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Dual Pivot Quicksort: Verification and Proof using KeY Jonas Schiffl Karlsruher Institut f¨ ur Technologie July 27th, 2016
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Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

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Page 1: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Dual Pivot Quicksort: Verification and Proofusing KeY

Jonas Schiffl

Karlsruher Institut fur Technologie

July 27th, 2016

Page 2: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Introduction

Why verify Dual Pivot Quicksort?

I Inspired by discovery of Timsort Bug

I Widely used standard library algorithm

I Complex enough

I Simple enough

Page 3: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Introduction

Why verify Dual Pivot Quicksort?

I Inspired by discovery of Timsort Bug

I Widely used standard library algorithm

I Complex enough

I Simple enough

Page 4: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Introduction

Why verify Dual Pivot Quicksort?

I Inspired by discovery of Timsort Bug

I Widely used standard library algorithm

I Complex enough

I Simple enough

Page 5: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Introduction

Why verify Dual Pivot Quicksort?

I Inspired by discovery of Timsort Bug

I Widely used standard library algorithm

I Complex enough

I Simple enough

Page 6: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Introduction

Why verify Dual Pivot Quicksort?

I Inspired by discovery of Timsort Bug

I Widely used standard library algorithm

I Complex enough

I Simple enough

Page 7: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Introduction

Why verify Dual Pivot Quicksort?

I Inspired by discovery of Timsort Bug

I Widely used standard library algorithm

I Complex enough

I Simple enough

Page 8: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Section 1

Algorithm Description

Page 9: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Quicksort

array index

value ofelementat index

Page 10: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Quicksort

array index

value ofelementat index

Page 11: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Quicksort

array index

value ofelementat index

Page 12: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Quicksort

array index

value ofelementat index

Page 13: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Quicksort

array index

value ofelementat index

Page 14: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Quicksort

array index

value ofelementat index

Page 15: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Quicksort

array index

value ofelementat index

Page 16: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Quicksort

array index

value ofelementat index

Page 17: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Quicksort

array index

value ofelementat index

Page 18: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Dual Pivot Quicksort

array index

value ofelementat index

Page 19: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Dual Pivot Quicksort

array index

value ofelementat index

Page 20: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Dual Pivot Quicksort

array index

value ofelementat index

Page 21: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Dual Pivot Quicksort

Why use Dual Pivot Quicksort?

I Theory: Average number of swaps reduced by 20%(Yaroslavskiy 2009)

I Practice: Multi-pivot Quicksorts are more cache-efficient(Kushagra 2014)

I Benchmarking shows it is faster

Page 22: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Dual Pivot Quicksort

Why use Dual Pivot Quicksort?

I Theory: Average number of swaps reduced by 20%(Yaroslavskiy 2009)

I Practice: Multi-pivot Quicksorts are more cache-efficient(Kushagra 2014)

I Benchmarking shows it is faster

Page 23: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Dual Pivot Quicksort

Why use Dual Pivot Quicksort?

I Theory: Average number of swaps reduced by 20%(Yaroslavskiy 2009)

I Practice: Multi-pivot Quicksorts are more cache-efficient(Kushagra 2014)

I Benchmarking shows it is faster

Page 24: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Dual Pivot Quicksort

Why use Dual Pivot Quicksort?

I Theory: Average number of swaps reduced by 20%(Yaroslavskiy 2009)

I Practice: Multi-pivot Quicksorts are more cache-efficient(Kushagra 2014)

I Benchmarking shows it is faster

Page 25: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Choosing a Sorting Algorithm

data type?

length?

byte

Counting Sort Insertion Sort

>29

<=29

length?

short, char

>3200 <47

Quicksort

else

length? highly structured?

int, long, float, double

<47

>285

elseno

Merge Sort

yes

Page 26: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Choosing a Sorting Algorithm

data type?

length?

byte

Counting Sort Insertion Sort

>29

<=29

length?

short, char

>3200 <47

Quicksort

else

length? highly structured?

int, long, float, double

<47

>285

elseno

Merge Sort

yes

Page 27: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Quicksort

Quicksort

Select 5 evenly spaced array elements

Sort elements in their positionsAll 5

elementsdistinct?

