Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Finding DFAs with maximal shortest
synchronizing word length
Henk Don
Radboud University Nijmegen
Joint work with Hans Zantema
LATA 2017, Umeå
1 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Table of contents
1 Introduction
2 Critical DFAs: goal and known results
3 New results for small number of states
4 DFAs on more than 6 states
5 Conclusions
2 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
State set: Q = {1, 2, 3, 4}. Number of states: n = 4.
3 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
State set: Q = {1, 2, 3, 4}. Number of states: n = 4.
Set of symbols: Σ = {a, b}.
3 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
State set: Q = {1, 2, 3, 4}. Number of states: n = 4.
Set of symbols: Σ = {a, b}. Words: e.g. abba.
3 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
State set: Q = {1, 2, 3, 4}. Number of states: n = 4.
Set of symbols: Σ = {a, b}. Words: e.g. abba.
Transitions: e.g. 1a = 2,
3 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
State set: Q = {1, 2, 3, 4}. Number of states: n = 4.
Set of symbols: Σ = {a, b}. Words: e.g. abba.
Transitions: e.g. 1a = 2, 1abba = 3
3 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
State set: Q = {1, 2, 3, 4}. Number of states: n = 4.
Set of symbols: Σ = {a, b}. Words: e.g. abba.
Transitions: e.g. 1a = 2, 1abba = 3 and {1, 4} b = 1.
3 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
Definition (Synchronizing word)
A DFA is synchronizing if there exists a word w ∈ Σ∗ and a
state qs ∈ Q such that qw = qs for all q ∈ Q.
The word w is called a synchronizing word.
4 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
Definition (Synchronizing word)
A DFA is synchronizing if there exists a word w ∈ Σ∗ and a
state qs ∈ Q such that qw = qs for all q ∈ Q.
The word w is called a synchronizing word.
Idea: wherever you start, following the path labelled by w leads to
the fixed state qs . Word w acts as reset button.
4 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Claim: w = baaabaaab is synchronizing.
Let’s check: read w starting from all states simultaneously.
Green means occupied.
5 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: baaabaaab.
6 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: baaabaaab.
7 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: baaabaaab.
8 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: baaabaaab.
9 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: baaabaaab.
10 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: baaabaaab.
11 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: baaabaaab.
12 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: baaabaaab.
13 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: w = baaabaaab, since qw = 1 for all
q = 1, 2, 3, 4.
14 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
a
1
b
2
a
b
4
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
b
a
a
3
b
Synchronizing word: w = baaabaaab, since qw = 1 for all
q = 1, 2, 3, 4.
|w | = 9 and there is no shorter synchronizing word.
14 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
Question: Worst case? How long can a shortest synchronizing
word be?
15 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
Question: Worst case? How long can a shortest synchronizing
word be?
Conjecture (Černý, 1964)
If a DFA is synchronizing, then the shortest synchronizing word has
length at most (n − 1)2 .
15 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Deterministic finite automata
Synchronization
Example by Černý: C4
Černý conjecture
Question: Worst case? How long can a shortest synchronizing
word be?
Conjecture (Černý, 1964)
If a DFA is synchronizing, then the shortest synchronizing word has
length at most (n − 1)2 .
Conjecture still open. Best upper bound is cubic in n.
15 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Critical DFAs
Trahtman’s analysis
Known critical DFAs on 3 states
Goal: identification of basic cricital DFAs
Critical = reaching (n − 1)2 bound
Basic = no identical symbols, no identity
16 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Critical DFAs
Trahtman’s analysis
Known critical DFAs on 3 states
Goal: identification of basic cricital DFAs
Critical = reaching (n − 1)2 bound
Basic = no identical symbols, no identity
Known results:
One construction for all n:
Černý’s sequence Cn .
Right picture: C4 .
4
Henk Don, Radboud University
2
a
b
16 / 36
a
1
b
b
a
a
3
b
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Critical DFAs
Trahtman’s analysis
Known critical DFAs on 3 states
Goal: identification of basic cricital DFAs
Critical = reaching (n − 1)2 bound
Basic = no identical symbols, no identity
Known results:
One construction for all n:
Černý’s sequence Cn .
