Accelerating lemma learning using joins LPAR 2008 – Doha, Qatar

Nikolaj Bjørner, Leonardo de MouraMicrosoft Research

Bruno DutertreSRI International

Satisfiability Modulo Theories (SMT)

Accelerating lemma learning using joins

SAT

Theories

SMT

ArithmeticBit-vectorsArrays…

Satisfiability Modulo Theories (SMT)

Accelerating lemma learning using joins

)1()2),3,,(((2 xyfyxawritereadfyx

Arithmetic

Satisfiability Modulo Theories (SMT)

Accelerating lemma learning using joins

)1()2),3,,(((2 xyfyxawritereadfyx

ArithmeticArray Theory

Satisfiability Modulo Theories (SMT)

Accelerating lemma learning using joins

)1()2),3,,(((2 xyfyxawritereadfyx

ArithmeticArray TheoryUninterpreted

Functions

SMT: Some Applications @ Microsoft

Accelerating lemma learning using joins

VCC

Hyper-V Terminator T-2

NModel

HAVOC

F7SAGE

Vigilante

SMT@Microsoft: Solver

Accelerating lemma learning using joins

Z3 is a new solver developed at Microsoft Research.Development/Research driven by internal customers.Free for academic research.Interfaces:

http://research.microsoft.com/projects/z3

Z3

TextC/C++.NET

OCaml

SMT = DPLL + Theories

Guessing (case-splitting)Deducing (BCP + Theory propagation)Conflict resolution Backtracking + Lemma

Accelerating lemma learning using joins

a=b f(a)=f(b), a < 5 a > 10, a > 6 b = 2

Most SMT solvers use only the literals from the given

formula!

Is SMT fast???

Accelerating lemma learning using joins

a[0] = 0if (c1) { a[1] = 0; } else { a[1] = 1; }…if (cn) { a[n] = 0; } else { a[n] = 1; }assert(a[0] == 0);

Is SMT fast???

Accelerating lemma learning using joins

a1= write(a0, 0, 0)(c1 a2 = write(a1,1,0))(c1 a2 = write(a1,1,1))…(cn an+1 = write(an,n,0))(cn an+1 = write(an,n,1))read(an+1,0) 0 It takes O(2n) time if

lemmas do not use new literals!

“Diamonds are eternal”

Accelerating lemma learning using joins

SP(E) calculus

Accelerating lemma learning using joins

Very slow in practice!

It can solve “diamonds” in polynomial time.

DPLL (E + )

New literals can be createdCase-splitting (guessing)Lemma Learning

Any SP(E) inference can be simulated by DPLL(E+)

Accelerating lemma learning using joins

How do we create ?

Accelerating lemma learning using joins

Look ahead

Accelerating lemma learning using joins

S

p

p

l

New literal

l

Look ahead

Accelerating lemma learning using joins

SS,

l

“The plan”

Accelerating lemma learning using joins

Define language L (of new literals). Examples:(Bounds) x > 5(Equality) x = y(Difference) x – y < 3

Theory propagation for LJoin operator for L

S

p

p

S1

S2

S1 S2

Join: Examples (Bounds)

Accelerating lemma learning using joins

p q, q x>5, p x>y, y > 4

Join: Examples (Bounds)

Accelerating lemma learning using joins

p

{ x > 5 }

p q, q x>5, p x>y, y > 4

Join: Examples (Bounds)

Accelerating lemma learning using joins

p

{ x > 5 }

p

{ x > 4 }

p q, q x>5, p x>y, y > 4

Join: Examples (Bounds)

Accelerating lemma learning using joins

{ x > 5 } { x > 4 } = {x > 4}

p q, q x>5, p x>y, y > 4

p

{ x > 5 }

p

{ x > 4 }

Join: Examples (Equalities)

Accelerating lemma learning using joins

{ x = y, y = z, x = z } { x = z, z = w, x = w } = {x = z}

Join: Examples (Difference constraints)

Accelerating lemma learning using joins

{ x - y < 3 } { x - y < 2, y - z < 1, x - z < 3} = {x - y < 3}

Join

Accelerating lemma learning using joins

Other examples:Linear arithmetic: polyhedral.Array partial equalities: a =i b (forall x: x = i a[x] = b[x])

k-look ahead.

Conclusion

Accelerating lemma learning using joins

SMT solvers are fast, but they may choke in simple formulas.DPLL(join) = SMT + “Abstract Interpretation”.Future work: new literals during conflict resolution.http://research.microsoft.com/projects/z3

Thank You!