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

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

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 glycolisis acetylcoa citrate 3Automated Problem Decomposition /33

Example Problem (Krebs Cycle) 4 Ag2 Ag0 Ag4 Ag1 Ag3 Ag Automated Problem Decomposition /33

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

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

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

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

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

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

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

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

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

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

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

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

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

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 c

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

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

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

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

Problems characteristics 23Automated Problem Decomposition /33

Results – Biological Networks ( ) 24Automated Problem Decomposition /33

Results – SAT problems 25Automated Problem Decomposition /33

Results – TPTP problems 26Automated Problem Decomposition /33

Results - summary 27Automated Problem Decomposition /33

Results - summary 28Automated Problem Decomposition /33

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

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

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

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

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