Faster Extraction of High-Level Minimal Unsatisfiable Cores Ryvchin Vadim and Ofer Strichman Technion, Israel SAT’11 Conference Ann Arbor, USA June 21, 2011

Agenda Introduction and motivation Optimizations Experimental results A. Partial Resolution B. Selective clause minimization C . Postponed IC-propagation E. Selective learning of IC-clauses G. Removal Strategy Experimental results Resolution vs. Selector variables

High-Level UC Given: A set of interesting constraints (IC)  = { IC1, IC2 , …, ICm }, and The remainder  The set    is a high-level UC (HLUC) if    is unsatisfiable HLUC is minimal (HLMUC) if removal of any IC makes    satisfiable

Examples Abstraction-refinement in model checking [MA’03] Latches in core define the next abstraction. Compositional Formal Equivalence Checking (FEC) [CGLNR’10] Decompose the compared circuits to blocks. Assume inputs to blocks are the same Assumptions in the core need to be proved. A very popular model checking technique is based on abstraction/refinement. A BDD-based model checker checks an abstract model, where the abstraction is defined by the set of latches that we replace with free inputs. B. If the property fails, a SAT solver is ran on the concrete model to the depth of the ce. C. If there is no BUG - the latches that participate in the proof define the abstraction in the next iteration. D: clearly, then we want to have as few latches in the proof as possible

Traditional UC Extraction: Stage 1: Translate to Clauses An interesting constraint The remainder (the rest of the formula) Each small square is a propositional clause, e.g. (a + b’)

Traditional UC Extraction: Stage 2: Extract a Clause-Level UC An interesting constraint The remainder (the rest of the formula) Colored squares belong to the clause-level UC

Traditional UC Extraction: Stage 3: Map UC back to ICs An interesting constraint The remainder (the rest of the formula) The UC contains three interesting constraints!

A Mismatch between Mainstream Research and the Needs of Real-World Applications Real-world applications: reduce # interesting constraints in the core Latches/gates for abstraction refinement Assumptions for compositional FEC Vast majority of existing algorithms: reduce # of clauses in the core 19/21 papers on UC extraction only consider clause-level UC extraction

Small/Minimal Clause-Level UC  Small/Minimal High-Level UC A small clause-level UC, but the high-level UC is the largest possible: A large clause-level UC, but the high-level UC is empty:

Resolution Refutation C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 C14 C15 C16 C1 C2 C3 C4 C5 C6 C7 C8 C9 Legend: Input clauses Derived clauses

Resolution Refutation C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 C14 C15 C16 C1 C2 C3 C4 C5 C6 C7 C8 C9 Legend: Input clauses Derived clauses Empty clause cone: { C4 C5 C6 C7 C13 C14 C19 C20 C23 } Unsat Core: { C4 C5 C6 C7 }

Resolution Refutation C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 C14 C15 C16 C1 C2 C3 C4 C5 C6 C7 C8 C9 Legend: Empty Clause Cone Unsat Core Empty clause cone: { C4 C5 C6 C7 C13 C14 C19 C20 C23 } Unsat Core: { C4 C5 C6 C7 }

Resolution with ICs C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 C14 Legend: Input Clauses IC1 IC2 Remainder Derived Clauses Derived clauses

Resolution with ICs C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 C14 Legend: Input Clauses IC1 IC2 Remainder Derived Clauses IC1 IC2 Remainder

HLUC C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 C14 C15 C16 C1 C2 Legend: IC1 IC2 Remainder HLUC: { IC2 }

HLMUC Algorithm [N’10] Solve  Æ  ’ = HLUC = ’  =  \ ICi UNSAT SAT Remove one ICi ϵ  that wasn’t already removed  No unchecked ICs  - remainder,  - ICs Initialization: ’ =  =  Assumption:  Æ  is UNSAT

Contribution of this Work Seven optimizations for single HLMUC. improved run time and smaller HLMUC Comparison between resolution and selector variables solvers.

A. Partial Resolution Observations: Suggestion: Result: IC-clauses usually between 5-15% of the problem clauses We do not need the whole resolution table Suggestion: Keep only clauses relevant to IC resolutions Result: The size of the resolution graph reduced Very effective on large CNFs

A. Partial Resolution C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 Legend: IC1 IC2 Remainder

A. Partial Resolution C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 Legend: IC1 IC2 Not Needed

A. Partial Resolution C22 C23=() C17 C20 C21 C10 C15 C16 C1 C7 C8 Legend: IC1 IC2

A. Partial Resolution - Summary C22 C23=() C17 C20 C21 C10 C15 C16 C1 C7 C8 Legend: IC1 IC2 Keep only the needed resolutions

B. Selective clause minimization Technique for shrinking conflict clauses The algorithm is based on traversing the resolution DAG backward from each literal in the learned clause The problem: May turn a non-IC-clause into a shorter IC-clause

