Extensions and Generalizations of Minimal Unsatisfiability Anton - PowerPoint PPT Presentation

Group-oriented MUSes [Liffiton&Sakallah, JAR’08; Nadel, FMCAD’10] F = G 1 ∪ G 2 ∪ G 3 ∈ UNSAT — partitioned into groups (sets) of clauses G 1 = { C 1 , C 2 } , G 2 = { C 3 , C 4 } , G 3 = { C 5 , C 6 } . C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) C 2 = ( q ) {G 1 , G 2 } is a subset-minimal set of groups sufficient to refute F . {G 1 , G 2 } is a group-MUS of (the partitioned) F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 5

Group-oriented MUSes [Liffiton&Sakallah, JAR’08; Nadel, FMCAD’10] F = G 1 ∪ G 2 ∪ G 3 ∈ UNSAT — partitioned into groups (sets) of clauses G 1 = { C 1 , C 2 } , G 2 = { C 3 , C 4 } , G 3 = { C 5 , C 6 } . C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) C 2 = ( q ) {G 1 , G 2 } is a subset-minimal set of groups sufficient to refute F . {G 1 , G 2 } is a group-MUS of (the partitioned) F . {G 1 , G 3 } is also a group-MUS of (the partitioned) F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 5

Group-oriented MUSes Def: Given a group-CNF formula F = G 0 ∪ G 1 ∪ · · · ∪ G n ∈ UNSAT, a group oriented MUS (GMUS) of F is a set {G i 1 , . . . , G i k } such that F ′ = G 0 ∪ � 1 ≤ j ≤ k G i j ∈ UNSAT, and for every 1 ≤ j ≤ k , F ′ \ G i j ∈ SAT. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 6

Group-oriented MUSes Def: Given a group-CNF formula F = G 0 ∪ G 1 ∪ · · · ∪ G n ∈ UNSAT, a group oriented MUS (GMUS) of F is a set {G i 1 , . . . , G i k } such that F ′ = G 0 ∪ � 1 ≤ j ≤ k G i j ∈ UNSAT, and for every 1 ≤ j ≤ k , F ′ \ G i j ∈ SAT. Example: a group = CNF representation of a gate in a circuit. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 6

Group-oriented MUSes Def: Given a group-CNF formula F = G 0 ∪ G 1 ∪ · · · ∪ G n ∈ UNSAT, a group oriented MUS (GMUS) of F is a set {G i 1 , . . . , G i k } such that F ′ = G 0 ∪ � 1 ≤ j ≤ k G i j ∈ UNSAT, and for every 1 ≤ j ≤ k , F ′ \ G i j ∈ SAT. Example: a group = CNF representation of a gate in a circuit. ◮ Note the special role of G 0 — the “background” clauses. Example: in PBA a group = CNF representation of a latch, G 0 is the rest of the circuit. ◮ G 0 ∈ UNSAT ⇐ ⇒ the GMUS is ∅ . ◮ MUS computation is a special case of group-MUS computation. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 6

Group-oriented MUSes Def: Given a group-CNF formula F = G 0 ∪ G 1 ∪ · · · ∪ G n ∈ UNSAT, a group oriented MUS (GMUS) of F is a set {G i 1 , . . . , G i k } such that F ′ = G 0 ∪ � 1 ≤ j ≤ k G i j ∈ UNSAT, and for every 1 ≤ j ≤ k , F ′ \ G i j ∈ SAT. Example: a group = CNF representation of a gate in a circuit. ◮ Note the special role of G 0 — the “background” clauses. Example: in PBA a group = CNF representation of a latch, G 0 is the rest of the circuit. ◮ G 0 ∈ UNSAT ⇐ ⇒ the GMUS is ∅ . ◮ MUS computation is a special case of group-MUS computation. Perhaps the most useful (from practical standpoint) and versatile generalization of MUSes. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 6

Computing Group-oriented MUSes All current MUS extraction algorithms can be lifted to group-oriented setting: ◮ A group G is necessary for GCNF F if F ∈ UNSAT and F \ G ∈ SAT. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 7

Computing Group-oriented MUSes All current MUS extraction algorithms can be lifted to group-oriented setting: ◮ A group G is necessary for GCNF F if F ∈ UNSAT and F \ G ∈ SAT. ◮ Fact: A group G is necessary for GCNF F iff ∃ τ , such that τ ( F \ G ) = 1 and τ ( G ) = 0. Note: τ need to falsify only some clauses of G . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 7

