slides - Loïc Paulevé
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks Loïc Paulevé1 , Geoffroy Andrieux2 , Heinz Koeppl1,3 1 ETH
Zürich 2 IRISA Rennes, France 3 IBM Research Zürich
25th International Conference on Computer Aided Verification July 13–19, 2013 - Saint Petersburg, Russia
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Biological Networks E.g., Signalling Networks
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Biological Networks E.g., Signalling Networks
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Components Genes, proteins, complexes, ...
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Activations Positive influence (a increase may promote c increase). Inhibitions Negative influence (b increase may promote d decrease).
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Typical settings 100 to +10,000 components Few information on kinetics
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
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No population ⇒ qualitative levels
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Automata networks - Sync / async - Boolean/multi-valued - ...
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Specify partial or complete cooperations.
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
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No population ⇒ qualitative levels
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
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No population ⇒ qualitative levels
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Automata networks - Sync / async - Boolean/multi-valued - ...
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Specify partial or complete cooperations.
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
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No population ⇒ qualitative levels
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Specify partial or complete cooperations.
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Automata networks - Sync / async - Boolean/multi-valued - ...
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
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No population ⇒ qualitative levels
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Automata networks - Sync / async - Boolean/multi-valued - ...
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Specify partial or complete cooperations.
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
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No population ⇒ qualitative levels
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Automata networks - Sync / async - Boolean/multi-valued - ...
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Specify partial or complete cooperations.
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1 How modify the system to prevent e1 reachability?
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets for Reachability a
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Set of local states that if all disabled break reachability from given initial states
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e.g. {c1 , d2 }
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets for Reachability a
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Set of local states that if all disabled break reachability from given initial states
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e.g. {c1 , d2 }
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets for Reachability a
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Set of local states that if all disabled break reachability from given initial states
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e.g. {c1 , d2 }
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Applications - Potential therapeutic targets - Refute model if reachability still occurs in the modified (real) system
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L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Challenges
Naive algorithm – M: automata network; ς: set of initial states CutSets ← ∅ For ω ∈ ℘(Local States) ordered by cardinality: if (@ω 0 ∈ CutSets : ω 0 ⊂ ω) and ∀s ∈ ς, (M ω, s) 2 EF zi : CutSets ← CutSets ∪ {ω}. Goal: scalable with biological networks complexity • numerous automata, all different; • few local states per automaton.
Contribution: under-approximation of cut sets • some will be missed, some will be too thick (non-minimal, for the model); • handle networks with more than 9,000 nodes. No candidate enumeration, no model-checking.
L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Challenges
Naive algorithm – M: automata network; ς: set of initial states CutSets ← ∅ For ω ∈ ℘(Local States) ordered by cardinality: if (@ω 0 ∈ CutSets : ω 0 ⊂ ω) and ∀s ∈ ς, (M ω, s) 2 EF zi : CutSets ← CutSets ∪ {ω}. Goal: scalable with biological networks complexity • numerous automata, all different; • few local states per automaton.
Contribution: under-approximation of cut sets • some will be missed, some will be too thick (non-minimal, for the model); • handle networks with more than 9,000 nodes. No candidate enumeration, no model-checking.
