Boundedness in languages of infinite words

We define a new class of languages of $\omega$-words, strictly extending $\omega$-regular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of $\omega$-regular expressions where two new variants of the Kleene star $L^*$ are added: $L^B$ and $L^S$. These new exponents are used to say that parts of the input word have bounded size, and that parts of the input can have arbitrarily large sizes, respectively. For instance, the expression $(a^Bb)^\omega$ represents the language of infinite words over the letters $a,b$ where there is a common bound on the number of consecutive letters $a$. The expression $(a^Sb)^\omega$ represents a similar language, but this time the distance between consecutive $b$'s is required to tend toward the infinite. We develop a theory for these languages, with a focus on decidability and closure. We define an equivalent automaton model, extending B\"uchi automata. The main technical result is a complementation lemma that works for languages where only one type of exponent---either $L^B$ or $L^S$---is used. We use the closure and decidability results to obtain partial decidability results for the logic MSOLB, a logic obtained by extending monadic second-order logic with new quantifiers that speak about the size of sets

Transformers Learn Shortcuts to Automata

Algorithmic reasoning requires capabilities which are most naturally understood through recurrent models of computation, like the Turing machine. However, Transformer models, while lacking recurrence, are able to perform such reasoning using far fewer layers than the number of reasoning steps. This raises the question: what solutions are learned by these shallow and non-recurrent models? We find that a low-depth Transformer can represent the computations of any finite-state automaton (thus, any bounded-memory algorithm), by hierarchically reparameterizing its recurrent dynamics. Our theoretical results characterize shortcut solutions, whereby a Transformer with $o(T)$ layers can exactly replicate the computation of an automaton on an input sequence of length $T$. We find that polynomial-sized $O(\log T)$-depth solutions always exist; furthermore, $O(1)$-depth simulators are surprisingly common, and can be understood using tools from Krohn-Rhodes theory and circuit complexity. Empirically, we perform synthetic experiments by training Transformers to simulate a wide variety of automata, and show that shortcut solutions can be learned via standard training. We further investigate the brittleness of these solutions and propose potential mitigations

Temporal Logic with Cyclic Counting and the Degree of Aperiodicity of Finite Automata

We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QA_M of finite automata whose degree of aperiodicity belongs to the division ideal generated by M is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings. These closure conditions define q-varieties of finite automata. We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages. We also characterize QA_M as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata. We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set M of moduli. It follows that when M is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in M

Temporal logic with cyclic counting and the degree of aperiodicity of finite automata

We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QAM of finite automata whose degree of aperiodicity belongs to the division ideal generated by M is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings. These closure conditions define q-varieties of finite automata. We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages. We also characterize QA M as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata. We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set M of moduli. It follows that when M is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in M

Beyond Zeno-behaviour

When modelling and analysing hybrid systems using techniques from computing science we may encounter problems with so-called Zeno-behaviour. This is the phenomenon that an innite number of events accumulates before a nite time (Zeno-time). When this happens the standard techniques from computing science fail to dis-tinguish between events that happen after that sequence of events. Many of those techniques have a semantics based on labelled transi-tion systems. In this article, we concentrate on those transition systems and try to nd a solution for the Zeno-problem. We rst introduce transi-tions over innite sequences, since an innite number of events needs to be described. Then we (re-)dene a notion of convergence over sequences in a metric space. Considering a transition system with a metric state space and transitions labelled by sequences we can dene a notion of prex- and accumulation-closedness. Finally within prex-and accumulation-closed transition systems, bisimilarity turns out to distinguish between various kinds of transnite behaviour. The bounc-ing ball, an example from hybrid system theory, is used to illustrate the relevance of these new denitions. 1