Introduction to SMT

Lecture 2, 2012Nikolaj Bjørner Microsoft ResearchDTU Winter course January 3rd 2012Organized by Hanne Riis Nielson, Flemming Nielson

Background Reading

September 2011

PlanGeneral overview of what is SMT

Compare with SAT, first-order theorem proving,..

Refresher on SAT and modern DPLL

Introduction to SMT solving techniques

Selected SMT applications (more Jan 4,5th)

Takeaways - this section should teach:The syntax and semantics of

SAT/SMT.

Algorithmic principles of modern SAT solvers

Algorithmic principles of modern SMT solvers

Some context

On Theorem Proving

Symbolic Engines: SAT, FTP and SMT

SAT: Propositional Satisfiability.(Tie Shirt) (Tie Shirt) (Tie

Shirt)

FTP: First-order Theorem Proving.X,Y,Z [X*(Y*Z) = (X*Y)*Z] X [X*inv(X) = e] X [X*e = e]

SMT: Satisfiability Modulo background Theories

b + 2 = c A[3] ≠ A[c-b+1]

SAT - Milestonesyear Milestone1960 Davis-Putnam procedure1962 Davis-Logeman-Loveland1984 Binary Decision Diagrams 1992 DIMACS SAT challenge1994 SATO: clause indexing1997 GRASP: conflict clause

learning1998 Search Restarts2001 zChaff: 2-watch literal,

VSIDS2005 Preprocessing techniques2007 Phase caching2008 Cache optimized indexing2009 In-processing, clause

management

2010 Blocked clause elimination

2002 2010

Problems impossible 10 years ago are trivial today

Concept

Millions of variables from HW designs Courtesy Daniel le Berre

FTP - MilestonesYear Milestone Who Year Milestone Who

1930 Hebrand's theorem Herbrand 1970 Completion and saturation procedures

many people and provers

1934 Sequent calculi Gentzen 1970 Knuth-Bendix ordering Knuth; Bendix1934 Inverse method Gentzen 1971 Selection function Kowalski; Kuehner1955 Semantic tableaux Beth 1972 Built-in equational theories Plotkin

1960 Herbrand-based theorem proving Wang Hao 1972 Prolog Colmerauer

1960 Ordered resolution Davis; Putnam 1974 Saturation algorithms Overbeek

1962 DLL Davis; Logemann; Loveland 1975 Completeness of paramodulation Brand

1963 First-order inverse method Maslov 1975 AC-unification Stickel1965 Unification J. Robinson 1976 Resolution as a decision procedure Joyner1965 First-order resolution J. Robinson 1979 Basic paramodulation Degtyarev1965 Subsumption J. Robinson 1980 Lexicographic path orderings Kamin; Levy1967 Orderings Slagle 1985 Theory resolution Stickel

1967 Demodulation or rewriting Wos; G. Robinson; Carson; Shalla 1986

Definitional clause form transformation Plaisted; Greenbaum

1968 Model elimination Loveland 1988 Superposition Zhang1969 Paramodulation G. Robinson; Wos 1988 Model construction Zhang

1989 Term indexing Stickel; Overbeek

1990 General theory of redundancy Bachmair; Ganzinger1992 Basic superposition Nieuwenhuis; Rubio1993 First instance-based methods Billon; Plaisted1993 Discount saturation algorithm Avenhaus; Denzinger1998 Finite model finding using SAT McCune2000 First-order DPLL Baumgartner2003 iProver method Ganzinger; Korovin2008 Sine selection Hoder

Some success stories:- Open Problems (of 25 years):

XCB: X ((X Y) (Z Y)) Z)is a single axiom for equivalence

- Knowledge Ontologies GBs of formulas Courtesy Andrei Voronkov, Manchester U

SMT - Milestonesyear Milestone1977 Efficient Equality Reasoning1979 Theory Combination

Foundations1979 Arithmetic + Functions 1982 Combining Canonizing

Solvers1992-8

Systems: PVS, Simplify, STeP, SVC

2002 Theory Clause Learning2005 SMT competition2006 Efficient SAT + Simplex2007 Efficient Equality Matching2009 Combinatory Array Logic, …

SATTheor

ySolver

sSMT

15KLOC + 285KLOC = Z3

Includes progress from SAT:

Simplify (of ’01) time

1sec

1

10

100

1000

Z3TimeOn VCCRegression

Nov 08 March 09

Z3(of ’07)TimeOn BoogieRegression

Introducing SMT byexamples

Satisfiability Modulo Theories (SMT)

Is formula satisfiable modulo theory T ?