Single Pivot Partition

no

Dual Pivot Partition

yes

Centralpart

large?Pivot Values Partition

yes

Recursion

no

Page 28: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Quicksort

Quicksort Select 5 evenly spaced array elements

Sort elements in their positionsAll 5

elementsdistinct?

Single Pivot Partition

no

Dual Pivot Partition

yes

Centralpart

large?Pivot Values Partition

yes

Recursion

no

Page 29: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Quicksort

Quicksort Select 5 evenly spaced array elements

Sort elements in their positions

All 5elementsdistinct?

Single Pivot Partition

no

Dual Pivot Partition

yes

Centralpart

large?Pivot Values Partition

yes

Recursion

no

Page 30: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Quicksort

Quicksort Select 5 evenly spaced array elements

Sort elements in their positionsAll 5

elementsdistinct?

Single Pivot Partition

no

Dual Pivot Partition

yes

Centralpart

large?Pivot Values Partition

yes

Recursion

no

Page 31: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Quicksort

Quicksort Select 5 evenly spaced array elements

Sort elements in their positionsAll 5

elementsdistinct?

Single Pivot Partition

no

Dual Pivot Partition

yes

Centralpart

large?Pivot Values Partition

yes

Recursion

no

Page 32: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Quicksort

Quicksort Select 5 evenly spaced array elements

Sort elements in their positionsAll 5

elementsdistinct?

Single Pivot Partition

no

Dual Pivot Partition

yes

Centralpart

large?Pivot Values Partition

yes

Recursion

no

Page 33: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Quicksort

Quicksort Select 5 evenly spaced array elements

Sort elements in their positionsAll 5

elementsdistinct?

Single Pivot Partition

no

Dual Pivot Partition

yes

Centralpart

large?

Pivot Values Partitionyes

Recursion

no

Page 34: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Quicksort

Quicksort Select 5 evenly spaced array elements

Sort elements in their positionsAll 5

elementsdistinct?

Single Pivot Partition

no

Dual Pivot Partition

yes

Centralpart

large?Pivot Values Partition

yes

Recursion

no

Page 35: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Quicksort

Quicksort Select 5 evenly spaced array elements

Sort elements in their positionsAll 5

elementsdistinct?

Single Pivot Partition

no

Dual Pivot Partition

yes

Centralpart

large?Pivot Values Partition

yes

Recursion

no

Page 36: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Single Pivot Partition

array index

value ofelementat index

Page 37: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Single Pivot Partition

array index

value ofelementat index

Page 38: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Dual Pivot Partition

array index

value ofelementat index

Page 39: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Dual Pivot Partition

array index

value ofelementat index

Page 40: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Swap Pivot Values Partition

array index

value ofelementat index

Page 41: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Swap Pivot Values Partition

array index

value ofelementat index

Page 42: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 43: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 44: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 45: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 46: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 47: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 48: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 49: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 50: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 51: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 52: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 53: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Java Implementation – Partitioning

less k great

Page 54: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Section 2

Specification and Proof

Page 55: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Work Flow

I Encapsulating source code in its own Java class

I Subdivision into three classes: One per partitioning style

I Writing specificationRunning KeYAdapting specification or source code

Page 56: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Work Flow

I Encapsulating source code in its own Java class

I Subdivision into three classes: One per partitioning style

I Writing specificationRunning KeYAdapting specification or source code

Page 57: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Work Flow

I Encapsulating source code in its own Java class

I Subdivision into three classes: One per partitioning style

I Writing specificationRunning KeYAdapting specification or source code

Page 58: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Work Flow

I Encapsulating source code in its own Java class

I Subdivision into three classes: One per partitioning style

I Writing specificationRunning KeYAdapting specification or source code

Page 59: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

General KeY Strategy

I Autopilot Strategy MacroI If proof fails:

I Confirm by generating counterexampleI Find violated specification conditionI Adapt specification (or source code)

I If no proof is found:I Increase number of steps (?)I Interactive Rule Apps (Quantifier Instantiation,

if-then-else-split)I Heap Simplification + SMT Solver

Page 60: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

General KeY Strategy

I Autopilot Strategy Macro

I If proof fails:I Confirm by generating counterexampleI Find violated specification conditionI Adapt specification (or source code)

