Descriptional and Computational Complexity of Finite Automata

2009 ◽  
pp. 23-42 ◽  
Author(s):  
Markus Holzer ◽  
Martin Kutrib
Keyword(s):  
Computational Complexity ◽  
Finite Automata

FROM EQUIVALENCE TO ALMOST-EQUIVALENCE, AND BEYOND: MINIMIZING AUTOMATA WITH ERRORS

2013 ◽  
Vol 24 (07) ◽  
pp. 1083-1097 ◽  
Author(s):  
MARKUS HOLZER ◽  
SEBASTIAN JAKOBI
Keyword(s):  
Computational Complexity ◽  
Finite Number ◽  
Finite Automata ◽  
Deterministic Finite Automata ◽  
Minimization Problems ◽  
Significant Difference ◽  
Straightforward Generalization

We introduce E-equivalence, which is a straightforward generalization of almost-equivalence. While almost-equivalence asks for ordinary equivalence up to a finite number of exceptions, in E-equivalence these exceptions or errors must belong to a (regular) set E. The computational complexity of deterministic finite automata (DFAs) minimization problems and their variants w.r.t. almost- and E-equivalence are studied. We show that there is a significant difference in the complexity of problems related to almost-equivalence, and those related to E-equivalence. Moreover, since hyper-minimal and E-minimal automata are not necessarily unique (up to isomorphism as for minimal DFAs), we consider the problem of counting the number of these minimal automata.

scholarly journals Minimal and Hyper-Minimal Biautomata

2016 ◽  
Vol 27 (02) ◽  
pp. 161-185
Author(s):  
Markus Holzer ◽  
Sebastian Jakobi
Keyword(s):  
Computational Complexity ◽  
Minimization Problem ◽  
Finite Automata ◽  
Deterministic Finite Automata ◽  
Np Complete ◽  
Carry Over

We compare deterministic finite automata (DFAs) and biautomata under the following two aspects: structural similarities between minimal and hyper-minimal automata, and computational complexity of the minimization and hyper-minimization problem. Concerning classical minimality, the known results such as isomorphism between minimal DFAs, and NL-completeness of the DFA minimization problem carry over to the biautomaton case. But surprisingly this is not the case for hyper-minimization: the similarity between almost-equivalent hyper-minimal biautomata is not as strong as it is between almost-equivalent hyper-minimal DFAs. Moreover, while hyper-minimization is NL-complete for DFAs, we prove that this problem turns out to be computationally intractable, i.e., NP-complete, for biautomata.

THE COMPLEXITY OF REGULAR(-LIKE) EXPRESSIONS

2011 ◽  
Vol 22 (07) ◽  
pp. 1533-1548 ◽  
Author(s):  
MARKUS HOLZER ◽  
MARTIN KUTRIB
Keyword(s):  
Computational Complexity ◽  
Finite Automata ◽  
Regular Expressions ◽  
Descriptional Complexity

We summarize results on the complexity of regular(-like) expressions and tour a fragment of the literature. In particular we focus on the descriptional complexity of the conversion of regular expressions to equivalent finite automata and vice versa, to the computational complexity of problems on regular-like expressions such as, e.g., membership, inequivalence, and non-emptiness of complement, and finally on the operation problem measuring the required size for transforming expressions with additional language operations (built-in or not) into equivalent ordinary regular expressions.

NONDETERMINISTIC FINITE AUTOMATA — RECENT RESULTS ON THE DESCRIPTIONAL AND COMPUTATIONAL COMPLEXITY

2009 ◽  
Vol 20 (04) ◽  
pp. 563-580 ◽  
Author(s):  
MARKUS HOLZER ◽  
MARTIN KUTRIB
Keyword(s):  
Computational Complexity ◽  
Finite Automata ◽  
Size Estimation ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Nondeterministic Finite Automata ◽  
Vast Literature ◽  
Recent Developments ◽  
Big Picture ◽  
Main Ideas

