5 research outputs found
Invariance: a Theoretical Approach for Coding Sets of Words Modulo Literal (Anti)Morphisms
Let be a finite or countable alphabet and let be literal
(anti)morphism onto (by definition, such a correspondence is determinated
by a permutation of the alphabet). This paper deals with sets which are
invariant under (-invariant for short).We establish an
extension of the famous defect theorem. Moreover, we prove that for the
so-called thin -invariant codes, maximality and completeness are two
equivalent notions. We prove that a similar property holds in the framework of
some special families of -invariant codes such as prefix (bifix) codes,
codes with a finite deciphering delay, uniformly synchronized codes and
circular codes. For a special class of involutive antimorphisms, we prove that
any regular -invariant code may be embedded into a complete one.Comment: To appear in Acts of WORDS 201
Synchronizing Strongly Connected Partial DFAs
We study synchronizing partial DFAs, which extend the classical concept of
synchronizing complete DFAs and are a special case of synchronizing unambiguous
NFAs. A partial DFA is called synchronizing if it has a word (called a reset
word) whose action brings a non-empty subset of states to a unique state and is
undefined for all other states. While in the general case the problem of
checking whether a partial DFA is synchronizing is PSPACE-complete, we show
that in the strongly connected case this problem can be efficiently reduced to
the same problem for a complete DFA. Using combinatorial, algebraic, and formal
languages methods, we develop techniques that relate main synchronization
problems for strongly connected partial DFAs with the same problems for
complete DFAs. In particular, this includes the \v{C}ern\'{y} and the rank
conjectures, the problem of finding a reset word, and upper bounds on the
length of the shortest reset words of literal automata of finite prefix codes.
We conclude that solving fundamental synchronization problems is equally hard
in both models, as an essential improvement of the results for one model
implies an improvement for the other.Comment: Full version of the paper at STACS 202
On maximal codes with bounded synchronization delay
AbstractWe give an algorithm to embed any rational code with bounded synchronization delay into a maximal one