propositional encoding - decision procedure

Propositional Encoding - Decision Procedure

Daniel Kroening, Ofer StrichmanPresented by Changki Hong

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Combination of theories Approach 1 : Combine decision procedures of the individ-

ual theories. Nelson-Oppen method

Approach 2 : Reduce all theories to a common logic if possible (e.g. Propositional logic) Combine decision procedure for individual theories with a proposi-

tional SAT solver.

Changki Hong @ pswlab Propositional Encodings

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In detail approach 2 Two encoding schemes in the category of approach 2

Lazy encoding- Keep interacting between SAT solver and decision procedures of each

theories.- Almost every tool that participated in the SMT competitions in 2005-

2007 belongs to this category of solvers.

Eager encoding- SAT solver is invoked only once with no further interaction with deci-

sion procedure of each theories.


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Contents Motivation Preliminaries Lazy encoding Eager encoding Conclusion


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Preliminaries

While (true){

if (Decide() == FALSE) return (SAT); while (BCP() == “conflict”) {

backtrack-level = Analyze_Conflict();if (backtrack-level < 0) return (UNSAT);else BackTrack(backtrack-level);

}}

Choose the next variable and value.Return False if all variables are assigned

Apply repeatedly the unit clause rule.Return “conflict” if reached a conflict

Backtrack until no conflict.Return -1 if impossible

Basic architecture of DPLL SAT solver

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Lazy encoding Two main engines

SAT solver : assigns truth values to literals in order to satisfy the Boolean structure of the formula

Decision procedure of the individual theories : checks whether this assignment is consistent in theory.

Definition 1. (Boolean encoder) Given a -literal l, we associate with it a unique Boolean variable

e(l), which we call the Boolean encoder of this literal. Given a -formula t, e(t) denotes the Boolean formula resulting

from substituting each -literal in t with its Boolean encoder. We also call it as propositional skeleton.


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Overview of lazy encoding Example

Let theory T be equality logic. - Á := x = y Æ ((y = z Æ x z) Ç x = z)

1. Compute propositional skeleton of the given formula- Á := e(x = y) Æ ((e(y = z) Æ e(x z)) Ç e(x = z)) - Let B := e(Á)

2. Pass B to a SAT solver - ® := {e(x = y) TRUE, e(y = z) TRUE, e(x z) TRUE, e(x = z) FALSE}

3. Decision procedure decides whether the conjunction of the literals cor-responding to this assignment (Th(®)) is satisfiable- Th(®) := x = y Æ y = z Æ x z Æ :(x = z) - blocking clause : e(:Th(®)) := :e(x = y) Ç :e(y = z) Ç :e(x z) Ç e(x = z)

4. Pass B Æ e(:Th(®)) to a SAT solver. - ® := {e(x = y) TRUE, e(y = z) TRUE, e(x z) FALSE, e(x = z) TRUE}


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Overview of lazy encoding


PropositionalSAT solver

DPT = A decision procedurefor theory T

® Th(®)

e(t) t

• ® - current assignment returned by SAT solver

• Th(®) - conjunction of the literal corresponding to current assignment and we de-fine each literal, denoted Th(liti, ®), as follows:

• t - t is returned by DPT and it is called blocking clause or lemma. This clause con-tradicts the current assignment, and hence blocks it from being repeated.

• e(t) - Boolean formula of the blocking clause.

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Lazy algorithm Algorithm 1. Lazy-basic


Input: A formula ÁOutput: “Satisfiable” if Á is satisfiable, and “Unsatisfiable” otherwise

1. function Lazy Basic (Á)2. B := e(Á);3. while (true) do4. <®, res> := SAT-Solver(B);5. if res =“Unsatisfiable” then return “Unsatisfiable”;6. else7. <t, res> := Deduction(Th(®)) ;8. if res =“Satisfiable” then return “Satisfiable”;9. B := B ∧e(t);

• Deduction • input - conjunction of the literal corresponding to current assignment• output - a tuple of the form <blocking clause, result> where the result is one of {“Satisfiable”,

“Unsatisfiable”}

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Integration into DPLL Algorithm 2. Lazy-DPLL


Input: A formula ϕOutput: “Satisfiable” if the formula is satisfiable, and “Unsatisfiable” otherwise

