On the random structure of behavioural transition systems. Jan Friso Groote, Remco van der Hofstad,...
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On the random structure of behavioural transition systems. Jan Friso Groote, Remco van der Hofstad, Matthias Raffelsieper
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Transcript of On the random structure of behavioural transition systems. Jan Friso Groote, Remco van der Hofstad,...
On the random structure of behavioural transition systems.
Jan Friso Groote, Remco van der Hofstad,
Matthias Raffelsieper
How do we count rabbits in nature?
/ Informatica PAGE 2
How do we count the number of states in software?
/ Informatica PAGE 3
How do we count bugs in software?
/ Informatica PAGE 4
Random state spaces.
• N number of states (4).
• λ fanout (2).
/ Informatica PAGE 5
Each state has λ outgoing states to a randomly chosen other state.
Predict the size of a state space
/ Informatica PAGE 6
• m rabbits seen (transitions): 1, 3, 5,....
• im unique rabbits are unique (states): 1, 2, 3,....
Predict N, the total number of states.
/ Informatica PAGE 7
Firewire data link layer
(IEEE 1394, Bas Luttik)
Typically findings:
/ Informatica PAGE 8
Parallel random state spaces.
/ Informatica PAGE 9
A realistic random state space is the
parallel composition of p random state spaces.
no communicationno communication
Estimation of ‘product state spaces’
Remco van der Hofstad
/ Informatica PAGE 10
Ball at distance j: BT(j).
Layer at distance j: ∂BT(j).
Estimation of ‘product state spaces’
Remco van der Hofstad
/ Informatica PAGE 11
Ball at distance j: BT(j).
Layer at distance j: ∂BT(j).
BT(0).
∂BT(0).
Estimation of ‘product state spaces’
Remco van der Hofstad
/ Informatica PAGE 12
Ball at distance j: BT(j).
Layer at distance j: ∂BT(j).
BT(1).
∂BT(1).
Estimation of ‘product state spaces’
Remco van der Hofstad
/ Informatica PAGE 13
Ball at distance j: BT(j).
Layer at distance j: ∂BT(j).
BT(2).
∂BT(2).
Ball sizes of product graphs.
/ Informatica PAGE 14
Expected ball size (single transition system):
/ Informatica PAGE 15
Estimation of the layer size (product graph)
/ Informatica PAGE 16
Take enough layers and estimate: λ1, λ2, N1, N2.
Can we predict (2 parallel systems)?
/ Informatica PAGE 17
Can we predict (3 parallel systems)?
/ Informatica PAGE 18
And now reality (firewire protocol):
/ Informatica PAGE 19
And now reality (firewire protocol):
/ Informatica PAGE 20
And now reality (CABP):
/ Informatica PAGE 21
A remark on debugging.
/ Informatica PAGE 22
1000 states 103 states α=0.05 probability that error remains
undetected with a test of length m.
Some open problems.
• Is the model really that good?
• How to reduce the extreme calculational effort to do the predictions?
• Can we predict the index of parallelism? What is the correct number of parallel processes to model a particular system?
• How to estimate the probability to hit an ‘erroneous state’ in a random state space [no, contrary what you think, this is not known...].
/ Informatica PAGE 23
