1. CSP-SAT Workshop, June 2011 IBM Watson Research Center, 2011 © 2011 IBM Corporation A General Nogood-Learning Framework for Pseudo-Boolean Multi-Valued SAT* Siddhartha JainBrown University Ashish SabharwalIBM Watson Meinolf SellmannIBM Watson * to appear at AAAI-2011

2. IBM Research © 2011 IBM Corporation SAT and CSP/CP Solvers [complete search]  CP Solvers: –work at a high level: multi-valued variables, linear inequalities, element constraints, global constraint (knapsack, alldiff, …) –have access to problem structure and specialized inference algorithms!  SAT: –problem crushed down to one, fixed, simple input format: CNF, Boolean variables, clauses –very simple inference at each node: unit propagation –yet, translating CSPs to SAT can be extremely powerful! E.g., SUGAR (winner of CSP Competitions 2008 & 2009) How do SAT solvers even come close to competing with CP? A key contributor: efficient and powerful “nogood learning” 2June 18, 2011 A General Nogood-Learning Framework

3. IBM Research © 2011 IBM Corporation 3June 18, 2011 A General Nogood-Learning Framework SAT Solvers as Search Engines  Systematic SAT solvers have become really efficient at searching fast and learning from “mistakes”  E.g., on an IBM model checking instance from SAT Race 2006, with ~170k variables, 725k clauses, solvers such as MiniSat and RSat roughly: –Make2000-5000decisions/second –Deduce600-1000 conflicts/second –Learn600-1000clauses/second (#clauses grows rapidly) –Restartevery 1-2 seconds (aggressive restarts)

4. IBM Research © 2011 IBM Corporation An Interesting Line of Work: SAT-X Hybrids  Goal: try to bring more structure to SAT and/or bring SAT-style techniques to CSP solvers  Examples: –Pseudo-Boolean solvers: inequalities over binary variables –SAT Module Theories (SMT): “attach” a theory T to a SAT solver –Lazy clause generation: record clausal reasons for domain changes in a CP solver –Multi-valued SAT: incorporate multi-valued semantics into SAT  preserve the benefits of SAT solvers in a more general context  strengthen unit propagation: unit implication variables (UIV)  strengthen nogood-learning! [Jain-O’Mahony-Sellmann, CP-2010] 4June 18, 2011 A General Nogood-Learning Framework Starting point for this work

5. IBM Research © 2011 IBM Corporation Conflict Analysis: Example  Consider a CSP with 5 variables: X 1, X 2, X 4  {1, 2, 3} X 3  {1, …, 7} X 5  {1, 2}  Pure SAT encoding: variables x 11, x 12, x 13, x 21, x 22, x 23, x 31, …, x 37, x 51, x 52  What happens when we set X 1 = 1, X 2 = 2, X 3 = 1? 5June 18, 2011 A General Nogood-Learning Framework x 11 = true x 22 = true x 31 = true x 51 = true C5 No more propagation, no conflict… really?

6. IBM Research © 2011 IBM Corporation Conflict Analysis: Incorporating UIVs  Unit implication variable (UIV): a multi-valued clause is “unit” as soon as all unassigned “literals” in it regard the same MV variable  SAT encoding, stronger propagation using UIVs: 6June 18, 2011 A General Nogood-Learning Framework x 11 = true x 22 = true x 31 = truex 51 = true x 41 ≠ true x 32 ≠ true C5 C2 C1 x 43 = true C4 x 42 = true C3 conflict

7. IBM Research © 2011 IBM Corporation What Shall We Learn?  Not a good idea to set x 41 ≠ true and x 31 = true  learn the nogood (x 41 = true || x 31 = false)  Problem? When we backtrack and set, say, X 3 to anything in {2, 3, 4, 5}, we end up performing exactly the same analysis again and again!  Solution: represent conflict graph nodes as variable inequations only 7June 18, 2011 A General Nogood-Learning Framework x 11 = true x 22 = true x 31 = truex 51 = true x 41 ≠ true x 32 ≠ true C5 C2 C1 x 43 = true C4 x 42 = true C3 conflict

8. IBM Research © 2011 IBM Corporation CMVSAT-1  Use UIV rather than UIP  Use variable inequations as the core representation: X 1 = 1, X 2 = 2, X 3 = 1 represented as X 1 ≠ 2, X 1 ≠ 3, …  finer granularity reasoning! X 3 doesn’t necessarily need to be 1 for a conflict, it just cannot be 6 or 7  Stronger learned multi-valued clause: (X 4 = 1 || X 3 = 6 || X 3 = 7) –Upon backtracking, we immediately infer X 3 ≠ 1, 2, 3, 4, 5 ! 8June 18, 2011 A General Nogood-Learning Framework

9. IBM Research © 2011 IBM Corporation This Work: Generalize This Framework  Core representation in implication graph –CMVSAT-1:variable inequations (X 4 ≠ 1) –in general:primitive constraints of the solver –example:linear inequalities (Y 2 ≤ 5, Y 3 ≥ 1)  Constraints –CMVSAT-1:multi-valued clauses (X 4 = 1 || X 3 = 6 || X 3 = 7) –in general:secondary constraints supported by the solver –example:(X 1 = true || Y 3 ≤ 4 || X 3 = Michigan)  Propagation of a secondary constraint C s : entailment of a new primitive constraint from C s and known primitives 9June 18, 2011 A General Nogood-Learning Framework

