slides - University of Edinburgh

Checking Equivalences and Partial Orders 

of One-Counter Processes 

Patrick Totzke 

LFCS, University of Edinburgh 

March 27, 2013

Background Monotonicity Infinite Branching bib 

Equivalences 

van Glabbeek’s Spectrum [Gla01]


Equivalences 

van Glabbeek’s Spectrum [Gla01]


Bisimulation Games 

. . . are played in rounds between Spoiler and Duplicator. If a player 

cannot move the other wins. Infinite plays are won by Duplicator.




cannot move the other wins. Infinite plays are won by Duplicator. 

In each round 

α vs. 

β 

1 Spoiler moves 

2 Duplicator responds 

a 

−→ from one process 

a 

−→ from the other 

3 the game continues from α ′ vs. β ′ .





In each round 

α vs. 

a 

α ′ 

β 



a 


a 







In each round 

α vs. β 

a a 

α ′ β ′ 



a 


a 







In each round 

α vs. β 

a a 

α ′ 

vs. 

β ′ 



a 


a 







In each round 

α vs. β 

a a 

α ′ 

vs. 

β ′ 



a 


a 


3 the game continues from α ′ vs. β ′ . 

Def: 

Bisimulation (∼) 

α ∼ β iff Duplicator has a strategy to win the game from α vs. β.


Simulation Games 



In each round 

α vs. β 

a a 

α ′ 

vs. 

β ′ 



a 

−→ from process α 

a 



Def: Simulation ( ≼ ) 

α ≼ β iff Duplicator has a strategy to win the game from α vs. β.


Trace Inclusion Games 


cannot move the other wins. 

At the beginning Spoiler announces a word a 0 a 1 . . . a k . 

In round i 

α vs. 

β 


a i 

−→ from process α 

a i a i 

vs. β ′ 

α ′ 


a i 



Def: 

Trace Inclusion (⊆) 

α ⊆ β iff Duplicator has a strategy to win the game from α vs. β.


One-Counter Automata 

(Q, Act, δ) δ ⊆ (Q × Act × {−1, 0, +1, = 0} × Q) 

a, +1 

p 

q



(Q, Act, δ) δ ⊆ (Q × Act × {−1, 0, +1, = 0} × Q) 

a, +1 

p 

q 

Induced LTS over Q × N 

q0 

p0 

a 

q1 q2 q3 

a a 

p1 p2 p3



(Q, Act, δ) δ ⊆ (Q × Act × {−1, 0, +1, = 0} × Q) 

a, +1 

b, −1 

p 

q 


b b b 

q0 q1 q2 q3 

p0 

a 

a a 

p1 p2 p3



(Q, Act, δ) δ ⊆ (Q × Act × {−1, 0, +1, = 0} × Q) 

a, +1 

b, −1 

p 

q 

c, 0 


b b b 

q0 q1 q2 q3 

p0 

c 

a 

c 

a 

c 

a 

c 

p1 p2 p3



(Q, Act, δ) δ ⊆ (Q × Act × {−1, 0, +1, = 0} × Q) 

d, = 0 

p 

a, +1 

b, −1 

q 

c, 0 


b b b 

q0 q1 q2 q3 

d 

p0 

c 

a 

c 

a 

c 

a 

c 

p1 p2 p3


One-Counter Nets 

(Q, Act, δ) δ ⊆ (Q × Act × {−1, 0, +1, = 0} × Q) 

d, = 0 

p 

a, +1 

b, −1 

q 

c, 0 


b b b 

q0 q1 q2 q3 

d 

p0 

c 

a 

c 

a 

c 

a 

c 

p1 p2 p3


One-Counter Nets 

(Q, Act, δ) δ ⊆ (Q × Act × {−1, 0, +1} × Q) 

a, +1 

b, −1 

p 

q 

c, 0 


b b b 

q0 q1 q2 q3 

p0 

c 

a 

c 

a 

c 

a 

c 

p1 p2 p3


Some Results 

PDA 

OCA 

OCN 

NFA


Some Results 

⊆ 

≼ 

∼ 

undecidable 

undecidable 

decidable [Sén98] 

PDA 

OCA 

OCN 

NFA


Some Results 

⊆ 

≼ 

∼ 

undecidable 

undecidable 


PDA 

⊆ undecidable [?] 

≼ undecidable [JMS99] 

∼ PSPACE-c [Srb09, BGJ10] 

OCA 

OCN 

NFA


Some Results 

⊆ 

≼ 

∼ 

undecidable 

undecidable 


PDA 




OCA 

⊆ 

≼ 

∼ 

undecidable [HMT13] 

decidable, [AC98, JKM00], 

PSPACE-hard [Srb09] 

PSPACE-c [Srb09, BGJ10] 

OCN 

NFA


Some Results 

⊆ 

≼ 

∼ 

undecidable 

undecidable 


PDA 




OCA 

⊆ 

≼ 

∼ 





OCN 

⊆ 

≼ 

∼ 

PSPACE-comp. 

P-comp. 

P-comp. 

NFA


Some Results 

⊆ 

≼ 

∼ 

undecidable 

undecidable 


PDA 

DPDA 

= decidable [Sén97], P-hard, 

primitive recursive [Sti02] 

⊆ 

undecidable 


≼ 

∼ 

⊆ 

≼ 

∼ 

⊆ 

≼ 

∼ 

undecidable [JMS99] 






PSPACE-comp. 

P-comp. 

P-comp. 

OCA 

OCN 

NFA 

DOCA 

DOCN 

= decidable [VP75] 

NL-complete, [BG11] 

⊆ 

undecidable [VP75] 

= NL-complete 

⊆ 

decidable, NL-hard 

DFA 

=, ⊆ NL-complete


Simulation on One-Counter Nets 

Example? 

Safe assumption: No deadlocks, Spoiler wins only by 

exhausting Duplicators counter. Energy Game with two 

energy levels. . . 

Two proofs for its decidability: [AC98, JM99]


Monotonicity in Nets 

a 

If p(m) −→q(n) Then p(m + 1) −→q(n + 1). 

a



a 


If m ′ ≥ m Then p(m) ≼ p(m ′ ). 

a



a 


If m ′ ≥ m Then p(m) ≼ p(m ′ ). 

If p(m) ≼ q(n) Then p(m ′ ) ≼ q(n ′ ) for all n ′ ≥ n, m ′ ≤ m. 

a


Monotonicity illustrated 

(m, n) is black iff pm ≼ qn 

n 

10 

5 

0 

q 

p 

0 5 10 15 20 

m




n 

10 

5 

0 

q 

p 

0 5 10 15 20 

m




n 

10 

5 

0 

q 

p 

0 5 10 15 20 

m


Frontiers and Belts 

n 

10 

5 

0 

q 

p 

0 5 10 15 20 

m



Belt Theorem 

n 

Every frontier lies in a belt with rational slope. 

10 

5 

0 

q 

p 

0 5 10 15 20 

m



Belt Theorem 


Observation 

Colour of a point is completely determined by its 1-neighbourhood.



Belt Theorem 


Observation 

Colour of a point is completely determined by its 1-neighbourhood. 

implies 

Theorem 

For any given OCN, ≼ is an effectively semilinear set.



n 

c 2 

c 3 

c 4 

q 

p 

c 1 

m


Infinite Branching 

Modeling may require infinite branching: 

randomness: ”guess number” 

inherently ∞-branching model, e.g. PA evolutions of vectors 

in N n .. 

abstraction from internal steps.


Weak Notions 

Weak Steps (a ≠ τ ∈ A) 

τ 

=⇒ := −→ τ ∗ 

a 

=⇒ := −→ τ ∗ 

a 

τ 

−→ −→ ∗


Weak Notions 

Weak Steps (a ≠ τ ∈ A) 

τ 

=⇒ := −→ τ ∗ 

a 

=⇒ := −→ τ ∗ 

a 

τ 

−→ −→ ∗ 

Definition (Weak Bismulation, Simulation and Trace Inclusion) 

Defined by 2-player games as before and let Duplicator move along 

weak steps. . . Write: ≈, ≦, .


Some Results - Strong Case 

⊆ 

≼ 

∼ 

undecidable 

undecidable 


PDA 

DPDA 

= decidable [Sén97], P-hard, 


⊆ undecidable 


≼ 

∼ 

⊆ 

≼ 

∼ 

⊆ 

≼ 

∼ 







PSPACE-comp. 

P-comp. 

P-comp. 

OCA 

OCN 

NFA 

DOCA 

DOCN 

= decidable [VP75] 

NL-complete, [BG11] 

⊆ 


= NL-complete 

⊆ 


DFA 

=, ⊆ NL-complete


Some Results - Weak Case 

 

≦ 

≈ 

undecidable 

undecidable 

undecidable 

PDA 

DPDA 

≡ 

 

decidable [Sén97], P-hard, 


undecidable 

 

≦ 

≈ 

undecidable 


undecidable 

OCA 

DOCA 

≡ 

 

decidable 


 

≦ 

≈ 


decidable [HMT13] 


undecidable [May03] 

OCN 

 

≦ 

≈ 

PSPACE-comp. 

P-comp. 

P-comp. 

NFA 

DOCN 

≡ decidable 


DFA ≡, NL-complete


Deciding ≦ on OCN-Processes [HMT13] 

1 reduce OCN ? ≦ OCN to OCN ? ≼ ωOCN.



1 reduce OCN ? ≦ OCN to OCN ? ≼ ωOCN. 

2 Definte sequence ≼ 0 ⊇ ≼ 1 ⊇ . . . of approximants using 

syntactic peculiarity of the ωOCN.





syntactic peculiarity of the ωOCN. 

3 Show finite convergence: ∃k ∈ N. ≼ k 

= ≼ k+1 

.





syntactic peculiarity of the ωOCN. 

3 Show finite convergence: ∃k ∈ N. ≼ k 

= ≼ k+1 

. 

4 Show that ≼ k 

is semilinear for finite k. (effectively reduce 

≼ k+1 

to ≼ k 

)


Bibliography 

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