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.

翻译：自20世纪70年代McNaughton、Papert和Schützenberger的开创性工作以来，已知正则语言可由一阶逻辑定义当且仅当其句法幺半群是非周期的。这一基本逻辑片段的代数刻画已被Droste和Gastin推广至量化情形，涉及多项式歧义加权自动机与加权一阶逻辑的受限片段。在量化框架下，完整加权一阶逻辑（未使用Droste与Gastin所施加的量词交替限制）比加权自动机表达能力更强，为此引入了配备双向导航、卵石或嵌套能力的自动机扩展。本文在自动机为多项式歧义的条件下，刻画了恰好能够识别完整加权一阶逻辑的这些扩展加权自动机的片段。