Right picture: C4 .
A couple of isolated examples
for n = 2, 3, 4, 5, 6 states.
Henk Don, Radboud University
2
a
b
4
16 / 36
a
1
b
b
a
a
3
b
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Critical DFAs
Trahtman’s analysis
Known critical DFAs on 3 states
Trahtman did an extensive analysis for n ≤ 10 and conjectured:
Conjecture (Trahtman, 2006)
Apart from the sequence Cn , only 8 critical DFAs exist for n ≥ 3:
three on 3 states,
three on 4 states,
one on 5 states,
one on 6 states.
Trahtman’s analysis had two restrictions:
Upper bound on the number of symbols: four.
Only considered minimal DFAs: not a proper extension of a
critical DFA.
17 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Critical DFAs
Trahtman’s analysis
Known critical DFAs on 3 states
C3 , synchronizing word: baab.
1
a, b
b
a
2
3
b
a
18 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Critical DFAs
Trahtman’s analysis
Known critical DFAs on 3 states
Trahtman’s first example, synchronizing word: daad.
1
a
a
d
d
2
3
a, d
19 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Critical DFAs
Trahtman’s analysis
Known critical DFAs on 3 states
Trahtman’s second example, synchronizing word: bceb.
e
1
b, c
c
e
b
2
3
b, c
e
20 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Critical DFAs
Trahtman’s analysis
Known critical DFAs on 3 states
Trahtman’s third example, synchronizing word: dced.
e
1
c
c
d
d, e
2
3
c
d, e
21 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Critical DFAs
Trahtman’s analysis
Known critical DFAs on 3 states
Trahtman conjectured: no more critical DFAs exist for n = 3.
Intuition: if you add symbols, it will be easier to synchronize.
But . . .
22 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
e
1
a, b, c
a
c
d
d, e
b
2
3
b, c
a, d, e
23 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
e
1
a, b, c
a
c
d
d, e
b
2
3
b, c
a, d, e
24 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
e
1
a, b, c
a
c
d
d, e
b
2
3
b, c
a, d, e
25 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
e
1
a,b, c
a
c
d
d,e
b
2
3
b, c
a, d,e
26 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
e
1
a, b,c
a
c
d
d, e
b
2
3
b,c
a,d, e
27 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
The automaton A3
is synchronizing, word length 4.
contains all previously known critical examples, the minimal
examples.
28 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
The automaton A3
is synchronizing, word length 4.
contains all previously known critical examples, the minimal
examples.
Theorem
supercritical DFAs don’t exist for n = 3,
28 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
The automaton A3
is synchronizing, word length 4.
contains all previously known critical examples, the minimal
examples.
Theorem
supercritical DFAs don’t exist for n = 3,
A3 contains all critical DFAs for n = 3,
28 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
The automaton A3
is synchronizing, word length 4.
contains all previously known critical examples, the minimal
examples.
Theorem
supercritical DFAs don’t exist for n = 3,
A3 contains all critical DFAs for n = 3,
A DFA contained in A3 is critical if and only if it contains one
of the four minimal examples.
28 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
The automaton A3
is synchronizing, word length 4.
contains all previously known critical examples, the minimal
examples.
Theorem
supercritical DFAs don’t exist for n = 3,
A3 contains all critical DFAs for n = 3,
A DFA contained in A3 is critical if and only if it contains one
of the four minimal examples.
This means we’ve found 11 new critical examples for n = 3.
28 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
For n = 4, detected all critical DFAs, 8 new examples.
29 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
For n = 4, detected all critical DFAs, 8 new examples.
For n = 5, 6, no new examples found.
29 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
For n = 4, detected all critical DFAs, 8 new examples.
For n = 5, 6, no new examples found.
For n ≤ 6 everything checked, unlimited alphabet (for n = 6
alphabet size can be 46655):
29 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Counterexample to Trahtman’s conjecture: A3 .
Conclusions for 3 states
Conclusions for at most 6 states
For n = 4, detected all critical DFAs, 8 new examples.