L. Paulevé, G. Andrieux, H. Koeppl
5/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Overview Automata network
Graph of Local Causality
Under-approximation of Cut Sets Prior work: over-/under-approximation of reachability in large-scale biological networks. [Paulevé et al. in Math. Struct. in Comp. Sci. 2012] L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
a1 →∗ a3
b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
c1 →∗c2
d1 →∗d2
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
Local state
a1 →∗ a3
b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
c1 →∗c2
d1 →∗d2
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
Local state Objective from initial context
a1 →∗ a3
b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
c1 →∗c2
d1 →∗d2
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
Local state Objective from initial context Solution - prior steps
a1 →∗ a3
b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
c1 →∗c2
d1 →∗d2
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
a1 →∗ a3
b3
b1
c2
c1 →∗c2
Soundness criteria ∗ ∗ bObjective →∗c2state of each 1 → b3 1 is impossible from any stateb1if→ at bleast onec2local solution is disabled. E.g. a1 →∗a3 is impossible in M {b3 , b1 } and in M {b3 , c2 } d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Computing GLC for Automata Networks a
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a1 →∗a3
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(ignore order, count, synchronism)
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d Complexity (construction + size of GLC)
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• polynomial in the total number of local states; • exponential in the number of local states
1
within one automaton
⇒ efficient with a small number of local states per automaton, whereas a very large number of automata can be handled. L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Computing GLC for Automata Networks a
b
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`1 `1
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`5
`2
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`2
b3
`6
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a1 →∗a3
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(ignore order, count, synchronism)
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d Complexity (construction + size of GLC)
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`3
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`5
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`6
• polynomial in the total number of local states; • exponential in the number of local states
1
within one automaton
⇒ efficient with a small number of local states per automaton, whereas a very large number of automata can be handled. L. Paulevé, G. Andrieux, H. Koeppl
8/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Computing GLC for Automata Networks a
b
3
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`4
`3
`1 `1
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`5
`2
2
`2
b3
`6
1
a1 →∗a3
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b1
c2
1
(ignore order, count, synchronism)
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d Complexity (construction + size of GLC)
2
2
`3
`4
`5
1
`6
• polynomial in the total number of local states; • exponential in the number of local states
1
within one automaton
⇒ efficient with a small number of local states per automaton, whereas a very large number of automata can be handled. L. Paulevé, G. Andrieux, H. Koeppl
8/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Computing GLC for Automata Networks a
b
3
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`4
`3
`1 `1
2
`5
`2
2
`2
b3
`6
1
a1 →∗a3
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b1
c2
1
(ignore order, count, synchronism)
c
d Complexity (construction + size of GLC)
2
2
`3
`4
`5
1
`6
• polynomial in the total number of local states; • exponential in the number of local states
1
within one automaton
⇒ efficient with a small number of local states per automaton, whereas a very large number of automata can be handled. L. Paulevé, G. Andrieux, H. Koeppl
8/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-Approximation Associate to each node sets of local states intersecting any trace from given context. V : nodes 7→ ℘(℘≤N (Obs)), Obs ⊂ LS ∗ ˜ V(a3 ) = V(a1 →∗a3 )×V(a 2 → a3 ) ∪ {{a3 }}
a3 (OR)
a1
→∗ a
a2 →∗a3
3
a1 →∗a3
2) ˜ V(a1 →∗a3 ) = V(sol 1 )×(sol
(OR)
V(sol 1 ) = V(b1 ) ∪ V(c2 ) (AND)
b1
c2 ∆
˜ 1 , . . . , f m } = {e i ∪ f j | i ∈ [1; n] ∧ j ∈ [1; m]} ; e i , f j ∈ ℘≤N (Obs) {e 1 , . . . , e n }×{f
L. Paulevé, G. Andrieux, H. Koeppl
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
c1 →∗c2
∅
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b3
c2
b1
b1 →∗b3
b1 →∗b1 ∅
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
c1 →∗c2
c2 → ∗ c2
∅
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
b1 →∗b1 ∅
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
c2
c1 →∗c2
c2 → ∗ c2
∅
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
{b1 }
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
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Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
{b1 }
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
{b1 }
d2
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 } L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
{b1 }
d2 {b1 }, {d2 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
b1 {b1 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
b1 {b1 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
b1 {b1 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c1 →∗c2
c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
b1 {b1 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c1 →∗c2
c2
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
{b1 }, {c2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3 {b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
{b1 }, {c2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
a3 {b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
{b1 }, {c2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
a3 {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
{b1 }, {c2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
a3 {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 }, {d2 }
a1 →∗a3
{b1 }, {c2 } {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
b3 {b1 }, {b3 }, {d2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Formal analysis of the whole PID Pathway Interaction Database http://pid.nci.nih.gov • Inductions, inhibitions, transcriptional regulation, complex formations, . . . • +9,000 interacting components.
Graph of Local Causality for (independent) reachability of active SNAIL, p15, p21 • From Process Hitting model (sub-class of Asynchronous ANs)
+21,000 concurrent automata (biological and logical); largest: 16 local states. • ≈20,000 nodes involving ≈1,600 biological components.
Extracted Cut Sets N 1 2 3 4 5 6
Visited nodes 29,022 36,602 44,174 54,322 68,214 90,902
Exec. time 0.9s 1.6s 5.4s 39s 8.3m 2.6h
SNAIL1 1 +6 +0 +30 +90 +930
p15INK4b1 1 +6 +92 +60 +80 +208
p21CIP11 1 +0 +0 +0 +0 +0
Implemented in PINT http://process.hitting.free.fr (OCaml); Dedicated data structures to efficiently compute cross products between million of sets.
L. Paulevé, G. Andrieux, H. Koeppl
11/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Discussion
Summary • Cut sets for transient reachability from a set of initial states
⇒ sets of local states necessary for reachability. • Tractable on very large-scale biological networks.
