Decidability of Weak Simulation
on One-Counter Nets
Piotr Hofman1
Richard Mayr2
University of Warsaw1
Patrick Totzke2
University of Edinburgh2
June 22, 2013
Context
Monotonicity
Proof Technique
One-Counter Nets
(Q, Act, δ)
δ ⊆ (Q × Act × {−1, 0, +1} × Q)
a, +1
p
q
Summary
Context
Monotonicity
Proof Technique
One-Counter Nets
(Q, Act, δ)
δ ⊆ (Q × Act × {−1, 0, +1} × Q)
a, +1
p
q
Induced LTS over Q × N
q0
q1
a
p0
q2
a
p1
q3
a
p2
p3
Summary
Context
Monotonicity
Proof Technique
One-Counter Nets
(Q, Act, δ)
δ ⊆ (Q × Act × {−1, 0, +1} × Q)
a, +1
b, −1
p
q
Induced LTS over Q × N
q0
b
q1
a
p0
b
q2
a
p1
b
q3
a
p2
p3
Summary
Context
Monotonicity
Proof Technique
Summary
One-Counter Nets
(Q, Act, δ)
δ ⊆ (Q × Act × {−1, 0, +1} × Q)
b, −1
a, +1
p
q
c, 0
Induced LTS over Q × N
q0
c
p0
b
a
q1
c
p1
b
a
q2
c
p2
b
a
q3
c
p3
Context
Monotonicity
Proof Technique
Summary
Simulation Games
. . . are played in rounds between Spoiler and Duplicator. If a player
cannot move the other wins. Infinite plays are won by Duplicator.
Context
Monotonicity
Proof Technique
Summary
Simulation Games
. . . are played in rounds between Spoiler and Duplicator. If a player
cannot move the other wins. Infinite plays are won by Duplicator.
In each round
α
vs.
β
1
Spoiler moves from α
2
Duplicator responds from β
3
game continues from α0 vs. β 0
Context
Monotonicity
Proof Technique
Summary
Simulation Games
. . . are played in rounds between Spoiler and Duplicator. If a player
cannot move the other wins. Infinite plays are won by Duplicator.
In each round
α
vs.
β
a
α0
1
Spoiler moves from α
2
Duplicator responds from β
3
game continues from α0 vs. β 0
Context
Monotonicity
Proof Technique
Summary
Simulation Games
. . . are played in rounds between Spoiler and Duplicator. If a player
cannot move the other wins. Infinite plays are won by Duplicator.
In each round
α
vs.
β
a
α0
a
β0
1
Spoiler moves from α
2
Duplicator responds from β
3
game continues from α0 vs. β 0
Context
Monotonicity
Proof Technique
Summary
Simulation Games
. . . are played in rounds between Spoiler and Duplicator. If a player
cannot move the other wins. Infinite plays are won by Duplicator.
In each round
α
vs.
β
a
α0
a
vs.
β0
1
Spoiler moves from α
2
Duplicator responds from β
3
game continues from α0 vs. β 0
Context
Monotonicity
Proof Technique
Summary
Simulation Games
. . . are played in rounds between Spoiler and Duplicator. If a player
cannot move the other wins. Infinite plays are won by Duplicator.
In each round
α
vs.
β
a
α0
a
vs.
β0
1
Spoiler moves from α
2
Duplicator responds from β
3
game continues from α0 vs. β 0
Def: Simulation ( )
α β iff Duplicator has a strategy to win from α vs. β.
Context
Monotonicity
Proof Technique
Summary
Simulation Approximant Games
. . . are played in rounds between Spoiler and Duplicator. If a player
cannot move the other wins.
In round from α, β, i
α
vs.
β
a
α0
a
vs.
β0
1
Spoiler moves from α; picks ordinal j < i
2
Duplicator responds from β
3
game continues from α0 , β 0 , j
Def: Simulation Approximant ( i )
α i β iff Duplicator has a strategy to win from α vs. β.
