State complexity of inversion operations

Cho, Da-Jung; Han, Yo-Sub; Ko, Sang‐Ki; Salomaa, Kai

doi:10.1016/j.tcs.2015.04.017

Cited by 3 publications

(6 citation statements)

References 24 publications

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“…We verify that the condition (2) can be satisfied by a function f such that f (i z ) = z and f ( j z ) = t. If i z appears in the sequence j 1 , …, j n , it must have a subindex different from z, and hence the function value f (i z ) = z is compatible with the condition (2). Similarly, if j z appears in the sequence j 1 , …, j z−1 , j z+1 , …, j n it has an index different from t (since j z = j t ) and hence also the function value f ( j z ) = t is compatible with the condition (2), and the function f with the required values can be defined.…”

Section: Corollarymentioning

confidence: 72%

“…The first component of the states of B simply simulates the computation of A, i.e., it keeps track of the state of A, assuming that up to the current position in the input a string of L 2 was not yet deleted. The second component of the states of B keeps track of all states that A could be in assuming that at some point in the preceding computation a string of L 2 was deleted from the input.…”

Section: Upper Bound For Deletionmentioning

confidence: 99%

“…It is known from Theorem 3 of [16] that the transition monoid of A (and syntactic monoid of L(A)) has n n elements. 2 Our DFA A has a different final state than the DFA used in [16] but this does not change the size of the transition monoid.…”

Section: Corollarymentioning

confidence: 99%

“…Similarly, if j z appears in the sequence j 1 , …, j z−1 , j z+1 , …, j n it has an index different from t (since j z = j t ) and hence also the function value f ( j z ) = t is compatible with the condition (2), and the function f with the required values can be defined.…”

Section: Corollarymentioning

confidence: 99%

“…The precise worst case state complexity of many basic language operations has been established; see e.g. [2,5,8,14,17,23,25,27,31,33,37]. Also there has been much work on the state complexity of combinations of basic language operations [3,7,9,10,18,32].…”

Section: Introductionmentioning

confidence: 99%

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State complexity of deletion and bipolar deletion

Han

Salomaa

2015

Acta Informatica

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It is well known that the language obtained by deleting an arbitrary language from a regular language is regular. We give an upper bound for the state complexity of deleting an arbitrary language from a regular language and a matching lower bound. We show that the state complexity of deletion is n · 2 n−1 [respectively, (n + 1 2 ) · 2 n − 2] when using complete (respectively, incomplete) deterministic finite automata. We show that the state complexity of bipolar deletion has an upper bound n n [respectively (n + 1) n − 1] when using complete (respectively, incomplete) deterministic finite automata. In both cases we give almost matching lower bounds.

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Section: Corollarymentioning

confidence: 72%

Section: Upper Bound For Deletionmentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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State complexity of deletion and bipolar deletion

Han

Salomaa

2015

Acta Informatica

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State Complexity of Pseudocatenation

Kari

2019

Language and Automata Theory and Applications

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A General Approach to State Complexity of Operations: Formalization and Limitations

Davies

2018

Developments in Language Theory

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The state complexity of the result of a regular operation is often positively correlated with the number of distinct transformations induced by letters in the minimal deterministic finite automaton of the input languages. That is, more transformations in the inputs means higher state complexity in the output. When this correlation holds, the state complexity of a unary operation can be maximized using languages in which there is one letter corresponding to each possible transformation; for operations of higher arity, we can use m-tuples of languages in which there is one letter corresponding to each possible m-tuple of transformations. In this way, a small set of languages can be used as witnesses for many common regular operations, eliminating the need to search for witnesses -though at the expense of using very large alphabets. We formalize this approach and examine its limitations. We define a class of "uniform" operations for which this approach works; the class is closed under composition and includes common operations such as star, concatenation, reversal, union, and complement. Our main result is that the worst-case state complexity of a uniform operation can be determined by considering a finite set of witnesses, and this set depends only on the arity of the operation and the state complexities of the inputs.

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State complexity of inversion operations

Cited by 3 publications

References 24 publications

State complexity of deletion and bipolar deletion

State complexity of deletion and bipolar deletion

State Complexity of Pseudocatenation

A General Approach to State Complexity of Operations: Formalization and Limitations

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