Prime Languages
نویسندگان
چکیده
We say that a deterministic finite automaton (DFA) A is composite if there are DFAs A1, . . . ,At such that L(A) = ⋂t i=1 L(Ai) and the index of every Ai is strictly smaller than the index of A. Otherwise, A is prime. We study the problem of deciding whether a given DFA is composite, the number of DFAs required in a decomposition, decompositions that are based on abstractions, methods to prove primality, and structural properties of DFAs that make the problem simpler or are retained in a decomposition. We also provide an algebraic view of the problem and demonstrate its usefulness for the special case of permutation DFAs.
منابع مشابه
On the existence of prime decompositions
We investigate factorizations of regular languages in terms of prime languages. A language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps. We give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a pr...
متن کاملPrime Decompositions of Regular Languages
We investigate factorizations of regular languages in terms of prime languages. A language is said to be strongly prime decomposable if any way of factorizing the language yields a prime decomposition in a finite number of steps. We give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabe...
متن کاملOn the decomposition of finite languages
Representations of nite languages as a product (catenation) of languages are investigated, where the factor languages are \prime", that is, cannot be decomposed further in a nontrivial manner. In general, such prime decompositions are not unique even the number of factors can vary exponentially. The paper investigates the uniqueness of prime decompositions, as well as the commuting of the facto...
متن کاملOutfix-Free Regular Languages and Prime Outfix-Free Decomposition
A string x is an outfix of a string y if there is a string w such that x1wx2 = y and x = x1x2. A set X of strings is outfix-free if no string in X is an outfix of any other string in X . Based on the properties of outfix strings, we develop a polynomial-time algorithm that determines outfix-freeness of regular languages. Note that outfix-free regular languages are always finite. We consider two...
متن کاملOverlap-Free Regular Languages
We define a language to be overlap-free if any two distinct strings in the language do not overlap with each other. We observe that overlap-free languages are a proper subfamily of infix-free languages and also a proper subfamily of comma-free languages. Based on these observations, we design a polynomial-time algorithm that determines overlapfreeness of a regular language. We consider two case...
متن کاملThe Copying Power of One-State Tree Transducers
One-state deterministic top-down tree transducers (or, tree homomorphisms) cannot handle “prime copying,” i.e., their class of output (string) languages is not closed under the operation L --) {$(w%~‘“’ 1 w E L, f(n) > 11, where f is any integer function whose range contains numbers with arbitrarily large prime factors (such as a polynomial). The exact amount of nonclosure under these copying o...
متن کامل