Membership Problem for Two-Dimensional General Row Jumping Finite Automata

2020 ◽  
Vol 31 (04) ◽  
pp. 527-538
Author(s):  
Grzegorz Madejski ◽  
Andrzej Szepietowski
Keyword(s):  
Computational Model ◽  
Turing Machine ◽  
Finite Automaton ◽  
Finite Automata ◽  
The Other ◽  
Membership Problem ◽  
Two Dimensional ◽  
Np Complete ◽  
Nondeterministic Turing Machine ◽  
Membership Problems

Two-dimensional general row jumping finite automata were recently introduced as an interesting computational model for accepting two-dimensional languages. These automata are nondeterministic. They guess an order in which rows of the input array are read and they jump to the next row only after reading all symbols in the previous row. In each row, they choose, also nondeterministically, an order in which segments of the row are read. In this paper, we study the membership problem for these automata. We show that each general row jumping finite automaton can be simulated by a nondeterministic Turing machine with space bounded by the logarithm. This means that the fixed membership problems for such automata are in NL, and so in P. On the other hand, we show that the uniform membership problem is NP-complete.

Space-bounded OTMs and REG ∞

Computability ◽  
2021 ◽  
pp. 1-16
Author(s):  
Merlin Carl
Keyword(s):  
Complexity Theory ◽  
Turing Machine ◽  
Finite Automata ◽  
Regular Languages ◽  
The Other ◽  
Deterministic Finite Automata ◽  
Logarithmic Space ◽  
Working Space ◽  
Important Theorem

An important theorem in classical complexity theory is that REG = LOGLOGSPACE, i.e., that languages decidable with double-logarithmic space bound are regular. We consider a transfinite analogue of this theorem. To this end, we introduce deterministic ordinal automata (DOAs) and show that they satisfy many of the basic statements of the theory of deterministic finite automata and regular languages. We then consider languages decidable by an ordinal Turing machine (OTM), introduced by P. Koepke in 2005 and show that if the working space of an OTM is of strictly smaller cardinality than the input length for all sufficiently long inputs, the language so decided is also decidable by a DOA, which is a transfinite analogue of LOGLOGSPACE ⊆ REG; the other direction, however, is easily seen to fail.

TWO TOPICS CONCERNING TWO-DIMENSIONAL AUTOMATA OPERATING IN PARALLEL

1992 ◽  
Vol 06 (02n03) ◽  
pp. 211-225 ◽  
Author(s):  
KATSUSHI INOUE ◽  
ITSUO SAKURAMOTO ◽  
MAKOTO SAKAMOTO ◽  
ITSUO TAKANAMI
Keyword(s):  
Turing Machine ◽  
Finite Automata ◽  
Space Complexity ◽  
Two Dimensional ◽  
Turing Machines ◽  
Small Space ◽  
Bottom Up ◽  
Constructible Function ◽  
Deterministic Turing Machine ◽  
Alternating Turing Machines

This paper deals with two topics concerning two-dimensional automata operating in parallel. We first investigate a relationship between the accepting powers of two-dimensional alternating finite automata (2-AFAs) and nondeterministic bottom-up pyramid cellular acceptors (NUPCAs), and show that Ω ( diameter × log diameter ) time is necessary for NUPCAs to simulate 2-AFAs. We then investigate space complexity of two-dimensional alternating Turing machines (2-ATMs) operating in small space, and show that if L (n) is a two-dimensionally space-constructible function such that lim n → ∞ L (n)/ loglog n > 1 and L (n) ≤ log n, and L′ (n) is a function satisfying L′ (n) =o (L(n)), then there exists a set accepted by some strongly L (n) space-bounded two-dimensional deterministic Turing machine, but not accepted by any weakly L′ (n) space-bounded 2-ATM, and thus there exists a rich space hierarchy for weakly S (n) space-bounded 2-ATMs with loglog n ≤ S (n) ≤ log n.

scholarly journals A stronger recognizability condition for two-dimensional languages

2013 ◽  
Vol Vol. 15 no. 2 (Automata, Logic and Semantics) ◽  
Author(s):  
Marcella Anselmo ◽  
Maria Madonia
Keyword(s):  
Finite Automata ◽  
The Other ◽  
Two Dimensional ◽  
Minimal Size ◽  
One Dimensional ◽  
Tiling System ◽  
International Audience ◽  
The One

Automata, Logic and Semantics International audience The paper presents a condition necessarily satisfied by (tiling system) recognizable two-dimensional languages. The new recognizability condition is compared with all the other ones known in the literature (namely three conditions), once they are put in a uniform setting: they are stated as bounds on the growth of some complexity functions defined for two-dimensional languages. The gaps between such functions are analyzed and examples are shown that asymptotically separate them. Finally the new recognizability condition results to be the strongest one, while the remaining ones are its particular cases. The problem of deciding whether a two-dimensional language is recognizable is here related to the one of estimating the minimal size of finite automata recognizing a sequence of (one-dimensional) string languages.

