The Equality Problem for Rational Series with Multiplicities in the Tropical Semiring is Undecidable

1 0 Introduction The tropical semiring is the semiring denoted by M which has support Nf+1g and operations ab = minfa; bg and ab = a+b. It was rst introduced in the context of cost minimization in Operations Research. However it appeared that M plays in fact a central role in several decision problems concerning rational languages (see 15] for a survey of the tropical semiring theory and of its applications). For instance, I. Simon showed that the nite power property for recognizable languages can be reduced to the limitedness problem for the tropical semiring (cf 15]). In the same way, series with multiplicities in the tropical semiring can also be used in order to analyse the non-deterministic behaviour of nite usual automata (cf 17]). One of the main open questions in the theory of the tropical semiring was to see if it is possible to decide whether two given M-rational series are equal or not (cf 15, 16]). We ooer here an answer to this problem since we show in this paper that the equality problem for M-rational series over an alphabet with at least two letters is undecidable. One should notice that most people thought that a decision procedure existed (cf 15] for instance) and our result is indeed based on a rather surprising encoding of a 10th Hilbert problem. It is also interesting to precise the structure of the proof of our undecidability result. Indeed it appears that we use as a main tool the tropical \ring" Z = (Z f+1g;min;+) which is just the extension of M to arbitrary integers. The importance of Z comes from the equivalence with respect to decidability of the equality problems for M and Z. According to this result, we can reduce our problem to showing that the equality