Efficient SAT Solving for Non- clausal Formulas using DPLL, Graphs, and Watched-cuts Himanshu Jain Edmund M. Clarke

Agenda  Motivation  Existing SAT solvers  Our approach  Negation Normal Form (NNF)  Graphs to represent NNF  Boolean Constraint Propagation (BCP)  Experimental Results 1

2  Applications in verification  Equivalence checking  Model checking  Theorem proving  Test generation  Static analysis  Circuits in practice  Thousands of inputs  Millions of gates  Structure sharing Boolean Satisfiability Boolean Circuit   ab aa bb aa b output  

3 Current SAT solvers  Davis-Putnam-Logemann-Loveland (DPLL) algorithm  Conversion of circuit to Conjunctive Normal Form (CNF)  Number of variables proportional to number of gates  Can be times more than the number of inputs in a circuit  Slowdown due to large number of new variables (and clauses)  Modern CNF solvers use pre-processing techniques  To reduce #variables and #clauses in CNF  But pre-processing has large memory requirements

4 Our SAT solving framework Boolean Circuit hpgraphvpgraph Negation Normal Form Fewer variables than CNF and more structure DPLL algorithm

5 Negation Normal Form (NNF) An NNF formula contains arbitrary nesting of AND (  ) and OR (  ) gates - Negations can appear at leaf level - No sharing of sub formulas      ab aa bb aa b yy  yy  aby 

6 How to obtain NNF from Boolean Circuits    ab aa bb aa b yy  yy  aby     ab aa bb aa b Circuit from converted to NNF by introducing one new variable y to remove sharing Boolean Circuit NNF out out’  

7 Why NNF: Number of variables CNF without pre-processing has 5-10X more variables

8 Why NNF: Number of variables 1705 points 428 points Note no pre-processing for NNF

9 Our SAT solving framework Boolean Circuit hpgraphvpgraph Negation Normal Form Peter Andrew’s Horizontal/Vertical Path Forms (1981) DPLL algorithm

10 Inductively creating hpgraph(  ) from NNF formula   is a literal m m Create new node  =  1   2 hpgraph(  1 ) R1R1 L1L1 hpgraph(  2 ) R2R2 L2L2 Take graph union  =  1   2 hpgraph(  1 ) R1R1 L1L1 hpgraph(  2 ) R2R2 L2L2 Add hyperedge from L 1 to R 2

11 p q r  s rr pp Example Formula F: (((p  q)   r   q)  (  p  (r   s)  q)) qq q hpgraph(F) R R R L L L R denotes a root node and L denotes a leaf node p  q   p p  q  r   s p  q  q  r   p  r  r   s  r  q  q   p  q  r   s  q  q CNF(F) horizontal path

12 Example Formula F: (((p  q)   r   q)  (  p  (r   s)  q)) p q rr qq q pp r  s vpgraph(F) RRR L L R denotes a root node and L denotes a leaf node p   r   q q   r   q  p  r  q  p   s  q DNF(F) vertical path

13 Our SAT solving framework Boolean Circuit hpgraph CNF-like vpgraph DNF-like Negation Normal Form DPLL algorithm Directed Acyclic Graphs (DAGs) Linear in size of original circuit

14 DPLL on hpgraph hpgraph Boolean Constraint Propagation (BCP) engine Decisions Top-level DPLL Algorithm Conflicts, Implied Literals

15 pp rr Meaning of BCP on hpgraph p q r  s qq q Conflict clause:  r   p R L Assignment  1 :={r=1,p=1} horizontal path = clause qq p q r  s rr pp q Unit clause:  q  r   s Implied literal: r R L Assignment  2 :={q=1,s=1} A horizontal path

16 Can we generalize the CNF watched literal scheme? - Watch two literals (nodes) on each clause (path) - But exponential number of clauses (paths)! p q r  s rr pp qq q R R R L L L Hpgraph Horizontal path = clause

17 Two Watched Cut Scheme Hpgraph p q r  s rr pp qq q R R R L L L 1.A node cut disconnects all horizontal paths 2.Watch two node cuts (allows observing two literals on each clause) 3.Minimal cuts (covered later) cut 1 cut 2 watch a node cut p  q  r   s

18 Actual picture…. An hpgraph hpgraph components 1.Two cuts for each hpgraph component 2.Can be updated locally during BCP 3.Non-chronological backtracking is cheap

19 Our algorithm generalizes CNF watched literal scheme An hpgraphSpecial hpgraph (CNF)

20 Acceptable cut p q r  s rr pp qq q R R R L L L cut 1 cut 2 A cut is acceptable if no literal appearing in it is false Cut 2 is: not acceptable for  := {p=1} Cut 1 is: acceptable for  := {p=1} Now let us see the use of cuts during BCP

21 BCP Case 1: Both cuts are acceptable and disjoint Hpgraph component p q r  s rr pp qq q R R R L L L cut 1 cut 2 No need to examine the hpgraph component  := {s=1} Intuitively, there will be no conflicts or implied literals

22 BCP Case 2 (conflict): No acceptable cut Hpgraph component p q r  s rr pp qq q R R R L L L cut 1 cut 2  := {p=1,q=1} An hpgraph has no acceptable cut if and only if current assignment falsifies the formula Conflict clause:  q   p

23 BCP Case 3 (Implications): Acceptable cuts but not disjoint Hpgraph component p q r  s rr pp qq q cut 1 cut 2 For  := {p=1} we can find two acceptable cuts. We cannot find two completely node-disjoint cuts Intuitively nodes common to both cuts contain implied literals In our example  r and  q are implied literals.

24 Finding and Maintaining Minimal Cuts p q r  s rr pp qq q R R R L L L minimal cut in hpgraph component = a path in vpgraph component hpgraph component p q rr qq q pp r  s RR R LL vpgraph component

25 Experimental Results  These techniques implemented in:  NFLSAT (Non-clausal FormuLas SATisfiability checker)  ~2500 Boolean circuits (industrial category)  Bounded model checking, k-induction, SW/HW verification  CNF obtained by adding new variables (one per AND gate in AIG)  Timeout of 600sec per problem  Comparing with state-of-the-art solvers  SAT 2009 competition winners: Precosat, Glucose  SAT-Race 2008 AIG track winners: MiniSAT++, Picoaigersat  Top three winners of SAT 2007 comp: RSAT, MiniSAT, PicoSAT

26 NFLSAT vs. Precosat NFLSAT solves 29 more problems x>y on 2018 points y>x on 306 points Total time: NFLSAT( sec), Precosat ( sec)

27 NFLSAT vs. Glucose NFLSAT solves 58 more problems y>x on 895 points x>y on 1382 points Total time: NFLSAT( sec), Glucose ( sec)

28 NFLSAT vs. MiniSAT++ (AIG) Total time: NFLSAT( sec), MiniSAT++ ( sec) Minisat++ solves 14 more problems

29 Summary Boolean Circuit Negation Normal Form vpgraphhpgraph Clause Database BCP engine Conflicts, Implied Literals Decisions Top-level DPLL Algorithm 1.Other features of modern SAT solvers 2. No pre-processing so far (circuit rewriting applicable) Linear time conversion

30 Questions