1. Decomposition for Reasoning with Biological Network Gauvain Bourgne, Katsumi Inoue ISSSB’11, Shonan Village, November 13 th -17 th 2011

2. Motivation In bioinformatics, need to reason on huge amount of data ◦ Huge networks (e.g. metabolic pathways, signaling pathways…) On such problems, centralized methods ◦ Long computation time ◦ Memory overflow Problem decomposition ◦ Divide into smaller problems or steps to recompose a global solution ◦ Need for (1) an automated process to decompose and (2) an algorithm to solve local problems and recompose global solution 2Automated Problem Decomposition /33

3. Example Problem (Krebs Cycle) 3 succinate formaldehyde creatinine creatine beta-alanine 2-oxe-glutarate l-lysine l-2-aminoadipate isocitrate trans-aconitate taurine nmnd nmna hippurate formate sarcosine l-as citrulline ornithine arginine urea methylamine tmao lactate glucose acetate acryloyl-coa pyruvate Fumarate fumarate 2.6.1.39 1.1.1.42 2.3.1.61 4.2.1.3 4.2.1.2 1.3.99.1 1.13.11.16 2.1.1.1 2.1.1.7 6.3.4.5 2.1.3.3 2.1.1.2 3.5.3.1 3.5.3.3 3.5.2.10 1.5.99.1 1.1.99.8 1.4.99.3 4.1.2.32 4.2.1.54 4.3.1.6 2.1.3.1 4.1.1.20 2.6.1.14 1.2.1.31 glycolisis 1.1.1.27 4.3.2.1 3.5.1.59 2.6.1.- acetylcoa 2.3.3.1 1.2.4.1 6.2.1.1 citrate 3Automated Problem Decomposition /33

4. Example Problem (Krebs Cycle) 4 Ag2 Ag0 Ag4 Ag1 Ag3 Ag5 4.2.1.2 1.1.1.42 4.1.1.20 1.1.1.42 4.1.1.20 2.3.3.1 4.3.1.6 2.3.3.1 4.3.1.6 2.1.3.1 2.1.3.3 3.5.3.1 2.1.3.3 3.5.3.1 1.5.99.1 1.3.99.1 4Automated Problem Decomposition /33

5. Overview Reasoning task Partition-based algorithm Automated decomposition Experimental evaluation Conclusion 5Automated Problem Decomposition /33

6. Overview Reasoning task Partition-based algorithm Automated decomposition Experimental evaluation Conclusion 6Automated Problem Decomposition /33

7. Logical representation Metabolic pathways: set of reactions R i : R i : m 1,m 2,…,m p  p 1,p 2,…,p n Such reactions can be represented as ◦ an activation rule  ¬m 1 v¬m 2 v…v¬m p v R i ◦ n production rules  ¬R i v p 1  ¬R i v p 2  …  ¬R i v p n  Clausal theory 7Automated Problem Decomposition /33

8. Problems (Conditional) accessibility problems  Sources (s i ), Conditional sources (c i ), Targets (t i )  Find which ti can be produced from si, possibly with the addition of ci as a new source ◦ Find all consequences of the form ¬c i v…v¬c k v t j Extraction of sub-networks Pathways completion (abduction) ◦ Find reactions (set of clauses) Hypothesis on state of reaction given experiments  Consequence finding (with specific form) 8Automated Problem Decomposition /33

9. Main reasoning task Consequence Finding (CF) in clausal theories ◦ Input  A clausal theory T  A production field P=  L is a list of literals  Cond is a condition (maximal length of the consequences, or number of occurrences of some literals) ◦ Output  All the consequences of T that are subsumption- minimal and belongs to P (formed with literals of L respecting condition Cond). Carc(T,P) 9Automated Problem Decomposition /33

10. Overview Reasoning task Partition-based algorithm Automated decomposition Experimental evaluation Conclusion 10Automated Problem Decomposition /33

11. Partition-based CF The task ◦ Consequence Finding (CF) in clausal theories  Input  A set of clausal theory T i such that UT i =T, and a set of reasoners a i associated with each partition  A production field P=  Output  Carc(T,P)  Where  The output should be produced through local computations and interactions between reasoners (message exchange) 11Automated Problem Decomposition /33

12. Partition-based Consequence Finding Generalization of Partition-based Theorem Proving [Amir & McIlraith, 2005] ◦ Based on Craig’s Interpolation Theorem: If C entails D, then there is a formula F involving only symbols common to C et D such that C entails F and F entails D. Principles Identify common symbols (communication languages) Build a tree structure (cycle-cut) Forward relevant consequences from leaf to root CDF 12Automated Problem Decomposition /33

