Prenex Normal Form PNF 1 Eliminate and transform

Prenex Normal Form (PNF) 1. Eliminate ! and $ (transform to Ç Æ : ) 2. Push negations inside using: : 8 x. $ 9 x. : : 9 x. $ 8 x. : 3. If there are name conflicts across scopes, solve with renaming. 4. Move quantifiers out by using recursively rules such as: 1. 2. 3. 4. 5. 6. Q 1 x. 1(x) Æ Q 2 y. 2(y) $ Q 1 x. Q 2 y. ( 1(x) Æ 2(y)) Qi 2{8, 9} Q 1 x. 1(x) Ç Q 2 y. 2(y) $ Q 1 x. Q 2 y. ( 1(x) Ç 2(y)) Qi 2{8, 9} 1 Æ 9 x. 2(x) $ 9 x. ( 1 Æ 2(x)) where x does not appear in 1 1 Æ 8 x. 2(x) $ 8 x. ( 1 Æ 2(x)) where x does not appear in 1 8 x. 1(x) Æ 8 x. 2(x) $ 8 x. ( 1(x) Æ 2(x)) 9 x. 1(x) Ç 9 x. 2(x) $ 9 x. ( 1(x) Ç 2(x)) Decision Procedures - An algorithmic point of view 1

Projection for non-CNF formulas: example 9 y 8 z 9 x. (y Ç (x Æ z)) 9 y 8 z. (y Ç (x Æ z))|x=0 Ç (y Ç (x Æ z))|x=1 9 y 8 z. ((y) Ç (y Ç z)) 9 y : 9 z. (: y Æ : z) 9 y. : ((: y Æ : z)|z=0 Ç (: y Æ : z)|z=1) 9 y. : (: y) True Decision Procedures - An algorithmic point of view 2

Basic DPLL Flow for QBF e u (e Ç u)(: e Ç : u) Unknown True (1) False(0) Decision Procedures - An algorithmic point of view 3

Basic DPLL Flow for QBF e u (e Ç u)(: e Ç : u) e=0 Unknown True (1) False(0) Decision Procedures - An algorithmic point of view 4

Basic DPLL Flow for QBF Existential quantification e u (e Ç u)(: e Ç : u) Universal quantification e=0 Satisfying Node Unknown True (1) u=1 False(0) Decision Procedures - An algorithmic point of view 5

Basic DPLL Flow for QBF e u (e Ç u)(: e Ç : u) e=0 Backtrack Unknown True (1) u=1 False(0) Decision Procedures - An algorithmic point of view 6

Basic DPLL Flow for QBF e u (e Ç u)(: e Ç : u) e=0 Unknown True (1) u=1 u=0 False(0) Decision Procedures - An algorithmic point of view 7

Basic DPLL Flow for QBF e u (e Ç u)(: e Ç : u) e=0 Unknown True (1) u=1 u=0 False(0) Decision Procedures - An algorithmic point of view 8

Basic DPLL Flow for QBF e u (e Ç u)(: e Ç : u) e=1 e=0 Unknown True (1) u=1 u=0 False(0) Decision Procedures - An algorithmic point of view 9

Basic DPLL Flow for QBF e u (e Ç u)(: e Ç : u) e=1 e=0 Unknown True (1) u=1 u=0 False(0) Decision Procedures - An algorithmic point of view 10

Basic DPLL Flow for QBF e u (e Ç u)(: e Ç : u) e=1 Unknown True (1) e=0 Conflicting Node u=1 u=0 False(0) Decision Procedures - An algorithmic point of view 11

Basic DPLL Flow for QBF e u (e Ç u)(: e Ç : u) e=1 e=0 Unknown True (1) u=1 u=0 False(0) Decision Procedures - An algorithmic point of view 12

Basic DPLL Flow for QBF False e u (e Ç u)(: e Ç : u) e=1 e=0 Unknown True (1) u=1 u=0 False(0) Decision Procedures - An algorithmic point of view 13

Basic DPLL Flow for QBF u e (u Ç e)(: u Ç : e) Unknown True (1) False(0) Decision Procedures - An algorithmic point of view 14

Basic DPLL Flow for QBF u e (u Ç e)(: u Ç : e) u=1 Unknown True (1) False(0) Decision Procedures - An algorithmic point of view 15

Basic DPLL Flow for QBF u e (u Ç e)(: u Ç : e) u=1 Unknown True (1) e=1 False(0) Decision Procedures - An algorithmic point of view 16

Basic DPLL Flow for QBF u e (u Ç e)(: u Ç : e) u=1 Unknown True (1) e=1 e=0 False(0) Decision Procedures - An algorithmic point of view 17

Basic DPLL Flow for QBF u e (u Ç e)(: u Ç : e) u=1 Unknown True (1) e=1 e=0 False(0) Decision Procedures - An algorithmic point of view 18

Basic DPLL Flow for QBF u e (u Ç e)(: u Ç : e) u=1 u=0 Unknown True (1) e=1 e=0 e=1 False(0) Decision Procedures - An algorithmic point of view 19

Basic DPLL Flow for QBF True u e (u Ç e)(: u Ç : e) u=1 u=0 Unknown True (1) e=1 e=0 e=1 False(0) Decision Procedures - An algorithmic point of view 20