Maximum Satisfiability Ruben Martins - PowerPoint PPT Presentation

Upper Bound Search for MaxSAT UNSAT Unsatisfiable Optimal subformula Solution Find upper bound k for #unsatisfied soft clauses ϕ SAT Solver Satisfying Refinement assignment SAT 20/41

Upper Bound Search for MaxSAT UNSAT Unsatisfiable Optimal subformula Solution Can we unsatisfy less than k clauses? ϕ SAT Solver Satisfying Refinement assignment SAT ϕ � ϕ ′ 20/41

Upper Bound Search for MaxSAT UNSAT Unsatisfiable Optimal subformula Solution Can we unsatisfy less than j ( < k ) clauses? ϕ ′ SAT Solver Satisfying Refinement assignment SAT 20/41

Upper Bound Search for MaxSAT UNSAT Unsatisfiable Optimal subformula Solution Can we unsatisfy less than j ( < k ) clauses? ϕ ′′ SAT Solver Satisfying Refinement assignment SAT 20/41

Linear Search Algorithms SAT-UNSAT Partial MaxSAT Formula: ϕ h (Hard): ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s (Soft): x 1 x 3 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 21/41

Linear Search Algorithms SAT-UNSAT Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 ◮ Relax all soft clauses ◮ Relaxation variables: ◮ V R = { r 1 , r 2 , r 3 , r 4 } ◮ If a soft clause ω i is unsatisfied , then r i = 1 ◮ If a soft clause ω i is satisfied , then r i = 0 21/41

Linear Search Algorithms SAT-UNSAT Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 V R = { r 1 , r 2 , r 3 , r 4 } ◮ Formula is satisfiable ◮ ν = { x 1 = 1 , x 2 = 0 , x 3 = 0 , r 1 = 0 , r 2 = 1 , r 3 = 1 , r 4 = 0 } ◮ Goal: Minimize number of relaxation variables assigned to 1 21/41

Can we unsatisfy less than 2 soft clauses? Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 µ = 2 V R = { r 1 , r 2 , r 3 , r 4 } ◮ r 2 and r 3 were assigned truth value 1: ◮ Current solution unsatisfies 2 soft clauses ◮ Can less than 2 soft clauses be unsatisfied? 21/41

Can we unsatisfy less than 2 soft clauses? Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( � r i ∈ V R r i ≤ 1) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 µ = 2 V R = { r 1 , r 2 , r 3 , r 4 } ◮ Add cardinality constraint that excludes solutions that unsatisfies 2 or more soft clauses: ◮ CNF( r 1 + r 2 + r 3 + r 4 ≤ 1) 21/41

Can we unsatisfy less than 2 soft clauses? No! Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( � r i ∈ V R r i ≤ 1) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 µ = 2 V R = { r 1 , r 2 , r 3 , r 4 } ◮ Formula is unsatisfiable: ◮ There are no solutions that unsatisfy 1 or less soft clauses 21/41

Can we unsatisfy less than 2 soft clauses? No! Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 1 x 3 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 µ = 2 V R = { r 1 , r 2 , r 3 , r 4 } ◮ Optimal solution : given by the last model and corresponds to unsatisfying 2 soft clauses: ◮ ν = { x 1 = 1 , x 2 = 0 , x 3 = 0 } 21/41

MaxSAT algorithms ◮ We have just seen a search on the upper bound ◮ What other kind of search can we do to find an optimal solution? 22/41

MaxSAT algorithms ◮ We have just seen a search on the upper bound ◮ What other kind of search can we do to find an optimal solution? ◮ What if we start searching from the lower bound ? 22/41

Lower Bound Search for MaxSAT SAT Satisfying Optimal assignment Solution Can we satisfy all soft clauses? ϕ SAT Solver Unsatisfiable Refinement subformula UNSAT 23/41

Lower Bound Search for MaxSAT SAT Satisfying Optimal assignment Solution Can we satisfy all soft clauses? ϕ SAT Solver Unsatisfiable Refinement subformula UNSAT ϕ � ϕ ′ 23/41