1. function Lazy-DPLL2. AddClauses(e(ϕ));3. if BCP() = “conflict” then return “Unsatisfiable”;4. while (true) do5. if ￢ Decide() then6. <t, res>:=Deduction(Th(®));7. if res=“Satisfiable” then return “Satisfiable”;8. AddClauses(e(t));9. while (BCP() = “conflict”) do10. backtrack-level := Analyze-Conflict();11. if backtrack-level < 0 then return “Unsatisfiable”;12. else BackTrack(backtrack-level);13. else14. while (BCP() = “conflict”) do15. backtrack-level := Analyze-Conflict();16. if backtrack-level < 0 then return “Unsatisfiable”;17. else BackTrack(backtrack-level);

If there is no more assignment to do

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Improvement Algorithm 2 does not call Deduction() until a full satisfying

assignment is found. Example

- Assume that the Decide() procedure assigns e(x1 ¸ 10) TRUE and e(x1 < 0) TRUE.

- Deduction() results in a contradiction.- Time taken to complete the assignment is wasted.

Algorithm 2 can be improved by running Deduction before a full assignment to the Boolean encoder is available. Contradictory partial assignment are ruled out early. Implications of literals that are still unassigned can be communi-

cated back to the SAT solver. - ex) once e(x1 ¸ 10) has been assigned TRUE, we can infer that

e(x1 < 0) must be FALSE and avoid conflict.


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Improved Lazy-DPLL Algorithm 3. DPLL(T)


Input: A formula ϕOutput: “Satisfiable” if the formula is satisfiable and “Unsatisfiable” otherwise

1. function DPLL(T)2. AddClauses(e(ϕ));3. if BCP() = “conflict” then return “Unsatisfiable”;4. while (true) do5. if ￢ Decide() then return “Satisfiable”; 6. repeat7. while (BCP() = “conflict”) do8. backtrack-level := Analyze-Conflict();9. if backtrack-level < 0 then return “Unsatisfiable”;10. else BackTrack(backtrack-level);11. <t, res>:=Deduction(Th(α));12. AddClauses(e(t));13. until res = Satisfiable

Full assignment

Partial assignment

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DPLL (T)


UNSAT

Decide

Deduction AddClauses

BackTrack

BCP Analyze-Conflict

Partial assign-ment

SATFull assignment

backtrack-level ¸ 0

backtrack-level < 0

Conflict

Satisfiable

®

Th(®) e(t) t

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Implementation details of DPLL(T)

Deduction Returning blocking clause

- If S is the set of literals that serve as the premises in the proof of unsatisfi-ability, then the blocking clause is

- Example- Th(®) := x = y Æ y = z Æ x z Æ :(x = z) - blocking clause - t := :(x = y) Ç :(y = z) Ç :(x z) Ç x = z


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Implementation details of DPLL(T)


Deduction Returning implied assignment instead of blocking clauses

- Th(®) implies a literal liti, then

- The encoded clause e(t) is of the form

- Example- Let e(x1 ¸ 10) TRUE, e(x1 < 0) is unassigned yet.- Deduction detects that :(x1 < 0) is implied.- t := :(x1 ¸ 10) Ç :(x1 < 0)- e(t) := (:(x1 ¸ 10) Ç :(x1 < 0))

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Eager encoding Eager encoding

Perform a full reduction from the problem of deciding -formulas to one of deciding propositional formulas.

All the necessary clauses are added to the propositional skeleton. SAT solver is invoked only once, with no further interaction with

decision procedure of each theories. Example

- Equality logic and Uninterpreted Functions- Substitute equality literals into Boolean variables and add constraints

- Array logic- Substitute array read operation into UF


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Eager encoding Algorithm 4. Eager-encoding


Input: A formula ϕOutput: “Satisfiable” if ϕ is satisfiable and “Unsatisfiable” otherwise

1. function Eager-Encoding(ϕ)2. e(P) := Deduction(lit(ϕ));3. ϕE := e(ϕ) Æ e(P);4. <α, res> := SAT-Solver(ϕE);5. if res =“Unsatisfiable” then return “Unsatisfiable”;6. else return “Satisfiable”;

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Conclusions The architecture of Yices


Linear arithmetic Bit vectors Arrays Pointer

EUF

DPLL-based SAT solver

Core decision pro-cedure

Satelite decision procedure

from tool paper describing Yices

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Conclusions Two encoding schemes

Lazy encoding- Keep interacting between SAT solver and decision procedure of each

theories.- Almost every tool that participated in the SMT competitions in 2005-

2007 belongs to this category of solvers.

Eager encoding- SAT solver is invoked only once with no further interaction with deci-

sion procedure of each theories.


propositional encoding - decision procedure

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