  Since the 1970s with the work of McNaughton, Papert and Sch\"utzenberger, a
regular language is known to be definable in the first-order logic if and only
if its syntactic monoid is aperiodic. This algebraic characterisation of a
fundamental logical fragment has been extended in the quantitative case by
Droste and Gastin, dealing with polynomially ambiguous weighted automata and a
restricted fragment of weighted first-order logic. In the quantitative setting,
the full weighted first-order logic (without the restriction that Droste and
Gastin use, about the quantifier alternation) is more powerful than weighted
automata, and extensions of the automata with two-way navigation, and pebbles
or nested capabilities have been introduced to deal with it. In this work, we
characterise the fragment of these extended weighted automata that recognise
exactly the full weighted first-order logic, under the condition that automata
are polynomially ambiguous.