Lower Bound Search for MaxSAT SAT Satisfying Optimal assignment Solution Can we satisfy all soft clauses but 1? ϕ ′ SAT Solver Unsatisfiable Refinement subformula UNSAT 23/41

Lower Bound Search for MaxSAT SAT Satisfying Optimal assignment Solution Can we satisfy all soft clauses but 1? ϕ ′ SAT Solver Unsatisfiable Refinement subformula UNSAT ϕ ′ � ϕ ′′ 23/41

Lower Bound Search for MaxSAT SAT Satisfying Optimal assignment Solution Can we satisfy all soft clauses but 2? ϕ ′′ SAT Solver Unsatisfiable Refinement subformula UNSAT 23/41

Lower Bound Search for MaxSAT SAT Satisfying Optimal assignment Solution Can we satisfy all soft clauses but 2? ϕ ′′ SAT Solver Unsatisfiable Refinement subformula UNSAT 23/41

Linear Search Algorithms UNSAT-SAT Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 ◮ Relax all soft clauses ◮ Relaxation variables: ◮ V R = { r 1 , r 2 , r 3 , r 4 } ◮ If a soft clause ω i is unsatisfied , then r i = 1 ◮ If a soft clause ω i is satisfied , then r i = 0 24/41

Can we satisfy all soft clauses? Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( � r i ∈ V R r i ≤ 0) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 µ = 2 V R = { r 1 , r 2 , r 3 , r 4 } ◮ Add cardinality constraint that excludes solutions that unsatisfies 1 or more soft clauses: ◮ CNF( r 1 + r 2 + r 3 + r 4 ≤ 0) 24/41

Can we satisfy all soft clauses but 1? Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( � r i ∈ V R r i ≤ 0) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 ◮ Formula is unsatisfiable: ◮ There are no solutions that unsatisfy 0 or less soft clauses ◮ Add cardinality constraint that excludes solutions that unsatisfies 2 or more soft clauses: ◮ CNF( r 1 + r 2 + r 3 + r 4 ≤ 1) 24/41

Can we satisfy all soft clauses but 2? Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( � r i ∈ V R r i ≤ 1) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 ◮ Formula is unsatisfiable: ◮ There are no solutions that unsatisfy 1 or less soft clauses ◮ Add cardinality constraint that excludes solutions that unsatisfies 3 or more soft clauses: ◮ CNF( r 1 + r 2 + r 3 + r 4 ≤ 2) 24/41

Can we satisfy all soft clauses but 2? Yes! Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( � r i ∈ V R r i ≤ 2) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 ◮ Formula is satisfiable: ◮ µ = { x 1 = 1 , x 2 = 0 , x 3 = 0 , r 1 = 0 , r 2 = 1 , r 3 = 1 , r 4 = 0 } ◮ Optimal solution unsatisfies 2 soft clauses 24/41

Unsatisfiability-based Algorithms ◮ What are the problems of this algorithm? (Hint) Number of relaxation variables? Size of the cardinality constraint? Other? 25/41

Unsatisfiability-based Algorithms ◮ What are the problems of this algorithm? (Hint) Number of relaxation variables? Size of the cardinality constraint? Other? ◮ We relax all soft clauses! ◮ The cardinality constraint contain as many literals as we have soft clauses! ◮ Can we do better? 25/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h (Hard): ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s (Soft): x 1 x 3 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 26/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 1 x 3 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 ◮ Formula is unsatisfiable 26/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 1 x 3 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 ◮ Formula is unsatisfiable ◮ Identify an unsatisfiable core 26/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + r 2 ≤ 1) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 ◮ Relax non-relaxed soft clauses in unsatisfiable core: ◮ Add cardinality constraint that excludes solutions that unsatisfies 2 or more soft clauses: ◮ CNF( r 1 + r 2 ≤ 1) ◮ Relaxation on demand instead of relaxing all soft clauses eagerly 26/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + r 2 ≤ 1) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 ◮ Formula is unsatisfiable 26/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + r 2 ≤ 1) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 ◮ Formula is unsatisfiable ◮ Identify an unsatisfiable core 26/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + . . . + r 4 ≤ 2) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 ◮ Relax non-relaxed soft clauses in unsatisfiable core: ◮ Add cardinality constraint that excludes solutions that unsatisfies 3 or more soft clauses: ◮ CNF( r 1 + r 2 + r 3 + r 4 ≤ 2) ◮ Relaxation on demand instead of relaxing all soft clauses eagerly 26/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + . . . + r 4 ≤ 2) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ∨ r 3 ¬ x 3 ∨ x 1 ∨ r 4 ◮ Formula is satisfiable: ◮ µ = { x 1 = 1 , x 2 = 0 , x 3 = 0 , r 1 = 0 , r 2 = 1 , r 3 = 1 , r 4 = 0 } ◮ Optimal solution unsatisfies 2 soft clauses 26/41

