Reordering Rule Makes OBDD Proof Systems Stronger Sam Buss 1 Dmitry - PowerPoint PPT Presentation

Reordering Rule Makes OBDD Proof Systems Stronger Sam Buss 1 Dmitry Itsykson 2 Alexander Knop 1 Dmitry Sokolov 3 1 University of California, San Diego 2 St. Petersburg Department of V.A. Steklov Institute of Mathematics 3 KTH Royal Institute of Technology Computational Complexity Conference June 23, 2018 Sokolov D. | OBDD Proof Systems 1/11

Ordered binary decision diagram ( OBDD ) x 1 OBDD s represent Boolean functions { 0 , 1 } n → { 0 , 1 } ; 0 1 π is an ordering of variables; x 2 x 2 if i < j then x π ( j ) cannot appear 0 0 before x π ( i ) . 1 1 x 3 x 3 0 0 1 1 0 1 Sokolov D. | OBDD Proof Systems 2/11

Ordered binary decision diagram ( OBDD ) OBDD proofs of unsatisfiability: x 1 0 1 sequence of OBDD s: D 1 , D 2 , D 3 , . . . , D m ; x 2 x 2 D m ≡ 0; 0 0 OBDD s for axioms. 1 1 Rules: x 3 x 3 ∧ (join): D i , D j ⇒ D k ≡ ( D i ∧ D j ) ; 0 0 w (weakening): D i ⇒ D j , D i | = D j ; 1 1 r (reordering): D i ⇒ D j , D j ≡ D i 0 1 but orders of variables are different. Join rule can be applied only for OBDD s in the same order. Sokolov D. | OBDD Proof Systems 2/11

Ordered binary decision diagram ( OBDD ) OBDD proofs of unsatisfiability: x 1 0 1 sequence of OBDD s: D 1 , D 2 , D 3 , . . . , D m ; x 2 x 2 D m ≡ 0; 0 0 OBDD s for axioms. 1 1 Rules: x 3 x 3 ∧ (join): D i , D j ⇒ D k ≡ ( D i ∧ D j ) ; 0 0 w (weakening): D i ⇒ D j , D i | = D j ; 1 1 r (reordering): D i ⇒ D j , D j ≡ D i 0 1 but orders of variables are different. Join rule can be applied only for OBDD s in the same order. Sokolov D. | OBDD Proof Systems 2/11

Ordered binary decision diagram ( OBDD ) OBDD proofs of unsatisfiability: x 1 0 1 sequence of OBDD s: D 1 , D 2 , D 3 , . . . , D m ; x 2 x 2 D m ≡ 0; 0 0 OBDD s for axioms. 1 1 Rules: x 3 x 3 ∧ (join): D i , D j ⇒ D k ≡ ( D i ∧ D j ) ; 0 0 w (weakening): D i ⇒ D j , D i | = D j ; 1 1 r (reordering): D i ⇒ D j , D j ≡ D i 0 1 but orders of variables are different. Join rule can be applied only for OBDD s in the same order. Sokolov D. | OBDD Proof Systems 2/11

Ordered binary decision diagram ( OBDD ) OBDD proofs of unsatisfiability: x 1 0 1 sequence of OBDD s: D 1 , D 2 , D 3 , . . . , D m ; x 2 x 2 D m ≡ 0; 0 0 OBDD s for axioms. 1 1 Rules: x 3 x 3 ∧ (join): D i , D j ⇒ D k ≡ ( D i ∧ D j ) ; 0 0 w (weakening): D i ⇒ D j , D i | = D j ; 1 1 r (reordering): D i ⇒ D j , D j ≡ D i 0 1 but orders of variables are different. Join rule can be applied only for OBDD s in the same order. Sokolov D. | OBDD Proof Systems 2/11

Ordered binary decision diagram ( OBDD ) OBDD proofs of unsatisfiability: x 1 0 1 sequence of OBDD s: D 1 , D 2 , D 3 , . . . , D m ; x 2 x 2 D m ≡ 0; 0 0 OBDD s for axioms. 1 1 Rules: x 3 x 3 ∧ (join): D i , D j ⇒ D k ≡ ( D i ∧ D j ) ; 0 0 w (weakening): D i ⇒ D j , D i | = D j ; 1 1 r (reordering): D i ⇒ D j , D j ≡ D i 0 1 but orders of variables are different. Join rule can be applied only for OBDD s in the same order. Sokolov D. | OBDD Proof Systems 2/11