Quality of under-approximation • Graph of Local Causality abstracts a lot of details around synchronisations. • The less sync the AN, the more accurate the cut sets. • Suited for qualitative biological networks.
Future work • Take into account the time scales of interactions. • Cut sets that do not break other dynamical properties. • Cut sets for other dynamical properties.
L. Paulevé, G. Andrieux, H. Koeppl
12/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
.
Thank you for your attention.
L. Paulevé, G. Andrieux, H. Koeppl
13/13
Zürich 2 IRISA Rennes, France 3 IBM Research Zürich
25th International Conference on Computer Aided Verification July 13–19, 2013 - Saint Petersburg, Russia
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Biological Networks E.g., Signalling Networks
a
+
c
-
+
e
+
+ +
b
d
f
L. Paulevé, G. Andrieux, H. Koeppl
2/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Biological Networks E.g., Signalling Networks
a
+
c
Components Genes, proteins, complexes, ...
-
+
e
+
+
+
b
Activations Positive influence (a increase may promote c increase). Inhibitions Negative influence (b increase may promote d decrease).
d
-
Typical settings 100 to +10,000 components Few information on kinetics
f
L. Paulevé, G. Andrieux, H. Koeppl
2/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
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L. Paulevé, G. Andrieux, H. Koeppl
3/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
c
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No population ⇒ qualitative levels
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+
a 0
e
c
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0
Automata networks - Sync / async - Boolean/multi-valued - ...
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e +
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`1
1
Specify partial or complete cooperations.
+
+
b
2
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`1 1
d
f
-
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L. Paulevé, G. Andrieux, H. Koeppl
3/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
c `2
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a
No population ⇒ qualitative levels
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Automata networks - Sync / async - Boolean/multi-valued - ...
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Specify partial or complete cooperations.
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L. Paulevé, G. Andrieux, H. Koeppl
3/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
c
1
a
+
0
No population ⇒ qualitative levels
`3
1
e
c
-
0
Automata networks - Sync / async - Boolean/multi-valued - ...
1
e +
0
b
+
d
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0
Specify partial or complete cooperations.
+
+
b
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-
`3 1
d
f
-
1
f
0 0
L. Paulevé, G. Andrieux, H. Koeppl
3/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
c
1
No population ⇒ qualitative levels
1
a
+
0
e
c
-
0
e +
`4
0
Specify partial or complete cooperations.
+
+
b
b
+
d
1
0
Automata networks - Sync / async - Boolean/multi-valued - ...
1
`4
2
1
d
f
-
1
f
0 0
L. Paulevé, G. Andrieux, H. Koeppl
3/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
c
1
No population ⇒ qualitative levels
1
a
+
0
e
`5c
-
0
Automata networks - Sync / async - Boolean/multi-valued - ...
`51
e +
0
b
+
d
1
0
Specify partial or complete cooperations.
+
+
b
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d
f
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L. Paulevé, G. Andrieux, H. Koeppl
3/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Qualitative Models for Biological Networks a
c
1
No population ⇒ qualitative levels
1
a
+
0
e
c
-
0
Automata networks - Sync / async - Boolean/multi-valued - ...
1
e +
0
b
+
d
1
0
Specify partial or complete cooperations.
+
+
b
2
1
d
f
-
1 How modify the system to prevent e1 reachability?
f
0 0
L. Paulevé, G. Andrieux, H. Koeppl
3/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets for Reachability a
c
1
Set of local states that if all disabled break reachability from given initial states
1
a
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e
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e.g. {c1 , d2 }
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L. Paulevé, G. Andrieux, H. Koeppl
4/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets for Reachability a
c
1
Set of local states that if all disabled break reachability from given initial states
1
a
+
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⇒
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e.g. {c1 , d2 }
e +
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L. Paulevé, G. Andrieux, H. Koeppl
4/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets for Reachability a
c
1
Set of local states that if all disabled break reachability from given initial states
1
a
+
0
e
c
-
⇒
0
1
e.g. {c1 , d2 }
e +
+
b
b
+
d
1
0
Applications - Potential therapeutic targets - Refute model if reachability still occurs in the modified (real) system
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+
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d
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L. Paulevé, G. Andrieux, H. Koeppl
4/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Challenges
Naive algorithm – M: automata network; ς: set of initial states CutSets ← ∅ For ω ∈ ℘(Local States) ordered by cardinality: if (@ω 0 ∈ CutSets : ω 0 ⊂ ω) and ∀s ∈ ς, (M ω, s) 2 EF zi : CutSets ← CutSets ∪ {ω}. Goal: scalable with biological networks complexity • numerous automata, all different; • few local states per automaton.
