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On the decidability of some problems about rational subsets of free partially commutative monoids
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Gibbons, Alan (Alan M.) and Rytter, Wojciech (1986) On the decidability of some problems about rational subsets of free partially commutative monoids. Coventry, UK: University of Warwick. Department of Computer Science. (Department of Computer Science Research Report). (Unpublished)
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Abstract
Let I=A+B be a partially commutative alphabet such that two letters commute if one of them belongs to A and the other one belongs to B. Let M=A* 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 ∩ Y= O?
Q2: X ⊆ Y?
DI X = Y?
Q4: X = M?
L25: M — X finite?
X is recognizable?
It was proved by Choffrut (see [2]) 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 (see [2], p 79) 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 01 is decidable, and problems Q2-06 are undecidable. Our paper is an application of results concerning reversal-bounded nondeterministic multicounter machines and nondeterministic general sequential machines.
| Item Type: | Report | ||||
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| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software | ||||
| Divisions: | Faculty of Science > Computer Science | ||||
| Library of Congress Subject Headings (LCSH): | Machine theory, Monoids | ||||
| Series Name: | Department of Computer Science Research Report | ||||
| Publisher: | University of Warwick. Department of Computer Science | ||||
| Place of Publication: | Coventry, UK | ||||
| Official Date: | August 1986 | ||||
| Dates: |
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| Number: | Number 79 | ||||
| Number of Pages: | 8 | ||||
| DOI: | CS-RR-079 | ||||
| Institution: | University of Warwick | ||||
| Theses Department: | Department of Computer Science | ||||
| Status: | Not Peer Reviewed | ||||
| Publication Status: | Unpublished | ||||
| Publisher Statement: | A.M. Gibbons and W. Rytter, “On the Decidability of some Problems about Rational Subsets of Free Partially Commutative Monoids”, <i>Theoretical Computer Science</i> <b>48</b>, pp. 329-337 (1986) | ||||
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