Learn about the process of deciding Existential Positive-First-Order Logic (EPR) formulas utilizing the DPLL algorithm and substitution sets, as discussed by Leonardo de Moura and Nikolaj Bjørner from Microsoft Research. Explore the significance of EPR in relation to SAT and QBF, and discover the contributions of EPR Calculus in achieving exponential speedups through hybrid substitution sets. Delve into the basic ideas, methodologies, and challenges involved in solving EPR instances.
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Presentation Transcript
Deciding EPR using DPLL and substitution sets Leonardo de Moura and Nikolaj Bj rner Microsoft Research
What , ( , , , ) p x y a b x y EPR ij i j ( ) , . x y p x a y ( , , ) ( , ) q x y ( , ) r b y ( ) ( , ) p z a . ( , ) q b z z Deciding EPR using DPLL + Substitution sets
Why? EPR is the next SAT QBF EPR .( .( ( ) . ( ) ) ... ) y x q y q x y q x x y q x y ... ... Limited success for QBF in BMC. Could a DPLL(EPR) solver do the trick? Deciding EPR using DPLL + Substitution sets
Why? EPR is the next SM(EPR+ ) . , , .( , .( , , .( A B C A : A A A B C A A B A A : : : ) : ) B B C A = C : : B B A A B : ) B : : B A C B C C Deciding EPR using DPLL + Substitution sets
How: Basic Idea Clause Set of Instances ( , ),( , ), ( , ),( , ) ( ) q x ( , ) r y x ( , ) x y p ( ) ( ) ( ) ( ) ( , ) ( , ) ( , ) ( , ) r p p p p q q q q r r Instead of r Deciding EPR using DPLL + Substitution sets
Basic Idea Clause Set of Substitutions ( , ),( , ), ( , ),( , ) ( ) q x ( , ) r y z p ;( , ) x x y z Hybrid substitution sets: standard substitution + Set Deciding EPR using DPLL + Substitution sets
Basic Idea Clause Set of Substitutions ( , ),( , ), ( , ),( , ) ( ) q x ( , ) r y z p ;( , ) x x y z { , , , } {11,10,01,00} a b = = = = , , x x x y y y 1 2 1 2 Hybrid substitution sets: standard substitution + BDD Deciding EPR using DPLL + Substitution sets
Basic Idea Clause Set of Substitutions x1 ( ) q x ( , ) r y z p ;( , ) x x y z y1 Hybrid substitution sets: standard substitution + BDD Deciding EPR using DPLL + Substitution sets
Basic Idea Clause Set of Substitutions ( , ),( , ), ( , ),( , ) ( ) q x ( , ) r y z p ;( , ) x x y z Use DPLL to split on atoms Use relational algebra to operate on Substitution sets = BDDs + unification Deciding EPR using DPLL + Substitution sets
Basic Example ( , ) { , , } { , , } x y a ( ) ( ) ( , ) p x y shape x shape y a shape x ( ) { } shape y ( ) { } = = ({ , , } { , , }) a = {( , )} { } { } x y a ( , ) {( , )} p x y Deciding EPR using DPLL + Substitution sets
Basic Example ( , ) { , , } { , , } x y a ( ) ( ) ( , ) p x y shape x shape y a shape x ( ) { , } shape y ( ) { , } ( , ),( , ), ( , ),( , ) ({ , , } { , , }) a = {*, } {*, } a x y ( , ),( , ), ( , ),( , ) ( , ) p x y Deciding EPR using DPLL + Substitution sets
Calculus Transitions Partial model || Set of clauses F ( ){ , } p x a b F F Decide Deciding EPR using DPLL + Substitution sets
Calculus Transitions Partial model || Set of clauses F ( ){ } p x a F p x , ( ) ( ) {( , ),( , ),( , )} Pr ( ){ } ( ){ , } p x a q y b c F q y a b a c b a Unit opagate Deciding EPR using DPLL + Substitution sets
Calculus Transitions Partial model || Set of clauses || Conflict Clause, Substitutions , F C ( ){ } p x a q F p x , ( ) Conflict { , , } a b c q ( ){ } p x a q F p x ( ) { , , },{ } a b c q a Deciding EPR using DPLL + Substitution sets
Calculus Transitions Partial model || Set of clauses || Conflict Clause, Substitutions , F C , ( ) F p x {( , ),( , ),( , )}, a a b a Factoring F p x q a c ( ) p y q c c ( ) { , },{ , } a c Deciding EPR using DPLL + Substitution sets
Theorems Soundness & Completeness well of course Stuck-freeness can always make progress Time: 2-EXP Space: EXP but then EPR is NTIME complete Deciding EPR using DPLL + Substitution sets
Calculus S-Transitions Recall basic example { , , } { , , } a : ( ) ( ) ( , ) p x y F shape x shape y a ( ) { } ( ) { , } shape x ( ) { } : q z a b shape x 4 propagations : ( ) { , } q z ( ) { , } 1 propagation a b shape x Propagate with all substitutions in simultaneously S-UnitPropagation S-Conflict Deciding EPR using DPLL + Substitution sets
Calculus S-Transitions Property S-UnitPropagation S-Conflict can be exponentially faster than basic instantiation or resolution. Deciding EPR using DPLL + Substitution sets
