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Symbolic program analysis as Satisfiability Modulo Theories. Nikolaj Bjørner Microsoft Research Based on joint work with Kryštof Hoder , Ken McMillan, Leonardo de Moura, Andrey Rybalchenko. Background: Z3  Efficient SMT Solver. Many custom solvers: Free f unctions
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Symbolic program analysis as Satisfiability Modulo Theories
Nikolaj Bjørner
Microsoft Research
Based on joint work with KryštofHoder, Ken McMillan, Leonardo de Moura, AndreyRybalchenko
Many custom solvers:
Free functions
Linear Arithmetic
Bitvectors
Algebraic datatypes
Arrays
Polynomials
Quantifiers
Several Applications:
Analysis, Testing, …
Leonardo de Moura, B, Christoph Wintersteiger
from http://rise4fun.com/z3
Engine
SLAyer
API
Simplifier
Cores
Models
SAGE
Proofs
Isabelle
HOL4
Methodology
FixedPoint
SLAyer
Sep. Logic
Abstract Interpretation
Logic Programming
GateKeeper
Simulation
Relation
Predicate Based MC
Summaries
SAGE
BDD MC
Abstraction
Refinement
Datalog
Havoc Poirot Corral
Houdini
Interpolating MC
Contract Checking
Pointsto analysis
Symbolic Software Checking
µZ3
PropertyDirected
Reachability solver
Services for other solvers
(Quantifier elimination,
Foldunfold simplification)
Datalog +
Relational domains
Some “anecdotal” experience
Recursive predicates:
Expressed as Horn clauses + query
µZ: Portfolio of solversand services for fixedpoints:
Bottomup Datalog Engine
 Finite Tables (e.g., Hashtables, BTrees)
 Symbolic Tables (e.g., BDDs)
 Composition of Relations:
 Abstract interpretation domains
 Reduced products
Symbolic Engine Modulo Theories
 Generalized Property Directed Reachability
GateKeeper
(sparse hashtables)
Magnus Madsen
Pointsto analysis
KOP2 database
(using magic sets)
Contract Checking
DKAL
(encoding Primal Infon Logic)
CAV 2011[Hoder, Bjørner,
de Moura]
Bebop benchmarks
(evaluate PDR
generalized to PDA)
Symbolic Software Checking
Corral samples
(evaluate PDR
Modulo Arithmetic)
SAT 2012[Hoder, Bjørner]
mc(x) = x10 if x > 100
mc(x) = mc(mc(x+11)) if x 100
assert (mc(x) 91)
Formulate as Horn clauses.
mc()
mc() mc() mc()
mc()
Solve for mc
Formulate as Predicate Transformer:
Check:
Instead of computing
then checking
Suffices to find postfixed point satisfying:
Program Verification (Safety)
as Solving least fixedpoints
asSatisfiability of Horn clauses
[Bjørner, McMillan, Rybalchenko, SMT workshop 2012]
Hilbert Sausage Factory: [Grebenshchikov, Lopes, Popeea, Rybalchenko et.al. PLDI 2012]
Should really not be a surprise:
Underappreciated:
The Quest: Horn Clause Satisfiability
Dafny
HAVOC
Program
annotated with
inductive invariants
Verification
condition
Inductive variable
selection
Slicing
Houdini
Corral
Dafny
HAVOC
Program
partially
annotated with
inductive invariants
Verification
condition
Verification Condition Generators can already produce Horn Clauses
Corral
Dafny
HAVOC
Program
partially
annotated with
inductive invariants
Why, LLVM
…
Horn Clauses
Kind
HSF
Leon
Duality
Aligator
UFO
IC3
Synergy
MCMTSAFARI
Summary as commands
Verifying procedure calls
[Predicate Abstraction and
Refinement for Verifying
MultiThreaded ProgramsGupta, Popeea, Rybalchenko,
POPL 2011]
Extract sufficient Horn Conditions
In a nutshell, solving partial correctness amounts to checking truth value of formulas of the form:
E.g., satisfiability of:
Handling background axioms:
Remark:
Abductive Logic Programming amounts to symbolic simulation:

 is consistent
eg. solve for negation of above formula:
PDR (aka. IC3) – Property Directed Reachability algorithm
Breakthroughin Symbolic Model Checking of Hardware [Aaron Bradley, VMCAI 2011]
Transition Decomposes main stepsSystem ÷ priority queueFormulation
ProceduresRegular vs. Push Down systems
Beyond Linear Real Arithmetic
Propositional  Timed Automata Decision ProcedureLogic Interpolants from models
Original Algorithm Description in code.
Tough to digest. Rule + strategy description could help deconstruct the steps.
Original Algorithm Applies to Hardware (Finite State Automata).
Software has procedure calls.
Original Algorithm is for Finite State Systems
Open question what it meant to incorporate
Infinite State systems (= theories)
[Hoder & Bjørner, SAT 2012]
Objective is to solve for R such that
Elements of PDR encoded as transitions:
Overapproximate reachable states
Search for counterexamples to
Resolve and Propagate conflicts
Objective is to solve for R such that
Initialize:
Main invariant:
Is valid?
Search for overapproximations of states
Is valid?
Initially: N = 0, start with
Is valid?
Unfold to the next level if
Is valid?
Main Invariant is established for N = 1
Is valid?
Model
Is valid?
C,
Is valid?
Unfold to the next level if
Is valid?
Etc.
Is valid?
Etc.
Is valid?
is a postfixed point
implies
Valid Formula is valid if
Is valid?
Induction w
Is valid?
Induction w
Is valid?
Induction w
Is valid?
Induction w
Is valid?
Monotonicity of F
Induction w
Is valid?
Induction w
Is valid?
Induction w
Is valid?
Decide
Is valid?
Decide
Recall:
Is feasible?
Start with summary
feasible?
Yes, e.g.,
Is reachable? (in
M(87)
= M(M(98))
= M(M(M(109)))
= M(M(99))
=M(M(M(110)))
= M(M(100))
= M(M(M(111)))
= M(M(101))
= M(91)
= M(M(102))
= M(92)
= M(M(103))
= M(93) …
Benchmarks from the SLAM
Research toolkit
Checking against controls depth, but potentially wide tree.
Our approach: build DAG by sharing states. Sharing is cheap, even no sharing works on Bebop
Mutual Exclusion
Clauses have model
Conflict Resolution
Conflict Propagation
Conflict Propagation
Conflict Resolution
Conflict Resolution
Conflict Propagation
Conflict Propagation
Objective:
synthesize inductive invariant proving property.
Reaching objective with interpolants: Synthesize interpolants, use for proving invariants. Be admired.Synthesize interpolants, evaluate on random formulas. Admire them.
Write papers about interpolants. Admire the theorems.Review papers about generating interpolants. Watch Kevin Bacon.
Reaching objective with PDR:…. Nevertheless, interpolants sneak in.
Suppose
A Craig Interpolant is formula
Horn version. Establish satisfiability of:
and find solution for
The question is:
Quantified Horn Clause SatisfiabilityModulo Theories
PDR Generalized:
http://rise4fun.com/Z3Py/tutorial/fixedpoints
Compilation
Restarts
Relational
Algebra
Abstract
Machine
x
0
1
y
z
0
1
Bounds
Intervals
+
=
+
Pentagons
=