BIBLIOGRAPHY,  synchronization @  TESTAS

  Download the freeware version for WINDOWS:
TESTAS (zip file, about 200 KB)


The package includes three main and some auxiliary programs. The main programs:
1) analyze an automaton of the language presented as oriented labeled graph (local testability and generalizations, piecewise testability, their levels);
2) find syntactic semigroup of the automaton, check the syntactic semigroup of automaton for the properties from 1)
3)view visual image of the graph (or subgraph) based on its structure properties (linear procedure),

4)check the synchronizabity, find synchronizing words,
5)find road coloring of DFA

The auxiliary programs (not included in DEMO version) find:
1) direct product of two transition graphs of the automaton;
2) direct product of two syntactic semigroups of the automaton 


      Bibliography  (Cerny conjecture, road coloring, synchronizing word)



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A word w is called synchronizing (recurrent, reset, directable) word of an automaton if w sends all states of the automaton on a unique state.

A deterministic finite automaton (DFA) with complete state transition graph G and transition semigroup S is considered. The problem of synchronization of DFA is natural and various aspects of this problem were touched upon the literature. Synchronization makes the behavior of an automaton resistant against input errors since, after detection of an error, a synchronizing word can reset the automaton back to its original state, as if no error had occurred. Therefore different problems of synchronization draw the attention.

A problem with a long story is the estimation of the minimal length of synchronizing word. Most known as a Cerny conjecture, it was aroused independently by distinct authors. Jan Cerny had found in 1964 n-state complete DFA with shortest synchronizing word of length (n-1)2. He had conjectured that it is an upper bound for the length of the shortest synchronizing word for any n-state complete DFA. The best known upper bound is now equal to (n3-n)/6. The conjecture holds true for wide class of automata, but in general the problem remains open. This simply looking conjecture is now one of the most longstanding open problems in the theory of finite automata.

The existence of some non-trivial subgroup in the transition semigroup of the automaton is essential in many investigation of Cerny conjecture. An opposite approach confirms the conjecture for automata having syntactic semigroup without non-trivial subgroups and proved to be fruitful:

The synchronizing DFA has a synchronizing word of length not greater than n(n-1)/2 for automata with syntactic semigroup having only trivial subgroups and therefore the Cerny conjecture holds true for such DFA.

The testing of synchronizing automata is an indispensable part of investigation in this area.
The best known to date algorithm of Eppstein finds a synchronizing word for n-state DFA in O(n3+n2q) time. The integer q denotes here the size of the input alphabet.

The C/C++ compact package TESTAS is mentioned on home pages of several prominent CONFERENCES among the systems for manipulating automata.
We realize in the package two modifications of Eppstein algorithm called cyclic and semigroup algorithms. For wide class of automata the time complexity of semigroup algorithms is O(n2q). There are no examples of automata for the time being such that the length of the shortest synchronizing word is greater than (n-1)2. The length of synchronizing word found by the algorithm was not greater than n2 in all considered to date cases including exotic examples of Cerny, Kari, Roman and all n-state automata for n<11.
In the class of automata for n<11 with alphabet of size 2 and for n<8 with alphabet of size < 5, all automata with shortest synchronizing word of (n-1)2 are found. There are 8 such automata out of Cerny sequence and five of them are found by the package TESTAS.
The theoretical possibility of the existence of examples with length of the minimal synchronizing word essentially greater than n2 is doubtful and connected mostly with disproof of the conjecture of Cerny. We consider the last algorithm as almost quadratic.

An algorithm to verify synchronizability of deterministic finite automaton has order O(n2) (in the worst case). The package TESTAS contains also Eppstein algorithm for finding synchronizing word, an algorithm for finding synchronizing word based on the results of Frankl and Klyachko-Rystsov-Spivak of O(n4q) time complexity and straightforward algorithm for finding synchronizing word of minimal length.

An automaton with transition graph G is synchronizing iff G2 has a sink state.

The last statement is a base for an algorithm to verify synchronizability of deterministic finite automaton of order O(n2) in the most worst case. The algorithm is subquadratic.

The comparison of the experimental data suggests that the length of the synchronizing word found by considered algorithms is normally much closer to the length of synchronizing word of the minimal length than to the above-mentioned theoretical upper bound. The results of the algorithms altogether correspond to the Cerny conjecture.