Calculus S-Transitions ( ) { } ( ) { , } shape x ( ) { } : q z a b shape x shape x shape x ( ) { } ( ) { } shape x shape x ( ) { } ( ) { } ( , ) {( , )} p x y { , , } { , , } a : ( ) ( ) ( , ) p x y F shape x shape y a Deciding EPR using DPLL + Substitution sets
Calculus S-Transitions ( ) { } ( ) { , } shape x ( ) { } : q z a b shape x shape x shape x ( ) { } ( ) { } ( , ) ( , ) shape x shape x ( ) { } ( ) { } ( , ) { p x y } { , , } { , , } a : ( ) ( ) ( , ) p x y F shape x shape y a Deciding EPR using DPLL + Substitution sets
Calculus S-Transitions ( ) { } ( ) { , } shape x ( ) { } : q z a b shape x shape x shape x ( ) { } ( ) { } ( , ) ( , ) ( , ) ( , ) p x y shape x shape x ( ) { } ( ) { } { , , } { , , } a : ( ) ( ) ( , ) p x y F shape x shape y a Deciding EPR using DPLL + Substitution sets
Calculus S-Transitions ( ) { } ( ) { , } shape x ( ) { } : q z a b shape x shape x shape x ( ) { } ( ) { } ( , ) ( , ) ( , ) ( , ) ( , ) p x y shape x shape x ( ) { } ( ) { } { , , } { , , } a : ( ) ( ) ( , ) p x y F shape x shape y a Deciding EPR using DPLL + Substitution sets
Calculus S-Transitions ( ) { } ( ) { , } shape x ( ) { } : q z a b shape x shape x shape x ( ) { } ( ) { } ( , ) ( , ) ( , ) ( , ) ( , ) p x y shape x shape x ( ) { } ( ) { } { , , } { , , } a : ( ) ( ) ( , ) p x y F shape x shape y a Deciding EPR using DPLL + Substitution sets
Calculus S-Transitions Property what is gained by S-UnitPropagation, S-Conflict can be lost during conflict resolution. Solutions (in paper): 1. Refine decisions, imprecise learning 2. Single instance learning Deciding EPR using DPLL + Substitution sets
Experimental evaluation Prototype ~6KLOC DPLL core, Hybrid substitution sets BuDDy (and CUDD) BuDDy caches substitution operations for speedup Examples: Concocted examples exponential speedup over instance and resolution based methods CASC-2007: 49/50 EPT, 46/50 EPS Deciding EPR using DPLL + Substitution sets
Beyond EPR(T) EPR(Theory of Equalities) Deciding EPR using DPLL + Substitution sets
Beyond EPR(T) EPR(Theory of Equalities) Core SM(EPR + ) EPR DPLL Deciding EPR using DPLL + Substitution sets
Beyond EPR(T) EPR(Theory of Equalities) Core SM(EPR + ) Some, but not all SAT techniques can be used in DPLL(EPR) EPR DPLL SAT DPLL BDD variants for substitutions Iterative squaring, resolution [see N &V next talk] Deciding EPR using DPLL + Substitution sets
Calculus -Example : ( ) : ( ) : ( ) q a ( ) ( ) ( ) s b F ( ) ( ) t b C C C p a p b q a s b r a 1 2 3 Deciding EPR using DPLL + Substitution sets
Calculus -Example : ( ) : ( ) : ( ) q x ( ) ( ) ( ) {( , )} s y F ( ) {( , , )} ( ) {( , , )} t z a b C C C p x p x q y s y r z a a a b b b 1 2 3 Deciding EPR using DPLL + Substitution sets
Calculus -Example q y s y : ( ) : ( ) : ( ) q x : ( ) : ( ) : ( ) ( ) ( ) ( ) {( , )} s y F ( ) ( ) {( , , )} ( ) ( ) {( , , )} ( ) {( , )} s y a b F ( ) {( , , )} ( ) {( , , )} t z a b r z a a a t z b b b C C C p x p x q y s y r z a a a b b b F 1 2 Decide 3 p x p x q x C C C ( ){ , , } r x a b c F 1 2 Decide F 3 ( ){ , , } ( ){ , } r x a b c t x b c Deciding EPR using DPLL + Substitution sets
Calculus -Example : ( ) : ( ) : ( ) q x ( ) ( ) ( ) {( , )} s y F ( ) {( , , )} ( ) {( , , )} t z a b C C C p x p x q y s y r z a a a b b b ( ){ , , } r x a b c F 1 2 Decide 3 ( ){ , , } ( ){ , } r x a b c t x b c F Decide } ( ){ , , } ( ) r x a b c t x b c p x a b F { , } ( ){ , Deciding EPR using DPLL + Substitution sets
Calculus -Example ( ) { , , } ( ) { , } r x a b c t x b c p x ( ) { , } a b F Pr Unit opagate ( ){ , , } ( ){ , } ( ){ , } ( ){ } p a x b r x a b c t x b c q F x a c b Pr b Unit ( ){ , } p x a opagate ( ) {( , , )} t z ( ) p x ( ) s y b b b r x a b c ( ){ , , } ( ){ , } ( ){ } ( ){ } q x a s x t x b F Deciding EPR using DPLL + Substitution sets
Calculus -Example ( ){ , , } ( ){ , } r x a b c t x b c F Decide ( ){ , , } ( ){ , } r x a b c t x b c p x a b F ( ){ , } Pr Unit opagate ( ) {( , , )} r z ( ) p x ( ) q y a a a ( ){ , , } ( ){ , } ( ){ } ( ) , b a x p r x a b c t x b c q x a F { } Deciding EPR using DPLL + Substitution sets
Calculus -Example ( ) { , , } ( ) { , } r x a b c t x b c p x ( ) { , } a b F Pr Unit opagate ( ){ , , } ( ){ , } ( ){ , } ( ){ } p a x b r x a b c t x b c q F x a c b Pr b Unit ( ){ , } p x a opagate ( ) {( , , )} t z ( ) p x ( ) s y b b b r x a b c ( ){ , , } ( ){ , } ( ){ } ( ){ } q x a s x t x b F Deciding EPR using DPLL + Substitution sets