13. Communication languages Graph induced from the partition Problem : eliminate cycles from it while ensuring a proper labeling. Cycle-cut While (G not acyclic) Take a minimal cycle S=(i 1,i 2 ),(i 2,i 3 ),…,(i p,i 1 ). Choose (i,j) in S s.t. is minimal For each (q,r)≠(i,j) in S, l(q,r)  l(q,r) U l(i,j) Remove (i,j) from E abc bfg ade acdf a ac b fad b b Automated Problem Decomposition /33 13

14. Forward Message-passing Algorithm (Sequential) Preprocessing ◦ Determine initial l(i,j) ◦ Apply Cut-cycles ◦ Determine P i  Non-root agents a i (with parent a j ): P i =  Root a k : P k =P Consequence-Finding ◦ From leaves to root  Determine Cn i =Carc(∑ i,P i )  Forward Cn i Carc 14Automated Problem Decomposition /33

15. Parallel Variant Carc Newcarc Newcarc Incremental computations: Newcarc(TUC,P)=Carc(TUC,P)\Carc(T,P) 15Automated Problem Decomposition /33

16. Overview Reasoning task Partition-based algorithm Automated decomposition Experimental evaluation Conclusion 16Automated Problem Decomposition /33

17. Decomposition of clausal theories Given a Clausal Theory T Find a set of partitions T i, such that ◦ UT i =T ◦ Reasoning is easier ie the application of partition-based algorithm to this decomposition is as efficient as possible.  Minimize the size of the communication languages  Ensure that some simplification can be done locally  Partitions should be cohesive and loosely coupled. 17Automated Problem Decomposition /33

18. c1: ¬b ∨ c ∨ e ∨ f c2: ¬a ∨ d ∨ e c3: ¬d ∨ g ∨ h c4: ¬e ∨ g c5: ¬g ∨ ¬h ∨ i c2 c1 c4 c3 c5 a a d d h h i i g g e e c c f f b b c2 c1 c4 c3 c5 a a d d h h i i g g e e c c f f b b c2 c1 c4 c3 c5 e e d g,h g Graph representation Clausal theory can be represented as graph Focus on common symbols 18 Automated Problem Decomposition /33 c2 c1 c4 c3 c5 1 1 1 2 1

19. Architecture Initial Theory.sol file Initial Theory.sol file Reduced graph representation Partitioned graph Partitioned clausal theory.dcf file Partitioned clausal theory.dcf file Root Solution kmetis Number of partitions Partition- based CF buildGraph graph2dcf Root choice heuristic Choose root with maximal average clause size 19Automated Problem Decomposition /33

20. Problem Decomposition ag1 ag3 ag2 ag5 ag4 ag0 20Automated Problem Decomposition /33

21. Overview Reasoning task Partition-based algorithm Automated decomposition Experimental evaluation Conclusion 21Automated Problem Decomposition /33

22. Benchmark Problems Biological networks TPTP problems ◦ Production field :  Vocabulary of conjecture (+ removing conjecture)  Full vocabulary with length limit SAT problems ◦ Production field  Based on frequency of literals  N% most/less frequent literals ◦ Size  Problems still not tractable as CF problems  Solving only a cohesive sub-problem (obtained by partition of the clause graph) 22Automated Problem Decomposition /33

23. Problems characteristics 23Automated Problem Decomposition /33

24. Results – Biological Networks 2 682 252 (3 321 857) 24Automated Problem Decomposition /33

25. Results – SAT problems 25Automated Problem Decomposition /33

26. Results – TPTP problems 26Automated Problem Decomposition /33

27. Results - summary 27Automated Problem Decomposition /33

28. Results - summary 28Automated Problem Decomposition /33

29. Results For almost all problems, decomposition can reduce the number of resolve operations needed. Especially, it can solve some problems that could not be solved Time is no often improved ◦ Due to communication time (parsing, and such) Approached decomposition with metis: ok. Root choice heuristic: still insufficient, though not bad for biological networks problems. Automated Problem Decomposition /33 29

30. Overview Reasoning task Partition-based algorithm Automated decomposition Experimental evaluation Conclusion 30Automated Problem Decomposition /33

31. Conclusion A sound and complete algorithm combined with automated problem decomposition ◦ Can increase efficiency (nb of operation) for almost all problems ◦ But, results dependent on the choice of root 31Automated Problem Decomposition /33

32. Future works Partition-based algorithm ◦ Variant for Newcarc computations ◦ Common Theories for 1 st order representations ◦ Ordered partitions to break cycle (without removing links) Decomposition ◦ Directly from metabolic pathway ◦ Root choice heuristic  Learning preference relation on root choice ◦ Choosing the number of partition 32Automated Problem Decomposition /33

33. Thank you for your attention Any question ? /33 33Automated Problem Decomposition