For n = 5, 6, no new examples found.
For n ≤ 6 everything checked, unlimited alphabet (for n = 6
alphabet size can be 46655):
Theorem
If a DFA on n ≤ 6 states is synchronizing, then its shortest
synchronizing word has length at most (n − 1)2 .
29 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Main result
Idea of the proof
Illustrative example
What about n ≥ 7?
30 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Main result
Idea of the proof
Illustrative example
What about n ≥ 7?
n
Number of DFAs on n states is huge: about 2n .
6
26 ≈ 1014044
30 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Main result
Idea of the proof
Illustrative example
What about n ≥ 7?
n
Number of DFAs on n states is huge: about 2n .
6
26 ≈ 1014044
Look for extensions of known critical DFAs.
30 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Main result
Idea of the proof
Illustrative example
What about n ≥ 7?
n
Number of DFAs on n states is huge: about 2n .
6
26 ≈ 1014044
Look for extensions of known critical DFAs.
Theorem
For n ≥ 5 every possible (basic) extension of Cn has a
synchronizing word of length strictly less than (n − 1)2 .
30 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
This is C10 .
8
3
Red = a, Blue = b.
w = (ban−1 )n−2 b
7
4
6
31 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Main result
Idea of the proof
Illustrative example
Now suppose we add a new symbol c.
To show that it synchronizes faster than (n − 1)2 : find
shortcut in process using c.
Assume c is a permutation (hardest case).
32 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
8
3
C10 extended.
Green = c.
7
4
6
33 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
C10 extended.
8
3
7
4
6
33 / 36
Green = c.
If long forward
c-edge:
Shortcut!
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
C10 extended.
8
3
7
4
6
33 / 36
Green = c.
If long forward
c-edge:
Shortcut!
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
C10 extended.
8
3
Green: c.
Typical example.
7
4
6
34 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
Still shortcut:
8
3
Call this configuration
S.
Start from S with b.
7
4
6
34 / 36
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
8
Still shortcut:
3
Call this configuration
S.
Start from S with b.
7
4
6
34 / 36
Then rotate with
3 × a.
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
2
8
Still shortcut:
3
Call this configuration
S.
Start from S with b.
7
4
6
34 / 36
Then rotate with
3 × a.
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
Still shortcut:
2
Call this configuration
S.
8
3
7
4
6
34 / 36
Start from S with b.
Then rotate with
3 × a.
Finally use 3 × c.
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
Still shortcut:
2
Call this configuration
S.
8
3
7
4
6
34 / 36
Start from S with b.
Then rotate with
3 × a.
Finally use 3 × c.
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
Still shortcut:
2
Call this configuration
S.
8
3
7
4
6
34 / 36
Start from S with b.
Then rotate with
3 × a.
Finally use 3 × c.
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
Still shortcut:
2
Call this configuration
S.
8
3
7
4
6
34 / 36
Start from S with b.
Then rotate with
3 × a.
Finally use 3 × c.
5
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
0
Main result
Idea of the proof
Illustrative example
1
9
Still shortcut:
Call this configuration
S.
2
Start from S with b.
8
3
Then rotate with
3 × a.
Finally use 3 × c.
7
4
6
34 / 36
5
Henk Don, Radboud University
Needed 7 steps
instead of 13 from S
to here.
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Several new critical DFAs detected for small n.
35 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Several new critical DFAs detected for small n.
All are extensions of known examples.
35 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Several new critical DFAs detected for small n.
All are extensions of known examples.
Known examples for n ≥ 5 don’t admit critical extensions.
35 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Several new critical DFAs detected for small n.
All are extensions of known examples.
Known examples for n ≥ 5 don’t admit critical extensions.
Černý conjecture true for n ≤ 6.
Challenge: known minimal critical DFAs have at most 3 symbols.
Prove an upper bound.
35 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length
Introduction
Critical DFAs: goal and known results
New results for small number of states
DFAs on more than 6 states
Conclusions
Thank you!
36 / 36
Henk Don, Radboud University
DFAs with maximal shortest synchronizing word length