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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 (Unpublished)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
Subjects: Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Divisions: Faculty of Science, Engineering and Medicine > 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:
DateEvent
August 1986Completion
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Status: Not Peer Reviewed
Publication Status: Unpublished
Reuse Statement (publisher, data, author rights): A.M.&nbsp;Gibbons and W.&nbsp;Rytter, &ldquo;On the Decidability of some Problems about Rational Subsets of Free Partially Commutative Monoids&rdquo;, <i>Theoretical Computer Science</i> <b>48</b>, pp.&nbsp;329-337 (1986)
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