On the Membership Problem of Permutation Grammars — A Direct Proof of NP-Completeness

2020 ◽  
Vol 31 (04) ◽  
pp. 515-525
Author(s):  
Benedek Nagy
Keyword(s):  
Direct Proof ◽  
Input Word ◽  
Membership Problem ◽  
Context Sensitive ◽  
Language Class ◽  
Language Classes ◽  
Sentential Form ◽  
Np Complete ◽  
Membership Problems ◽  
Context Free

One of the most essential classes of problems related to formal languages is the membership problem (also called word problem), i.e., to decide whether a given input word belongs to the language specified, e.g., by a generative grammar. For context-free languages the problem is solved efficiently by various well-known parsing algorithms. However, there are several important languages that are not context-free. The membership problem of the context-sensitive language class is PSPACE-complete, thus, it is believed that it is generally not solvable in an efficient way. There are various language classes between the above mentioned two classes having membership problems with various complexity. One of these classes, the class of permutation languages, is generated by permutation grammars, i.e., context-free grammars extended with permutation rules, where a permutation rule allows to interchange the position of two consecutive nonterminals in the sentential form. In this paper, the membership problem for permutation languages is studied. A proof is presented to show that this problem is NP-complete.

scholarly journals COMPUTATIONAL COMPLEXITY OF GENERATORS AND NONGENERATORS IN ALGEBRA

2002 ◽  
Vol 12 (05) ◽  
pp. 719-735 ◽  
Author(s):  
CLIFFORD BERGMAN ◽  
GIORA SLUTZKI
Keyword(s):  
Computational Complexity ◽  
Algebraic Structure ◽  
Membership Problem ◽  
Generating Set ◽  
Np Complete ◽  
Membership Problems

We discuss the computational complexity of several problems concerning subsets of an algebraic structure that generate the structure. We show that the problem of determining whether a given subset X generates an algebra A is P-complete, while determining the size of the smallest generating set is NP-complete. We also consider several questions related to the Frattini subuniverse, Φ(A), of an algebra A. We show that the membership problem for Φ(A) is co-NP-complete, while the membership problems for Φ(Φ(A)), Φ(Φ(Φ(A))),… all lie in the class P‖(NP).

scholarly journals Synchronizing series-parallel deterministic finite automata with loops and related problems

2021 ◽  
Vol 55 ◽  
pp. 7
Author(s):  
Jens Bruchertseifer ◽  
Henning Fernau
Keyword(s):  
Word Problem ◽  
Finite Automaton ◽  
Parameterized Complexity ◽  
Finite Automata ◽  
Parameterized Algorithms ◽  
Deterministic Finite Automaton ◽  
Deterministic Finite Automata ◽  
Np Complete ◽  
Synchronizing Word

We study the problem DFA-SW of determining if a given deterministic finite automaton A possesses a synchronizing word of length at most k for automata whose (multi-)graphs are TTSPL, i.e., series-parallel, plus allowing some self-loops. While DFA-SW remains NP-complete on TTSPL automata, we also find (further) restrictions with efficient (parameterized) algorithms. We also study the (parameterized) complexity of related problems, for instance, extension variants of the synchronizing word problem, or the problem of finding smallest alphabet-induced synchronizable sub-automata.

scholarly journals Mechanical interactions between adjacent airways in the lung

2014 ◽  
Vol 116 (6) ◽  
pp. 628-634 ◽  
Author(s):  
Baoshun Ma ◽  
Jason H. T. Bates
Keyword(s):  
Computational Model ◽  
Continuum Model ◽  
Network Models ◽  
The Other ◽  
Two Dimensional ◽  
Elastic Continuum ◽  
Spring Network Model ◽  
Triangular Network ◽  
One Airway ◽  
The Continuum

The forces of mechanical interdependence between the airways and the parenchyma in the lung are powerful modulators of airways responsiveness. Little is known, however, about the extent to which adjacent airways affect each other's ability to narrow due to distortional forces generated within the intervening parenchyma. We developed a two-dimensional computational model of two airways embedded in parenchyma. The parenchyma itself was modeled in three ways: 1) as a network of hexagonally arranged springs, 2) as a network of triangularly arranged springs, and 3) as an elastic continuum. In all cases, we determined how the narrowing of one airway was affected when the other airway was relaxed vs. when it narrowed to the same extent as the first airway. For the continuum and triangular network models, interactions between airways were negligible unless the airways lay within about two relaxed diameters of each other, but even at this distance the interactions were small. By contrast, the hexagonal spring network model predicted that airway-airway interactions mediated by the parenchyma can be substantial for any degree of airway separation at intermediate values of airway contraction forces. Evidence to date suggests that the parenchyma may be better represented by the continuum model, which suggests that the parenchyma does not mediate significant interactions between narrowing airways.

