3 research outputs found
The state complexity of random DFAs
The state complexity of a Deterministic Finite-state automaton (DFA) is the
number of states in its minimal equivalent DFA. We study the state complexity
of random -state DFAs over a -symbol alphabet, drawn uniformly from the
set of all such automata. We show that, with
high probability, the latter is for a certain
explicit constant
Ergodicity of Random Walks on Random DFA
Given a DFA we consider the random walk that starts at the initial state and
at each time step moves to a new state by taking a random transition from the
current state. This paper shows that for typical DFA this random walk induces
an ergodic Markov chain. The notion of typical DFA is formalized by showing
that ergodicity holds with high probability when a DFA is sampled uniformly at
random from the set of all automata with a fixed number of states. We also show
the same result applies to DFA obtained by minimizing typical DFA
Diameter and Stationary Distribution of Random -out Digraphs
Let be a random -out regular directed multigraph on the set of
vertices . In this work, we establish that for every ,
there exists such that
. Our techniques also allow us to
bound some extremal quantities related to the stationary distribution of a
simple random walk on . In particular, we determine the asymptotic
behaviour of and , the maximum and the minimum values
of the stationary distribution. We show that with high probability and . Our proof shows that the
vertices with near to lie at the top of "narrow, slippery
towers", such vertices are also responsible for increasing the diameter from
to .Comment: 31 page