Contribution: under-approximation of cut sets • some will be missed, some will be too thick (non-minimal, for the model); • handle networks with more than 9,000 nodes. No candidate enumeration, no model-checking.
L. Paulevé, G. Andrieux, H. Koeppl
5/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Challenges
Naive algorithm – M: automata network; ς: set of initial states CutSets ← ∅ For ω ∈ ℘(Local States) ordered by cardinality: if (@ω 0 ∈ CutSets : ω 0 ⊂ ω) and ∀s ∈ ς, (M ω, s) 2 EF zi : CutSets ← CutSets ∪ {ω}. Goal: scalable with biological networks complexity • numerous automata, all different; • few local states per automaton.
Contribution: under-approximation of cut sets • some will be missed, some will be too thick (non-minimal, for the model); • handle networks with more than 9,000 nodes. No candidate enumeration, no model-checking.
L. Paulevé, G. Andrieux, H. Koeppl
5/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Overview Automata network
Graph of Local Causality
Under-approximation of Cut Sets Prior work: over-/under-approximation of reachability in large-scale biological networks. [Paulevé et al. in Math. Struct. in Comp. Sci. 2012] L. Paulevé, G. Andrieux, H. Koeppl
6/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
a1 →∗ a3
b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
c1 →∗c2
d1 →∗d2
7/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
Local state
a1 →∗ a3
b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
c1 →∗c2
d1 →∗d2
7/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
Local state Objective from initial context
a1 →∗ a3
b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
c1 →∗c2
d1 →∗d2
7/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
Local state Objective from initial context Solution - prior steps
a1 →∗ a3
b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
c1 →∗c2
d1 →∗d2
7/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Graph of Local Causality • Causality of a3 . • Initial context ς = {a 7→ {1}; b 7→ {1}; c 7→ {1, 2}; d 7→ {2}}.
a3
a1 →∗ a3
b3
b1
c2
c1 →∗c2
Soundness criteria ∗ ∗ bObjective →∗c2state of each 1 → b3 1 is impossible from any stateb1if→ at bleast onec2local solution is disabled. E.g. a1 →∗a3 is impossible in M {b3 , b1 } and in M {b3 , c2 } d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
7/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Computing GLC for Automata Networks a
b
3
?
`4
`3
`1 `1
2
`5
`2
2
`2
b3
`6
1
a1 →∗a3
3
1
(ignore order, count, synchronism)
c
d Complexity (construction + size of GLC)
2
2
`3
`4
`5
1
`6
• polynomial in the total number of local states; • exponential in the number of local states
1
within one automaton
⇒ efficient with a small number of local states per automaton, whereas a very large number of automata can be handled. L. Paulevé, G. Andrieux, H. Koeppl
8/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Computing GLC for Automata Networks a
b
3
?
`4
`3
`1 `1
2
`5
`2
2
`2
b3
`6
1
a1 →∗a3
3
1
(ignore order, count, synchronism)
c
d Complexity (construction + size of GLC)
2
2
`3
`4
`5
1
`6
• polynomial in the total number of local states; • exponential in the number of local states
1
within one automaton
⇒ efficient with a small number of local states per automaton, whereas a very large number of automata can be handled. L. Paulevé, G. Andrieux, H. Koeppl
8/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Computing GLC for Automata Networks a
b
3
?
`4
`3
`1 `1
2
`5
`2
2
`2
b3
`6
1
a1 →∗a3
3
b1
c2
1
(ignore order, count, synchronism)
c
d Complexity (construction + size of GLC)
2
2
`3
`4
`5
1
`6
• polynomial in the total number of local states; • exponential in the number of local states
1
within one automaton
⇒ efficient with a small number of local states per automaton, whereas a very large number of automata can be handled. L. Paulevé, G. Andrieux, H. Koeppl
8/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Computing GLC for Automata Networks a
b
3
?