A NOTE ON THREE-WAY TWO-DIMENSIONAL PROBABILISTIC TURING MACHINES

2000 ◽  
Vol 14 (04) ◽  
pp. 477-500
Author(s):  
TOKIO OKAZAKI ◽  
KATSUSHI INOUE ◽  
AKIRA ITO ◽  
YUE WANG
Keyword(s):  
Turing Machine ◽  
Error Probability ◽  
Finite Automata ◽  
Nondecreasing Function ◽  
Two Dimensional ◽  
Turing Machines ◽  
One Dimensional ◽  
Nondeterministic Finite Automata ◽  
Deterministic Turing Machine ◽  
Function Of Two Variables

This paper introduces a three-way two-dimensional probabilistic Turing machine (tr2-ptm), and investigates several properties of the machine. The tr2-ptm is a two-dimensional probabilistic Turing machine (2-ptm) whose input head can only move left, right, or down, but not up. Let 2-ptms (resp. tr2-ptms) denote a 2-ptm (resp. tr2-ptm) whose input tape is restricted to square ones, and let 2-PTMs(S(n)) (resp. TR2-PTMs(S(n))) denote the class of sets recognized by S(n) space-bounded 2-ptms's (resp. tr2-ptms's) with error probability less than ½, where S(n): N→N is a function of one variable n (= the side-length of input tapes). Let TR2-PTM(L(m,n)) denote the class of sets recognized by L(m,n) space-bounded tr2-ptm's with error probability less than ½, where L(m,n): N2→N is a function of two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). The main results of this paper are: (1) 2-NFAs - TR2-PTMs(S(n))≠ϕ for any S(n)=o(log n), where 2-NFAs denotes the class of sets of square tapes accepted by two-dimensional nondeterministic finite automata, (2) TR2-PTMsS(n)[Formula: see text]2-PTMs(S(n)) for any S(n)=o(log n), and (3) for any function g(n)=o(log n) (resp. g(n)=o(log n/log log n)) and any monotonic nondecreasing function f(m) which can be constructed by some one-dimensional deterministic Turing machine, TR2-PTM(f(m)+g(n)) (resp. TR2-PTM(f(m)×g(n))) is not closed under column catenation, column closure, and projection. Additionally, we show that two-dimensional nondeterministic finite automata are equivalent to two-dimensional probabilistic finite automata with one-sided error in accepting power.

scholarly journals Finite Automata Implementations Considering CPU Cache

10.14311/1008 ◽  
2007 ◽  
Vol 47 (6) ◽  
Author(s):  
J. Holub
Keyword(s):  
Mathematical Models ◽  
Finite Automaton ◽  
Finite Automata ◽  
The Other ◽  
Practical Implementation ◽  
Deterministic Finite Automaton ◽  
Input Text ◽  
Finite State ◽  
Nondeterministic Finite Automaton

The finite automata are mathematical models for finite state systems. More general finite automaton is the nondeterministic finite automaton (NFA) that cannot be directly used. It is usually transformed to the deterministic finite automaton (DFA) that then runs in time O(n), where n is the size of the input text. We present two main approaches to practical implementation of DFA considering CPU cache. The first approach (represented by Table Driven and Hard Coded implementations) is suitable forautomata being run very frequently, typically having cycles. The other approach is suitable for a collection of automata from which various automata are retrieved and then run. This second kind of automata are expected to be cycle-free. 

THE SIMULATION OF TWO-DIMENSIONAL ONE-MARKER AUTOMATA BY THREE-WAY TURING MACHINES

1989 ◽  
Vol 03 (03n04) ◽  
pp. 393-404 ◽  
Author(s):  
AKIRA ITO ◽  
KATSUSHI INOUE ◽  
ITSUO TAKANAMI
Keyword(s):  
Turing Machine ◽  
Two Dimensional ◽  
Turing Machines ◽  
Nondeterministic Turing Machine ◽  
Necessary And Sufficient ◽  
Sufficient Space

We denote a two-dimensional deterministic (nondeterministic) one-marker automaton by 2-DM1 (2-NM1), and a three-way two-dimensional deterministic (nondeterministic) Turing machine by TR2-DTM (TR2-NTM). In this paper, we show that the necessary and sufficient space for TR2-NTMs to simulate 2-DM1s (2-NM1s) is n log n (n2), and the necessary and sufficient space for TR2-DTMs to simulate 2-DM1s (2-NM1s) is 2O(n log n) (2 O(n2)), where n is the number of columns of rectangular input tapes.

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