  FO transductions, aperiodic deterministic two-way transducers, as well as
aperiodic streaming string transducers are all equivalent models for first
order definable functions. In this paper, we solve the long standing open
problem of expressions capturing first order definable functions, thereby
generalizing the seminal SF=AP (star free expressions = aperiodic languages)
result of Sch\"utzenberger. Our result also generalizes a lesser known
characterization by Sch\"utzenberger of aperiodic languages by SD-regular
expressions (SD=AP). We show that every first order definable function over
finite words captured by an aperiodic deterministic two-way transducer can be
described with an SD-regular transducer expression (SDRTE). An SDRTE is a
regular expression where Kleene stars are used in a restricted way: they can
appear only on aperiodic languages which are prefix codes of bounded
synchronization delay. SDRTEs are constructed from simple functions using the
combinators unambiguous sum (deterministic choice), Hadamard product, and
unambiguous versions of the Cauchy product and the k-chained Kleene-star, where
the star is restricted as mentioned. In order to construct an SDRTE associated
with an aperiodic deterministic two-way transducer, (i) we concretize
Sch\"utzenberger's SD=AP result, by proving that aperiodic languages are
captured by SD-regular expressions which are unambiguous and stabilising; (ii)
by structural induction on the unambiguous, stabilising SD-regular expressions
describing the domain of the transducer, we construct SDRTEs. Finally, we also
look at various formalisms equivalent to SDRTEs which use the function
composition, allowing to trade the k-chained star for a 1-star.