Unsatisfiability-based Algorithms ◮ What are the problems of this algorithm? (Hint) Number of relaxation variables? Size of the cardinality constraint? Other? 27/41

Unsatisfiability-based Algorithms ◮ What are the problems of this algorithm? (Hint) Number of relaxation variables? Size of the cardinality constraint? Other? ◮ We must translate cardinality constraints into CNF! ◮ If the number of literals is large than we may generate a very large formula! ◮ Can we do better? 27/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h (Hard): ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s (Soft): x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 x 1 x 3 28/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 x 1 x 3 ◮ Formula is unsatisfiable 28/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 x 1 x 3 ◮ Formula is unsatisfiable ◮ Identify an unsatisfiable core 28/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + r 2 ≤ 1) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 ◮ Relax unsatisfiable core: ◮ Add relaxation variables ◮ Add AtMost1 constraint 28/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + r 2 ≤ 1) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 ◮ Formula is unsatisfiable 28/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + r 2 ≤ 1) ϕ s : x 1 ∨ r 1 x 3 ∨ r 2 x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 ◮ Formula is unsatisfiable ◮ Identify an unsatisfiable core 28/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + r 2 ≤ 1) CNF( r 3 + . . . + r 6 ≤ 1) ϕ s : x 1 ∨ r 1 ∨ r 3 x 3 ∨ r 2 ∨ r 4 x 2 ∨ ¬ x 1 ∨ r 5 ¬ x 3 ∨ x 1 ∨ r 6 ◮ Relax unsatisfiable core: ◮ Add relaxation variables ◮ Add AtMost1 constraint ◮ Soft clauses may be relaxed multiple times 28/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 CNF( r 1 + r 2 ≤ 1) CNF( r 3 + . . . + r 6 ≤ 1) ϕ s : x 1 ∨ r 1 ∨ r 3 x 3 ∨ r 2 ∨ r 4 x 2 ∨ ¬ x 1 ∨ r 5 ¬ x 3 ∨ x 1 ∨ r 6 ◮ Formula is satisfiable ◮ An optimal solution would be: ◮ ν = { x 1 = 1 , x 2 = 0 , x 3 = 0 } 28/41

Unsatisfiability-based Algorithms Partial MaxSAT Formula: ϕ h : ¬ x 2 ∨ ¬ x 1 x 2 ∨ ¬ x 3 ϕ s : x 2 ∨ ¬ x 1 ¬ x 3 ∨ x 1 x 1 x 3 ◮ Formula is satisfiable ◮ An optimal solution would be: ◮ ν = { x 1 = 1 , x 2 = 0 , x 3 = 0 } ◮ This assignment unsatisfies 2 soft clauses 28/41

Challenges for Unsatisfiability-based MaxSAT Algorithms ◮ Unsatisfiable cores found by the SAT solver are not minimal 29/41

Challenges for Unsatisfiability-based MaxSAT Algorithms ◮ Unsatisfiable cores found by the SAT solver are not minimal Formula ϕ ϕ 29/41

Challenges for Unsatisfiability-based MaxSAT Algorithms ◮ Unsatisfiable cores found by the SAT solver are not minimal Formula ϕ ϕ c Unsatisfiable core ϕ c ϕ 29/41

