Checking Whether an Automaton Is Monotonic Is NP

Introduction
NP-completeness
Checking Whether an Automaton Is Monotonic
Is NP-complete
Marek Szykuła
University of Wrocław, Poland
CIAA, 18.08.2015
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Monotonic automata
We deal with complete (semi)automata A = (Q, Σ, δ).
Monotonic automaton
An automaton is monotonic if there exists a linear order on Q
such that:
for every p, q ∈ Q and a ∈ Σ, if p q then δ(p, a) δ(q, a).
Immediately we have also that p q implies δ(p, w ) δ(q, w ),
for every word w ∈ Σ∗ .
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Monotonic automata
We deal with complete (semi)automata A = (Q, Σ, δ).
Monotonic automaton
An automaton is monotonic if there exists a linear order on Q
such that:
for every p, q ∈ Q and a ∈ Σ, if p q then δ(p, a) δ(q, a).
Immediately we have also that p q implies δ(p, w ) δ(q, w ),
for every word w ∈ Σ∗ .
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Monotonic automata
We deal with complete (semi)automata A = (Q, Σ, δ).
Monotonic automaton
An automaton is monotonic if there exists a linear order on Q
such that:
for every p, q ∈ Q and a ∈ Σ, if p q then δ(p, a) δ(q, a).
Immediately we have also that p q implies δ(p, w ) δ(q, w ),
for every word w ∈ Σ∗ .
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Examples of monotonic automata
b
a
a
1
a
2
3
b
b
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Examples of monotonic automata
b
b
a
a
1
a
2
a
3
b
X
a
Y
b
b
b
Marek Szykuła
Monotonic Is NP-complete
a
Z
Introduction
NP-completeness
Monotonic automata
Oriented automata
Examples of monotonic automata
b
b
a
a
1
a
a
3
2
a
X
b
Y
b
b
b
a
b
a
a
b
b
Marek Szykuła
a
Monotonic Is NP-complete
Z
Introduction
NP-completeness
Monotonic automata
Oriented automata
Examples of monotonic automata
b
b
a
a
1
a
a
3
2
a
X
b
a
Y
b
b
b
X
a
b
a
a
Y
Z
b
b
X ≺ Y then (by a) Y ≺ Z , but X ≺ Z implies (by b) Z ≺ Y .
Marek Szykuła
Monotonic Is NP-complete
Z
Introduction
NP-completeness
Monotonic automata
Oriented automata
Monotonic automata
A proper subclass of aperiodic automata
(aperiodic – transformations induced by words do not have
non-trivial cycles).
(Gomes, Howie 1992) Maximal transition semigroups are
isomorphic to the transition semigroup of:
a
b
a
a
1
2
b
This contains all
b
2n−1
n
...
a
n
b
order-preserving transformations.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Oriented automata
Introduced by Eppstein (1990) under the term “monotonic”.
Automata with a cyclic order of the states rather than linear.
This is a wider class containing monotonic automata.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Oriented automata
Definition of an oriented automaton
An automaton is oriented if there is a cyclic order of the states
that is preserved by the actions of all letters.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Oriented automata
Definition of an oriented automaton
An automaton is oriented if there is a cyclic order of the states
that is preserved by the actions of all letters.
[1, 2, 3, 4, 5, 6, 7, 8]
1
8
2
[4, 4, 6, 6, 6, 7, 1, 2]
7
3
6
4
[4, 6, 7, 1, 2]
subsequence of the cyclic order
5
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Oriented automata
Definition of an oriented automaton
An automaton is oriented if there is a cyclic order of the states
that is preserved by the actions of all letters.
1
8
2
7
3
6
This transformation does not
preserve the cyclic order.
4
5
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Motivation
Monotonic automata appears in several contexts (e.g. as
bulding blocks in largest aperiodic semigroups, infiniteness
of the dot-depth hierarchy).
Recognizing languages recognized by monotonic
automata.
Applications in synchronizing automata
Tight bounds for the length of the shortest reset words:
Monotonic: n − 1 (Ananichev, Volkov 2004).
Oriented: (n − 1)2 (Eppstein 1990).
Polynomial algorithms finding a shortest reset word of a
given automaton with a preserved order, which is a hard
problem in general.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Motivation
Monotonic automata appears in several contexts (e.g. as
bulding blocks in largest aperiodic semigroups, infiniteness
of the dot-depth hierarchy).
Recognizing languages recognized by monotonic
automata.
Applications in synchronizing automata
Tight bounds for the length of the shortest reset words:
Monotonic: n − 1 (Ananichev, Volkov 2004).
Oriented: (n − 1)2 (Eppstein 1990).
Polynomial algorithms finding a shortest reset word of a
given automaton with a preserved order, which is a hard
problem in general.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Motivation
Monotonic automata appears in several contexts (e.g. as
bulding blocks in largest aperiodic semigroups, infiniteness
of the dot-depth hierarchy).
Recognizing languages recognized by monotonic
automata.
