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**First-order and counting theories of \(\omega\)-automatic structures.**
*(English)*
Zbl 1141.03015

Let \(\mathcal A\) be a relational structure of finite signature with universe \(A\). This structure is said to be \(\omega\)- automatic if there is a triple \((\Gamma, L,h)\) where \(\Gamma\) is a finite alphabet, \(L\subseteq \Gamma^{\omega}\), and \(h\) is a surjection from \(L\) to \(A\) with the property that the “inverse image” under \(h\) of each \(n\)-ary relation of \(\mathcal A\) is a regular \(\omega\)-language, i.e., accepted by some \(n\)-dimensional Büchi automaton. If in addition \(h\) is injective, \(\mathcal A\) is said to be injectively \(\omega\)-automatic.

The authors state as their objective the extension of results for first-order logic with the addition of “counting” quantifiers from automatic structures to \(\omega\)-automatic structures. Initially they consider first-order logic extended by modulo-counting quantifiers and extended by exact counting quantifiers for infinite cardinalities. They show that any injectively \(\omega\)-automatic structure has a decidable theory in these logics. The proof uses Muller automata.

The authors then go on to study more powerful logics. The logic \(\mathcal L(\mathcal Q_u)\) has quantifiers of the form \(\mathcal Q_{\mathcal C}y: (\psi_1(y),\dots, \psi_n (y))\), where \(y\) is a first-order variable and \(\mathcal C\) is an \(n\)-ary relation on cardinals. Such a formula is true in \(\mathcal A\) if the \(n\)-tuple of the cardinalities of the sets defined in \(\mathcal A\) by the \(\psi_i (y)\) are in the relation \(\mathcal C\).

Let \(\mathcal L\) be a fragment of \(\mathcal L(\mathcal Q_u)\) that contains only countably many generalized quantifiers, and let \(\mathcal A\) be an injectively \(\omega\)-automatic structure whose Gaifman-graph has bounded degree. The authors prove that the \(\mathcal L\)-theory of \(\mathcal A\) can be decided by a Turing machine with oracle access to the relations \(\mathcal C\) appearing in the quantifiers of \(\mathcal A\). The authors show that both of the assumptions on \(\mathcal A\) are necessary for this result.

The authors state as their objective the extension of results for first-order logic with the addition of “counting” quantifiers from automatic structures to \(\omega\)-automatic structures. Initially they consider first-order logic extended by modulo-counting quantifiers and extended by exact counting quantifiers for infinite cardinalities. They show that any injectively \(\omega\)-automatic structure has a decidable theory in these logics. The proof uses Muller automata.

The authors then go on to study more powerful logics. The logic \(\mathcal L(\mathcal Q_u)\) has quantifiers of the form \(\mathcal Q_{\mathcal C}y: (\psi_1(y),\dots, \psi_n (y))\), where \(y\) is a first-order variable and \(\mathcal C\) is an \(n\)-ary relation on cardinals. Such a formula is true in \(\mathcal A\) if the \(n\)-tuple of the cardinalities of the sets defined in \(\mathcal A\) by the \(\psi_i (y)\) are in the relation \(\mathcal C\).

Let \(\mathcal L\) be a fragment of \(\mathcal L(\mathcal Q_u)\) that contains only countably many generalized quantifiers, and let \(\mathcal A\) be an injectively \(\omega\)-automatic structure whose Gaifman-graph has bounded degree. The authors prove that the \(\mathcal L\)-theory of \(\mathcal A\) can be decided by a Turing machine with oracle access to the relations \(\mathcal C\) appearing in the quantifiers of \(\mathcal A\). The authors show that both of the assumptions on \(\mathcal A\) are necessary for this result.

Reviewer: J. M. Plotkin (East Lansing)

### MSC:

03C80 | Logic with extra quantifiers and operators |

03B25 | Decidability of theories and sets of sentences |

03D05 | Automata and formal grammars in connection with logical questions |

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\textit{D. Kuske} and \textit{M. Lohrey}, J. Symb. Log. 73, No. 1, 129--150 (2008; Zbl 1141.03015)

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