SAT Solvers Ranjit Jhala, UC San Diego April 9, 2013 Decision - - PowerPoint PPT Presentation

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SAT Solvers Ranjit Jhala, UC San Diego April 9, 2013 Decision - - PowerPoint PPT Presentation

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SAT Solvers Ranjit Jhala, UC San Diego April 9, 2013 Decision Procedures We will look very closely at the following 1. Propositional Logic 2. Theory of Equality 3. Theory of Uninterpreted Functions 4. Theory of Difference-Bounded Arithmetic

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SAT Solvers

Ranjit Jhala, UC San Diego April 9, 2013

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Decision Procedures

We will look very closely at the following

  • 1. Propositional Logic
  • 2. Theory of Equality
  • 3. Theory of Uninterpreted Functions
  • 4. Theory of Difference-Bounded Arithmetic

Decision Problem: Satisfaction

◮ Does eval s p return True for some assignment s ? ◮ “Can we assign the variables to make the formula true” ?

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Decision Procedures

We will look very closely at the following

  • 1. Propositional Logic
  • 2. Theory of Equality
  • 3. Theory of Uninterpreted Functions
  • 4. Theory of Difference-Bounded Arithmetic

Why?

◮ Representative ◮ Have “efficient” algorithms

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Decision Procedures

We will look very closely at the following

  • 1. Propositional Logic
  • 2. Theory of Equality
  • 3. Theory of Uninterpreted Functions
  • 4. Theory of Difference-Bounded Arithmetic

Plan

◮ First in isolation ◮ Then in combination ◮ Very slick SW-Eng, based on logic

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Decision Procedures: Propositional Logic

Popularly called SAT Solvers

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Decision Procedures: Propositional Logic

Basics

◮ Propositional Logic 101 ◮ Conjunctive Normal Form ◮ Resolution

Algorithms

◮ Resolution ◮ Backtracking Search ◮ Boolean Constraint Propagation ◮ Conflict Driven Learning & Backjumping

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Decision Procedures: Propositional Logic

Basics

◮ Propositional Logic 101 ◮ Conjunctive Normal Form ◮ Resolution

Algorithms

◮ Resolution ◮ Backtracking Search ◮ Boolean Constraint Propagation ◮ Conflict Driven Learning & Backjumping

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Propositional Logic 101

Propositional Variables

data PVar

Propositional Formulas

data Formula = Prop PVar | Not Formula | Formula ‘And‘ Formula | Formula ‘Or‘ Formula

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Decision Procedures: Propositional Logic

Basics

◮ Propositional Logic 101 ◮ Conjunctive Normal Form ◮ Resolution

Algorithms

◮ Resolution ◮ Backtracking Search ◮ Boolean Constraint Propagation ◮ Conflict Driven Learning & Backjumping

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Conjunctive Normal Form

Restricted representation of Formula

Literals: Variables or Negated Variables

data Literal = Pos PVar | Neg PVar

Clauses: Disjunctions (Or) of Literals

data Clauses = [Literal]

CNF Formulas: Conjunctions (And) of Clauses

data CnfFormula = [Clauses]

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Conjunctive Normal Form: Example

Consider a Formula

(x1 ∨ x2) ∧ (¬x1 ∨ x3) ∧ ¬x3

Represented as a Formula

(Prop 1 ‘Or‘ Prop 2) ‘And‘ (Not (Prop 1) ‘Or‘ Prop 3) ‘And‘ (Not (Prop 3) )

Represented as a CnfFormula

[ [Pos 1 , Pos 2] , [Neg 1 , Pos 3] , [Neg 3 ] ]

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Conjunctive Normal Form Conversion

Theorem There is a poly-time function toCNF :: Formula -> CnfFormula toCNF = error "Exercise For The Reader" Such that any f is satisfiable iff (toCNF f) is satisfiable.

◮ toCNF adds new variables for sub-formulas ◮ otherwise, an exponential blowup in CnfFormula size

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Conjunctive Normal Form Conversion

Theorem There is a poly-time function toCNF :: Formula -> CnfFormula toCNF = error "Exercise For The Reader" Such that any f is satisfiable iff (toCNF f) is satisfiable. Henceforth Only consider formulas in Conjunctive Normal Form Formulas

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Decision Procedures: Propositional Logic

Basics

◮ Propositional Logic 101 ◮ Conjunctive Normal Form

Algorithms

◮ Resolution ◮ Backtracking Search ◮ Boolean Constraint Propagation ◮ Conflict Driven Learning & Backjumping

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Properties of CNF

Pure Variable

◮ One which appears only +ve or −ve in a CnfFormula

Empty Clause

◮ If a CnfFormula has some Clause without Literals ◮ Then the CnfFormula is UNSAT

Trivial Formula

◮ If a CnfFormula has no Clause ◮ Or every variable is pure ◮ Then the CnfFormula is SAT

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Goal

Determine satisfaction by reducing CnfFormula to one of

◮ Empty Clause (ie UNSAT), or ◮ Trivial Formula (ie SAT).