SMT solvers have specialized algorithms for

T

Satisfiability Modulo Theories (SMT)

ArithmeticArray Theory Uninterpreted Functions

𝑥+2=𝑦⇒ 𝑓 (𝑠𝑒𝑙𝑒𝑐𝑡 (𝑠𝑡𝑜𝑟𝑒 (𝑎 ,𝑥 ,3 ) , 𝑦−2 ) )= 𝑓 (𝑦−𝑥+1)

Satisfiability Modulo Theories (SMT)

b + 2 = c and f(select(store(a,b,3), c-2)) ≠ f(c-b+1)

Satisfiability Modulo Theories (SMT)

Arithmetic

b + 2 = c and f(select(store(a,b,3), c-2)) ≠ f(c-b+1)

Satisfiability Modulo Theories (SMT)

ArithmeticArray Theory

b + 2 = c and f(select(store(a,b,3), c-2)) ≠ f(c-b+1)

Satisfiability Modulo Theories (SMT)

ArithmeticArray TheoryUninterpreted Functions

b + 2 = c and f(select(store(a,b,3), c-2)) ≠ f(c-b+1)

Satisfiability Modulo Theories (SMT)

b + 2 = c and f(select(store(a,b,3), c-2)) ≠ f(c-b+1)

Substituting c by b+2

Satisfiability Modulo Theories (SMT)

b + 2 = c and f(select(store(a,b,3), b+2-2)) ≠ f(b+2-b+1)

Simplifying

Satisfiability Modulo Theories (SMT)

b + 2 = c and f(select(store(a,b,3), b)) ≠ f(3)

Satisfiability Modulo Theories (SMT)

b + 2 = c and f(select(store(a,b,3), b)) ≠ f(3)

Applying array theory axiom

Satisfiability Modulo Theories (SMT)

b + 2 = c and f(3) ≠ f(3)

Inconsistent/Unsatisfiable

Job Shop Scheduling

Machines

Jobs

P = NP? Laundry 𝜁 (𝑠 )=0⇒ 𝑠=12 +𝑖𝑟

Tasks

Job Shop SchedulingConstraints:

Precedence: between two tasks of the same job

Resource: Machines execute at most one job at a time

413

2

[ 𝑠𝑡𝑎𝑟 𝑡2 , 2 ..𝑒𝑛𝑑2 , 2 ]∩ [𝑠𝑡𝑎𝑟 𝑡 4 , 2 ..𝑒𝑛𝑑4 , 2 ]=∅

Job Shop SchedulingConstraints:Encoding:

Precedence: - start time of job 2 on mach 3 - duration of job 2 on mach 3Resource:

413

2

[ 𝑠𝑡𝑎𝑟 𝑡2,2 ..𝑒𝑛𝑑2,2 ]∩ [ 𝑠𝑡𝑎𝑟 𝑡4,2 ..𝑒𝑛𝑑4,2 ]=∅

Not convex

Job Shop Scheduling

Job Shop Scheduling

case split

case split

Efficient solvers:- Floyd-Warshal algorithm- Ford-Fulkerson algorithm

𝑧−𝑧=5 – 2 –3 – 2=−2<0

SAT solving

Modern DPLL in a nutshellInitialize Decide Propagate

Conflict Resolve

Learn Backjump

Unsat Sat Restart

Adapted and modified from [Nieuwenhuis, Oliveras, Tinelli J.ACM 06]

SMT solving

DPLL(T) solver interactionT- Propagate

T- Conflict

h𝑤 𝑒𝑟𝑒𝑎>𝑏 ,𝑏>𝑐 ,𝑎≤𝑐⊆𝑀T- Conflict

T- Propagate

Main components of modern SMT Solvers

2 ( ( ( , ,3), 2)) ( 1)x y f read write a x y f y x

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

x y f v f y xread write a x y v

Purification

Note: read is just another name for select, write is just another name for store

Main components of modern SMT Solvers

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

x y f v f y xread write a x y v

2 1( ( , ,3), 2)

( ) ( )

x y u y xread write a x y vf v f u

Purification

Main components of modern SMT Solvers

2 1( ( , ,3), 2)

( ) ( )

x y u y xread write a x y vf v f u

Purification

2 1 3 2( ( , , ), )

( ) ( )

x y u y x z w yread write a x z w vf v f u

ArithmeticArrraysFunctions

Main components of modern SMT SolversPropositional Abstraction

3

1 2 4

5

6

2 1 3 2

( ( , , ), )

( ) ( )

pp p p

p

p

x y u y x z w y

read write a x z w v

f v f u

2 1 3 2( ( , , ), )

( ) ( )

x y u y x z w yread write a x z w vf v f u

Main components of modern SMT SolversPropositional Assignment

3

1 2 4

5

6

2 1 3 2

( ( , , ), )

( ) ( )

pp p p

p

p

x y u y x z w y

read write a x z w v

f v f u

1

2

3

4

5

6

,,,,,

p truep truep truep truep truep false

Using SAT solver

Main components of modern SMT SolversTheory Solving

2 1 3 2x y u y x z w y

( ( , , ), )read write a x z w v

( ) ( )f v f u

w x

z v

3u z

Arithmetic

Arrays

Free functions

Theories exchange equalities between shared variables.

But how?

Model-based CombinationThe running example was easy. But what about:

Either or . or (are integers)

Arithmetic module needs to somehow

learnthat . - Integer linear arithmetic is non-convex.

Model-based combinationDelayed Theory Combination solution

[2006 Bruttomesso et.al.]Add equality literals for every pair of shared variables:

: Solvers work completely independently.: Works with non-convex theories.: O(n2) up-front cost. No use of

propagation.

( ), ( ), ( )x y x y x z x z y z y z

Model-based combinationIdea:

Have solvers produce models.Use models to introduce equalities on demand.

If Then guess

: No up-front O(n2) cost of adding equalities: Works with non-convex theories: Models are conservative approximations then

Model-based - Example

Model-based - Example

Model-based - Example

Model-based - Example

Model-based - Example

Model-based - Example