Asymptotic Monadic Second-Order Logic [chapter]

Achim Blumensath, Olivier Carton, Thomas Colcombet
<span title="">2014</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this paper we introduce so-called asymptotic logics, logics that are meant to reason about weights of elements in a model in a way inspired by topology. Our main subject of study is Asymptotic Monadic Second-Order Logic over infinite words. This is a logic talking about ωwords labelled by integers. It contains full monadic second-order logic and can express asymptotic properties of integers labellings. We also introduce several variants of this logic and investigate their relationship to the
more &raquo; ... logic MSO+U. In particular, we compare their expressive powers by studying the topological complexity of the different models. Finally, we introduce a certain kind of tiling problems that is equivalent to the satisfiability problem of the weak fragment of asymptotic monadic second-order logic, i.e., the restriction with quantification over finite sets only. it induces a topology. However, there is no such assumption in general (and d may even be non-binary). Nevertheless, we can consider the topology over the non-negative reals in which the open sets are the neighbourhoods of 0 (as well as ∅ of course). Then the quantifiers ∀ε, ∃δ,. . . can be replaced by quantifiers ranging over open sets, and tests of the form d(−) < ε by membership tests of d(−) in an open set. Furthermore, these tests respect the positivity assumption as defined by Flum and Ziegler. However, this relationship of our logic with those from the literature does not seem to help with solving the questions raised in the present paper. Monadic second-order logic and asymptotic monadic second-order logic. In this paper, we consider the asymptotic variant of monadic second-order logic, though certainly this notion of asymptoticity can be combined with other formalisms. Let us recall that monadic second-order logic is the extension of first-order logic by set quantifiers. There is a long history of works dealing with the decidability of monadic second-order logic over some classes of structures, the prominent examples being the results over ω by Büchi [7] and over the infinite binary tree by Rabin [15]. These results can be regarded as foundations for a theory of 'regular languages' of infinite words and trees. We are interested in knowing whether this logic can be 'made asymptotic' while keeping these strong decidability properties. We have good hopes that -at least some of -these results can be generalised to more general ones, in which monadic logic is extended with asymptotic capabilities. Before continuing, let us formalise what is 'asymptotic monadic second-order logic' (AMSO for short). The first aspect is that weight maps range over the elements of the structure, and not tuples. This is a design choice, our goal being to concentrate our attention on the simplest situation. The second aspect is cosmetic: instead of considering quantities ranging over R + , we consider quantities ranging over N. Essentially, this amounts for weights to exchange d(−) with 1/d(−) and, for quantifiers ∃ε, ∀δ ranging over R + , to exchange them for ∃r, ∀s ranging over N. As a consequence, existentially quantified numbers are used as upper bounds, while universally quantified ones are used as lower bounds. Hence, the syntax of 'asymptotic monadic second-order logic' is the one of MSO, extended by number quantifiers ∃r, ∀s ranging over N and by predicates of the form d(x) < r and d(x) ≥ s, where x is a first-order variable, under the assumption that there is an even number of negations between the quantifier and the use. Let us give some examples. The structure here is ω and f : ω → N is a weight map. The convention is that variables x, y, z range over elements of ω, upper case variables X, Y, Z over subsets of ω, and r, s over N.
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