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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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How do we count rabbits in nature? / Informatica PAGE 1

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How do we count the number of states in software? / Informatica PAGE 2

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How do we count bugs in software? / Informatica PAGE 3

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Random state spaces. N number of states (4). λ fanout (2). / Informatica PAGE 4 Each state has λ outgoing states to a randomly chosen other state.

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Predict the size of a state space / Informatica PAGE 5 m rabbits seen (transitions):1, 3, 5,.... i m unique rabbits are unique (states):1, 2, 3,....

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Predict N, the total number of states. / Informatica PAGE 6 Firewire data link layer (IEEE 1394, Bas Luttik)

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Typically findings: / Informatica PAGE 7

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Parallel random state spaces. / Informatica PAGE 8 A realistic random state space is the parallel composition of p random state spaces. no communication

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Estimation of ‘product state spaces’ Remco van der Hofstad / Informatica PAGE 9 Ball at distance j: B T (j). Layer at distance j:∂B T (j).

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Estimation of ‘product state spaces’ Remco van der Hofstad / Informatica PAGE 10 Ball at distance j: B T (j). Layer at distance j:∂B T (j). B T (0). ∂B T (0).

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Estimation of ‘product state spaces’ Remco van der Hofstad / Informatica PAGE 11 Ball at distance j: B T (j). Layer at distance j:∂B T (j). B T (1). ∂B T (1).

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Estimation of ‘product state spaces’ Remco van der Hofstad / Informatica PAGE 12 Ball at distance j: B T (j). Layer at distance j:∂B T (j). B T (2). ∂B T (2).

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Ball sizes of product graphs. / Informatica PAGE 13

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Expected ball size (single transition system): / Informatica PAGE 14

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Estimation of the layer size (product graph) / Informatica PAGE 15 Take enough layers and estimate: λ 1, λ 2, N 1, N 2.

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Can we predict (2 parallel systems)? / Informatica PAGE 16

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Can we predict (3 parallel systems)? / Informatica PAGE 17

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And now reality (firewire protocol): / Informatica PAGE 18

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And now reality (firewire protocol): / Informatica PAGE 19

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And now reality (CABP): / Informatica PAGE 20

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A remark on debugging. / Informatica PAGE 21 1000 states 10 3 states α=0.05 probability that error remains undetected with a test of length m.

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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 22

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