Computing Group-oriented MUSes All current MUS extraction algorithms can be lifted to group-oriented setting: ◮ A group G is necessary for GCNF F if F ∈ UNSAT and F \ G ∈ SAT. ◮ Fact: A group G is necessary for GCNF F iff ∃ τ , such that τ ( F \ G ) = 1 and τ ( G ) = 0. Note: τ need to falsify only some clauses of G . ◮ Now one can do deletion-based, insertion-based, dichotomic, etc. group-MUS extraction. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 7

Computing Group-oriented MUSes All current MUS extraction algorithms can be lifted to group-oriented setting: ◮ A group G is necessary for GCNF F if F ∈ UNSAT and F \ G ∈ SAT. ◮ Fact: A group G is necessary for GCNF F iff ∃ τ , such that τ ( F \ G ) = 1 and τ ( G ) = 0. Note: τ need to falsify only some clauses of G . ◮ Now one can do deletion-based, insertion-based, dichotomic, etc. group-MUS extraction. ◮ Some low-level modification to a SAT solver are possible to optimize SAT solver use [Ryvchin and Strichman, SAT’11] . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 7

Computing Group-oriented MUSes: optimizations Group-set refinement: ◮ When F \ G ∈ UNSAT, get an unsatisfiable core U from the SAT solver. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 8

Computing Group-oriented MUSes: optimizations Group-set refinement: ◮ When F \ G ∈ UNSAT, get an unsatisfiable core U from the SAT solver. ◮ Remove any group G i for which U ∩ G i = ∅ . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 8

Computing Group-oriented MUSes: optimizations Group-set refinement: ◮ When F \ G ∈ UNSAT, get an unsatisfiable core U from the SAT solver. ◮ Remove any group G i for which U ∩ G i = ∅ . ◮ In assumption-based implementations, one can get a “group-core” right away. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 8

Computing Group-oriented MUSes: optimizations Group-set refinement: ◮ When F \ G ∈ UNSAT, get an unsatisfiable core U from the SAT solver. ◮ Remove any group G i for which U ∩ G i = ∅ . ◮ In assumption-based implementations, one can get a “group-core” right away. ◮ Very effective. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 8

Computing Group-oriented MUSes: optimizations Model rotation: ◮ When F \ G ∈ SAT, get a model τ — this is a witness for necessity G . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 9

Computing Group-oriented MUSes: optimizations Model rotation: ◮ When F \ G ∈ SAT, get a model τ — this is a witness for necessity G . ◮ As with MUSes, try to modify τ into a witness for another group G ′ . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 9

Computing Group-oriented MUSes: optimizations Model rotation: ◮ When F \ G ∈ SAT, get a model τ — this is a witness for necessity G . ◮ As with MUSes, try to modify τ into a witness for another group G ′ . ◮ Issue: τ may satisfy more than one clause. Solutions: A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 9

Computing Group-oriented MUSes: optimizations Model rotation: ◮ When F \ G ∈ SAT, get a model τ — this is a witness for necessity G . ◮ As with MUSes, try to modify τ into a witness for another group G ′ . ◮ Issue: τ may satisfy more than one clause. Solutions: ◮ ignore (i.e. do not rotate) - actually, “works” quite well. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 9

Computing Group-oriented MUSes: optimizations Model rotation: ◮ When F \ G ∈ SAT, get a model τ — this is a witness for necessity G . ◮ As with MUSes, try to modify τ into a witness for another group G ′ . ◮ Issue: τ may satisfy more than one clause. Solutions: ◮ ignore (i.e. do not rotate) - actually, “works” quite well. ◮ use group structure - e.g. a group is a set of unrelated equivalences. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 9

Computing Group-oriented MUSes: optimizations Model rotation: ◮ When F \ G ∈ SAT, get a model τ — this is a witness for necessity G . ◮ As with MUSes, try to modify τ into a witness for another group G ′ . ◮ Issue: τ may satisfy more than one clause. Solutions: ◮ ignore (i.e. do not rotate) - actually, “works” quite well. ◮ use group structure - e.g. a group is a set of unrelated equivalences. ◮ use mini-SLS. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 9

