State complexity of basic language operations combined with reversal (original) (raw)
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.
Complexity in Prefix-Free Regular Languages
Computing Research Repository, 2010
We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next provide the tight bounds on state complexity of symmetric difference, and deterministic and nondeterministic state complexity of difference and cyclic shift of prefix-free languages.
Incomplete transition complexity of some basic operations
2013
Abstract. Y. Gao et al. studied for the first time the transition complexity of Boolean operations on regular languages based on not necessarily complete DFAs. For the intersection and the complementation, tight bounds were presented, but for the union operation the upper and lower bounds differ by a factor of two. In this paper we continue this study by giving tight upper bounds for the concatenation, the Kleene star and the reversal operations.
On inverse operations and their descriptional complexity
We investigate the descriptional complexity of some inverse language operations applied to languages accepted by finite automata. For instance, the inverse Kleene star operation for a language L asks for the smallest language S such that S * is equal to L, if it exists [J. Brzozowski. Roots of star events. J. ACM 14, 1967]. Other inverse operations based on the chop operation or on insertion/deletion operations can be defined appropriately. We present a general framework, that allows us to give an easy characterization of inverse operations, whenever simple conditions on the originally considered language operation are fulfilled. It turns out, that in most cases we obtain exponential upper and lower bounds that are asymptotically close, for the investigated inverse language operation problems.
Estimation of state complexity of combined operations
Theoretical Computer Science, 2009
It appears that the state complexity of each combined operation has its own special features. Thus, it is important and practical to obtain good estimates for some commonly used general cases. In this paper, we consider the state complexity of combined Boolean operations and give an exact bound for all of them in the case when the alphabet is not fixed. Moreover, we show that for any fixed alphabet, this bound can be reached in infinitely many cases. We also consider the state complexity of multiple catenations. The state complexities are obtained in the cases of the catenations of three and four languages. An estimate for the catenation of an arbitrary number of languages is given, which is very close to the state complexities in the three and four languages cases.
On the State Complexity of Operations on Two-Way Finite Automata
Developments in Language Theory
The paper investigates the effect of basic language-theoretic operations on the number of states in two-way deterministic finite automata (2DFAs). If m and n are the number of states in the 2DFAs recognizing the arguments of the following operations, then their result requires the following number of states: at least m + n − o(m + n) and at most 4m + n + const for union; at least m + n − o(m + n) and at most m + n + 1 for intersection; at least (m n) + 2 (n) log m and at most 2m m+1 • 2 n n+1 for concatenation; at least 1 n 2 n 2 −1 and at most 2 O (n n+1) for Kleene star, square and projections; between n + 1 and n + 2 for reversal; exactly 2n for inverse homomorphisms. All results are obtained by first establishing high lower bounds on the number of states in any 1DFAs recognizing these languages, and then using these bounds to reason about the size of any equivalent 2DFAs.