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We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or…
For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognize this language with a non-deterministic, alternating, or weak alternating parity automaton. These questions are known as,…
We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic and machine-independent…
Using a genealogically ordered infinite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to…
The class of Basic Feasible Functionals BFF is the second-order counterpart of the class of first-order functions computable in polynomial time. We present several implicit characterizations of BFF based on a typed programming language of…
We look at classes of languages associated to the fragment of first-order logic B{\Sigma}1 which disallows quantifier alternations. Each class is defined by choosing the set of predicates on positions that may be used. Two key such…
We consider the problem of computing the probability of regular languages of infinite trees with respect to the natural coin-flipping measure. We propose an algorithm which computes the probability of languages recognizable by \emph{game…
There is a growing concern about typically opaque decision-making with high-performance machine learning algorithms. Providing an explanation of the reasoning process in domain-specific terms can be crucial for adoption in risk-sensitive…
An index for a finite automaton is a powerful data structure that supports locating paths labeled with a query pattern, thus solving pattern matching on the underlying regular language. In this paper, we solve the long-standing problem of…
We show that the first-order theory of Sturmian words over Presburger arithmetic is decidable. Using a general adder recognizing addition in Ostrowski numeration systems by Baranwal, Schaeffer and Shallit, we prove that the first-order…
As was shown recently, many important AI problems require counting the number of models of propositional formulas. The problem of counting models of such formulas is, according to present knowledge, computationally intractable in a worst…
Border basis detection (BBD) is described as follows: given a set of generators of an ideal, decide whether that set of generators is a border basis of the ideal with respect to some order ideal. The motivation for this problem comes from a…
Dynamical systems at the edge of chaos, which have been considered as models of self-organization phenomena, are marked by their ability to perform nontrivial computations. To distinguish them from systems with limited computing power, we…
Partially observable Markov decision processes (POMDPs) form a prominent model for uncertainty in sequential decision making. We are interested in constructing algorithms with theoretical guarantees to determine whether the agent has a…
The general sequential decision-making problem, which includes Markov decision processes (MDPs) and partially observable MDPs (POMDPs) as special cases, aims at maximizing a cumulative reward by making a sequence of decisions based on a…
In general, the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we show how a (unique) string representation for (complete) initially-connected deterministic automata…
Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{\lambda-1}, b^{\lambda}-1\right]$, we denote by $R_\lambda (n)$ the reverse of $n$ in base $b$, obtained by reversing the order of the digits of $n$. We…
We investigate finite deterministic automata in sets with non-homogeneous atoms: integers with successor. As there are uncount- ably many deterministic finite automata in this setting, we restrict our attention to automata with semilinear…
We introduce \emph{patterned numbers}, a digit--divisor-based classification of integers motivated by recreational mathematics. A number is defined to be patterned if at least one of its positive divisors appears as a digit in its base-10…
Counters that hold natural numbers are ubiquitous in modeling and verifying software systems; for example, they model dynamic creation and use of resources in concurrent programs. Unfortunately, such discrete counters often lead to…