Computing Group-oriented MUSes: optimizations Model rotation: ◮ When F \ G ∈ SAT, get a model τ — this is a witness for necessity G . ◮ As with MUSes, try to modify τ into a witness for another group G ′ . ◮ Issue: τ may satisfy more than one clause. Solutions: ◮ ignore (i.e. do not rotate) - actually, “works” quite well. ◮ use group structure - e.g. a group is a set of unrelated equivalences. ◮ use mini-SLS. ◮ Problem: if G 0 is really large, the next step will get into G 0 , and rotation will get “lost” inside it. No satisfactory solution yet. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 9

Computing Group-oriented MUSes: optimizations Model rotation: ◮ When F \ G ∈ SAT, get a model τ — this is a witness for necessity G . ◮ As with MUSes, try to modify τ into a witness for another group G ′ . ◮ Issue: τ may satisfy more than one clause. Solutions: ◮ ignore (i.e. do not rotate) - actually, “works” quite well. ◮ use group structure - e.g. a group is a set of unrelated equivalences. ◮ use mini-SLS. ◮ Problem: if G 0 is really large, the next step will get into G 0 , and rotation will get “lost” inside it. No satisfactory solution yet. ◮ Effectiveness varies, depending on the structure of GCNFs. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 9

Computing Group-oriented MUSes: optimizations Redundancy removal: ◮ Instead of SAT checking F \ G , check ( F \ G ) ∪ CNF( ¬G )). A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 10

Computing Group-oriented MUSes: optimizations Redundancy removal: ◮ Instead of SAT checking F \ G , check ( F \ G ) ∪ CNF( ¬G )). ◮ Can build CNF( ¬G ) as follows (Plaisted-Greenbaum transform): A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 10

Computing Group-oriented MUSes: optimizations Redundancy removal: ◮ Instead of SAT checking F \ G , check ( F \ G ) ∪ CNF( ¬G )). ◮ Can build CNF( ¬G ) as follows (Plaisted-Greenbaum transform): ◮ For each C ∈ G create a new variable u C , and A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 10

Computing Group-oriented MUSes: optimizations Redundancy removal: ◮ Instead of SAT checking F \ G , check ( F \ G ) ∪ CNF( ¬G )). ◮ Can build CNF( ¬G ) as follows (Plaisted-Greenbaum transform): ◮ For each C ∈ G create a new variable u C , and for each literal l ∈ C create a clause ( ¬ u C ∨ ¬ l ), i.e. u C → ¬ l . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 10

Computing Group-oriented MUSes: optimizations Redundancy removal: ◮ Instead of SAT checking F \ G , check ( F \ G ) ∪ CNF( ¬G )). ◮ Can build CNF( ¬G ) as follows (Plaisted-Greenbaum transform): ◮ For each C ∈ G create a new variable u C , and for each literal l ∈ C create a clause ( ¬ u C ∨ ¬ l ), i.e. u C → ¬ l . ◮ Create a clause � C ∈G u c . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 10

Computing Group-oriented MUSes: optimizations Redundancy removal: ◮ Instead of SAT checking F \ G , check ( F \ G ) ∪ CNF( ¬G )). ◮ Can build CNF( ¬G ) as follows (Plaisted-Greenbaum transform): ◮ For each C ∈ G create a new variable u C , and for each literal l ∈ C create a clause ( ¬ u C ∨ ¬ l ), i.e. u C → ¬ l . ◮ Create a clause � C ∈G u c . ◮ Same problem with tainted cores as in MUS. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 10

Computing Group-oriented MUSes: optimizations Redundancy removal: ◮ Instead of SAT checking F \ G , check ( F \ G ) ∪ CNF( ¬G )). ◮ Can build CNF( ¬G ) as follows (Plaisted-Greenbaum transform): ◮ For each C ∈ G create a new variable u C , and for each literal l ∈ C create a clause ( ¬ u C ∨ ¬ l ), i.e. u C → ¬ l . ◮ Create a clause � C ∈G u c . ◮ Same problem with tainted cores as in MUS. ◮ Effectiveness is not clear. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 10

Variable MUSes [Zhen-Yu Chen and De-Cheng Ding, TAMC’06] F = { C 1 , . . . , C 6 } , Var ( F ) = { p , q , r } C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) C 2 = ( q ) What is a subset-minimal set of variables sufficient to refute F ? A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 11

