10 research outputs found
Chrobak Normal Form Revisited, with Applications
Abstract. It is well known that any nondeterministic finite automata over a unary alphabet can be represented in a certain normal form called the Chrobak normal form [1]. We present a very simple conversion pro-cedure working in O(n3) time. Then we extend the algorithm to improve two trade-offs concerning conversions between different representations of unary regular languages. Given an n-state NFA, we are able to find a regular expression of size O ( n2 logn) describing the same language (which improves the previously known O(n2) size bound [8]) and a context-free grammar in Chomsky normal form with O(√n logn) nonterminals (which improves the previously known O(n2/3) bound [3]). As a byproduct of our conversion procedure, we get an alternative proof of the Chrobak normal form theorem. We believe that its efficiency and simplicity make the effort of reproving an already known result worth-while. Key-words: unary automata, descriptional complexity
Deterministic Automata for Unordered Trees
Automata for unordered unranked trees are relevant for defining schemas and
queries for data trees in Json or Xml format. While the existing notions are
well-investigated concerning expressiveness, they all lack a proper notion of
determinism, which makes it difficult to distinguish subclasses of automata for
which problems such as inclusion, equivalence, and minimization can be solved
efficiently. In this paper, we propose and investigate different notions of
"horizontal determinism", starting from automata for unranked trees in which
the horizontal evaluation is performed by finite state automata. We show that a
restriction to confluent horizontal evaluation leads to polynomial-time
emptiness and universality, but still suffers from coNP-completeness of the
emptiness of binary intersections. Finally, efficient algorithms can be
obtained by imposing an order of horizontal evaluation globally for all
automata in the class. Depending on the choice of the order, we obtain
different classes of automata, each of which has the same expressiveness as
CMso.Comment: In Proceedings GandALF 2014, arXiv:1408.556
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
Optimal state reductions of automata with partially specified behaviors
Nondeterministic finite automata with don't care states, namely states which neither accept nor reject, are considered. A characterization of deterministic automata compatible with such a device is obtained. Furthermore, an optimal state bound for the smallest compatible deterministic automata is provided. It is proved that the problem of minimizing deterministic don't care automata is NP-complete and PSPACE-hard in the nondeterministic case. The restriction to the unary case is also considered
Approximate Membership for Regular Languages modulo the Edit Distance
International audienceWe present a probabilistic algorithm for testing approximate membership of words to regular languages modulo the edit distance. The time complexity of our algorithm, which is independent of the size of the input word, is polynomial in the size of the input automaton and the inverse error precision. All previous property testing algorithms for regular languages, whether they consider approximations modulo the Hamming distance or the edit distance with moves, run in exponential time if not fixing one of these parameters
Automata for Unordered Trees
International audienceWe present a framework for defining automata for unordereddata trees that is parametrized by the way in which multisets of children nodes are described. Presburger tree automata and alternatingPresburger tree automata are particular instances. We establish the usual equivalence in expressiveness of tree automata and MSO for the automata defined inour framework.We then investigate subclasses of automata for unordered treesfor which testing language equivalence is in P-time. For this we start from automata in our framework that describe multisets of childrenby finite automata, and propose two approaches of how todo this deterministically. We show that a restriction to confluent horizontal evaluation leads to polynomial-time emptiness and universality, but still suffers fromcoNP-completeness of the emptiness of binary intersections. Finally, efficient algorithms can be obtained by imposing an order of horizontal evaluation globally for all automata in the class. Depending onthe choice of the order, we obtain different classes of automata, eachof which has the same expressiveness as Counting MSO
Z3str4: A Solver for Theories over Strings
Satisfiability Modulo Theories (SMT) solvers supporting rich theories of strings have facilitated numerous industrial applications with the need to reason about string operations and predicates that are present in many popular programming languages. Constraints encountered in practical applications have immense value in inspiring new algorithms and heuristics that string solvers can take advantage of to tackle new, more difficult problems. This is especially relevant as the combinations of operators typically supported by string solvers, or that are encountered in program analysis constraints, quickly result in theories whose satisfiability problems are undecidable.
I present a number of theoretical and practical contributions in the domain of string solving. On the theoretical side, I illustrate decidability and undecidability results related to different relevant theories which include strings. On the practical side, I describe a collection of algorithms and heuristics designed to address challenges encountered in applications of string solvers, culminating with the introduction of Z3str4, a state-of-the-art solver for theories over strings. Z3str4 incorporates many improvements over its predecessor Z3str3, including an algorithm selection architecture that takes advantage of multiple solving algorithms in order to leverage the strengths of diverse string solving procedures against formulas they are predicted to be able to solve efficiently. I also present a back-end model construction algorithm for Z3str4 which is a hybrid between word-based and unfolding-based algorithms. Furthermore, I showcase the power of Z3str4 against other state-of-the-art tools in an empirical evaluation over a large and diverse collection of benchmarks. Additionally, I describe algorithms and heuristics specific to solving regular expression constraints, and demonstrate their effectiveness in a detailed and focused empirical evaluation
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems