## Codes and equations on trees.(English)Zbl 0974.68095

Summary: The objective of this paper is to study, by new formal methods, the notion of tree code introduced by M. Nivat in [Nivat, Maurice (ed.) et al., Tree automata and languages. Amsterdam etc.: North-Holland. Stud. Comput. Sci. Artif.Intell. 10, 1-19 (1992; Zbl 0798.68083)]. In particular, we consider the notion of stability for sets of trees closed under concatenation. This allows us to give a characterization of tree codes which is very close to the algebraic characterization of word codes in terms of free monoids. We further define the stable hull of a set of trees and derive a defect theorem for trees, which generalizes the analogous result for words. As a consequence, we obtain some properties of tree codes having two elements. Moreover, we propose a new algorithm to test whether a finite set of trees is a tree code. The running time of the algorithm is polynomial in the size of the input. We also introduce the notion of tree equation as complementary view to tree codes. The main problem emerging in this approach is to decide the satisfiability of tree equations: it is a special case of second-order unification, and it remains still open.

### MSC:

 68Q45 Formal languages and automata 68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science)

tree code

Zbl 0798.68083
Full Text:

### References:

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