10. IBM Research © 2011 IBM Corporation This Work: Generalize This Framework  Learned nogoods –CMVSAT-1:multi-valued clauses (X 4 = 1 || X 3 = 6 || X 3 = 7) –in general:disjunctions of negations of primitives (with certain desirable properties) –example:(X 1 = true || Y 3 ≤ 4 || X 3 = Michigan)  Desirable properties of nogoods? –The “unit” part of the learned nogood that is implied upon backtracking must be representable as a set of primitives! –E.g., if Y 3 is meant to become unit upon backtracking, then (X 1 = true || Y 3 ≤ 4 || Y 3 ≥ 6 || X 3 = Michigan) is NOT desirable cannot represent Y 3 ≤ 4 || Y 3 ≥ 6 as a conjunction of primitives   upon backtracking, cannot propagate the learned nogood 10June 18, 2011 A General Nogood-Learning Framework

11. IBM Research © 2011 IBM Corporation Sufficient Conditions 1.System distinguishes between primitive and secondary constraints 2.Secondary constraint propagators: –Entail new primitive constraints –Efficiently provide a set of primitives constraints sufficient for the entailment 3.Can efficiently detect conflicting sets of primitives –and represent the disjunction of their negations (the “nogood”) as a secondary constraint 4.Certain sets of negated primitives (e.g., those arising from propagation upon backtracking) succinctly representable as a set of primitives 5.Branching executed as the addition of one or more* primitives Under these conditions, we can efficiently learn strong nogoods! 11June 18, 2011 A General Nogood-Learning Framework [details in AAAI-2011 paper]

12. IBM Research © 2011 IBM Corporation Abstract Implication Graph (under sufficient conditions) 12June 18, 2011 A General Nogood-Learning Framework C p : primary constraints branched upon or entailed at various decision levels Learned nogood: disjunction of negations of primitives in shaded nodes

13. IBM Research © 2011 IBM Corporation General Framework: Example  X 1  {true, false} X 3  {NY, TX, FL, CA} X 5  {r, g, b} X 2, X 4, X 6, X 7  {1, …, 100}  Branch on X 1 ≠ true, X 2 ≤ 50: Learn: (X 3 = FL || X 4 ≥ 31) 13June 18, 2011 A General Nogood-Learning Framework Notes: 1.The part of the nogood that is unit upon backtracking need not always regard the same variable! (e.g., when X ≤ Y is primitive) 2.Neither is regarding the same variable sufficient for being a valid nogood! (e.g., X ≤ 4 || X ≥ 11 wouldn’t be representable)

14. IBM Research © 2011 IBM Corporation Empirical Evaluation  Ideas implemented in CMVSAT-2  Currently supported: –usual domain variables, and range variables –linear inequalities, e.g. (X 1 + 5 X 2 – 13 X 3 ≤ 6) –disjunctions of equations and range constraints e.g. (X 1  [5…60] || X 2  [37…74] || X 5 = b || X 10 ≤ 15)  Comparison against: –SAT solver:Minisat[Een-Sorensson 2004, version 2.2.0] –CSP solver:Mistral[Hebrard 2008] –MIP solver:SCIP[Achterberg 2004, version 2.0.1] Encodings generated using Numberjack [Hebrard et al 2010, version 0.1.10-11-24] 14June 18, 2011 A General Nogood-Learning Framework

15. IBM Research © 2011 IBM Corporation Empirical Evaluation  Benchmark domains (100 instances of each) –QWH-C: weighted quasi-group with holes / Latin square completion random cost c ik  {1, …, 10} assigned to cell (i,k) cost constraint: sum ik (c ik X ik ) ≤ (sum of all costs) / 2 size 25 x 25, 40% filled entries, all satisfiable –MSP-3: market split problem notoriously hard for constraint solvers partition 20 weighted items into 3 equal sets, 10% satisfiable –NQUEENS-W: weighted n-queens, 30x30 average weight of occupied cells ≥ 70% of weight max size 30 x 30, random weights  {1,…,10} 15June 18, 2011 A General Nogood-Learning Framework

16. IBM Research © 2011 IBM Corporation Empirical Evaluation: Results  CMVSAT-2 shows good performance across a variety of domains –Solved all 300 instances in < 4 sec on average  MiniSat not suitable for domains like QWH-C –Encoding (even “compact” ones like in Sugar) too large to generate or solve –20x slower on MSP-3  Mistral explores a much larger search space than needed –Lack of nogood learning becomes a bottleneck: e.g.: 3M nodes for QWH-C (36% solved), compared to 231 nodes for CMVSAT-2  SCIP takes 100x longer on QWH-c, 10x longer on NQUEENS-W 16June 18, 2011 A General Nogood-Learning Framework SAT solverCSP solverMIP solver

17. IBM Research © 2011 IBM Corporation Summary  A generalized framework for SAT-style nogood-learning –extends CMVSAT-1 –low-overhead process, retains efficiency of SAT solvers –sufficient conditions: primitive constraints secondary constraints, propagation as entailment of primitives valid cutsets in conflict graphs: facilitate propagation upon backtracking other efficiency / representation criteria  CMVSAT-2: robust performance across a variety of problem domains –compared to a SAT solver, a CSP solver, a MIP solver –open: more extensive evaluation and comparison against, e.g., lazy clause generation, pseudo-Boolean solvers, SMT solvers –nonetheless, a promising and fruitful direction! 17June 18, 2011 A General Nogood-Learning Framework