On the random structure of behavioural transition systems. Jan Friso Groote, Remco van der Hofstad, Matthias Raffelsieper
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On the random structure of behavioural transition systems.
Jan Friso Groote, Remco van der Hofstad,
Matthias Raffelsieper
How do we count rabbits in nature?
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How do we count the number of states in software?
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How do we count bugs in software?
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Random state spaces.
• N number of states (4).
• λ fanout (2).
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Each state has λ outgoing states to a randomly chosen other state.
Predict the size of a state space
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• m rabbits seen (transitions): 1, 3, 5,....
• im unique rabbits are unique (states): 1, 2, 3,....
Predict N, the total number of states.
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Firewire data link layer
(IEEE 1394, Bas Luttik)
Typically findings:
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Parallel random state spaces.
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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
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Ball at distance j: BT(j).
Layer at distance j: ∂BT(j).
Estimation of ‘product state spaces’
Remco van der Hofstad
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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
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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
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Ball at distance j: BT(j).
Layer at distance j: ∂BT(j).
BT(2).
∂BT(2).
Ball sizes of product graphs.
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Expected ball size (single transition system):
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Estimation of the layer size (product graph)
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Take enough layers and estimate: λ1, λ2, N1, N2.
Can we predict (2 parallel systems)?
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Can we predict (3 parallel systems)?
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And now reality (firewire protocol):
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And now reality (firewire protocol):
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And now reality (CABP):
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A remark on debugging.
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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...].
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