In this paper, we solve the long standing openproblem of expressions capturing first order definable functions . FO transductions, aperiodic deterministic two-way transducers, as well asaperiodic streaming string transducers are all equivalent models for firstorder definable function . SDRTEs are constructed from simple functions using thecombinators unambiguous sum (deterministic choice) and Hadamard product, and unambiguous versions of the Cauchy product and the k-chained Kleene-star, where the star is restricted as mentioned . An SDETE is aregular expression where Kleene stars are used in a restricted way: they canappear only on a periodic languages which are prefix codes of boundedsynchronization delay . We alsolook at various formalisms which use the functioncomposition, allowing to trade the k.-chained star for a 1-star . Finally, we al Sol Sol Solvook at different formalisms equivalent to SDEs which use functioncompositions which use

Author(s) : Luc Dartois, Paul Gastin, Shankara Narayanan Krishna

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Keywords : star - restricted - aperiodic - kleene - chained -

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