Context
Monotonicity
Proof Technique
Summary
Weak Notions
Weak Steps (a 6= τ ∈ Act)
τ
τ
=⇒ := −→ ∗
a
τ
a
τ
=⇒ := −→ ∗ −→ −→ ∗
Context
Monotonicity
Proof Technique
Summary
Weak Notions
Weak Steps (a 6= τ ∈ Act)
τ
τ
=⇒ := −→ ∗
a
τ
a
τ
=⇒ := −→ ∗ −→ −→ ∗
Def: Weak Simulation 5 and Approximants 5i
by 2-player games as before where Duplicator makes weak steps. . .
Context
Monotonicity
Proof Technique
Summary
Example
Countdown game
a, 0
a, −1
S
D
a, −1
τ, +1
τ, 0
a, 0
C
B
Context
Monotonicity
Proof Technique
Summary
Example
Countdown game
a, 0
a, −1
S
D
τ, 0
Strong Simulation:
S0 0 D0
a, −1
τ, +1
a, 0
C
B
Context
Monotonicity
Proof Technique
Summary
Example
Countdown game
a, 0
a, −1
S
D
τ, 0
Strong Simulation:
S0 0 D0
S0 61 D0
a, −1
τ, +1
a, 0
C
B
Context
Monotonicity
Proof Technique
Summary
Example
Countdown game
a, 0
a, −1
S
D
τ, 0
Strong Simulation:
S0 0 D0
S0 61 D0
a, −1
τ, +1
a, 0
C
B
Weak Simulation:
S0 5ω D0
Context
Monotonicity
Proof Technique
Summary
Example
Countdown game
a, 0
a, −1
S
D
τ, 0
Strong Simulation:
a, −1
τ, +1
a, 0
C
B
Weak Simulation:
S0 0 D0
S0 5ω D0
S0 61 D0
S0 65ω+1 D0
Context
Monotonicity
Proof Technique
Summary
Example
Countdown game
a, 0
a, −1
S
D
τ, 0
Strong Simulation:
a, −1
τ, +1
a, 0
C
B
Weak Simulation:
S0 0 D0
S0 5ω D0
S0 61 D0
S0 65ω+1 D0
S0 65 D0
Context
Monotonicity
Proof Technique
Our Main Contribution
We show decidability of the
OCN Weak Simulation Problem
Input:
A net N = (Q, Act, δ) and configurations pm, qn.
Question:
pm 5 qn?
Summary
Context
Monotonicity
Proof Technique
Our Main Contribution
We show decidability of the
OCN Weak Simulation Problem
Input:
A net N = (Q, Act, δ) and configurations pm, qn.
Question:
pm 5 qn?
Theorem
For a given net, the relation 5 is effectively semilinear.
Summary
Context
Monotonicity
Proof Technique
Why should you care?
In practice, modelling might use both ∞-states and branching:
network protocols/queues keeping track of their workload
random guesses
Theoretically, surprising:
rare positive result for behavioral preorder that is not finitely
approximable 5 6= 5ω .
goes against the usual ‘finer is easier’ trend
Summary
Context
Monotonicity
Proof Technique
Some Context – Strong Case
PDA
Petri Nets
OCA
OCN
NFA
Summary
Context
Monotonicity
Proof Technique
Summary
Some Context – Strong Case
⊆
undecidable
⊆
undec.
[Hüt94]
undecidable
undec.
[Hüt94]
∼
decidable
∼
undec.
[PJ95]
PDA
Petri Nets
[Sén98]
⊆
undecidable
undecidable
∼
PSPACE-c [Srb09, BGJ10]
OCA
[JMS99]
⊆
undecidable
decidable [AC98],
PSPACE-hard [Srb09]
∼
PSPACE-c [Srb09, BGJ10]
[HMT13]
OCN
NFA
⊆
PSPACE-comp.