Variable MUSes [Zhen-Yu Chen and De-Cheng Ding, TAMC’06] F = { C 1 , . . . , C 6 } , Var ( F ) = { p , q , r } C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) C 2 = ( q ) What is a subset-minimal set of variables sufficient to refute F ? { p , q } is such a subset of variables. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 11

Variable MUSes [Zhen-Yu Chen and De-Cheng Ding, TAMC’06] F = { C 1 , . . . , C 6 } , Var ( F ) = { p , q , r } C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) C 2 = ( q ) What is a subset-minimal set of variables sufficient to refute F ? { p , q } is such a subset of variables. { p , q } is a variable-MUS of F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 11

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 12

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 2 = ( q ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 12

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 2 = ( q ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) ◮ The subformula induced by { p , q } is F| { p , q } = { C 1 , C 2 , C 3 , C 4 } . ◮ variable r is “removed” from F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 12

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 2 = ( q ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) ◮ The subformula induced by { p , q } is F| { p , q } = { C 1 , C 2 , C 3 , C 4 } . ◮ variable r is “removed” from F . ◮ The subformula induced by { p } , F| { p } = { C 1 } . ◮ variables q , r are “removed” from F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 12

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . Def: A set V ⊆ Var ( F ) is a variable-MUS (VMUS) of F if F| V ∈ UNSAT, and for any V ′ ⊂ V , F| V ′ ∈ SAT. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 13

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . Def: A set V ⊆ Var ( F ) is a variable-MUS (VMUS) of F if F| V ∈ UNSAT, and for any V ′ ⊂ V , F| V ′ ∈ SAT. C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 2 = ( q ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 13

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . Def: A set V ⊆ Var ( F ) is a variable-MUS (VMUS) of F if F| V ∈ UNSAT, and for any V ′ ⊂ V , F| V ′ ∈ SAT. C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 2 = ( q ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) ◮ F| { p , q } = { C 1 , C 2 , C 3 , C 4 } ∈ UNSAT. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 13

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . Def: A set V ⊆ Var ( F ) is a variable-MUS (VMUS) of F if F| V ∈ UNSAT, and for any V ′ ⊂ V , F| V ′ ∈ SAT. C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 2 = ( q ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) ◮ F| { p } = { C 1 } ∈ SAT. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 13

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . Def: A set V ⊆ Var ( F ) is a variable-MUS (VMUS) of F if F| V ∈ UNSAT, and for any V ′ ⊂ V , F| V ′ ∈ SAT. C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 2 = ( q ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) ◮ F| { q } = { C 2 } ∈ SAT. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 13

Variable MUSes Def: The subformula of CNF F induced by V ⊆ Var ( F ) is the formula F| V = { C | C ∈ F and Var ( C ) ⊆ V } . I.e. F| V includes only those clauses of F whose variables are in V . Def: A set V ⊆ Var ( F ) is a variable-MUS (VMUS) of F if F| V ∈ UNSAT, and for any V ′ ⊂ V , F| V ′ ∈ SAT. C 1 = ( p ) C 3 = ( ¬ p ∨ ¬ q ) C 5 = ( ¬ p ∨ r ) C 2 = ( q ) C 4 = ( p ∨ q ) C 6 = ( ¬ q ∨ ¬ r ) ◮ Hence, { p , q } is a VMUS of F . Notation: { p , q } ∈ VMUS( F ). A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 13

Computing Variable MUSes Basic algorithm is similar to deletion-based MUS extraction algorithm: based on detection of necessary variables. Notation: for v ∈ Var ( F ), F v = { C | C ∈ F and v ∈ Var ( C ) } . Definition v ∈ Var ( F ) is necessary for F if F ∈ UNSAT and F \ F v ∈ SAT. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 14

Computing Variable MUSes Basic algorithm is similar to deletion-based MUS extraction algorithm: based on detection of necessary variables. Notation: for v ∈ Var ( F ), F v = { C | C ∈ F and v ∈ Var ( C ) } . Definition v ∈ Var ( F ) is necessary for F if F ∈ UNSAT and F \ F v ∈ SAT. Properties: 1. V ∈ VMUS( F ) if and only if every v ∈ V is necessary for F| V . 2. If v is necessary for F , then v is necessary for any unsatisfiable F ′ ⊆ F . 3. Fact: v is necessary for F iff for some τ , τ ( F \ F v ) = 1, and τ ( F v ) = 0. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 14

Computing Variable MUSes: optimizations [Belov et al, SAT’12] Model rotation — needs to be modified: ◮ When F \ F v ∈ SAT, the model falsifies some clauses with v . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 15

Computing Variable MUSes: optimizations [Belov et al, SAT’12] Model rotation — needs to be modified: ◮ When F \ F v ∈ SAT, the model falsifies some clauses with v . ◮ If we flip v (as in MUS extraction) the only clauses that will get falsified are those with variable v as well (in the opposite polarity). Does this give us anything ? A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 15

Computing Variable MUSes: optimizations [Belov et al, SAT’12] Model rotation — needs to be modified: ◮ When F \ F v ∈ SAT, the model falsifies some clauses with v . ◮ If we flip v (as in MUS extraction) the only clauses that will get falsified are those with variable v as well (in the opposite polarity). Does this give us anything ? ◮ Fact: For any τ , any variable shared among clauses of Unsat ( F , τ ) is necessary for F . Note: In particular, when Unsat ( F , τ ) = { C } is a singleton – all variables of C are necessary. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 15

Computing Variable MUSes: optimizations [Belov et al, SAT’12] Model rotation — needs to be modified: ◮ When F \ F v ∈ SAT, the model falsifies some clauses with v . ◮ If we flip v (as in MUS extraction) the only clauses that will get falsified are those with variable v as well (in the opposite polarity). Does this give us anything ? ◮ Fact: For any τ , any variable shared among clauses of Unsat ( F , τ ) is necessary for F . Note: In particular, when Unsat ( F , τ ) = { C } is a singleton – all variables of C are necessary. ◮ For the assignment returned by SAT solver, v satisfies this condition, but maybe other variables as well. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 15

Computing Variable MUSes: optimizations [Belov et al, SAT’12] Model rotation — needs to be modified: ◮ When F \ F v ∈ SAT, the model falsifies some clauses with v . ◮ If we flip v (as in MUS extraction) the only clauses that will get falsified are those with variable v as well (in the opposite polarity). Does this give us anything ? ◮ Fact: For any τ , any variable shared among clauses of Unsat ( F , τ ) is necessary for F . Note: In particular, when Unsat ( F , τ ) = { C } is a singleton – all variables of C are necessary. ◮ For the assignment returned by SAT solver, v satisfies this condition, but maybe other variables as well. ◮ So, flip v , but collect variables shared falsified clauses. Continue recursively. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 15

Computing Variable MUSes: optimizations Variable-set refinement: ◮ If a variable is not in the unsatisfiable core, remove it. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 16

Computing Variable MUSes: optimizations Variable-set refinement: ◮ If a variable is not in the unsatisfiable core, remove it. Redundancy removal — same trick as for groups, but with CNF ( ¬F v ). A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 16

Computing Variable MUSes: optimizations Variable-set refinement: ◮ If a variable is not in the unsatisfiable core, remove it. Redundancy removal — same trick as for groups, but with CNF ( ¬F v ). All optimizations are very effective in this setting. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 16

Computing Variable MUSes: optimizations Variable-set refinement: ◮ If a variable is not in the unsatisfiable core, remove it. Redundancy removal — same trick as for groups, but with CNF ( ¬F v ). All optimizations are very effective in this setting. VMUSes can often be computed faster than MUSes (not always !). A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 16

Computing Variable MUSes: optimizations Variable-set refinement: ◮ If a variable is not in the unsatisfiable core, remove it. Redundancy removal — same trick as for groups, but with CNF ( ¬F v ). All optimizations are very effective in this setting. VMUSes can often be computed faster than MUSes (not always !). Another idea: translate to group-MUS. ◮ Doesn’t work well, compared to the dedicated algorithm. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 16

Minimal Equivalent Subformulas Let M be the set of all models of CNF formula F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 17

Minimal Equivalent Subformulas Let M be the set of all models of CNF formula F . What happens to M if we remove some clause C from F ? A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 17

Minimal Equivalent Subformulas Let M be the set of all models of CNF formula F . What happens to M if we remove some clause C from F ? Case 1: M grows — the clause C is irredundant in F . C “blocks” some assignments that no other clause does. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 17

Minimal Equivalent Subformulas Let M be the set of all models of CNF formula F . What happens to M if we remove some clause C from F ? Case 1: M grows — the clause C is irredundant in F . C “blocks” some assignments that no other clause does. Case 2: M does not change — the clause C is redundant in F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 17