Challenges for Unsatisfiability-based MaxSAT Algorithms ◮ Unsatisfiable cores found by the SAT solver are not minimal Formula ϕ ϕ c ϕ m Unsatisfiable core ϕ c ϕ Minimal core ϕ m 29/41

Challenges for Unsatisfiability-based MaxSAT Algorithms ◮ Unsatisfiable cores found by the SAT solver are not minimal Formula ϕ ϕ c ϕ m Unsatisfiable core ϕ c ϕ Minimal core ϕ m ◮ Minimizing unsatisfiable cores is computationally hard 29/41

Partitioning in MaxSAT ◮ Partitioning in MaxSAT: ◮ Partition the soft clauses into disjoint sets ◮ Iteratively increase the size of the MaxSAT formula ϕ ϕ 1 ϕ 2 ϕ 3 ϕ n . . . ◮ Advantages: ◮ Easier formulas for the SAT solver ◮ Smaller unsatisfiable cores at each iteration 30/41

Framework for Partitioning-based MaxSAT Algorithms SAT Satisfying Optimal assignment Solution ϕ SAT Solver Unsatisfiable Refinement subformula UNSAT 31/41

Framework for Partitioning-based MaxSAT Algorithms ϕ 1 SAT Satisfying Optimal assignment Solution ϕ 2 SAT Solver Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms ϕ 1 SAT Satisfying Optimal assignment Solution ϕ 2 ϕ 1 SAT Solver Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms ϕ 1 SAT Satisfying Optimal assignment Solution ϕ 2 ϕ 1 SAT Solver Unsatisfiable Refinement subformula UNSAT ϕ 3 ϕ 1 � ϕ ′ 1 31/41

Framework for Partitioning-based MaxSAT Algorithms ϕ 1 SAT Satisfying Optimal assignment Solution ϕ ′ ϕ 2 SAT Solver 1 Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms Solution may not be optimal! ϕ 1 SAT Satisfying Optimal assignment Solution ϕ ′ ϕ 2 SAT Solver 1 Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms Yes ϕ 1 SAT Satisfying Partitions? assignment No ϕ ′ ϕ 2 SAT Solver 1 Optimal Solution Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms Yes ϕ 1 SAT Satisfying Partitions? assignment No ϕ ′ 1 ∪ ϕ 2 ϕ 2 SAT Solver Optimal Solution Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms Yes ϕ 1 SAT Satisfying Partitions? assignment No ϕ ′ 1 ∪ ϕ 2 ϕ 2 SAT Solver Optimal Solution Unsatisfiable Refinement subformula UNSAT ϕ 3 ϕ ′ 1 ∪ ϕ 2 � ϕ ′′ 1 ∪ ϕ ′ 2 31/41

Framework for Partitioning-based MaxSAT Algorithms Yes ϕ 1 SAT Satisfying Partitions? assignment No ϕ ′′ 1 ∪ ϕ ′ ϕ 2 SAT Solver 2 Optimal Solution Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms Yes ϕ 1 SAT Satisfying Partitions? assignment No ϕ ′′ 1 ∪ ϕ ′ ϕ 2 SAT Solver 2 Optimal Solution Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms Yes ϕ 1 SAT Satisfying Partitions? assignment No ϕ ′′ 1 ∪ ϕ ′ 2 ∪ ϕ 3 ϕ 2 SAT Solver Optimal Solution Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms Yes ϕ 1 SAT Satisfying Partitions? assignment No ϕ ′′ 1 ∪ ϕ ′ 2 ∪ ϕ 3 ϕ 2 SAT Solver Optimal Solution Unsatisfiable Refinement subformula UNSAT ϕ 3 ϕ ′′ 1 ∪ ϕ ′ 2 ∪ ϕ 3 � ϕ ′′′ 1 ∪ ϕ ′′ 2 ∪ ϕ ′ 3 31/41

Framework for Partitioning-based MaxSAT Algorithms Yes ϕ 1 SAT Satisfying Partitions? assignment No ϕ ′′′ 1 ∪ ϕ ′′ 2 ∪ ϕ ′ ϕ 2 SAT Solver 3 Optimal Solution Unsatisfiable Refinement subformula UNSAT ϕ 3 31/41