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Reducing Formulas By Resolution

(“Reduce” is, perhaps, not the best word. . . ) Resolution: For any A, B and variable x, the formula (A ∨ x) ∧ (B ∨ ¬x) is equivalent to the formula (A ∨ B)

◮ The variable x is called a pivot variable

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General Resolution

Resolution: For any Ai, Bj and variable x, the formula

  • i

(Ai ∨ x) ∧

  • j

(Bj ∨ ¬x) is equivalent to the formula

  • i,j

(Ai ∨ Bj)

◮ Pivot variable x is eliminated by resolution

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Davis-Putnam Algorithm: Example 1

Input Formula

◮ (x1 ∨ x2 ∨ x3) ∧ (x2 ∨ ¬x3 ∨ x5) ∧ (¬x2 ∨ x4))

Pivot on x2

◮ (x1 ∨ x3 ∨ x4) ∧ (¬x3 ∨ x5 ∨ x4)

Pivot on x3

◮ (x1 ∨ x4 ∨ x5)

All variables are pure . . . hence, SAT

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Davis-Putnam Algorithm: Example 2

Input Formula

◮ (x1 ∨ x2) ∧ (x1 ∨ ¬x2) ∧ (¬x1 ∨ x3) ∧ (¬x1 ∨ ¬x3)

Pivot on x2

◮ (x1) ∧ (¬x1 ∨ x3) ∧ (¬x1 ∨ ¬x3)

Pivot on x3

◮ (x1) ∧ (¬x1)

Pivot on x1

◮ ()

Empty clause . . . hence, UNSAT

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Davis-Putnam Algorithm

Algorithm

  • 1. Select pivot and perform resolution
  • 2. Repeat until SAT or UNSAT

Issues?

◮ Space blowup (formula size blows up on resolution)

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Decision Procedures: Propositional Logic

Basics

◮ Propositional Logic 101 ◮ Conjunctive Normal Form

Algorithms

◮ Resolution ◮ Backtracking Search ◮ Boolean Constraint Propagation ◮ Conflict Driven Learning & Backjumping

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Decision Tree: Describes Space of All Assignments

Figure: SAT Decision Tree (Courtesy: Lintao Zhang)

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Decision Tree: SAT via Depth First Search

Figure: DFS On Decision Tree (Courtesy: Lintao Zhang)

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Backtracking Search

Don’t build whole tree, but lazily search solutions

◮ Choose a variable x, set to True ◮ Remove constraints where x appears ◮ Recurse on remaining constraints ◮ Backtrack if a contradiction is found

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Backtracking Search (1/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (2/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (3/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (4/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (5/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (6/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (7/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (8/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (9/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (10/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (11/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (12/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (13/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (14/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (15/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (16/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (17/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (18/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (19/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (20/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search (21/21)

Figure: Basic DLL (Courtesy: Lintao Zhang)

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Backtracking Search

Don’t build whole tree, but lazily search solutions

◮ Choose a variable x, set to True ◮ Remove constraints where x appears ◮ Recurse on remaining constraints ◮ Backtrack if a contradiction is found

(whew!)

◮ DFS avoids space blowup (only need to save stack) . . . ◮ . . . but not time (natch)

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Decision Procedures: Propositional Logic

Basics

◮ Propositional Logic 101 ◮ Conjunctive Normal Form

Algorithms

◮ Resolution ◮ Backtracking Search ◮ Boolean Constraint Propagation ◮ Conflict Driven Learning & Backjumping

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Boolean Constraint Propagation

Often, we don’t really have a choice. . .

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Boolean Constraint Propagation

Unit Clause Rule

◮ If an (unsatisfied) Clause has one unassigned Literal ◮ Then that Literal must be True in any SAT assignment

Example

◮ Formula (x1 ∨ ¬x2 ∨ x3) ∧ (x2 ∨ ¬x3) ∧ (¬x1 ∨ ¬x3) ◮ Assignment x1 = T, x2 = T ◮ The last clause is a unit clause ◮ Any SAT assigment must set ¬x3 = T (i.e. x3 = F)

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Boolean Constraint Propagation

Unit Clause Rule

◮ If an (unsatisfied) Clause has one unassigned Literal ◮ Then that Literal must be True in any SAT assignment

BCP or Unit Propagation

◮ Repeat applying unit clause rule ◮ Until no unit clause remains.

