## On partitions separating words.(English)Zbl 1239.68039

The paper continues and improves a result from [J. Brzozowski, E. Grant and J. Shallit, “Closures in formal languages and Kuratowski’s theorem”, Lect. Notes Comput. Sci. 5583, 125–144 (2009; Zbl 1247.68129)] (this being the only recent reference; all the others are before 1998), namely, it proves the following theorem for a finite alphabet $$A$$: Let $$m$$ be a positive integer and let $$u_1,u_2,\dots,u_m\in A^{+}$$. There exists a closed partition of $$A^{+}$$ separating these words if and only if for all distinct $$i,j\in\{1,\dots,m\}$$, the words $$u_i$$ and $$u_j$$ do not commute.
Although the proof addresses only the binary case $$A=\{0,1\}$$, the result is very interesting. Moreover, the authors show that the languages $$L_1,\dots,L_m$$ defined here as the partition of $$A^+$$ (with $$u_i\in L_i$$) are regular while the languages from the original theorem [loc. cit., Theorem 14] are context-sensitive. I appreciate the construction of finite automata which recognize these languages as a remarkable result.
Using the notion of a Parikh vector in order to treat these problems from a geometrical point of view, the paper proposes an interesting conjecture: Let $$m\in\mathbb{N}_+$$ and $$u_1,\dots,u_m\in A^+$$ be words such that the Parikh vectors of any two distinct words are linearly independent. Then there exists a closed partition of $$A^+$$ into $$m$$ commutative context-free languages separating $$u_1,\dots,u_m$$.
This assertion is proved only for the case $$m=2$$.
The paper is well constructed, with a surprising interference between formal languages theory and number theory. Although some assertions create problems for the reader (for example, on page 1309, a sequence $$z=123$$ can be interpreted and evaluated in four different modes: $$b((1)(2)(3))=11,\; b((1)(23))=25,\; b((12)(3))=27,\; b((123))=123$$), in general, all information is described in detail, and adequate examples are provided to clarify various situations. I recommend this paper in particular because of the possibility of interesting implications for future research.

### MSC:

 68Q45 Formal languages and automata 68R15 Combinatorics on words 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)

Zbl 1247.68129
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### References:

  DOI: 10.1016/0022-0000(87)90018-3 · Zbl 0627.68063  Brzozowski J., LNCS 5583 pp 125–  Boudet A., LNCS 1059 pp 30–  Ginsburg S., The Mathematical Theory of Context Free Languages (1966) · Zbl 0184.28401  Hammer P. C., Nieuw Archief v. Wiskunde 7 pp 74–  DOI: 10.1006/jcss.1997.1553 · Zbl 0914.68119  Kuratowski C., Fund. Math. 3 pp 182–  Lothaire M., Combinatorics on Words (1983) · Zbl 0514.20045  DOI: 10.1016/0012-365X(84)90055-4 · Zbl 0542.54001
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