`4
`3
`1 `1
2
`5
`2
2
`2
b3
`6
1
a1 →∗a3
3
b1
c2
1
(ignore order, count, synchronism)
c
d Complexity (construction + size of GLC)
2
2
`3
`4
`5
1
`6
• polynomial in the total number of local states; • exponential in the number of local states
1
within one automaton
⇒ efficient with a small number of local states per automaton, whereas a very large number of automata can be handled. L. Paulevé, G. Andrieux, H. Koeppl
8/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-Approximation Associate to each node sets of local states intersecting any trace from given context. V : nodes 7→ ℘(℘≤N (Obs)), Obs ⊂ LS ∗ ˜ V(a3 ) = V(a1 →∗a3 )×V(a 2 → a3 ) ∪ {{a3 }}
a3 (OR)
a1
→∗ a
a2 →∗a3
3
a1 →∗a3
2) ˜ V(a1 →∗a3 ) = V(sol 1 )×(sol
(OR)
V(sol 1 ) = V(b1 ) ∪ V(c2 ) (AND)
b1
c2 ∆
˜ 1 , . . . , f m } = {e i ∪ f j | i ∈ [1; n] ∧ j ∈ [1; m]} ; e i , f j ∈ ℘≤N (Obs) {e 1 , . . . , e n }×{f
L. Paulevé, G. Andrieux, H. Koeppl
9/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b3
b1
c2
b1 →∗b3
b1 →∗b1
c2 → ∗ c2
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
c1 →∗c2
∅
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b3
c2
b1
b1 →∗b3
b1 →∗b1 ∅
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
c1 →∗c2
c2 → ∗ c2
∅
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
b1 →∗b1 ∅
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
c2
c1 →∗c2
c2 → ∗ c2
∅
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
{b1 }
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
{b1 }
d2
L. Paulevé, G. Andrieux, H. Koeppl
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
{b1 }
d2
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 } L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3
{b1 }
d2 {b1 }, {d2 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b1 {b1 }
b3
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
b1 {b1 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
b1 {b1 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
c2
c1 →∗c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
b1 {b1 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c1 →∗c2
c2
c2 → ∗ c2
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
b1 {b1 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c1 →∗c2
c2
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
{b1 }, {c2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. a3 {b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
{b1 }, {c2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
a3 {b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
{b1 }, {c2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
a3 {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 }, {d2 }
a1 →∗a3 b3 {b1 }, {b3 }, {d2 }
{b1 }, {c2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Cut Sets Under-approximation Example Sketch • Follow the topological order of GLC. • SCCs: arbitrary/random order for updating nodes having child modified. • Always converges. {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
a3 {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 , c2 }, {c2 , d2 }
{b1 }, {b3 }, {d2 }
a1 →∗a3
{b1 }, {c2 } {a3 }, {b1 }, {b3 , c2 }, {c2 , d2 }
b3 {b1 }, {b3 }, {d2 }
b1 {b1 }
c1 →∗c2
c2 {c2 }
b1 →∗b3 {b1 }, {d2 }
d2 {b1 }, {d2 }
{b1 }
d1 →∗d2 {b1 }
b1 →∗b1 ∅
∅
c2 → ∗ c2 ∅
∅
{b1 }, {d2 }
L. Paulevé, G. Andrieux, H. Koeppl
10/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Formal analysis of the whole PID Pathway Interaction Database http://pid.nci.nih.gov • Inductions, inhibitions, transcriptional regulation, complex formations, . . . • +9,000 interacting components.
Graph of Local Causality for (independent) reachability of active SNAIL, p15, p21 • From Process Hitting model (sub-class of Asynchronous ANs)
+21,000 concurrent automata (biological and logical); largest: 16 local states. • ≈20,000 nodes involving ≈1,600 biological components.
Extracted Cut Sets N 1 2 3 4 5 6
Visited nodes 29,022 36,602 44,174 54,322 68,214 90,902
Exec. time 0.9s 1.6s 5.4s 39s 8.3m 2.6h
SNAIL1 1 +6 +0 +30 +90 +930
p15INK4b1 1 +6 +92 +60 +80 +208
p21CIP11 1 +0 +0 +0 +0 +0
Implemented in PINT http://process.hitting.free.fr (OCaml); Dedicated data structures to efficiently compute cross products between million of sets.
L. Paulevé, G. Andrieux, H. Koeppl
11/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
Discussion
Summary • Cut sets for transient reachability from a set of initial states
⇒ sets of local states necessary for reachability. • Tractable on very large-scale biological networks.
Quality of under-approximation • Graph of Local Causality abstracts a lot of details around synchronisations. • The less sync the AN, the more accurate the cut sets. • Suited for qualitative biological networks.
Future work • Take into account the time scales of interactions. • Cut sets that do not break other dynamical properties. • Cut sets for other dynamical properties.
L. Paulevé, G. Andrieux, H. Koeppl
12/13
Under-approximating Cut Sets for Reachability in Large-Scale Automata Networks
.
Thank you for your attention.
L. Paulevé, G. Andrieux, H. Koeppl
13/13