Ordered binary decision diagram ( OBDD ) OBDD proofs of unsatisfiability: x 1 0 1 sequence of OBDD s: D 1 , D 2 , D 3 , . . . , D m ; x 2 x 2 D m ≡ 0; 0 0 OBDD s for axioms. 1 1 Rules: x 3 x 3 ∧ (join): D i , D j ⇒ D k ≡ ( D i ∧ D j ) ; 0 0 w (weakening): D i ⇒ D j , D i | = D j ; 1 1 r (reordering): D i ⇒ D j , D j ≡ D i 0 1 but orders of variables are different. Join rule can be applied only for OBDD s in the same order. Sokolov D. | OBDD Proof Systems 2/11

Ordered binary decision diagram ( OBDD ) OBDD proofs of unsatisfiability: x 1 0 1 sequence of OBDD s: D 1 , D 2 , D 3 , . . . , D m ; x 2 x 2 D m ≡ 0; 0 0 OBDD s for axioms. 1 1 Rules: x 3 x 3 ∧ (join): D i , D j ⇒ D k ≡ ( D i ∧ D j ) ; 0 0 w (weakening): D i ⇒ D j , D i | = D j ; 1 1 r (reordering): D i ⇒ D j , D j ≡ D i 0 1 but orders of variables are different. Join rule can be applied only for OBDD s in the same order. Sokolov D. | OBDD Proof Systems 2/11

Ordered binary decision diagram ( OBDD ) OBDD proofs of unsatisfiability: x 1 0 1 sequence of OBDD s: D 1 , D 2 , D 3 , . . . , D m ; x 2 x 2 D m ≡ 0; 0 0 OBDD s for axioms. 1 1 Rules: x 3 x 3 ∧ (join): D i , D j ⇒ D k ≡ ( D i ∧ D j ) ; 0 0 w (weakening): D i ⇒ D j , D i | = D j ; 1 1 r (reordering): D i ⇒ D j , D j ≡ D i 0 1 but orders of variables are different. Join rule can be applied only for OBDD s in the same order. Sokolov D. | OBDD Proof Systems 2/11

Orders are important f : { 0 , 1 } n × { 0 , 1 } m → { 0 , 1 } ; Alice knows x 1 , . . . , x n ∈ { 0 , 1 } , Bob knows y 1 , . . . , y m ∈ { 0 , 1 } ; they want to compute f ( x , y ) ; assume that f has an OBDD of size S in some order in that all x i ’s preceed all y j ’s; communication complexity of f is at most log S + 1; EQ : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } , EQ ( x , y ) = 1 ⇔ x = y ; if all x i ’s preceed all y j ’s in π , then size of any π - OBDD for EQ ( x , y ) is at least 2 n ; ∃ short OBDD for EQ ( x , y ) in the order x 1 , y 1 , x 2 , y 2 , . . . , x n , y n . Sokolov D. | OBDD Proof Systems 3/11

Orders are important f : { 0 , 1 } n × { 0 , 1 } m → { 0 , 1 } ; Alice knows x 1 , . . . , x n ∈ { 0 , 1 } , Bob knows y 1 , . . . , y m ∈ { 0 , 1 } ; they want to compute f ( x , y ) ; assume that f has an OBDD of size S in some order in that all x i ’s preceed all y j ’s; communication complexity of f is at most log S + 1; EQ : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } , EQ ( x , y ) = 1 ⇔ x = y ; if all x i ’s preceed all y j ’s in π , then size of any π - OBDD for EQ ( x , y ) is at least 2 n ; ∃ short OBDD for EQ ( x , y ) in the order x 1 , y 1 , x 2 , y 2 , . . . , x n , y n . Sokolov D. | OBDD Proof Systems 3/11

Orders are important f : { 0 , 1 } n × { 0 , 1 } m → { 0 , 1 } ; Alice knows x 1 , . . . , x n ∈ { 0 , 1 } , Bob knows y 1 , . . . , y m ∈ { 0 , 1 } ; they want to compute f ( x , y ) ; assume that f has an OBDD of size S in some order in that all x i ’s preceed all y j ’s; communication complexity of f is at most log S + 1; EQ : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } , EQ ( x , y ) = 1 ⇔ x = y ; if all x i ’s preceed all y j ’s in π , then size of any π - OBDD for EQ ( x , y ) is at least 2 n ; ∃ short OBDD for EQ ( x , y ) in the order x 1 , y 1 , x 2 , y 2 , . . . , x n , y n . Sokolov D. | OBDD Proof Systems 3/11

