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**On the decidability of some problems about rational subsets of free partially commutative monoids.**
*(English)*
Zbl 0638.68084

Let \(I=A\cup B\) be a partially commutative alphabet such that two letters commute iff one of them belongs to A and the other one belongs to B. Let \(M=A^*\times B^*\) denote the free partially commutative monoid generated by I. We consider the following six problems for rational (given by regular expressions) subsets \(X,Y\) of \(M\):
\[
(Q1): X\cap Y=\emptyset? \quad (Q2): X\subseteq Y? \quad (Q3): X=Y? \quad (Q4): X=M? \quad (Q5): M-X \text{ finite?} \quad (Q6): X\text{ is recognizable?}
\]
It is known [see J. Berstel, Transductions and context-free languages (1979; Zbl 0424.68049)] that all these problems are undecidable if Card\(A>1\) and Card\(B>1\), and they are decidable if Card\(A=\) Card\(B=1\) (Card\(U\) denotes the cardinality of U).

It was conjectured by C. Choffrut that these problems are decidable in the remaining cases, where Card\(A=1\) and Card\(B>1\). In this paper we show that if Card\(A=1\) and Card\(B>1\), then the problem (Q1) is decidable,and problems (Q2)-(Q6) are undecidable. Our paper is an application of results concerning reversal-bounded, nondeterministic, multicounter machines and nondeterministic, general sequential machines.

It was conjectured by C. Choffrut that these problems are decidable in the remaining cases, where Card\(A=1\) and Card\(B>1\). In this paper we show that if Card\(A=1\) and Card\(B>1\), then the problem (Q1) is decidable,and problems (Q2)-(Q6) are undecidable. Our paper is an application of results concerning reversal-bounded, nondeterministic, multicounter machines and nondeterministic, general sequential machines.

### MSC:

68Q45 | Formal languages and automata |

20M35 | Semigroups in automata theory, linguistics, etc. |

20M05 | Free semigroups, generators and relations, word problems |

03D05 | Automata and formal grammars in connection with logical questions |

### References:

[1] | Berstel, J., Transductions and Context-free Languages (1979), Teubner: Teubner Stuttgart · Zbl 0424.68040 |

[2] | Bertoni, A.; Mauri, G.; Sabadini, N., Equivalence and membership problem for regular trace languages, (ICALP ’82. ICALP ’82, Lecture Notes in Computer Science, 140 (1982), Springer: Springer Berlin), 61-71 · Zbl 0486.68079 |

[3] | Choffrut, C., Free partially commutative monoids, Tech. Rept. Laboratoire Informatique Théorique et Programmation 86.20 (March 1986) |

[4] | Chrobak, M.; Rytter, W., The unique decipherability problem with partially commutative alphabet, (Proc. Math. Found. of Comput. Science. Proc. Math. Found. of Comput. Science, Lecture Notes in Computer Science, 233 (1986), Springer: Springer Berlin), 256-263 · Zbl 0618.68063 |

[5] | Ibarra, O. H., Reversal-bounded multicounter machines and their decision problems, J. ACM, 25, 1, 106-133 (1978) · Zbl 0365.68059 |

[6] | Ibarra, O. H., The unsolvability of the equivalence problem for \(e\)-free ngsm’s with unary input (output) alphabet and applications, SIAM J. Comput., 7, 4, 524-532 (1978) · Zbl 0386.68054 |

[7] | Mazurkiewicz, A., Concurrent program schemes and their interpretations, (DAIMIPB 78 (1977), Aarhus University) |

[8] | Mazurkiewicz, A., Traces, histories, graphs: instances of processes monoid, (Lecture Notes in Computer Science, 176 (1984), Springer: Springer Berlin), 115-133 · Zbl 0577.68061 |

[9] | Rytter, W., Some properties of trace languages, Fund. Inform., VII, 1, 107-127 (1984) · Zbl 0546.68064 |

[10] | Szijarto, M., A classification and closure properties of languages for describing concurrent systems behaviours, Fund. Inform., IV, 3, 531-550 (1981) · Zbl 0486.68074 |

[11] | Tarlecki, A., Notes on the implementability of formal languages by concurrent systems, ICS PAS Repts. 481 (1982), Warsaw |

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