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
