46,457 research outputs found

    Operations on Automata with All States Final

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    We study the complexity of basic regular operations on languages represented by incomplete deterministic or nondeterministic automata, in which all states are final. Such languages are known to be prefix-closed. We get tight bounds on both incomplete and nondeterministic state complexity of complement, intersection, union, concatenation, star, and reversal on prefix-closed languages.Comment: In Proceedings AFL 2014, arXiv:1405.527

    Transition Complexity of Incomplete DFAs

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    In this paper, we consider the transition complexity of regular languages based on the incomplete deterministic finite automata. A number of results on Boolean operations have been obtained. It is shown that the transition complexity results for union and complementation are very different from the state complexity results for the same operations. However, for intersection, the transition complexity result is similar to that of state complexity.Comment: In Proceedings DCFS 2010, arXiv:1008.127

    Operational State Complexity of Deterministic Unranked Tree Automata

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    We consider the state complexity of basic operations on tree languages recognized by deterministic unranked tree automata. For the operations of union and intersection the upper and lower bounds of both weakly and strongly deterministic tree automata are obtained. For tree concatenation we establish a tight upper bound that is of a different order than the known state complexity of concatenation of regular string languages. We show that (n+1) ( (m+1)2^n-2^(n-1) )-1 vertical states are sufficient, and necessary in the worst case, to recognize the concatenation of tree languages recognized by (strongly or weakly) deterministic automata with, respectively, m and n vertical states.Comment: In Proceedings DCFS 2010, arXiv:1008.127

    State Complexity of Combined Operations on Finite Languages

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    State complexity is a descriptive complexity measure for regular languages. It is a fundamental topic in automata and formal language theory. The state complexity of a regular language is the number of states in the minimal complete deterministic finite automaton accepting the language. During the last few decades, many publications have focused and studied the state complexity of many individual as well as combined operations on regular languages. Also, the state complexity of some basic operations on finite languages has been studied. But until now there has been no study on the state complexity of combined operations on finite languages. In this thesis, we will first study the state complexity of the combined operation, star of union, on finite languages and give an exact bound. Then we will investigate the state complexity of star of catenation and show its approximation with a good ratio bound and finally, we will prove an upper bound for star of intersection

    Nondeterministic State Complexity for Suffix-Free Regular Languages

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    We investigate the nondeterministic state complexity of basic operations for suffix-free regular languages. The nondeterministic state complexity of an operation is the number of states that are necessary and sufficient in the worst-case for a minimal nondeterministic finite-state automaton that accepts the language obtained from the operation. We consider basic operations (catenation, union, intersection, Kleene star, reversal and complementation) and establish matching upper and lower bounds for each operation. In the case of complementation the upper and lower bounds differ by an additive constant of two.Comment: In Proceedings DCFS 2010, arXiv:1008.127

    State complexity of union and intersection of star on k regular languages

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    AbstractIn this paper, we continue our study on state complexity of combined operations. We study the state complexities of L1∗∪L2∗, ⋃i=1kLi∗, L1∗∩L2∗, and ⋂i=1kLi∗ for regular languages Li, 1≤i≤k. We obtain the exact bounds for these combined operations and show that the bounds are different from the mathematical compositions of the state complexities of their component individual operations

    Advanced Topics on State Complexity of Combined Operations

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    State complexity is a fundamental topic in formal languages and automata theory. The study of state complexity is also strongly motivated by applications of finite automata in software engineering, programming languages, natural language and speech processing and other practical areas. Since many of these applications use automata of large sizes, it is important to know the number of states of the automata. In this thesis, we firstly discuss the state complexities of individual operations on regular languages, including union, intersection, star, catenation, reversal and so on. The state complexity of an operation on unary languages is usually different from that of the same operation on languages over a larger alphabet. Both kinds of state complexities are reviewed in the thesis. Secondly, we study the exact state complexities of twelve combined operations on regular languages. The state complexities of most of these combined operations are not equal to the compositions of the state complexities of the individual operations which make up these combined operations. We also explore the reason for this difference. Finally, we introduce the concept of estimation and approximation of state complexity. We show close estimates and approximations of the state complexities of six combined operations on regular languages which are good enough to use in practice

    Unrestricted State Complexity of Binary Operations on Regular and Ideal Languages

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    We study the state complexity of binary operations on regular languages over different alphabets. It is known that if LmL'_m and LnL_n are languages of state complexities mm and nn, respectively, and restricted to the same alphabet, the state complexity of any binary boolean operation on LmL'_m and LnL_n is mnmn, and that of product (concatenation) is m2n2n1m 2^n - 2^{n-1}. In contrast to this, we show that if LmL'_m and LnL_n are over different alphabets, the state complexity of union and symmetric difference is (m+1)(n+1)(m+1)(n+1), that of difference is mn+mmn+m, that of intersection is mnmn, and that of product is m2n+2n1m2^n+2^{n-1}. We also study unrestricted complexity of binary operations in the classes of regular right, left, and two-sided ideals, and derive tight upper bounds. The bounds for product of the unrestricted cases (with the bounds for the restricted cases in parentheses) are as follows: right ideals m+2n2+2n1m+2^{n-2}+2^{n-1} (m+2n2m+2^{n-2}); left ideals mn+m+nmn+m+n (m+n1m+n-1); two-sided ideals m+2nm+2n (m+n1m+n-1). The state complexities of boolean operations on all three types of ideals are the same as those of arbitrary regular languages, whereas that is not the case if the alphabets of the arguments are the same. Finally, we update the known results about most complex regular, right-ideal, left-ideal, and two-sided-ideal languages to include the unrestricted cases.Comment: 30 pages, 15 figures. This paper is a revised and expanded version of the DCFS 2016 conference paper, also posted previously as arXiv:1602.01387v3. The expanded version has appeared in J. Autom. Lang. Comb. 22 (1-3), 29-59, 2017, the issue of selected papers from DCFS 2016. This version corrects the proof of distinguishability of states in the difference operation on p. 12 in arXiv:1609.04439v
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