigates the relations between the following four properties of codes C over a finite alphabet X: (i) For all words $$u,v\in X^*$$ and for every $$w\in C^*:$$ $$wu,uv,vu\in C^*$$ implies $$u,v\in C^*$$. (ii) Every one-sided infinite product of words of C is unambiguous. (iii) C has finite decoding delay. (iv) C has bounded decoding delay.
As is well-known, these four properties are equivalent for finite codes. The paper shows that in general only (iv)$$\to (iii)\to (ii)\to (i)$$ holds, whereas none of the reverse implications is true.
To this end several characterizations of the classes of codes defined by the above conditions are given. Counterexamples to the implications (iii)$$\to (iv)$$ and (ii)$$\to (iii)$$ are derived by topological arguments utilizing properties of $$C^{\omega}$$ in the space $$X^{\omega}$$ of all $$\omega$$-words over X. The last section deals with infinite regular codes for which the reverse implications (i)$$\to (ii)$$ and (iii)$$\to (iv)$$ are proved to be true, whereas (ii)$$\to (iii)$$ does not hold.

##### MSC:
 68Q45 Formal languages and automata 94A45 Prefix, length-variable, comma-free codes 20M35 Semigroups in automata theory, linguistics, etc.
##### Keywords:
infinite products; $$\omega$$-languages; decoding delay
Full Text:
##### References:
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