Minimal Equivalent Subformulas Let M be the set of all models of CNF formula F . What happens to M if we remove some clause C from F ? Case 1: M grows — the clause C is irredundant in F . C “blocks” some assignments that no other clause does. Case 2: M does not change — the clause C is redundant in F . C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 17

Minimal Equivalent Subformulas Let M be the set of all models of CNF formula F . What happens to M if we remove some clause C from F ? Case 1: M grows — the clause C is irredundant in F . C “blocks” some assignments that no other clause does. Case 2: M does not change — the clause C is redundant in F . C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) C 1 is irredundant in F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 17

Minimal Equivalent Subformulas Let M be the set of all models of CNF formula F . What happens to M if we remove some clause C from F ? Case 1: M grows — the clause C is irredundant in F . C “blocks” some assignments that no other clause does. Case 2: M does not change — the clause C is redundant in F . C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) C 1 is irredundant in F . C 2 is redundant in F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 17

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . A CNF formula F is irredundant if it does not have redundant clauses. ◮ I.e. every clause “serves a purpose” C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . A CNF formula F is irredundant if it does not have redundant clauses. ◮ I.e. every clause “serves a purpose” Keep on removing redundant clauses, until the remainder is irredundant: C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . A CNF formula F is irredundant if it does not have redundant clauses. ◮ I.e. every clause “serves a purpose” Keep on removing redundant clauses, until the remainder is irredundant: ◮ C 2 is redundant in F (subsumed by C 1 ) C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . A CNF formula F is irredundant if it does not have redundant clauses. ◮ I.e. every clause “serves a purpose” Keep on removing redundant clauses, until the remainder is irredundant: ◮ C 2 is redundant in F (subsumed by C 1 ) : F ′ = F \ { C 2 } C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . A CNF formula F is irredundant if it does not have redundant clauses. ◮ I.e. every clause “serves a purpose” Keep on removing redundant clauses, until the remainder is irredundant: ◮ C 2 is redundant in F (subsumed by C 1 ) : F ′ = F \ { C 2 } ◮ C 6 is redundant in F ′ (resolve C 3 and C 5 ) C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . A CNF formula F is irredundant if it does not have redundant clauses. ◮ I.e. every clause “serves a purpose” Keep on removing redundant clauses, until the remainder is irredundant: ◮ C 2 is redundant in F (subsumed by C 1 ) : F ′ = F \ { C 2 } ◮ C 6 is redundant in F ′ (resolve C 3 and C 5 ) : F ′′ = F ′ \ { C 6 } C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . A CNF formula F is irredundant if it does not have redundant clauses. ◮ I.e. every clause “serves a purpose” Keep on removing redundant clauses, until the remainder is irredundant: ◮ C 2 is redundant in F (subsumed by C 1 ) : F ′ = F \ { C 2 } ◮ C 6 is redundant in F ′ (resolve C 3 and C 5 ) : F ′′ = F ′ \ { C 6 } ◮ the rest of clauses are irredundant in F ′′ C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) F ′′ ⊂ F is irredundant and logically equivalent to F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . A CNF formula F is irredundant if it does not have redundant clauses. ◮ I.e. every clause “serves a purpose” Keep on removing redundant clauses, until the remainder is irredundant: ◮ C 2 is redundant in F (subsumed by C 1 ) : F ′ = F \ { C 2 } ◮ C 6 is redundant in F ′ (resolve C 3 and C 5 ) : F ′′ = F ′ \ { C 6 } ◮ the rest of clauses are irredundant in F ′′ C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) F ′′ ⊂ F is irredundant and logically equivalent to F . F ′′ is a Minimal Equivalent Subformula (MES) of F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas [Kleine B¨ uning and Zhao, AMAI’05; Liberatore, AIJ’05] A clause C ∈ F is redundant in F if F \ { C } ≡ F . Alternatively: C ∈ F is redundant if F \ { C } � C . A CNF formula F is irredundant if it does not have redundant clauses. ◮ I.e. every clause “serves a purpose” Keep on removing redundant clauses, until the remainder is irredundant: ◮ C 2 is redundant in F (subsumed by C 1 ) : F ′ = F \ { C 2 } ◮ C 6 is redundant in F ′ (resolve C 3 and C 5 ) : F ′′ = F ′ \ { C 6 } ◮ the rest of clauses are irredundant in F ′′ C 1 = ( x ) C 3 = ( y ∨ z ) C 5 = ( ¬ z ∨ t ) C 2 = ( x ∨ t ) C 4 = ( ¬ y ∨ ¬ z ) C 6 = ( y ∨ t ) F ′′ ⊂ F is irredundant and logically equivalent to F . F ′′ is a Minimal Equivalent Subformula (MES) of F . There is another. A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 18

