Simulation Games
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Transcript Simulation Games
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/