Minimal DFAs for Testing Divisibility

We present and prove a theorem answering the question "how many states does a minimal deterministic finite automaton (DFA) that recognizes the set of base-b numbers divisible by k have?"Comment: LaTeX, 7 pages (corrected typo in new version

Separating Words from Every Start State with Horner Automata

We show that a well-known family of deterministic finite automata can be used to distinguish distinct binary strings of the same length from every start state. Further, we establish almost matching lower and upper bounds on the number of states of such automata necessary to achieve this type of separation. Our result improves the currently best known linear upper bound for arbitrary DFA.Comment: In Proceedings AFL 2023, arXiv:2309.0112

Modelling and Optimizing Imperfect Maintenance of Coatings on Steel Structures

Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. In this paper we consider the problem of finding the set of actions that minimizes the expected maintenance costs over a bounded horizon. To this end we model the size of the area affected by corrosion by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, maintenance may also change the parameters of the gamma deterioration process. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with a numerical example.non-stationary gamma process;condition-based maintenance;degradation modelling;imperfect maintenance;life-cycle management;renewal-type dynamic programming equation

Numeration Systems: a Link between Number Theory and Formal Language Theory

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions. We discuss the notion of numeration systems, recognizable sets of integers and automatic sequences. We briefly sketch some results about transcendence related to the representation of real numbers. We conclude with some applications to combinatorial game theory and verification of infinite-state systems and present a list of open problems.Comment: 21 pages, 3 figures, invited talk DLT'201

Testing Simon’s congruence

Piecewise testable languages are a subclass of the regular languages. There are many equivalent ways of defining them; Simon’s congruence ∼kis one of the most classical approaches. Two words are ∼k-equivalent if they have the same set of (scattered) subwords of length at most k. A language L is piecewise testable if there exists some k such that L is a union of ∼k-classes. For each equivalence class of ∼k, one can define a canonical representative in shortlex normal form, that is, the minimal word with respect to the lexicographic order among the shortest words in ∼k. We present an algorithm for computing the canonical representative of the ∼k-class of a given word w ∈ A∗of length n. The running time of our algorithm is in O(|A|n) even if k ≤ n is part of the input. This is surprising since the number of possible subwords grows exponentially in k. The case k > n is not interesting since then, the equivalence class of w is a singleton. If the alphabet is fixed, the running time of our algorithm is linear in the size of the input word. Moreover, for fixed alphabet, we show that the computation of shortlex normal forms for ∼kis possible in deterministic logarithmic space. One of the consequences of our algorithm is that one can check with the same complexity whether two words are ∼k-equivalent (with k being part of the input)