14 research outputs found
Brzozowski Algorithm Is Generically Super-Polynomial Deterministic Automata
International audienceWe study the number of states of the minimal automaton of the mirror of a rational language recognized by a random deterministic automaton with n states. We prove that, for any d > 0, the probability that this number of states is greater than nd tends to 1 as n tends to infinity. As a consequence, the generic and average complexities of Brzozowski minimization algorithm are super-polynomial for the uniform distribution on deterministic automata
A New Technique for Reachability of States in Concatenation Automata
We present a new technique for demonstrating the reachability of states in
deterministic finite automata representing the concatenation of two languages.
Such demonstrations are a necessary step in establishing the state complexity
of the concatenation of two languages, and thus in establishing the state
complexity of concatenation as an operation. Typically, ad-hoc induction
arguments are used to show particular states are reachable in concatenation
automata. We prove some results that seem to capture the essence of many of
these induction arguments. Using these results, reachability proofs in
concatenation automata can often be done more simply and without using
induction directly.Comment: 23 pages, 1 table. Added missing affiliation/funding informatio
On the Average Complexity of Moore's State Minimization Algorithm
We prove that, for any arbitrary finite alphabet and for the uniform
distribution over deterministic and accessible automata with n states, the
average complexity of Moore's state minimization algorithm is in O(n log n).
Moreover this bound is tight in the case of unary utomata
Asymptotic enumeration of Minimal Automata
We determine the asymptotic proportion of minimal automata, within n-state
accessible deterministic complete automata over a k-letter alphabet, with the
uniform distribution over the possible transition structures, and a binomial
distribution over terminal states, with arbitrary parameter b. It turns out
that a fraction ~ 1-C(k,b) n^{-k+2} of automata is minimal, with C(k,b) a
function, explicitly determined, involving the solution of a transcendental
equation.Comment: 12+5 pages, 2 figures, submitted to STACS 201
State Complexity of Combined Operations on Finite Languages
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
Advanced Topics on State Complexity of Combined Operations
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