The equivalence problem of multidimensional multitape automata

Grigorian, Haik; Shoukourian, S.

doi:10.1016/j.jcss.2008.02.006

Cited by 7 publications

(8 citation statements)

References 8 publications

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“…It is natural to consider the application of the obtained result to multitape finite automata -we are implicitly tied to it with the introduced alternative coding [5,12]. In the case of mutitape automata, for each pair of symbols from one tape, these symbols are not commutative, while for each pair of symbols from different tapes, these symbols are commutative.…”

Section: Lemma 4 (Projection Lemma) [11]mentioning

confidence: 99%

“…The second result is based on the solution of the equivalence problem of multidimensional multitape automata [12]. Definition of a regular expression for a multidimensional multitape finite automaton via a set of newly introduced macros that represent a movement direction of the tape heads for a given multidimensional multitape automata brings to a solution of the following problems.…”

Section: Introductionmentioning

confidence: 99%

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Analytics for AI Related Applications of Multidimensional Multitape Finite Automata

Shoukourian,

Grigoryan,

Amirkhanyan

2024

Frontiers in Artificial Intelligence and Applications

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A special binary representation/coding of an element in a free partially commutative monoid initially introduced in 1980 has been successfully used as a basis for the effective/polynomial solution of the following long standing open problems: functional equivalence of program schemata with non-degenerate operators, equivalence of deterministic multitape finite automata (MFA), equivalence of deterministic multidimensional multitape finite automata (MMFA), regular expressions for MFA, systems of equations for MFA regular sets. In addition, the consideration of the coding leads to an alternative characterization of commutation classes in free partially commutative monoids, which, in comparison with the already known characterization implying from the projection lemma, brings to a better efficiency when checking the equality of traces with lengths longer than a certain number or with alphabets containing more than two symbols. Regular expressions for languages of MFA and MMFA are also defined based on the mentioned coding. A brief overview of relevant AI applications concludes considerations.

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Section: Lemma 4 (Projection Lemma) [11]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analytics for AI Related Applications of Multidimensional Multitape Finite Automata

Shoukourian,

Grigoryan,

Amirkhanyan

2024

Frontiers in Artificial Intelligence and Applications

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“…Let N = {0, 1, …}. We call a set N n an n-dimensional tape [6,7], elements of the set N n , i.e., n-tuples of the form ( ,... , ) a a 1 n , cells of the tape, and numbers a a 1 ,... , n coordinates of the corresponding cell. The cell (0, …, 0) is called initial.…”

Section: Equivalence Problem For Regular Expressions Over a Commutatimentioning

confidence: 99%

“…This investigation is based on the use of some results of the theory of partially commutative semigroups [5] and automata with multidimensional tapes [6,7].…”

mentioning

confidence: 99%

Equivalence of regular expressions over a partially commutative alphabet

Shoukourian

2009

Cybern Syst Anal

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The equivalence problem is considered for regular expressions over a partially commutative alphabet. The alphabet is decomposed into disjoint subsets of noncommutative elements. The special case of the problem when the cardinal number of only one subset is larger than 1 and the cardinal numbers of the other subsets are equal to 1 is proved to be algorithmically solvable.The algorithmic decidability of the equivalence problem for regular expressions over a commutative alphabet is proved by V. N. Red'ko in 1964. It follows from the result obtained by V. A. Tuzov in 1971 that, in the general case, the equivalence problem for regular expressions over a partially commutative alphabet is algorithmically undecidable. A partially commutative alphabet is understood to be the union of disjoint subsets in which any two symbols from the same subset are not permutable and any two symbols from different subsets are permutable. In a definite sense, this alphabet is a generalization of a commutative alphabet that corresponds to the case when the cardinality of each subset is equal to unity. V. A. Tuzov showed that if there are at least two subsets of cardinality greater than one, then the equivalence problem is algorithmically undecidable. Thus, it is only one special case in which it was unknown whether the equivalence problem is decidable, namely, the case when the cardinality of one of subsets in a partially commutative alphabet is more than unity and the cardinality of all other subsets is equal to unity. It is proved in this article that, in this case, the equivalence problem for regular expressions is also algorithmically decidable.Let Y be a finite alphabet, and let Y Y n 1 , , ¼ be a partition of the alphabet Y with respect to Q into disjoint subsets. If Y is a set, then we denote the cardinality of this set by | | Y . Let R 1 and R 2 be regular expressions [1] in the alphabet Y . If, for each w R Î 1 , there is a word w R ¢Î 2 coincident with w to within the permutation of its adjacent symbols belonging to different partition classes of the alphabet Y and, on the contrary, for each word w R ¢Î 2 , there is a word w R Î 1 coincident with ¢ w to within the permutation of its adjacent symbols belonging to partition classes of the alphabet Y , then we call the regular expressions R 1 and R 2 Q-equivalent and denote by R R 1 2 ( ) Q .On the one hand, V. N. Redko showed [2] that if all partition classes of an alphabet are one-element, then the problem of Q-equivalence is algorithmically decidable. On the other hand, as follows from the result of V. A. Tuzov [3], if a partition of an alphabet contains at least two classes with the number of symbols that is not smaller than two, then the problem of Q-equivalence is reduced to the problem of equivalence of nondeterministic multitape automata [4] and, hence, it is algorithmically undecidable.In this paper, an "intermediate case" is considered when all partition classes of the alphabet Y except for one are one-element. This investigation is based on the use of some results of the ...

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“…A new representation of languages for multitape finite automata (MFA), based on a special binary coding of elements in a free partially commutative semigroup were considered in [2]. This coding was also used for the solution of several problems in the theory of automata, which were previously open [2][3][4].…”

mentioning

confidence: 99%

Measurement of Distance Between Regular Events for Multitape Automata Based on a New Characterization of Equivalence Classes

Grigoryan¹,

Hayrapetyan²

2021

Proc. YSU A: Phys. Math. Sci.

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In this paper several problems related to the implementation of the method for the approximate calculation of distance between regular events for multitape finite automata are considered and resolved. An algorithm of matching for the considered regular expressions is suggested and results of the algorithm application to some specific regular expressions are adduced. The proposed method can be used not only for the mentioned implementation, but also separately.

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The equivalence problem of multidimensional multitape automata

Cited by 7 publications

References 8 publications

Analytics for AI Related Applications of Multidimensional Multitape Finite Automata

Analytics for AI Related Applications of Multidimensional Multitape Finite Automata

Equivalence of regular expressions over a partially commutative alphabet

Measurement of Distance Between Regular Events for Multitape Automata Based on a New Characterization of Equivalence Classes

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