Minimal Equivalent Subformulas Def: A Minimal Equivalent Subformula (MES) of a CNF F is an irredundant subformula of F logically equivalent to F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 19

Minimal Equivalent Subformulas Def: A Minimal Equivalent Subformula (MES) of a CNF F is an irredundant subformula of F logically equivalent to F . ◮ F ′ ⊆ F and F ′ ≡ F and ∀ C ∈ F ′ , F ′ \ { C } �≡ F A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 19

Minimal Equivalent Subformulas Def: A Minimal Equivalent Subformula (MES) of a CNF F is an irredundant subformula of F logically equivalent to F . ◮ F ′ ⊆ F and F ′ ≡ F and ∀ C ∈ F ′ , F ′ \ { C } �≡ F Note: If F ∈ UNSAT, then an MES is a Minimal Unsatisfiable Subformula (MUS) of F . A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 19

Minimal Equivalent Subformulas Def: A Minimal Equivalent Subformula (MES) of a CNF F is an irredundant subformula of F logically equivalent to F . ◮ F ′ ⊆ F and F ′ ≡ F and ∀ C ∈ F ′ , F ′ \ { C } �≡ F Note: If F ∈ UNSAT, then an MES is a Minimal Unsatisfiable Subformula (MUS) of F . Computation of MESes is about removing redundant clauses. Why would we want to do this ? A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 19

Minimal Equivalent Subformulas Def: A Minimal Equivalent Subformula (MES) of a CNF F is an irredundant subformula of F logically equivalent to F . ◮ F ′ ⊆ F and F ′ ≡ F and ∀ C ∈ F ′ , F ′ \ { C } �≡ F Note: If F ∈ UNSAT, then an MES is a Minimal Unsatisfiable Subformula (MUS) of F . Computation of MESes is about removing redundant clauses. Why would we want to do this ? ◮ In some applications redundancy is undesirable: ◮ knowledge bases [P. Liberatore, AIJ 2005] ◮ conformant and contingent planning [S.T. To, et al, AAAI 2010-11] ◮ probabilistic reasoning systems [M. Niepert, et al, J. of Approx. Reasoning, 2010] A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 19

Minimal Equivalent Subformulas Def: A Minimal Equivalent Subformula (MES) of a CNF F is an irredundant subformula of F logically equivalent to F . ◮ F ′ ⊆ F and F ′ ≡ F and ∀ C ∈ F ′ , F ′ \ { C } �≡ F Note: If F ∈ UNSAT, then an MES is a Minimal Unsatisfiable Subformula (MUS) of F . Computation of MESes is about removing redundant clauses. Why would we want to do this ? ◮ In some applications redundancy is undesirable: ◮ knowledge bases [P. Liberatore, AIJ 2005] ◮ conformant and contingent planning [S.T. To, et al, AAAI 2010-11] ◮ probabilistic reasoning systems [M. Niepert, et al, J. of Approx. Reasoning, 2010] ◮ Construction of concise CNF encodings A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 19

Is Redundancy Common ? A sample of 300 benchmarks from various applications of SAT A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 20

Is Redundancy Common ? A sample of 300 benchmarks from various applications of SAT ◮ Number of instances ( x ) with less than y % redundant clauses A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 20

Is Redundancy Common ? A sample of 300 benchmarks from various applications of SAT ◮ Number of instances ( x ) with less than y % redundant clauses ◮ 2/3 of instances have 20%-50% redundant clauses A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 20

Is Redundancy Common ? A sample of 300 benchmarks from various applications of SAT ◮ Number of instances ( x ) with less than y % redundant clauses ◮ 2/3 of instances have 20%-50% redundant clauses ◮ 1/3 has > 50%, in some cases > 90%, of redundant clauses ! A. Belov MU and MUSes: Extensions and Generalizations EPCL Training Camp, 2012 # 20