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

DANIEL KROB
1994 International journal of algebra and computation  
problem for Z-rational series over an alphabet with at least two letters is undecidable. To prove this last result, we show in fact that the decidability of the equality problem for Z is equivalent to the decidability of the local inequality problem for Z. Using this equivalence and a reduction to a 10th Hilbert problem, we prove then the undecidability of the equality problem for Z-rational series over an alphabet with at least two letters. Hence this allows us to obtain the undecidability of
more » ... he same problem for M-rational series. Moreover our methods give us also immediately other decidability and undecidability results for connected problems. 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 = (N f?1g;max;+) is undecidable. In the same way, we obtained also as an immediate byproduct of our results, a new proof of a di cult undecidability result of Ibarra (cf 9]). Let us nally add that a rst version of our result appeared in 11]. Unfortunately there was a large technical gap in the corresponding proof. This explains the purpose of this new paper where we give a correct proof, based however on the same idea than in 11].
doi:10.1142/s0218196794000063 fatcat:2kdnex362vav3gsmuobp72cesy