P-comp.
∼
P-comp.
Context
Monotonicity
Proof Technique
Summary
Some Context – Weak Case
j
undecidable
5
undecidable
≈
undecidable
j
undecidable
5
undecidable
≈
undecidable
PDA
Petri Nets
j
undec.
[Hüt94]
5
undec.
[Hüt94]
≈
undec.
[PJ95]
OCA
[JMS99]
j
undecidable
5
decidable [HMT13]
PSPACE-hard [Srb09]
≈
undecidable
[HMT13]
OCN
[May03]
NFA
j
PSPACE-comp.
5
P-comp.
≈
P-comp.
Context
Monotonicity
Proof Technique
Monotonicity in Nets
a
a
If pm −→qn Then p(m + 1) −→q(n + 1).
Summary
Context
Monotonicity
Proof Technique
Monotonicity in Nets
a
a
If pm −→qn Then p(m + 1) −→q(n + 1).
If m0 ≤ m Then pm0 pm.
Summary
Context
Monotonicity
Proof Technique
Monotonicity in Nets
a
a
If pm −→qn Then p(m + 1) −→q(n + 1).
If m0 ≤ m Then pm0 pm.
If m0 ≤ m, pm qn and n ≤ n0 Then pm0 qn0 .
Summary
Context
Monotonicity
Proof Technique
Summary
Monotonicity illustrated
(m, n) is black iff pm qn
n
10
5
0
q
p 0
5
10
15
20
m
Context
Monotonicity
Proof Technique
Summary
Monotonicity illustrated
(m, n) is black iff pm qn
n
10
5
0
q
p 0
5
10
15
20
m
Context
Monotonicity
Proof Technique
Summary
Monotonicity illustrated
(m, n) is black iff pm qn
n
10
5
0
q
p 0
5
10
15
20
m
Context
Monotonicity
Proof Technique
Summary
Belt Theorem [JKM00, AC98]
“Every frontier lies in a belt with rational slope”.
n
c2
c3
c4
q
p
c1
m
Context
Monotonicity
Proof Technique
Strong Simulation for OCN
Theorem [JKM00, AC98]
For any given OCN, is an effectively semilinear set.
Summary
Context
Monotonicity
Proof Technique
Summary
Proof of the main result
Symbolic infinite branching
Reduce (OCN 5 OCN)
1
(OCN ω-Net)
Context
Monotonicity
Proof Technique
Summary
Proof of the main result
Symbolic infinite branching
Reduce (OCN 5 OCN)
1
(OCN ω-Net)
Approximants for the new game
∃ finite sequence 0 ⊇ 1 ⊇ 2 ⊇ · · · ⊇ k = 2
Context
Monotonicity
Proof Technique
Summary
Proof of the main result
Symbolic infinite branching
Reduce (OCN 5 OCN)
1
(OCN ω-Net)
Approximants for the new game
2
∃ finite sequence 0 ⊇ 1 ⊇ 2 ⊇ · · · ⊇ k = Compute approximants for finite k
Recursively compute k by reduction to (OCN OCN)
3
Context
Monotonicity
Proof Technique
Summary
Proof of the main result
Symbolic infinite branching
Reduce (OCN 5 OCN)
1
(OCN ω-Net)
Approximants for the new game
2
∃ finite sequence 0 ⊇ 1 ⊇ 2 ⊇ · · · ⊇ k = Compute approximants for finite k
Recursively compute k by reduction to (OCN OCN)
3
Context
Monotonicity
Proof Technique
Symbolic Infinite Branching
ω-Net N = (Q, Act, δ) with transitions
δ ⊆ Q × Act × {−1, 0, 1, ω} × Q
. . . induces LTS over Q × N like OCN. A transition
a, ω
p
q
a
introduces strong steps pm −→qn for any n ≥ m.
