scholarly journals Finite automata with undirected state graphs

2021 ◽  
Author(s):  
Martin Kutrib ◽  
Andreas Malcher ◽  
Christian Schneider
Keyword(s):  
Lower Bounds ◽  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Deterministic Case ◽  
Boolean Operations ◽  
State Complexity ◽  
Descriptional Complexity ◽  
Language Classes ◽  
Different Types ◽  
State Graphs

AbstractWe investigate finite automata whose state graphs are undirected. This means that for any transition from state p to q consuming some letter a from the input there exists a symmetric transition from state q to p consuming a letter a as well. So, the corresponding language families are subregular, and in particular in the deterministic case, subreversible. In detail, we study the operational descriptional complexity of deterministic and nondeterministic undirected finite automata. To this end, the different types of automata on alphabets with few letters are characterized. Then, the operational state complexity of the Boolean operations as well as the operations concatenation and iteration is investigated, where tight upper and lower bounds are derived for unary as well as arbitrary alphabets under the condition that the corresponding language classes are closed under the operation considered.

DESCRIPTIONAL COMPLEXITY OF SPLICING SYSTEMS

2008 ◽  
Vol 19 (04) ◽  
pp. 813-826 ◽  
Author(s):  
REMCO LOOS ◽  
ANDREAS MALCHER ◽  
DETLEF WOTSCHKE
Keyword(s):  
Lower Bounds ◽  
Finite Automata ◽  
Regular Languages ◽  
Upper And Lower Bounds ◽  
Descriptional Complexity ◽  
Nondeterministic Finite Automata ◽  
Total Length ◽  
Splicing Systems

In this paper, the descriptional complexity of extended finite splicing systems is studied. These systems are known to generate exactly the class of regular languages. Upper and lower bounds are shown relating the size of these splicing systems, defined as the total length of the rules and the initial language of the system, to the size of their equivalent minimal nondeterministic finite automata (NFA). In addition, an accepting model of extended finite splicing systems is studied. Using this variant one can obtain systems which are more than polynomially more succinct than the equivalent NFA or generating extended finite splicing system.

Descriptional Complexity of the Forever Operator

2019 ◽  
Vol 30 (01) ◽  
pp. 115-134 ◽  
Author(s):  
Michal Hospodár ◽  
Galina Jirásková ◽  
Peter Mlynárčik
Keyword(s):  
Regular Language ◽  
Finite Automata ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Descriptional Complexity ◽  
Trade Offs ◽  
Number Formula ◽  
Deterministic Automata ◽  
Initial States ◽  
Nondeterministic State

We examine the descriptional complexity of the forever operator, which assigns the language [Formula: see text] to a regular language [Formula: see text], and we investigate the trade-offs between various models of finite automata. We consider complete and partial deterministic finite automata, nondeterministic finite automata with single or multiple initial states, alternating, and Boolean finite automata. We assume that the argument and the result of this operation are accepted by automata belonging to one of these six models. We investigate all possible trade-offs and provide a tight upper bound for 32 of 36 of them. The most interesting result is the trade-off from nondeterministic to deterministic automata given by the Dedekind number [Formula: see text]. We also prove that the nondeterministic state complexity of [Formula: see text] is [Formula: see text] which solves an open problem stated by Birget [The state complexity of [Formula: see text] and its connection with temporal logic, Inform. Process. Lett. 58 (1996) 185–188].

scholarly journals On the number of excursion sets of planar Gaussian fields

2020 ◽  
Vol 178 (3-4) ◽  
pp. 655-698
Author(s):  
Dmitry Beliaev ◽  
Michael McAuley ◽  
Stephen Muirhead
Keyword(s):  
Plane Wave ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Gaussian Fields ◽  
Important Special Case ◽  
Level Densities ◽  
Excursion Sets ◽  
Nodal Set ◽  
Different Types ◽  
Special Case

Abstract The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

SIMULATIONS OF UNARY ONE-WAY MULTI-HEAD FINITE AUTOMATA

2014 ◽  
Vol 25 (07) ◽  
pp. 877-896 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER ◽  
MATTHIAS WENDLANDT
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Unary Languages ◽  
Order Of Magnitude ◽  
Simulation Results ◽  
Special Case ◽  
Nondeterministic Finite Automaton

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case upper and lower bounds that are tight in the order of magnitude are shown. For the latter case we obtain an upper bound of O(n2k) and a lower bound of Ω(nk) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. In addition, we study how the conversion costs vary in the special case of finite and, in particular, of singleton unary lanuages. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

EXACT GENERATION OF MINIMAL ACYCLIC DETERMINISTIC FINITE AUTOMATA

2008 ◽  
Vol 19 (04) ◽  
pp. 751-765 ◽  
Author(s):  
MARCO ALMEIDA ◽  
NELMA MOREIRA ◽  
ROGÉRIO REIS
Keyword(s):  
Normal Form ◽  
Lower Bounds ◽  
Finite Automata ◽  
Canonical Representation ◽  
Upper And Lower Bounds ◽  
Generation Algorithm ◽  
Deterministic Finite Automata

