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### The equality problem for rational series with multiplicities in the tropical semiring is undecidable [chapter]

Daniel Krob
1992 Lecture Notes in Computer Science
The equality problem for rational series with multiplicities in the tropical semiring is undecidable.  ...

### THE EQUALITY PROBLEM FOR RATIONAL SERIES WITH MULTIPLICITIES IN THE TROPICAL SEMIRING IS UNDECIDABLE

DANIEL KROB
1994 International journal of algebra and computation
In particular, we solve also another open question (cf 16]) by showing that the equality problem for rational series over an alphabet with at least two letters and with multiplicities in the semiring N  ...  problem for Z-rational series over an alphabet with at least two letters is undecidable.  ...  Weber : section 4.3 is in fact a consequence of several discussions we had together.  ...

### What's Decidable about Weighted Automata? [chapter]

Shaull Almagor, Udi Boker, Orna Kupferman
2011 Lecture Notes in Computer Science
Our results thus draw a sharper picture about the decidability of decision problems for weighted automata, in both the front of containment vs. universality and the front of the ∪ {∞} vs. the ¡ ∪ {∞} domains  ...  ¢ ≥0 ∪ {∞}, is PSPACE-complete.  ...  In [11] , Krob proved that the universality problem for rational series is undecidable for the tropical semiring with domain ¤ ∪ {∞}, and that this implies undecidability of the containment problem for  ...

### Some consequences of a Fatou property of the tropical semiring

Daniel Krob
1994 Journal of Pure and Applied Algebra
This property allows us to give partial decidability results for the equality problem for rational series with multiplicities in the tropical semiring.  ...  We show that the equatorial semiring 2min = (Z u { + co}, mitt, + ) is a Fatou extension of the tropical semiring M = (N u { + cm}, min, + ).  ...  An important problem in the tropical semiring theory was to see if it is possible to decide whether two M-rational series are equal or not.  ...

### Sequential?

Sylvain Lombardy, Jacques Sakarovitch
2006 Theoretical Computer Science
We review results on sequentiality in the cases of series of finite image, of series with multiplicity in fields, and of series with multiplicity in idempotent semirings.  ...  This paper is a survey where we try to organise the known answers to the question whether a given finite automaton with multiplicity in a semiring K is equivalent to a sequential, or input deterministic  ...  The comments of two careful referees were of great help in the writing of the second version.  ...

### Page 477 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews
Krob, Daniel (F-PARIS7-BP; Paris) The equality problem for rational series with multiplicities in the tropical semiring is undecidable. Internat. J. Algebra Comput. 4 (1994), no. 3, 405-425.  ...  Thus the author solves an open problem by showing that the equality problem for rational series over the semiring (NU —oo, max, +) is undecidable.  ...

### Page 7467 of Mathematical Reviews Vol. , Issue 94m [page]

1994 Mathematical Reviews
Do Long Van (VN-HMI; Hanoi) 94m:68101 68Q45 03D05 03D35 Krob, Daniel (F-ROUENS-LI; Mont-Saint-Aignan) The equality problem for rational series with multiplicities in the tropical semiring is undecidable  ...  “One of the main open questions in the theory of the tropical semiring was whether it is possible to decide whether two given #- rational series are equal.  ...

### Page 6952 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews
One of the authors has proved in an ICALP’92 paper that the equality problem for rational series with multiplicities in the tropical semiring is undecidable in the case of an alphabet with at least two  ...  of identities for one-letter rational expressions with multiplicities in the tropical semiring.  ...

### Series which are both max-plus and min-plus rational are unambiguous

Sylvain Lombardy, Jean Mairesse
2005 RAIRO - Theoretical Informatics and Applications
The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture.  ...  We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand.  ...  The equality problem is already undecidable in max and for two letters alphabet [9] , but it is decidable for finitely ambiguous automata over Ê max [6, 17] .  ...

### Page 310 of Computational Linguistics Vol. 23, Issue 2 [page]

1997 Computational Linguistics
The equality problem for rational series with multiplicities in the tropical semiring is undecidable. Journal of Algebra and Computation, 4. Kuich, Werner and Arto Salomaa. 1986.  ...  In Proceedings of the 34th Annual Meeting, Santa Cruz, California. Association for Computational Linguistics. Nerode, Anil. 1958. Linear automaton transformations. In Proceedings of AMS, volume 9.  ...

### Axiomatizing Tropical Semirings [chapter]

Luca Aceto, Zoltán Ésik, Anna Ingólfsdóttir
2001 Lecture Notes in Computer Science
It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that  ...  Two prime examples of such structures are the (max, +) semiring and the tropical semiring.  ...  We would like to thank the anonymous referees whose suggestions helped us improve the paper.  ...

### A complete system of identities for one-letter rational expressions with multiplicities in the tropical semiring

Agnès Bonnier-Rigny, Daniel Krob
1994 Theoretical Computer Science
identities for the equational theory of the algebra of one-letter rational expressions with multiplicities in the tropical semiring. . 0304-3975/94/%07.00 0 1994-Elsevier Science B.V.  ...  Krob, A complete system of identities for one-letter rational expressions with multiplicities in the tropical semiring, Theoretical Computer Science 134 (1994) 27-50 We construct a complete system of rational  ...  Acknowledgments Special thanks are due to the anonymous referee for his careful reading of the paper and for several valuable comments.  ...

### Max-plus algebra in the history of discrete event systems

J. Komenda, S. Lahaye, J.-L. Boimond, T. van den Boom
2018 Annual Reviews in Control
This paper is a survey of the history of max-plus algebra and its role in the field of discrete event systems during the last three decades.  ...  It is based on the perspective of the authors but it covers a large variety of topics, where max-plus algebra plays a key role.  ...  In particular, it has been shown in Krob (1992) that equalities and inequalities of rational max-plus formal power series are undecidable.  ...

### Axiomatizing rational power series over natural numbers

S.L. Bloom, Z. Ésik
2009 Information and Computation
We show that the semirings N rat * of rational power series with coefficients in the semiring N of natural numbers are the free partial iteration semirings.  ...  Moreover, we characterize the semirings N rat ∞ * as the free semirings in the variety of iteration semirings defined by three additional simple identities, where N ∞ is the completion of N obtained by  ...  Acknowledgment The authors would like thank two anonymous referees for their insightful comments and suggestions which have improved this paper.  ...

### Rational and Recognisable Power Series [chapter]

Jacques Sakarovitch
2009 Monographs in Theoretical Computer Science
It is undecidable whether the supports of two Z-rational series on A * are equal.  ...  The equivalence of recognisable series over A * with coefficients in the semiring M = N ∞ , min, + is undecidable.  ...  The original work for reduction of representations is again from Schützenberger [44] . The presentation here follows roughly [5] but as in a background and owes much to my discussions with S.  ...
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