Complementing Semi-Deterministic
Büchi Automata
František Blahoudek1 Matthias Heizmann2 Sven Schewe3
Jan Strejček1 Ming-Hsien Tsai4
1
Masaryk University, Brno, Czech Republic
2
University of Freiburg, Germany
3
University of Liverpool, UK
4
Institute of Information Science, Academia Sinica, Taiwan
September 9th 2016
HIGHLIGHTS 2016
Motivation
I termination analysis in Ultimate Büchi Automizer
1/4
Motivation
I termination analysis in Ultimate Büchi Automizer
A
Σω
L (A)
L (A)
1/4
Motivation
I termination analysis in Ultimate Büchi Automizer
A {A
all runs of A
are rejecting
L (A) = L (A)
Σω
L (A)
1/4
Motivation
I termination analysis in Ultimate Büchi Automizer
A { A (0.76n)n
all runs of A
are rejecting
L (A) = L (A)
Σω
L (A)
I complementation of BA is expensive
1/4
Motivation
I termination analysis in Ultimate Büchi Automizer
A { A (0.76n)n
all runs of A
are rejecting
L (A) = L (A)
Σω
L (A)
I complementation of BA is expensive
I special shape of some automata
More effective
complementation?
1/4
Semi-Detetrministic Büchi Automata (SDBA)
δt
Q2 , δ2
Q1 , δ1
2/4
Semi-Detetrministic Büchi Automata (SDBA)
δt
Q2 , δ2
Q1 , δ1
no accepting states
deterministic
2/4
Semi-Detetrministic Büchi Automata (SDBA)
δt
Q2 , δ2
Q1 , δ1
no accepting states
deterministic
A run of an SDBA
1
blocks,
2/4
Semi-Detetrministic Büchi Automata (SDBA)
δt
Q2 , δ2
Q1 , δ1
no accepting states
deterministic
A run of an SDBA
1
blocks,
2
stays in Q1 ,
2/4
Semi-Detetrministic Büchi Automata (SDBA)
δt
Q2 , δ2
Q1 , δ1
no accepting states
deterministic
A run of an SDBA
1
blocks,
2
stays in Q1 ,
3
is trapped in non-accepting states of Q2 , or
2/4
Semi-Detetrministic Büchi Automata (SDBA)
δt
Q2 , δ2
Q1 , δ1
no accepting states
deterministic
A run of an SDBA
1
blocks,
2
stays in Q1 ,
3
is trapped in non-accepting states of Q2 , or
Safe
2/4
Semi-Detetrministic Büchi Automata (SDBA)
δt
Q2 , δ2
Q1 , δ1
no accepting states
deterministic
A run of an SDBA
1
blocks,
2
stays in Q1 ,
3
is trapped in non-accepting states of Q2 , or
4
is accepting.
Safe
2/4
NCSB complementation for SDBA
follows all possible runs of A
1
blocks
2
stays in Q1
3
becomes safe
4
accepting
3/4
NCSB complementation for SDBA
follows all possible runs of A
1
blocks
2
stays in Q1
3
becomes safe
4
accepting
I merely extended breakpoint construction
I less then 4n states
I low degree of non-determinism
3/4
NCSB complementation for SDBA
follows all possible runs of A
1
blocks
2
stays in Q1
3
becomes safe
4
accepting
I
I
I
I
I
merely extended breakpoint construction
less then 4n states
low degree of non-determinism
unambiguous automata
suitable for on-the-fly implementation
3/4
NCSB complementation for SDBA
follows all possible runs of A
1
blocks
2
stays in Q1
3
becomes safe
4
accepting
I
I
I
I
I
I
merely extended breakpoint construction
less then 4n states
low degree of non-determinism
unambiguous automata
suitable for on-the-fly implementation
improves the performance of Ultimate Büchi Automizer
3/4
NCSB complementation for SDBA
follows all possible runs of A
1
blocks
2
stays in Q1
3
becomes safe
4
accepting
I
I
I
I
I
I
I
merely extended breakpoint construction
less then 4n states
low degree of non-determinism
unambiguous automata
suitable for on-the-fly implementation
improves the performance of Ultimate Büchi Automizer
can be seen as highly optimized rank-based construction
3/4
Evaluation
I GOAL
I 91 SDBA from the Termination category of SV-COMP 2015
construction
Ramsey-based
Rank-based
Determinization-based
Slice-based
NCSB
states
transitions
4/4
Evaluation
I GOAL
I 91 SDBA from the Termination category of SV-COMP 2015
construction
Ramsey-based
Rank-based
Determinization-based
Slice-based
NCSB
states
16909
2703
1841
1392
950
transitions
848969
21095
24964
14783
8003
4/4