We give a canonical representation for minimal acyclic deterministic finite automata (MADFA) with n states over an alphabet of k symbols. Using this normal form, we present a method for the exact generation of MADFAs. This method avoids a rejection phase that would be needed if a generation algorithm for a larger class of objects that contains the MADFAs were used. We give upper and lower bounds for MADFAs enumeration and some exact formulas for small values of n.

scholarly journals Trans-Dichotomous Algorithms without Multiplication —some Upper and Lower Bounds

1997 ◽  
Vol 4 (12) ◽  
Author(s):  
Andrej Brodnik ◽  
Peter Bro Miltersen ◽  
J. Ian Munro
Keyword(s):  
Linear Space ◽  
Lower Bounds ◽  
Query Time ◽  
Upper And Lower Bounds ◽  
Boolean Operations ◽  
And Shifts ◽  
Necessary And Sufficient ◽  
The Way

We show that on a RAM with addition, subtraction, bitwise<br />Boolean operations and shifts, but no multiplication, there is a<br />trans-dichotomous solution to the static dictionary problem using<br />linear space and with query time sqrt(log n(log log n)^(1+o(1))). On<br />the way, we show that two w-bit words can be multiplied in<br />time (log w)^(1+o(1)) and that time Omega(log w) is necessary, and that<br />Theta(log log w) time is necessary and sufficient for identifying the<br />least significant set bit of a word.

Minimal Reversible Deterministic Finite Automata

2018 ◽  
Vol 29 (02) ◽  
pp. 251-270 ◽  
Author(s):  
Markus Holzer ◽  
Sebastian Jakobi ◽  
Martin Kutrib
Keyword(s):  
Lower Bounds ◽  
Finite Automata ◽  
Transition Function ◽  
Upper And Lower Bounds ◽  
State Graph ◽  
Deterministic Finite Automata ◽  
Injective Mapping ◽  
Pattern Approach ◽  
Almost All

We study reversible deterministic finite automata (REV-DFAs), that are partial deterministic finite automata whose transition function induces an injective mapping on the state set for every letter of the input alphabet. We give a structural characterization of regular languages that can be accepted by REV-DFAs. This characterization is based on the absence of a forbidden pattern in the (minimal) deterministic state graph. Again with a forbidden pattern approach, we also show that the minimality of REV-DFAs among all equivalent REV-DFAs can be decided. Both forbidden pattern characterizations give rise to [Formula: see text]-complete decision algorithms. In fact, our techniques allow us to construct the minimal REV-DFA for a given minimal DFA. These considerations lead to asymptotic upper and lower bounds on the conversion from DFAs to REV-DFAs. Thus, almost all problems that concern uniqueness and the size of minimal REV-DFAs are solved.

scholarly journals NONDETERMINISTIC DESCRIPTIONAL COMPLEXITY OF REGULAR LANGUAGES

2003 ◽  
Vol 14 (06) ◽  
pp. 1087-1102 ◽  
Author(s):  
MARKUS HOLZER ◽  
MARTIN KUTRIB
Keyword(s):  
Finite Automata ◽  
Regular Languages ◽  
Boolean Operations ◽  
Worst Case ◽  
Descriptional Complexity ◽  
Exact Number ◽  
Nondeterministic Finite Automata ◽  
Order Of Magnitude

We investigate the descriptional complexity of operations on finite and infinite regular languages over unary and arbitrary alphabets. The languages are represented by nondeterministic finite automata (NFA). In particular, we consider Boolean operations, catenation operations – concatenation, iteration, λ-free iteration – and the reversal. Most of the shown bounds are tight in the exact number of states, i.e. the number is sufficient and necessary in the worst case. Otherwise tight bounds in the order of magnitude are shown.

On a Conjecture by Christian Choffrut

2017 ◽  
Vol 28 (05) ◽  
pp. 483-501 ◽  
Author(s):  
Aleksandrs Belovs ◽  
J. Andres Montoya ◽  
Abuzer Yakaryılmaz
Keyword(s):  
Lower Bounds ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Minimum Amount ◽  
Deterministic Finite Automaton ◽  
Open Problems

It is one of the most famous open problems to determine the minimum amount of states required by a deterministic finite automaton to distinguish a pair of strings, which was stated by Christian Choffrut more than thirty years ago. We investigate the same question for different automata models and we obtain new upper and lower bounds for some of them including alternating, ultrametric, quantum, and affine finite automata.

scholarly journals STATE COMPLEXITY AND THE MONOID OF TRANSFORMATIONS OF A FINITE SET

2005 ◽  
Vol 16 (03) ◽  
pp. 547-563 ◽  
Author(s):  
BRYAN KRAWETZ ◽  
JOHN LAWRENCE ◽  
JEFFREY SHALLIT
Keyword(s):  
Lower Bounds ◽  
Formal Languages ◽  
The State ◽  
Upper And Lower Bounds ◽  
State Complexity ◽  
Finite Set

In this paper we consider the state complexity of an operation on formal languages, root(L). This naturally entails the discussion of the monoid of transformations of a finite set. We obtain good upper and lower bounds on the state complexity of root(L) over alphabets of all sizes. As well, we present an application of these results to the problem of 2DFA-DFA conversion.

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