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

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

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]

2011
*
Lecture Notes in Computer Science
*

Our results thus draw a sharper picture about

doi:10.1007/978-3-642-24372-1_37
fatcat:wul25bahdrauxgyqhjqozviiwu
*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

1994
*
Journal of Pure and Applied Algebra
*

This property allows us to give partial decidability results

doi:10.1016/0022-4049(94)90090-6
fatcat:bi2qwrlwbjfodbj5rkfxyfitxy
*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?

2006
*
Theoretical Computer Science
*

We review results on sequentiality

doi:10.1016/j.tcs.2006.01.028
fatcat:gft4vcqg65cbvmbwhswma2kr2u
*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

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]

2001
*
Lecture Notes in Computer Science
*

It

doi:10.1007/3-540-45315-6_3
fatcat:m66w4ara3ndh7pa54cl4klj63e
*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

1994
*
Theoretical Computer Science
*

identities

doi:10.1016/0304-3975(94)90276-3
fatcat:p4qnaj7jlfdudjlugnqi2ruiz4
*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

2018
*
Annual Reviews in Control
*

This paper

doi:10.1016/j.arcontrol.2018.04.004
fatcat:37jjipig2jatvpd4ec47afcxya
*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

2009
*
Information and Computation
*

We show that

doi:10.1016/j.ic.2009.02.003
fatcat:tvqp6ttherh4zep2fotkpsuzou
*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]

2009
*
Monographs in Theoretical Computer Science
*

It

doi:10.1007/978-3-642-01492-5_4
fatcat:dki7cvharbcv5hk2xgpr3hhqmq
*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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