Summary
Context
Monotonicity
Proof Technique
Reduction to Strong Simulation (OCN vs. ω-Net)
Lemma
For a OCN N one can construct a OCN M ⊇ N and an ω-net
M 0 ⊇ N where for all configurations pm, qn holds that
pm 5 qn w.r.t. N ⇐⇒ pm qn w.r.t. M, M 0 .
Summary
Context
Monotonicity
Proof Technique
Summary
Reduction to Strong Simulation (OCN vs. ω-Net)
ω-Countdown net
a, −1
τ, 0
D
a, −1
τ, +1
a, 0
C
B
Context
Monotonicity
Proof Technique
Summary
Reduction to Strong Simulation (OCN vs. ω-Net)
ω-Countdown net
a, −1
a, −1
τ, +1
τ, 0
D
a, 0
C
B
a, ω
a, −1
a, −1
a, ω
D
a, ω
C
B
Context
Monotonicity
Proof Technique
Summary
Reduction to Strong Simulation (OCN vs. ω-Net)
ω-Countdown net
a, −1
a, −1
τ, +1
τ, 0
D
a, 0
C
B
a, ω, 0
a, −1, 0
a, −1, 0
a, ω, 1
D
a, ω, 0
C
B
Context
Monotonicity
Proof Technique
Summary
Reduction to Strong Simulation (OCN vs. ω-Net)
ω-Countdown net
a, −1
a, −1
τ, +1
τ, 0
a, 0
D
C
B
a, ω, 0
a, −1, 0
a, −1, 0
a, ω, 1
a, ω, 0
D
x, 0
x, 0
C
a, −1
D
B
x, 0
x, −1
x, +1
a, ω
C
a, −1
a, ω
x, 0
x, 0
x, 0
a, ω
B
x, 0
x, 0
Context
Monotonicity
Proof Technique
Summary
Proof of the main result
Symbolic infinite branching
Reduce (OCN 5 OCN)
1
(OCN ω-Net)
Approximants for the new game
2
∃ finite sequence 0 ⊇ 1 ⊇ 2 ⊇ · · · ⊇ k = Compute approximants for finite k
Recursively compute k by reduction to (OCN OCN)
3
Context
Monotonicity
Proof Technique
Summary
Proof of the main result
Symbolic infinite branching
Reduce (OCN 5 OCN)
1
(OCN ω-Net)
Approximants for the new game
2
∃ finite sequence 0 ⊇ 1 ⊇ 2 ⊇ · · · ⊇ k = Compute approximants for finite k
Recursively compute k by reduction to (OCN OCN)
3
Context
Monotonicity
Proof Technique
Approximants for strong simulation (OCN vs. ω-Net)
β
α
Summary
Context
Monotonicity
Proof Technique
Approximants for strong simulation (OCN vs. ω-Net)
β
α
. . . holds if Duplicator can guarantee to either
survive α (ordinal) rounds or
make an ω-move at least β times.
Summary
Context
Monotonicity
Proof Technique
Summary
Approximants for strong simulation (OCN vs. ω-Net)
β
α
. . . holds if Duplicator can guarantee to either
survive α (ordinal) rounds or
make an ω-move at least β times.