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6)

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) 1-UIP based conflict analysis: c6= (¬v1 Ç ¬v3 Ç ¬v4) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) v1  v3 ¬v3  ¬v1

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c6= (¬v1 Ç ¬v3 Ç ¬v4) c6= (¬v1 Ç ¬v4) v1  v3 ¬v3  ¬v1

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c6= (¬v1 Ç ¬v4) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) (IC) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

B. Selective clause minimization c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) (IC) With minimization using c2: c6= (¬v1 Ç ¬v4) (IC) Without minimization: c6= (¬v1 Ç ¬v3 Ç ¬v4) (remainder)

B. Selective clause minimization Suggested solution: Disable minimization if it adds dependency on IC-clause. c6= (¬v1 Ç ¬v3 Ç ¬v4) instead of c6= (¬v1 Ç ¬v4) Disabling minimization  reduces #derived IC-clauses  reduces #IC-clauses in UC and finds HLMUC faster

C. Postpone IC-propagations Change BCP order Run BCP Conflict Analyze Conflict no implications Next Operations

C. Postpone IC-propagations Change BCP order Run BCP over non IC-clause Conflict Analyze Conflict found implication no implications Propagate a single IC-clause Conflict no implications Next Operations

C. Postpone IC-propagations Increase chances to get conflicts in remainder Decreases number of derived IC-clauses Decreases number of IC-clauses in UC.

E. Selective Learning implication IC-clause implication X @2 @2 @5 @5 @3 @3

E. Selective Learning implication IC-clause implication X @2 @2 @5 @5 @3 @3

E. Selective Learning implication IC-clause implication X @2 @2 @5 @5 @5 @5 @5 @5 @5 X @5 @5 @5 @3 @3 Learnt clause should be marked “IC-clause”

E. Selective Learning We refrain from learning IC-clauses Instead, do not learn it learn a (non-asserting) remainder clause make a decision How ?

E. Selective Learning How ? Treat the last IC-clause implication as decision Perform new 1-UIP conflict analysis The learnt clause is ‘remainder’

E. Selective Learning implication IC-clause implication X @2 @2 @5 @5 @3 @3

E. Selective Learning implication IC-clause implication X @2 @2 @5 @5 @6 @5 @5 @6 X @5 @5 @6 @3 @3

E. Selective Learning implication IC-clause implication X @2 @2 @5 @5 @6 @5 @5 @6 X @5 @5 @6 @3 @3

E. Selective Learning implication IC-clause implication X @2 @2 @5 @5 @6 @5 @5 @6 X @5 @5 @6 @3 @3

G. Removal Strategy Recall: Solve  Æ  ’ = HLUC = ’  =  \ ICi UNSAT SAT Remove one ICi ϵ  that wasn’t already removed  No unchecked ICs Recall: In each iteration one IC is chosen to be removed.

G. Removal Strategy What is the effect of the removal order? Which IC should we remove first?

G. Removal Strategy Criterion: #clauses in UC Choose the one that contains least clauses in UC If UNSAT (not necessary), will converge faster If UNSAT (not necessary), will likely allow further removals that contains most clauses in UC If SAT (necessary), clauses are added as ‘remainder’ fast

Experimental Results Benchmark Set: Machines: Industrial set of problems from Intel Average #clauses = 2,572,270 Average #ICs = 3804 Average #IC-clauses = 96568 (6% of #clauses) Machines: Intel® Xeon® 4Ghz 32Gb of memory

Experimental Results

Experimental Results (zoom)

Experimental Results Improves N’10: Time: 44913 sec vs. 20453 sec (x2.19 speedup) HLMUC size: 18995 vs. 5090 (73.2% reduction)

Resolution vs. Selector variables Main cause for performance gap: clause minimization. When clause minimization is off: similar results (time + size of HLMUC). Selector variables: minimization does not turn non-IC- clause into IC-clause Selector variable prevents the minimization

Resolution vs. Selector variables c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) (IC) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

Resolution vs. Selector variables c1= (¬v1 Ç v2) c2= (¬v2 Ç v3 Ç ¬s) (IC) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) Selector variables are pure  cannot be removed  every IC-clause have at least one c1 c1 c2 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

Resolution vs. Selector variables Resolution-based: c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) (IC) Selector-variables-based: c2= (¬v2 Ç v3 Ç ¬s) (IC) v1  v3 ¬v3  ¬v1 Cannot be resolved away v1  v3

Resolution vs. Selector variables

Resolution vs. Selector variables

Resolution vs. Selector variables Comparison to selector-variables: Time: 28032 sec vs. 20453 sec (x1.37 speedup) HLMUC size: 12090 vs. 5090 (57.8% reduction)

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