8 research outputs found
DFAs and PFAs with Long Shortest Synchronizing Word Length
It was conjectured by \v{C}ern\'y in 1964, that a synchronizing DFA on
states always has a shortest synchronizing word of length at most ,
and he gave a sequence of DFAs for which this bound is reached. Until now a
full analysis of all DFAs reaching this bound was only given for ,
and with bounds on the number of symbols for . Here we give the full
analysis for , without bounds on the number of symbols.
For PFAs the bound is much higher. For we do a similar analysis as
for DFAs and find the maximal shortest synchronizing word lengths, exceeding
for . For arbitrary n we give a construction of a PFA on
three symbols with exponential shortest synchronizing word length, giving
significantly better bounds than earlier exponential constructions. We give a
transformation of this PFA to a PFA on two symbols keeping exponential shortest
synchronizing word length, yielding a better bound than applying a similar
known transformation.Comment: 16 pages, 2 figures source code adde
On the Number of Synchronizing Colorings of Digraphs
We deal with -out-regular directed multigraphs with loops (called simply
\emph{digraphs}). The edges of such a digraph can be colored by elements of
some fixed -element set in such a way that outgoing edges of every vertex
have different colors. Such a coloring corresponds naturally to an automaton.
The road coloring theorem states that every primitive digraph has a
synchronizing coloring.
In the present paper we study how many synchronizing colorings can exist for
a digraph with vertices. We performed an extensive experimental
investigation of digraphs with small number of vertices. This was done by using
our dedicated algorithm exhaustively enumerating all small digraphs. We also
present a series of digraphs whose fraction of synchronizing colorings is equal
to , for every and the number of vertices large enough.
On the basis of our results we state several conjectures and open problems.
In particular, we conjecture that is the smallest possible fraction of
synchronizing colorings, except for a single exceptional example on 6 vertices
for .Comment: CIAA 2015. The final publication is available at
http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1
Synchronization Problems in Automata without Non-trivial Cycles
We study the computational complexity of various problems related to
synchronization of weakly acyclic automata, a subclass of widely studied
aperiodic automata. We provide upper and lower bounds on the length of a
shortest word synchronizing a weakly acyclic automaton or, more generally, a
subset of its states, and show that the problem of approximating this length is
hard. We investigate the complexity of finding a synchronizing set of states of
maximum size. We also show inapproximability of the problem of computing the
rank of a subset of states in a binary weakly acyclic automaton and prove that
several problems related to recognizing a synchronizing subset of states in
such automata are NP-complete.Comment: Extended and corrected version, including arXiv:1608.00889.
Conference version was published at CIAA 2017, LNCS vol. 10329, pages
188-200, 201
Proposals for tendering Gourock to Dunoon ferry services
Title from cover. Two items in a folder: 'Proposals for tendering Gourock to Dunoon ferry services: a consultation paper' and ' Proposals for tendering Gourock to Dunoon ferry services: draft invitation to tender for consultation'Available from British Library Document Supply Centre- DSC:m03/19822 / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo