Simulation Games
Michael Maurer
Overview
Motivation
 4 Different (Bi)simulation relations and
their rules to determine the winner
 Problem with delayed simulation
 Parity Games
 Construction of (Bi)simulations as Parity
Games

Motivation

Capability of mimicking the behavior of
another automaton (structural similarities,
language containment)

Efficiently reducing the size of finite-state
automata (known as quotienting)
Simulation Games

4 different Simulation Game Definitions for a
given Büchi automaton A
:
1) ordinary simulation game,
2) direct (strong) simulation game,
3) delayed simulation game,
4) fair simulation game,
Simulation Games




Played by 2 players: Spoiler and Duplicator
At the start: two pebbles (Red and Blue) are
placed on two vertices q0 and q’0
Spoiler chooses a transition
and moves Red to qi+1
Duplicator also chooses a transition
and moves Blue to q‘i+1
If Duplicator can‘t move, the game halts and
Spoiler wins
Who will be the winner?

Either the game halts, in which case Spoiler
wins

Or the game produces two infinite runs:
and

For each of the 4 simulation games there exist
different rules to determine the winner
Rules for the winner

Ordinary simulation:
 Duplicator
wins in any case
 Fairness conditions are ignored
Duplicator wins as long as the game does not halt

Direct simulation:
D
wins iff for all i, if
then
Rules for the winner

Delayed simulation:
D
wins iff for all i, if
that

then there exists j ≥ i such
Fair simulation:
D
wins iff there are infinitely many j such that
or only finitely many i such that
 In other words: if there are infinitely many i such that
, then there are also infinitely many j such that
Simulation Relation




A state q‘ ordinary, direct, delayed, fair simulates
a state q, if there is a winning strategy for D
The simulation relation is denoted by
,
where * stands for one of the 4 simulations
The relations are ordered by containment:
(preorder)
For di, de, f: if
then
Bisimulation Games



For all of the mentioned simulations
corresponding notions of bisimulation via
modification of the game
S can choose in each round which pebble he will
move and D has to respond with the other one
Bisimulations define an equivalence relation
Bisimulation winning rules

Fair: an accept state appears infinitely often on
one of the 2 runs π and π‘ an accept state
must appear infinitely often on the other as well

Delayed: an accept state at position i of either
run an accept state at j ≥ i of the other run

Direct: an accept state at position i of either run
an accept state at position i of both runs
Problem with delayed simulation

Quotienting: states that simulate each other are
merged

Difficult to find a working definition of a
simulation preserving quotient with respect to
delayed simulation

Not at all clear how such a quotient should be
defined
Problem with delayed simulation

Example for the quotienting problem:
c
1
Quotienting
b
A
a
2

3
b
a
c
1‘
B
b
2‘
B accepts aω, but A does not
Removing transition (1‘,a,1‘) would provide a
simulation-equivalent quotient for A
Parity Games

A parity game graph
has two
disjoint sets of vertices V0 and V1, their union is V

It also has an edge set
function
to each vertex

Played by two players, Zero and One and the game
starts by placing a pebble on
and a priority
that assigns a priority
Parity Games

Rule for moving the pebble: pebble on vi,
Zero (One) moves the pebble to vi+1, such that

If a player can not move, the other one wins
Otherwise the game produces an infinite run


Considering the minimum priority kπ that occurs
infinitely often in the run π; Zero wins, if kπ is
even, One otherwise
(Bi)Simulations from Parity Games


Example: Parity game graph
fair simulation game
The set of vertices for Zero:

The set of vertices for One:
The set of the edges for Zero and One:

The priority function:

for the
(Bi)Simulations from Parity Games

Example Büchi automaton:
b
1

a
a
3
2
kjhjk
c
V0f = {(2,1,a),(2,2,a),(2,3,a),(2,1,b),(2,2,b),(2,3,b),(2,1,c),(2,2,c),(2,3,c),
(3,1,a),(3,2,a),(3,3,a)}

Jhkjh
V1f = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}

Hghjg
Player 0
Player 1
EAf={((2,1,a),(2,2)),((3,1,a),(3,2)),((2,2,b),(2,2)),((2,2,a),(2,3)),..} U
{((1,1),(2,1,a)),((1,2),(2,2,a)),((2,2),(2,3,b)),..}
(Bi)Simulations from Parity Games

Example Büchi automaton:
b
1
a
a
3
2
c

kjhjk
pAf ((2,1,a)) = 2 ;
pAf ((2,3,c)) = 0 ;
pAf ((3,1)) = 1 ;
pAf ((1,3)) = 0 ;
(Bi)Simulations from Paritiy Games

Parity Game constructed:
 Zero
has a winning strategy from (q,q’), iff q is
fairly simulated by q’
 Jurdzinkis algorithm as fast algorithm for
computing fair (bi)simulation relations and
delayed simulations
 Other relations can be constructed from the
fair simulation formulas (Handout)
References

Carsten Fritz, Thomas Wilke: Simulation Relations for Alternating
Parity Automata and Parity Games. DLT 2006, LNCS 4036, pp. 5970, Springer-Verlag (2006)

Kousha Etessami, Thomas Wilke, Rebecca A. Schuller: Fair
Simulation Relations, Parity Games and State Space Reduction for
Büchi Automata. ICALP 2001, LNCS 2076, pp. 694-707, SpringerVerlag (2001)

Carsten Fritz: Simulation-Based Simplification of omega-Automata.
PhD thesis, Technische Fakultät der Christian Albrecht Universität
zu Kiel (2005) available at http:/e-diss.uni-kiel.de/diss_1644/