1 Coinduction Principle for Games Michel de Rougemont Université Paris II & LRI.
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Transcript of 1 Coinduction Principle for Games Michel de Rougemont Université Paris II & LRI.
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Coinduction Principle for Games
Michel de Rougemont
Université Paris II & LRI
2
Coinduction (inspired by D. Kozen)
• Naive notionInduction: F monotone, F has a unique least-fixed point.Coinduction: F is antimonotone, unique smallest greatest fixed-point.Example: Even numbers over N
• General PrincipleFunction F defined by an equation, admits a fixed-point F*.We want to show that F* satisfies a property: it suffices to show that the property is preserved by the equation.
• Foundation: Let B a Banach Space, R is a linear Operator, of spectral radius <1. It admits a fixed point.
• Applications to Computer Science:1. Evolutionary Games2. Combinatorics3. Stochastic processes
3
1. Evolutionary Dynamics
1 2+a 0
0 1 2+a
2+a 0 1
Example: Rock-Scissor-Paper:
Mixed strategy= density of agents playing pure strategiesReplicator Strategy:
( , ) ( , )ii i i
xx g e x g x x x
t
( , ) (0,...0,1,0,..0) . .tig e x A x
1 1 2 1
2 2 3 1
1 3 1 1
(2 )
(2 )
(2 )
t
t
t
x x a x x Ax x
x x a x x Ax x
x x a x x Ax x
0Case a
( , ) ( ) *
1 1 1* ( , , )
3 3 3
tT x Min x t x
x
2
c Show T(x, ) ?
4
2. Enumerative Combinatorics
=( (0), (1), (2),....)= (0)+X ' Coinductive Counting (J. Rutten), Stream Calculus
Male Bees=Drones D (Q,D)QFemale Bees=Queens Q QD
How many Q ancestors at level k?
Q D
Q
Q D
D
Q
Q
D
Q
D
Q
kq0 q1
0 0 1 (0)=0 ; ' =
1 1 1 0 (0)=1 ; ' =
1 02 2
1 X = ; =
1-X-X 1-X-X
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3. Probabilistic Processes
•Equivalence of Markov Chains ?
•Metric Analogue of Bisimulation (Desharnais and al.)
•D-bar measure in Statistics
•Approximate Equivalence
•Property of Markov Chains:
….
p
1-p
( ) E [ ( , ) ]c
T n Time s tn
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Coinduction to compare 2 processes
1 1 2 2 E [ ( , ) ]; E [ ( , ) ]T Time s t in P T Time s t in P
•Property of Markov Chains:
Examples from D. Kozen, Lics 2006
Generation of a biased coin (q) given a biased coin (p).
Consider 3 different processes.
1 2 We want to show that T T
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Biased coin simulation
Algorithm : qflip(q):If q >p( if pflip=head) return head else return qflip(q-p/1-p) )
else ( if pflip=head) return qflip(q/p) else return tail )
p q0
1
q’0
1
Given: pflip a biased coin (head, tail) with probability (p,1-p).
Task(q) : Generate a biased coin with probability (q,1-q).
q p0
1
q’0
p q
q p
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Strategy : qflip(q)
Convergence: output with probability Min(p,1-p) at each step. Halts with probability 1.
H(q)=Probability qflip(q)=head. ( / ) if
( )
(1 ). (1 (1 /1 )) if
p H q p q p
H q
p p H q p q p
p q0
1
q’0
1
*( ) is a solution.H q q
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Time estimation of qflip(q)
Estimated Time:
0
0
0
1 . ( / ) if
( )
1 (1 ). (1 (1 /1 )) if
p E q p q p
E q
p E q p q p
p q0
1
Algorithm : qflip(q):If q >p( if pflip=head) return head else return qflip(q-p/1-p) )
else ( if pflip=head) return qflip(q/p) else return tail )
q’0
1
0
1 *( )
1
q qE q
p p
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Strategy 1: qflip(q)
Estimated Time:
1
q/p if
1- ( ) 1- if 1-
1-
11 if 1-
q p
qf q p q p
p
qp q
p
p q0 1
Algorithm 1 : qflip(q): If ( q>1-p) ( if pflip=head) return qflip((q-1+p)/p) else return head
Else if q >p( if pflip=head) return head else return qflip(q-p/1-p) )else ( if pflip=head) return qflip(q/p) else return tail )
1-p
1 if p<q 1-p ( )
o.w.
pr q
p
1 1 1 ( ) 1 ( ) ( ( ))E q r q E f q
1-p p
11
Comparison between Strategies 0 and 1
1
4p
1
4p
0 1 Experimental Graph *( ), *( ) 0 1E q E q q
1 Difficulty: no analytical representation of *( ) E q
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Bounded Linear Operators
1
1
( ) 1 ( ) ( ( )) .