α =
T
β
βα
β =
T
α
βα
Context
Monotonicity
Proof Technique
Summary
Approximants illustrated
00
10
20
01
11
21
0
02
12
22
1
2
1
2
∅
0
Context
Monotonicity
Proof Technique
Summary
Example
(ω · 2)-Countdown game
a, 0
a, −1
S
D
a, −1
a, ω
a, −1
a, ω
C
B
Context
Monotonicity
Proof Technique
Summary
Example
(ω · 2)-Countdown game
a, 0
a, −1
S
D
a, −1
a, ω
S0 2 D0
a, −1
a, ω
C
B
Context
Monotonicity
Proof Technique
Summary
Example
(ω · 2)-Countdown game
a, 0
a, −1
S
D
a, −1
a, ω
S0 2 D0
S0 ω·2 D0
a, −1
a, ω
C
B
Context
Monotonicity
Proof Technique
Summary
Example
(ω · 2)-Countdown game
a, 0
a, −1
S
D
a, −1
a, ω
S0 2 D0
S0 ω·2 D0
S0 63ω·2+1 D0
a, −1
a, ω
C
B
Context
Monotonicity
Proof Technique
Summary
Example
(ω · 2)-Countdown game
a, 0
a, −1
S
D
a, −1
a, ω
a, −1
a, ω
C
B
S0 2 D0
S0 ω·2 D0
S0 63ω·2+1 D0
S0 63 D0
Context
Monotonicity
Proof Technique
Summary
Example
(ω · 2)-Countdown game
a, 0
a, −1
S
D
a, −1
a, ω
a, −1
a, ω
C
B
S0 2 D0
S0 ω·2 D0
S0 63ω·2+1 D0
S0 63 D0
= 3
Context
Monotonicity
Proof Technique
Summary
Example
(ω · 2)-Countdown game
a, 0
a, −1
S
D
a, −1
a, ω
a, −1
a, ω
C
B
S0 2 D0
S0 ω·2 D0
S0 63 D0
S0 63ω·2+1 D0
= 3
Lemma
For any OCN N and ω-Net M, there is k ∈ N such that
= k
Context
Monotonicity
Proof Technique
Summary
Proof of the main result
Symbolic infinite branching
Reduce (OCN 5 OCN)
1
(OCN ω-Net)
Approximants for the new game
2
∃ finite sequence 0 ⊇ 1 ⊇ 2 ⊇ · · · ⊇ k = Compute approximants for finite k
Recursively compute k by reduction to (OCN OCN)
3
Context
Monotonicity
Proof Technique
Summary
Proof of the main result
Symbolic infinite branching
Reduce (OCN 5 OCN)
1
(OCN ω-Net)
Approximants for the new game
2
∃ finite sequence 0 ⊇ 1 ⊇ 2 ⊇ · · · ⊇ k = Compute approximants for finite k
Recursively compute k by reduction to (OCN OCN)
3
Context
Monotonicity
Proof Technique
Computing k+1
Observation
If a response via −→ ω leads to (game) position pm 6k qn then
pm 6k qn0 for all n0 ∈ N.
Summary
Context
Monotonicity
Proof Technique
Summary
Computing k+1
Observation
If a response via −→ ω leads to (game) position pm 6k qn then
pm 6k qn0 for all n0 ∈ N.
For any pair p, q of states there is a minimal sufficient value m with
pm 6k qn for all n
Context
Monotonicity
Proof Technique
Computing k+1
Compute minimal sufficient values ∈ N ∪ {∞} for all (p, q)
Build gadget nets that test if Spoiler’s counter is sufficient.
Summary
Context
Monotonicity
Proof Technique
Computing k+1
Compute minimal sufficient values ∈ N ∪ {∞} for all (p, q)
Build gadget nets that test if Spoiler’s counter is sufficient.
Use Defenders Forcing to substitute ω-transitions by the
ability to move into testing gadgets.
Summary
Context
Monotonicity
Proof Technique
Computing k+1
Compute minimal sufficient values ∈ N ∪ {∞} for all (p, q)
Build gadget nets that test if Spoiler’s counter is sufficient.
Use Defenders Forcing to substitute ω-transitions by the
ability to move into testing gadgets.
Strong simulation game OCN vs. OCN.
Summary
Context
Monotonicity
Proof Technique
Summary
Proof of the main result
Symbolic infinite branching
Reduce (OCN 5 OCN)
1
(OCN ω-Net)
Approximants for the new game
2
∃ finite sequence 0 ⊇ 1 ⊇ 2 ⊇ · · · ⊇ k = Compute approximants for finite k
Recursively compute k by reduction to (OCN OCN)
3
Context
Monotonicity
Proof Technique
Conclusion
Weak Simulation is decidable for One-Counter Nets
Our proof crucially depends on monotonicity! We
symbolically capture ∞ branching,
derive finite sequence of approximants and
use semilinearity of OCN OCN to compute approximants
and check convergence.