Nondeterministic finite automata (NFAs) were introduced in [68], where their equivalence to deterministic finite automata was shown. Over the last 50 years, a vast literature documenting the importance of finite automata as an enormously valuable concept has been developed. In the present paper, we tour a fragment of this literature. Mostly, we discuss recent developments relevant to NFAs related problems like, for example, (i) simulation of and by several types of finite automata, (ii) minimization and approximation, (iii) size estimation of minimal NFAs, and (iv) state complexity of language operations. We thus come across descriptional and computational complexity issues of nondeterministic finite automata. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved.

scholarly journals Two Extensions of Cover Automata

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 338
Author(s):  
Cezar Câmpeanu
Keyword(s):  
Computational Complexity ◽  
Finite Automata ◽  
The State ◽  
Finite Cover ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Compact Representations ◽  
Minimization Algorithms ◽  
Deterministic Automata ◽  
Representation Transformation

Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get even more compact representations of finite languages. DFCAs are extended by accepting either “do not care” symbols or considering multiple entry DFCAs. We study for each of the two models the existence of the minimization or simplification algorithms and their computational complexity, the state complexity of these representations compared with other representations of the same language, and the bounds for state complexity in case we perform a representation transformation. Minimization for both models proves to be NP-hard. A method is presented to transform minimization algorithms for deterministic automata into simplification algorithms applicable to these extended models. DFCAs with “do not care” symbols prove to have comparable state complexity as Nondeterministic Finite Cover Automata. Furthermore, for multiple entry DFCAs, we can have a tight estimate of the state complexity of the transformation into equivalent DFCA.

scholarly journals Search for a substring of characters using the theory of non-deterministic finite automata and vector-character architecture

2020 ◽  
Vol 9 (3) ◽  
pp. 1238-1250
Author(s):  
Dmitry V. Pashchenko ◽  
Dmitry A. Trokoz ◽  
Alexey I. Martyshkin ◽  
Mihail P. Sinev ◽  
Boris L. Svistunov
Keyword(s):  
Computational Complexity ◽  
Hardware Implementation ◽  
Finite Automata ◽  
Target Line ◽  
Computing Systems ◽  
Deterministic Finite Automata ◽  
Low Level ◽  
Hardware Implementations ◽  
The Future

The paper proposed an algorithm which purpose is searching for a substring of characters in a string. Principle of its operation is based on the theory of non-deterministic finite automata and vector-character architecture. It is able to provide the linear computational complexity of searching for a substring depending on the length of the searched string measured in the number of operations with hyperdimensional vectors when repeatedly searching for different strings in a target line. None of the existing algorithms has such a low level of computational complexity. The disadvantages of the proposed algorithm are the fact that the existing hardware implementations of computing systems for performing operations with hyperdimensional vectors require a large number of machine instructions, which reduces the gain from this algorithm. Despite this, in the future, it is possible to create a hardware implementation that can ensure the execution of operations with hyperdimensional vectors in one cycle, which will allow the proposed algorithm to be applied in practice.

scholarly journals Subset Synchronization and Careful Synchronization of Binary Finite Automata

2016 ◽  
Vol 27 (05) ◽  
pp. 557-577 ◽  
Author(s):  
Vojtěch Vorel
Keyword(s):  
Computational Complexity ◽  
Lower Bound ◽  
Finite Automata ◽  
Strongly Connected ◽  
Deterministic Automata ◽  
Finite Domains ◽  
Exponential Lower Bound

We present a strongly exponential lower bound that applies both to the subset synchronization threshold for binary deterministic automata and to the careful synchronization threshold for binary partial automata. In the later form, the result finishes the research initiated by Martyugin (2013). Moreover, we show that both the thresholds remain strongly exponential even if restricted to strongly connected binary automata. In addition, we apply our methods to computational complexity. Existence of a subset reset word is known to be PSPACE-complete; we show that this holds even under the restriction to strongly connected binary automata. The results apply also to the corresponding thresholds in two more general settings: D1- and D3-directable nondeterministic automata and composition sequences over finite domains.

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