Since the 1970s with the work of McNaughton, Papert and Schützenberger, 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.

        @inproceedings{manuel2021algebraic,
        title={An Automata Theoretic Characterization of Weighted First-Order Logic},
        author={Nevatia, Dhruv and Benjamin, Monmege},
        booktitle={2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)},
        pages={1--13},
        year={2021},
        organization={IEEE}
        }
    

We give an algebraic characterisation of first-order logic with the neighbour relation, on finite words. For this, we consider languages of finite words over alphabets with an involution on them. The natural algebras for such languages are involution semigroups. To characterise the logic, we define a special kind of semidirect product of involution semigroups, called the locally hermitian product. The characterisation theorem for FO with neighbour states that a language is definable in the logic if and only if it is recognised by a locally hermitian product of an aperiodic commutative involution semigroup, and a locally trivial involution semigroup. We then define the notion of involution varieties of languages, namely classes of languages closed under Boolean operations, quotients, involution, and inverse images of involutory morphisms. An Eilenberg-type correspondence is established between involution varieties of languages and pseudovarieties of involution semigroups.

        @inproceedings{manuel2021algebraic,
        title={An algebraic characterisation of first-order logic with neighbour},
        author={Manuel, Amaldev and Nevatia, Dhruv},
        booktitle={2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)},
        pages={1--13},
        year={2021},
        organization={IEEE}
        }