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Boolean Constraint Propagation: Example

Revisit Example With BCP

Figure: Boolean Constraint Propagation (Courtesy: Lintao Zhang)

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Boolean Constraint Propagation

DPLL = Backtracking Search + BCP

◮ Backtracking: Avoids space blowup ◮ BCP: Avoid doing obvious work ◮ Still repeatedly explore all choices (e.g. whole left subtree)

Wanted

◮ Means to learn to repeat dead ends ◮ Key to scaling to practical problems

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Decision Procedures: Propositional Logic

Basics

◮ Propositional Logic 101 ◮ Conjunctive Normal Form

Algorithms

◮ Resolution ◮ Backtracking Search ◮ Boolean Constraint Propagation ◮ Conflict Driven Learning & Backjumping

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Conflict Driven Learning

Key Insight

◮ On finding conflict, don’t (just) backtrack ◮ Learn new clause to prevent same conflict in future

Major breakthrough

◮ J. P. Marques-Silva and K. A. Sakallah, “GRASP – A New

Search Algorithm for Satisfiability,” Proc. ICCAD 1996.

◮ R. J. Bayardo Jr. and R. C. Schrag “Using CSP look-back

techniques to solve real world SAT instances.” Proc. AAAI, 1997

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Conflict Driven Learning

◮ Resolve on conflict variable to learn new conflict clause ◮ Add clause to set of clauses ◮ Backjump using conflict clause

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Conflict Driven Learning

Revisit Example With CDL

◮ Learn, Add, Backjump ◮ Vastly faster search

Figure: Boolean Constraint Propagation (Courtesy: Lintao Zhang)

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Backtracking Only (01/26)

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Backtracking Only (02/26)

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Backtracking Only (03/26)

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Backtracking Only (04/26)

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Backtracking Only (05/26)

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Backtracking Only (06/26)

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Backtracking Only (07/26)

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Backtracking Only (08/26)

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Backtracking Only (09/26)

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Backtracking Only (10/26)

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Backtracking Only (11/26)

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Backtracking Only (12/26)

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Backtracking Only (13/26)

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Backtracking Only (14/26)

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Backtracking Only (15/26)

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Backtracking Only (16/26)

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Backtracking Only (17/26)

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Backtracking Only (18/26)

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Backtracking Only (19/26)

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Backtracking Only (20/26)

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Backtracking Only (21/26)

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Backtracking Only (22/26)

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Backtracking Only (23/26)

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Backtracking Only (24/26)

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Backtracking Only (25/26)

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Backtracking Only (26/26)

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Boolean Constraint Propagation (01/23)

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Boolean Constraint Propagation (02/23)

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Boolean Constraint Propagation (03/23)

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Boolean Constraint Propagation (04/23)

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Boolean Constraint Propagation (05/23)

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Boolean Constraint Propagation (06/23)

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Boolean Constraint Propagation (07/23)

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Boolean Constraint Propagation (08/23)

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Boolean Constraint Propagation (09/23)

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Boolean Constraint Propagation (10/23)

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Boolean Constraint Propagation (11/23)

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Boolean Constraint Propagation (12/23)

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Boolean Constraint Propagation (13/23)

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Boolean Constraint Propagation (14/23)

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Boolean Constraint Propagation (15/23)

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Boolean Constraint Propagation (16/23)

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Boolean Constraint Propagation (17/23)

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Boolean Constraint Propagation (18/23)

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Boolean Constraint Propagation (19/23)

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Boolean Constraint Propagation (20/23)

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Boolean Constraint Propagation (21/23)

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Boolean Constraint Propagation (22/23)

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Boolean Constraint Propagation (23/23)

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Conflict Driven Learning (01/21)

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Conflict Driven Learning (02/21)

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Conflict Driven Learning (03/21)

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Conflict Driven Learning (04/21)

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Conflict Driven Learning (05/21)

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Conflict Driven Learning (06/21)

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Conflict Driven Learning (07/21)

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Conflict Driven Learning (08/21)

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Conflict Driven Learning (09/21)

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Conflict Driven Learning (10/21)

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Conflict Driven Learning (11/21)

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Conflict Driven Learning (12/21)

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Conflict Driven Learning (13/21)

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Conflict Driven Learning (14/21)

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Conflict Driven Learning (15/21)

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Conflict Driven Learning (16/21)

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Conflict Driven Learning (17/21)

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Conflict Driven Learning (18/21)

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Conflict Driven Learning (19/21)

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Conflict Driven Learning (20/21)

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Conflict Driven Learning (21/21)

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More Details about SAT Solvers

Lectures By Lintao Zhang (ZChaff)

◮ 1 ◮ 2

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Next Time: SMT = SAT + Theories

  • 1. Propositional Logic
  • 2. Combining Theories

◮ Equality + Uninterpreted Functions ◮ Difference-Bounded Arithmetic

  • 3. Combining SAT + Theories