State complexity of basic language operations combined with reversal (original) (raw)
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State Complexity of Basic Operations on Non-returning Regular Languages
Lecture Notes in Computer Science, 2013
We consider the state complexity of basic operations on non-returning regular languages. For a non-returning minimal DFA, the start state does not have any in-transitions. We establish the precise state complexity of four Boolean operations (union, intersection, difference, symmetric difference), catenation, reverse, and Kleene-star for non-returning regular languages. Our results are usually smaller than the state complexities for general regular languages and larger than the state complexities for suffix-free regular languages. In the case of catenation and reversal, we define witness languages over a ternary alphabet. Then we provide lower bounds for a binary alphabet. For every operation, we also study the unary case.
On the state complexity of reversals of regular languages
2004
We compare the number of states between minimal deterministic ÿnite automata accepting a regular language and its reversal (mirror image). In the worst case the state complexity of the reversal is 2 n for an n-state language. We present several classes of languages where this maximal blow-up is actually achieved and study the conditions for it. In the case of ÿnite languages the maximal blow-up is not possible but still a surprising variety of di erent growth types can be exhibited.
Incomplete Transition Complexity of Basic Operations on Finite Languages
Lecture Notes in Computer Science, 2013
The state complexity of basic operations on finite languages (considering complete DFAs) has been in studied the literature. In this paper we study the incomplete (deterministic) state and transition complexity on finite languages of boolean operations, concatenation, star, and reversal. For all operations we give tight upper bounds for both descriptional measures. We correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right automaton is larger than the left one. For all binary operations the tightness is proved using family languages with a variable alphabet size. In general the operational complexities depend not only on the complexities of the operands but also on other refined measures.
State complexity of some operations on binary regular languages
Theoretical Computer Science, 2005
We investigate the state complexity of some operations on binary regular languages. In particular, we consider the concatenation of languages represented by deterministic finite automata, and the reversal and complementation of languages represented by nondeterministic finite automata. We prove that the upper bounds on the state complexity of these operations, which were known to be tight for larger alphabets, are tight also for binary alphabets.
State Complexity of Basic Operations Combined with Reversal
2007
We study the state complexity of boolean operations, concatenation, and star, with one or two of the argument languages reversed. We derive tight upper bounds for the symmetric differences and differences of such languages. We prove that the previously discovered bounds for union, intersection, concatenation and star of such languages can all be met by the recently introduced universal witness and its variants.
Incomplete operational transition complexity of regular languages
Information and Computation, 2015
The state complexity of basic operations on regular languages considering complete deterministic finite automata (DFA) has been extensively studied in the literature. But, if incomplete DFAs are considered, transition complexity is also an significant measure. In this paper we study the incomplete (deterministic) state and transition complexity of some operations for regular and finite languages. For regular languages we give a new tight upper bound for the transition complexity of the union, which refutes the conjecture presented by Y. Gao et al.. For finite languages, we correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right operand is larger than the left one. We also present some experimental results to test the behaviour of those operations on the average case, and we conjecture that for many operations and in practical applications the worst-case complexity is seldom reached.
The Operational Incomplete Transition Complexity on Finite Languages
2013
The state complexity of basic operations on finite languages (considering complete DFAs) has been extensively studied in the literature. In this paper we study the incomplete (deterministic) state and transition complexity on finite languages of boolean operations, concatenation, star, and reversal. For all operations we give tight upper bounds for both descriptional measures. We correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right automaton is larger than the left one. For all binary operations the tightness is proved using family languages with a variable alphabet size. In general the operational complexities depend not only on the complexities of the operands but also on other refined measures.
A Review on State Complexity of Individual Operations
Abstract. The state complexity of a regular language is the number of states of its minimal determinitisc finite automaton. The complexity of a language operation is the complexity of the resulting language seen as a function of the complexities of the operation arguments. In this report we review some of the results of state complexity of individual operations for regular and some subregular languages.
A Survey on Operational State Complexity
arXiv (Cornell University), 2015
Descriptional complexity is the study of the conciseness of the various models representing formal languages. The state complexity of a regular language is the size, measured by the number of states of the smallest, either deterministic or nondeterministic, finite automaton that recognises it. Operational state complexity is the study of the state complexity of operations over languages. In this survey, we review the state complexities of individual regularity preserving language operations on regular and some subregular languages. Then we revisit the state complexities of the combination of individual operations. We also review methods of estimation and approximation of state complexity of more complex combined operations.
Lecture Notes in Computer Science, 2013
The tight upper bound on the state complexity of the reverse of R-trivial and J -trivial regular languages of the state complexity n is 2 n−1 . The witness is ternary for R-trivial regular languages and (n − 1)ary for J -trivial regular languages. In this paper, we prove that the bound can be met neither by a binary R-trivial regular language nor by a J -trivial regular language over an (n − 2)-element alphabet. We provide a characterization of tight bounds for R-trivial regular languages depending on the state complexity of the language and the size of its alphabet. We show the tight bound for J -trivial regular languages over an (n − 2)-element alphabet and a few tight bounds for binary J -trivial regular languages. The case of J -trivial regular languages over an (n−k)element alphabet, for 2 ≤ k ≤ n − 3, is open.