Co-transcriptional splicing generates RNA sequences from a DNA template by deleting subsequences nondeterministically. Recent work showed how to encode an NFA into such a template, but the construction requires deleting subsequences whose length grows with the distance between states, which makes such deletions unlikely under the local nature of co-transcriptional splicing. We introduce k-bandwidth NFAs, in which transitions span at most k states. These automata form a strict hierarchy of language classes. For finite languages, bandwidth 2 suffices, and bandwidth 1 can be decided in polynomial-time when the language is presented as a list of words. Minimizing the bandwidth is NP-hard even for fixed k ≥ 2.
In this paper, we settle a series of algorithmic problems related to the (sets of) subsequences occurring in the strings of a given formal language, represented by a deterministic finite automaton with translucent letters accepting it, which were left open in [Fazekas et al., ISAAC 2024]. Firstly, we show that one can decide in polynomial time whether all elements of a given language have a given string as subsequence. We continue by considering the problems of deciding for a given language L⊂Σ ^* and a given positive integer k>0 (respectively, for all integers k>0 ) whether there exists a k-universal string in L (i.e., a string which has all possible length-k strings over Σ as subsequences). For both problems we show that they are NP-hard, and in the case of the second problem we give an NP-upper bound and thus achieve NP-completeness. For the other problem, i.e. for the question whether there exists a k-universal word in L given a language L⊂Σ ^* and a positive integer k>0 , we present a PSPACE algorithm.
Automata on linear orderings are a form of finite-state automata introduced by Bruyère and Carton, that generalize the concepts of finite-word, infinite-word, and transfinite-word automata. They recognize words indexed by linear orderings, defined as mappings from the elements of such orderings to a finite alphabet. Some theoretical developments involving automata on linear orderings are hindered by the fact that their definition does not allow epsilon transitions, i.e., transitions with an empty label. In this paper, we define a variant of automata on linear orderings that allows such transitions in their transition relation, and show that this extension does not modify the expressive power of the automata. We motivate the usefulness of epsilon automata on linear orderings by using them for correcting an erroneous construction in the original proof of equivalence between automata on linear orderings and the corresponding notion of regular expressions.
This paper is the second in a two-part series presenting "MeSCaL", a program dedicated to testing properties of regular languages. It provides users with an interface that allows them to define regular languages through regular expressions or finite automata. They can then perform property tests on the defined languages. These properties consist of checking membership in predefined classes of regular languages. MeSCaL supports about a hundred classes. The first paper in the series details some of its data structures, explains why existing algorithms from the literature are unlikely to run in reasonable time even for simple input languages, and describes optimizations that make the software more efficient than those algorithms. It applies to 20 different classes. This second article continues that work, but remains largely independent of the first: it considers 44 new classes for which membership tests require the precomputation of new objects, called orbits. The computation of these orbits uses some results from the first part as a black box. The classes to which the presented optimizations apply are levels of language hierarchies, which come in two types: concatenation hierarchies and navigational hierarchies. While the first part focused on level one of these hierarchies, this second article addresses level two, as well as intermediate levels between one and two. It also explains how to handle hierarchies whose base level (level zero) was not covered in the first article.
In this paper, we present an L* -style algorithm for actively learning a bidirectional deterministic finite automaton (biDFA) in polynomial time using three types of oracles. We show how the W-method for the equivalence oracle can be adapted to our algorithm and present a novel heuristic for choosing the orientation of states. With this algorithm, one can identify automata for a subset of the linear languages that includes but is not limited to the regular languages. Since the equivalence oracle is an important part of the algorithm, we also discuss complexity bounds for different versions of the language equivalence problem for biDFAs. These results, together with our algorithm, also prove complexity bounds for the biDFA minimisation problem. Finally, we provide an implementation of the algorithm and experimentally show its performance with different approximation heuristics.
