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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.
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