Abstract Finite automata are equivalent to right-linear context-free grammars, also capturing the lowest level of the Chomsky-hierarchy. A literature review introduce the importance of deterministic, nondeterministic, and alternating finite automata. This paper discuss the problems of finite automata. We are interested in the descriptional and computational complexity issues of finite automata in this paper. IntroductionThe concept of alternation was developed which is known as alternating finite automata (AFAs) that is equivalent to DFAs. It is known that NFAs can offer exponential time in the number of states compared with DFAs.

And AFAs can simulated by DFAs with a tight double exponential state bound of 22n. Most of the work done on the descriptional complexity of simulation and by different types of automata and on the computational complexity of decision problems relevant to the finite automata. 2. Descriptional complexity of finite automata simulationsAs regular languages have many representations in the world of finite automata, so it is natural to investigate their representation by different types of automata in order to optimize the space requirements.

We measure the costs of representations in terms of the states of automaton accepting a language. For this simulation problem is defined in this paper, few of them are listed below:In particular, we are interested in simulations between DFAs, NFAs, and AFAs.Any NFA can construct an equivalent DFA which is called as power set construction.

For NFA by DFA simulation: Let n ? 1 and A be the n-state NFA. Then 2n states are sufficient and necessary in the worst case for a DFA to accept L (A).For Boolean automata by AFA simulation: Let n ? 1 and A be the n-state Boolean automaton. Then n + 1 states are sufficient for an AFA to accept L (A).For AFA by DFA simulation. Let n ? 1 and A be the n-state AFA or Boolean automaton. Then 22n states are sufficient and necessary in the worst case for a DFA to accept L (A).For AFA by NNFA simulation: Let n ? 1 and A be the n-state AFA or Boolean automaton.

Then 2n states are sufficient and necessary in the worst case for an NNFA to accept L (A).For Reversed AFA by DFA simulation: Let n ? 1 and A be the n-state AFA or Boolean automaton. Then 2n + 1 states are sufficient and necessary in the worst case for a DFA to accept the reversal of L (A). The reversal of every language accepted by an n-state DFA is accepted by a Boolean automaton with log2 (n) states. The theorem provides basis that the reversal of every n-state DFA language is also accepted by some AFA with log2 (n) states. However, it reveals that log2 (n) + 1 states are sufficient for this purpose.

The NNFA can be simulated by a DFA with at most 2m states, we conclude 2m ? 22n , and that is, the NNFA has at least m ? 2n states.The nondeterministic finite automata with multiple entry states is known as NNFA which can be simulated by an NFA having one more state. The additional state is used as new sole initial state which is connected to the old initial states. For AFA by NFA simulation: Let n ? 1 and A be the n-state AFA or Boolean automaton. Then 2n + 1 states are sufficient for an NFA to accept L (A). It reveals that for reversals of n-state DFA languages we can always achieve an exponential saving of states. Interestingly, this potential gets lost when we consider the n-state DFA languages itself (instead of their reversals). We dealt with simulations whose costs optimality is witnessed by regular languages which may be built over alphabets with two or more letters.

For the particular case of unary regular languages, that is, languages over a single letter alphabet, the situation turned out to be significantly different. For state complexity issues of unary finite automata Landau’s function F (n) = max {lcm (x1. . .

xk) | x1, . . . , xk ? 1 and x1 +• • •+xk = n }, which gives the maximal order of the cyclic subgroups of the symmetric group on n elements, plays a crucial role. Here, lcm denotes the least common multiple. Since F depends on the irregular distribution of the prime numbers.

The asymptotic growth rate are:limn?? (ln F (n)/?n • ln n) = 1was determined, which implies the rough estimate F (n) ? e? (?n•ln n). 3. Computational complexity of some decision problems for finite automataThis paper considered problem in this section are all decidable. These problems have finite automata as inputs. Therefore an appropriate coding function is needed which maps a finite automaton A and a string w to a word (A, w) over a fixed alphabet.3.1. The fixed and general membership problemThe former problem is device independent by definition and is commonly referred to in the literature as the fixed membership problem for regular languages:Fix a finite automaton A.

A natural generalization is the general membership problem, in which a finite automaton A and a word w, i.e. a suitable coding (A,w).The fixed membership problem for regular languages reduces to the general membership problem for any suitable class of automata. The complexity of the general membership problem may depend on the given language descriptor. The problem is NL-complete for NFAs and becomes P-complete for AFAs For General membership: The general membership problem for DFAs is L-complete with respect to constant depth reducibility. Moreover, the problem is NL-complete for NFAs and becomes P-complete for AFAs.

For Fixed membership: The fixed membership problem for regular languages isNC1-complete with respect to constant depth reducibility.The fixed membership problem for regular languages recognized by aperiodic monoids belongs to AC0.The fixed membership problem for regular languages recognized by solvable monoids belongs to ACC0.3.2. Emptiness, universality, equivalence, and related problemsSome non-trivial properties for problems on DFAs, NFAs, and AFAs. The non-emptiness problem for NFAs .It is easy to see that emptiness reduces to non-universality if the automata class are log space effectively closed under arbitrary homomorphism and concatenation with regular languages.

The complexity of the emptiness problem for finite automata, if automata are log space effectively closed under intersection with regular sets, then the general membership log space many-one reduces to the nonemptiness problem for the same type of automata.The non-emptiness problem for NFAs and DFAs is NL-complete, and it is PSPACE-complete for AFAs. The results remain valid for automata accepting unary languages, except for DFAs accepting unary languages, whose non-emptiness problem becomes L-complete.The universality problem for DFAs is NL-complete, and for NFAs and AFAs it is PSPACE-complete. For automata accepting unary languages, the universality problem is L-complete for DFAs, co NP-complete for NFAs, and PSPACE complete for AFAs.The equivalence problem for DFAs is NL-complete, and for NFAs and AFAs it is PSPACE-complete. For automata accepting unary languages, the equivalence problem is L-complete for DFAs, coNP-complete for NFAs, and PSPACE complete for AFAs.Finally, we summarize some results on the operation problem from the computational complexity perspective.

Let For Minimization: The DFA-to-DFA minimization problem is NL-complete, while the NFA to NFA minimization problem is PSPACE-complete, even if the input is given as a deterministic finite automaton. The AFA-to-AFA minimization problem is PSPACE-complete, too.References1. Elena Czeizler, Eugen Czeizler, On the descriptional complexity of Watson_Crick automata, in 2009.

2. Tania K. Roblot, Finite-State Descriptional Complexity, in 2009.