. . ( ) ( ( ))
E q r q E f q a R E
R E q r q E f q
B, Banach Space, R is a linear Operator, of spectral radius <1.Affine Operator τ(e) =a+ReΦ closed non empty region preserved by τ.
Conclusion: there exists a fixed point e* (τ(e*) =a+Re*) s.t e* is in Φ.
Example: E(q) is bounded.
closed non-empty region preserved by
B
Φ
E*
is the property we want to prove on E*
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Coinduction principle
Co induction principle:
*
( ) ( ) ( ( ))
---------------------------------------
( )
e e e e e
e
(e)=a+Re where R has Spectral radius < 1
closed region preservev by B
* Unique solution of e=a+Re satisfisfies e
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Application
Co induction principle1
( ) : ( )1
q qE q E q
p p
1 1 ( ) .(1 ( ) ( ( ))) radius 1-pE q r q E f q
*1
1 We want to show: ( )
1
q qq E q
p p
1(1) E ( ) take E= q.0
1
q qq E q
p p
1 1(2) E[ ( ) ) ( ( )( ) ))]
1 1
q q q qq E q q E q
p p p p
1 11 1
( ) 1 ( ) 1(4) ( ( )) ) 1 ( ) ( ( ))
1 1
f q f q q qE f q r q E f q
p p p p
1 11
( ) 1 ( ) 1(3) It suffices to show: ( ( )) ) ( )( )
1 1
f q f q q qE f q E q
p p p p
Take cases : , 1 ,1q p p q p p q
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Case 1
Co induction setting1
( ) : ( )1
q qE q E q
p p
1 1 ( ) .(1 ( ) ( ( ))) radius 1-pE q r q E f q
*1
1 We want to show: ( )
1
q qq E q
p p
1 11 1
( ) 1 ( ) 1(4) ( ( )) ) 1 ( ) ( ( ))
1 1
f q f q q qE f q r q E f q
p p p p
1Case 1: ( ) / and ( )q p f q q p r q p
2
1 / 1(6) 1 ( / ) 1 ( )
1 1
q q p q qpE q p p
p p p p
2
11
(5) ( ) 1 ( )1 1
qq q q q qp
E pEp p p p p p
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Strategy 2: qflip(q)
Estimated Time:
1
q/p if
1-1- if 1-
1-
( ) if 1/2 1
1
11 if 1-
q p
qp q p
p
f q qq p
p
qp q
p
p q0 1
Algorithm 2 : qflip(q): If ( q>1-p) ( if pflip=head) return qflip((q-1+p)/p) else return head Else If ( q>0.5) ( if pflip=head) return tail else qflip(q/1-p) Else if q >p( if pflip=head) return head else return qflip(q-p/1-p) )
else ( if pflip=head) return qflip(q/p) else return tail )
1-p
1 if p<q 1-p ( )
o.w.
pr q
p
2 2 2 ( ) 1 ( ) ( ( ))E q r q E f q
1-p p
1/2
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New Application
2 2 2 Second trial: break point at (1- ,1 (1 ) ) if (1 )c Max p p p p
Co induction principle on pairs (E,E’)
First trial. ( , ') : ( ) '( ). !!E E q E q E q Incorrect
( , ') [ .(1 ( ) ( ( ))), .(1 ( ) ' ( '( )))]E E q r q E f q q r q E f q
* *2 1 We want to show: ( ) ( )q E q E q
Second trial. ( , ') : ( ) '( ) .....E E q E q E q
* * * *2 1 2 1 ( ), ( ) are nowhere differentiable but ( ) ( )E q E q E q E q
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Probabilistic Processes
•Equivalence of Markov Chains ? Are M1 ,M2 ε-close ?
•Metric Analogue of Bisimulation (Desharnais and al.)•Distance d between distributions obtained by iterations, also the maximum fixed-point of a Functional F.
•Property of Markov Chains:
….
p-ε
1-p+ ε
….
p
1-p
M1
M2
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Conclusion
General Principle: Given τ linear bounded operator of spectral radius <1,Φ closed non empty region preserved by τ, we conclude that there exists a fixed
point e* in Φ.
Applications:1. Stochastic processes. Compare Expected time between two fractal
processes.
2. Evolutionary Games. Compare convergence time.
3. Analysis of Streams.