### Editors: Nicolas Ollinger and Heribert Vollmer; Article No. 19

Achim Blumensath, Thomas Colcombet, Paweł Parys
14 Leibniz International Proceedings in Informatics Schloss Dagstuhl-Leibniz-Zentrum für Informatik   unpublished
We prove that satisfiability over infinite words is decidable for a fragment of asymptotic monadic second-order logic. In this fragment we only allow formulae of the form ∃t∀s∃r ϕ(r, s, t), where ϕ does not use quantifiers over number variables, and variables r and s can be only used simultaneously , in subformulae of the form s < f (x) ≤ r. This paper continues a line of research trying to find logics where satisfiability is decidable over infinite words (and infinite trees). The most
more » ... n logic of this kind is monadic second-order logic (MSO) considered in the seminal work of Büchi [8]. Extending MSO by the ability of comparing some quantities quickly leads to undecidability. The idea behind the logic MSO+U and a more recently introduced logic called asymptotic monadic second-order logic (AMSO) is to extend MSO by the ability to express boundedness properties of some sequences of numbers. In MSO+U this is realized by an additional quantifier U stating that there are arbitrarily large finite sets satisfying the given formula. AMSO does not have a built in ability to refer to the size of sets. Instead, it describes weighted structures (in particular weighted infinite words), which are structures in which the elements are labelled by natural numbers, called their weights. More precisely, AMSO extends MSO by quantifiers over variables of a new kind, ranging over natural numbers. These variables can be compared with weights in the word, but only under a certain positivity requirement: existentially quantified numbers can only serve as upper bounds, while universally quantified numbers can only serve as lower bounds. The two logics MSO+U and AMSO happen to be inter-reducible as far as the decidability of satisfiability is concerned [1], and, unfortunately, this means that both are undecidable over infinite words [5]. Nevertheless, some natural fragments of these logics remain decidable.