16 research outputs found
Iterated uniform finite-state transducers
A deterministic iterated uniform finite-state transducer (for short, iufst) operates the same length-preserving transduction on several left-to-right sweeps. The first sweep occurs on the input string, while any other sweep processes the output of the previous one. We focus on constant sweep bounded iufsts. We study their descriptional power vs. deterministic finite automata, and the state cost of implementing language operations. Then, we focus on non-constant sweep bounded iufsts, showing a nonregular language hierarchy depending on sweep complexity
Descriptional Complexity of Finite Automata -- Selected Highlights
The state complexity, respectively, nondeterministic state complexity of a
regular language is the number of states of the minimal deterministic,
respectively, of a minimal nondeterministic finite automaton for . Some of
the most studied state complexity questions deal with size comparisons of
nondeterministic finite automata of differing degree of ambiguity. More
generally, if for a regular language we compare the size of description by a
finite automaton and by a more powerful language definition mechanism, such as
a context-free grammar, we encounter non-recursive trade-offs. Operational
state complexity studies the state complexity of the language resulting from a
regularity preserving operation as a function of the complexity of the argument
languages. Determining the state complexity of combined operations is generally
challenging and for general combinations of operations that include
intersection and marked concatenation it is uncomputable
Optimal Regular Expressions for Palindromes of Given Length
The language P_n (P?_n, respectively) consists of all words that are palindromes of length 2n (2n-1, respectively) over a fixed binary alphabet. We construct a regular expression that specifies P_n (P?_n, respectively) of alphabetic width 4? 2?-4 (3? 2?-4, respectively) and show that this is optimal, that is, the expression has minimum alphabetic width among all expressions that describe P_n (P?_n, respectively). To this end we give optimal expressions for the first k palindromes in lexicographic order of odd and even length, proving that the optimal bound is 2n+4(k-1)-2 S?(k-1) in case of odd length and 2n+3(k-1)-2 S?(k-1)-1 for even length, respectively. Here S?(n) refers to the Hamming weight function, which denotes the number of ones in the binary expansion of the number n
Lower bounds for the state complexity of probabilistic languages and the language of prime numbers
This paper studies the complexity of languages of finite words using automata
theory. To go beyond the class of regular languages, we consider infinite
automata and the notion of state complexity defined by Karp. Motivated by the
seminal paper of Rabin from 1963 introducing probabilistic automata, we study
the (deterministic) state complexity of probabilistic languages and prove that
probabilistic languages can have arbitrarily high deterministic state
complexity. We then look at alternating automata as introduced by Chandra,
Kozen and Stockmeyer: such machines run independent computations on the word
and gather their answers through boolean combinations. We devise a lower bound
technique relying on boundedly generated lattices of languages, and give two
applications of this technique. The first is a hierarchy theorem, stating that
there are languages of arbitrarily high polynomial alternating state
complexity, and the second is a linear lower bound on the alternating state
complexity of the prime numbers written in binary. This second result
strengthens a result of Hartmanis and Shank from 1968, which implies an
exponentially worse lower bound for the same model.Comment: Submitted to the Journal of Logic and Computation, Special Issue on
LFCS'2016) (Logical Foundations of Computer Science). Guest Editors: S.
Artemov and A. Nerode. This journal version extends two conference papers:
the first published in the proceedings of LFCS'2016 and the second in the
proceedings of LICS'2018. arXiv admin note: substantial text overlap with
arXiv:1607.0025