We also consider (weak) trace inclusion for OCN and (weak)
Simulation between OCN and NFA.
Summary
Context
Monotonicity
Proof Technique
Summary
Bibliography
Parosh Aziz Abdulla and Karlis Cerans. Simulation is decidable for one-counter nets. In CONCUR,
pages 253–268, 1998.
Stanislav Böhm, Stefan Göller, and Petr Jančar. Bisimilarity of one-counter processes is
pspace-complete. In CONCUR, pages 177–191, 2010.
P. Hofman, R. Mayr, and P. Totzke. Decidability of weak simulation on one-counter nets. Technical
Report EDI-INF-RR-1415, University of Edinburgh, 2013.
Hans Hüttel. Undecidable equivalences for basic parallel processes. In TACS, pages 454–464, 1994.
Petr Jančar, Antonı́n Kučera, and Faron Moller. Simulation and bisimulation over one-counter
processes. In Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science,
STACS ’00, pages 334–345, London, UK, 2000. Springer-Verlag.
Petr Jančar, Faron Moller, and Zdenek Sawa. Simulation problems for one-counter machines. In
SOFSEM, pages 404–413, 1999.
Richard Mayr. Undecidability of weak bisimulation equivalence for 1-counter processes. In Proceedings
of the 30th international conference on Automata, languages and programming, ICALP’03, pages
570–583, Berlin, Heidelberg, 2003. Springer-Verlag.
P. Petr Jančar. Undecidability of bisimilarity for petri nets and some related problems. Theor. Comput.
Sci., 148(2):281–301, 1995.
Géraud Sénizergues. Decidability of bisimulation equivalence for equational graphs of finite out-degree.
In FOCS, pages 120–129, 1998.
Jirı́ Srba. Beyond language equivalence on visibly pushdown automata. Logical Methods in Computer
Science, 5(1):1–22, 2009.
Context
Monotonicity
Proof Technique
Summary
Bibliography
Parosh Aziz Abdulla and Karlis Cerans. Simulation is decidable for one-counter nets. In CONCUR,
pages 253–268, 1998.
Stanislav Böhm, Stefan Göller, and Petr Jančar. Bisimilarity of one-counter processes is
pspace-complete. In CONCUR, pages 177–191, 2010.
P. Hofman, R. Mayr, and P. Totzke. Decidability of weak simulation on one-counter nets. Technical
Report EDI-INF-RR-1415, University of Edinburgh, 2013.
Hans Hüttel. Undecidable equivalences for basic parallel processes. In TACS, pages 454–464, 1994.
Petr Jančar, Antonı́n Kučera, and Faron Moller. Simulation and bisimulation over one-counter
processes. In Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science,
STACS ’00, pages 334–345, London, UK, 2000. Springer-Verlag.
Questions?
Petr Jančar, Faron Moller, and Zdenek Sawa. Simulation problems for one-counter machines. In
SOFSEM, pages 404–413, 1999.
Richard Mayr. Undecidability of weak bisimulation equivalence for 1-counter processes. In Proceedings
of the 30th international conference on Automata, languages and programming, ICALP’03, pages
570–583, Berlin, Heidelberg, 2003. Springer-Verlag.
P. Petr Jančar. Undecidability of bisimilarity for petri nets and some related problems. Theor. Comput.
Sci., 148(2):281–301, 1995.
Géraud Sénizergues. Decidability of bisimulation equivalence for equational graphs of finite out-degree.
In FOCS, pages 120–129, 1998.
Jirı́ Srba. Beyond language equivalence on visibly pushdown automata. Logical Methods in Computer
Science, 5(1):1–22, 2009.