Applications in synchronizing automata
Tight bounds for the length of the shortest reset words:
Monotonic: n − 1 (Ananichev, Volkov 2004).
Oriented: (n − 1)2 (Eppstein 1990).
Polynomial algorithms finding a shortest reset word of a
given automaton with a preserved order, which is a hard
problem in general.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
Monotonic automata
Oriented automata
Motivation
Monotonic automata appears in several contexts (e.g. as
bulding blocks in largest aperiodic semigroups, infiniteness
of the dot-depth hierarchy).
Recognizing languages recognized by monotonic
automata.
Applications in synchronizing automata
Tight bounds for the length of the shortest reset words:
Monotonic: n − 1 (Ananichev, Volkov 2004).
Oriented: (n − 1)2 (Eppstein 1990).
Polynomial algorithms finding a shortest reset word of a
given automaton with a preserved order, which is a hard
problem in general.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
Theorem
The problem of checking whether a given automaton over at
least binary alphabet is monotonic is NP-complete.
Proof
1
First we reduce from Monotone Not-All-Equal 3SAT.
Like 3SAT, but the formula is satisfied if and only if every
clause contains at least one true and one false literal.
Literals are positive occurrences of variables (there are no
negations).
2
Then we reduce to the problem of recognizing binary
monotonic automata.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
Theorem
The problem of checking whether a given automaton over at
least binary alphabet is monotonic is NP-complete.
Proof
1
First we reduce from Monotone Not-All-Equal 3SAT.
Like 3SAT, but the formula is satisfied if and only if every
clause contains at least one true and one false literal.
Literals are positive occurrences of variables (there are no
negations).
2
Then we reduce to the problem of recognizing binary
monotonic automata.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
Theorem
The problem of checking whether a given automaton over at
least binary alphabet is monotonic is NP-complete.
Proof
1
First we reduce from Monotone Not-All-Equal 3SAT.
Like 3SAT, but the formula is satisfied if and only if every
clause contains at least one true and one false literal.
Literals are positive occurrences of variables (there are no
negations).
2
Then we reduce to the problem of recognizing binary
monotonic automata.
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
The clause gadget
pf
qh
aj
cj
xj
qf
ph
aj
yj
bj
zj
cj
bj
pg
qg
The clause gadget for a j-th clause (vf , vg , vh ).
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
The construction
p1
q1
...
...
pf
qf
...
...
x1
...
xj
...
xk
y1
...
yj
...
yk
z1
...
zj
...
zk
pn
s
qn
The action of the letter aj , where vf is the first variable in Cj .
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
Restriction to binary alphabets
q11
q21
...
qn1
b = a1
a
a
a
a
q12
q22
...
qn2
a
...
a
...
a
a
...
a
a
...
a
a
a2
Marek Szykuła
...
q1k
a
a
q2k
a
...
a
qnk
ak
Monotonic Is NP-complete
s
a
Introduction
NP-completeness
The proof
Conclusions and problems
Theorem
The problem of checking whether a given automaton over at
least binary alphabet is oriented is NP-complete.
1
2
n
3
4
...
5
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
Theorem
The problem of checking whether a given automaton over at
least binary alphabet is oriented is NP-complete.
Σ
1
n+1
2
n
3
This is oriented if and only
the original automaton is monotonic.
4
...
5
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
A note about unary alphabets
Unary alphabets
A unary automaton is monotonic if and only the letter has
no non-trivial cycle.
A unary automaton is oriented if and only if all cycles of the
letter have the same length.
2
3
1
4
5
9
6
8
7
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
Inclusion relations with other subclasses
RESPECTING INTERVALS OF A DIGRAPH
PARTIALLY ORDERABLE
APERIODIC
WEAKLY MONOTONIC
GENERALIZED MONOTONIC
MONOTONIC
Marek Szykuła
Monotonic Is NP-complete
ORIENTED
Introduction
NP-completeness
The proof
Conclusions and problems
Inclusion relations with other subclasses
RESPECTING INTERVALS OF A DIGRAPH
?
PARTIALLY ORDERABLE
P
APERIODIC
PSPACE-complete
WEAKLY MONOTONIC
?
GENERALIZED MONOTONIC
?
MONOTONIC
NP-complete
Marek Szykuła
Monotonic Is NP-complete
ORIENTED
NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
Summary
From the algorithmic point of view, usefulness of
monotonicity and orientability is limited if do not know the
preserved order.
Complexity of recognizing generalized and weakly
monotonic?
Complexity of recognizing automata respecting intervals of
a digraph?
Tack så mycket!
Marek Szykuła
Monotonic Is NP-complete
Introduction
NP-completeness
The proof
Conclusions and problems
Summary
From the algorithmic point of view, usefulness of
monotonicity and orientability is limited if do not know the
preserved order.
Complexity of recognizing generalized and weakly
monotonic?
Complexity of recognizing automata respecting intervals of
a digraph?
Tack så mycket!
Marek Szykuła
Monotonic Is NP-complete

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