Orders are important f : { 0 , 1 } n × { 0 , 1 } m → { 0 , 1 } ; Alice knows x 1 , . . . , x n ∈ { 0 , 1 } , Bob knows y 1 , . . . , y m ∈ { 0 , 1 } ; they want to compute f ( x , y ) ; assume that f has an OBDD of size S in some order in that all x i ’s preceed all y j ’s; communication complexity of f is at most log S + 1; EQ : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } , EQ ( x , y ) = 1 ⇔ x = y ; if all x i ’s preceed all y j ’s in π , then size of any π - OBDD for EQ ( x , y ) is at least 2 n ; ∃ short OBDD for EQ ( x , y ) in the order x 1 , y 1 , x 2 , y 2 , . . . , x n , y n . Sokolov D. | OBDD Proof Systems 3/11

OBDD ( ∧ , w ) -proofs [Atserias, Kolaitis, Vardi 04] OBDD ( ∧ , w ) simulates CP ∗ ⇒ PHP n + 1 = has proofs of poly size; n unsatisfiable linear systems over F 2 have short proofs; [Segerlind 07] 2 n Ω( 1 ) lower bound for tree-like OBDD ( ∧ , w ) -proofs; cek 08] 2 n Ω( 1 ) lower bound for dag-like [Kraj´ ıˇ OBDD ( ∧ , w ) -proofs; [this paper] OBDD ( ∧ , w ) is exponentially stronger than CP ∗ . [this paper] OBDD ( ∧ , w , r ) is exponentially stronger than OBDD ( ∧ , w ) . Sokolov D. | OBDD Proof Systems 4/11

OBDD ( ∧ , w ) -proofs [Atserias, Kolaitis, Vardi 04] OBDD ( ∧ , w ) simulates CP ∗ ⇒ PHP n + 1 = has proofs of poly size; n unsatisfiable linear systems over F 2 have short proofs; [Segerlind 07] 2 n Ω( 1 ) lower bound for tree-like OBDD ( ∧ , w ) -proofs; cek 08] 2 n Ω( 1 ) lower bound for dag-like [Kraj´ ıˇ OBDD ( ∧ , w ) -proofs; [this paper] OBDD ( ∧ , w ) is exponentially stronger than CP ∗ . [this paper] OBDD ( ∧ , w , r ) is exponentially stronger than OBDD ( ∧ , w ) . Sokolov D. | OBDD Proof Systems 4/11

OBDD ( ∧ , w ) -proofs [Atserias, Kolaitis, Vardi 04] OBDD ( ∧ , w ) simulates CP ∗ ⇒ PHP n + 1 = has proofs of poly size; n unsatisfiable linear systems over F 2 have short proofs; [Segerlind 07] 2 n Ω( 1 ) lower bound for tree-like OBDD ( ∧ , w ) -proofs; cek 08] 2 n Ω( 1 ) lower bound for dag-like [Kraj´ ıˇ OBDD ( ∧ , w ) -proofs; [this paper] OBDD ( ∧ , w ) is exponentially stronger than CP ∗ . [this paper] OBDD ( ∧ , w , r ) is exponentially stronger than OBDD ( ∧ , w ) . Sokolov D. | OBDD Proof Systems 4/11

OBDD ( ∧ , w ) -proofs [Atserias, Kolaitis, Vardi 04] OBDD ( ∧ , w ) simulates CP ∗ ⇒ PHP n + 1 = has proofs of poly size; n unsatisfiable linear systems over F 2 have short proofs; [Segerlind 07] 2 n Ω( 1 ) lower bound for tree-like OBDD ( ∧ , w ) -proofs; cek 08] 2 n Ω( 1 ) lower bound for dag-like [Kraj´ ıˇ OBDD ( ∧ , w ) -proofs; [this paper] OBDD ( ∧ , w ) is exponentially stronger than CP ∗ . [this paper] OBDD ( ∧ , w , r ) is exponentially stronger than OBDD ( ∧ , w ) . Sokolov D. | OBDD Proof Systems 4/11