I If no proof is found:I Increase number of steps (?)I Interactive Rule Apps (Quantifier Instantiation,

if-then-else-split)I Heap Simplification + SMT Solver

Page 61: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

General KeY Strategy

I Autopilot Strategy MacroI If proof fails:

I Confirm by generating counterexampleI Find violated specification conditionI Adapt specification (or source code)

I If no proof is found:I Increase number of steps (?)I Interactive Rule Apps (Quantifier Instantiation,

if-then-else-split)I Heap Simplification + SMT Solver

Page 62: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

General KeY Strategy

I Autopilot Strategy MacroI If proof fails:

I Confirm by generating counterexampleI Find violated specification conditionI Adapt specification (or source code)

I If no proof is found:I Increase number of steps (?)I Interactive Rule Apps (Quantifier Instantiation,

if-then-else-split)I Heap Simplification + SMT Solver

Page 63: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Feasibility – Problems with KeY

I Computation timeI Method extractionI Exact LocalizationI SMT SolverI Block Contracts

I Error in specification or lack of resources?

I Localizability

I Stability

I Responsiveness

Page 64: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Feasibility – Problems with KeY

I Computation time

I Method extractionI Exact LocalizationI SMT SolverI Block Contracts

I Error in specification or lack of resources?

I Localizability

I Stability

I Responsiveness

Page 65: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Feasibility – Problems with KeY

I Computation timeI Method extractionI Exact LocalizationI SMT SolverI Block Contracts

I Error in specification or lack of resources?

I Localizability

I Stability

I Responsiveness

Page 66: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Feasibility – Problems with KeY

I Computation timeI Method extractionI Exact LocalizationI SMT SolverI Block Contracts

I Error in specification or lack of resources?

I Localizability

I Stability

I Responsiveness

Page 67: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Feasibility – Problems with KeY

I Computation timeI Method extractionI Exact LocalizationI SMT SolverI Block Contracts

I Error in specification or lack of resources?

I Localizability

I Stability

I Responsiveness

Page 68: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Feasibility – Problems with KeY

I Computation timeI Method extractionI Exact LocalizationI SMT SolverI Block Contracts

I Error in specification or lack of resources?

I Localizability

I Stability

I Responsiveness

Page 69: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Feasibility – Problems with KeY

I Computation timeI Method extractionI Exact LocalizationI SMT SolverI Block Contracts

I Error in specification or lack of resources?

I Localizability

I Stability

I Responsiveness

Page 70: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Violation of Single Pivot Partition Invariant

less k great

Page 71: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Violation of Single Pivot Partition Invariant

less k great

Page 72: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Violation of Single Pivot Partition Invariant

while (a[great] > pivot2) {

if (great -- == k) {

break outer;

}

}

while (a[great] == pivot2) {

if (great -- == k) {

break outer;

}

}

while (a[great] > pivot) {

--great;

}

...

Page 73: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Violation of Single Pivot Partition Invariant

less great k

... ... ...

< = > = >

Page 74: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Section 3

Conclusive Remarks

Page 75: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Conclusive Remarks

I Verifying a large, complex, real-world Java program with KeYis feasable, but not without challenges

I Correct sorting, but invariant is violated

Page 76: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Conclusive Remarks

I Verifying a large, complex, real-world Java program with KeYis feasable, but not without challenges

I Correct sorting, but invariant is violated

Page 77: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Conclusive Remarks

I Verifying a large, complex, real-world Java program with KeYis feasable, but not without challenges

I Correct sorting, but invariant is violated

Page 78: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Conclusive Remarks

I Verifying a large, complex, real-world Java program with KeYis feasable, but not without challenges

I Correct sorting, but invariant is violated

Page 79: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Further Work

I Prove permutation property

I Prove method as-is

I Prove entire sort(int[]) method

I Prove entire sort method

Page 80: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Further Work

I Prove permutation property

I Prove method as-is

I Prove entire sort(int[]) method

I Prove entire sort method

Page 81: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Further Work

I Prove permutation property

I Prove method as-is

I Prove entire sort(int[]) method

I Prove entire sort method

Page 82: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Further Work

I Prove permutation property

I Prove method as-is

I Prove entire sort(int[]) method

I Prove entire sort method

Page 83: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Further Work

I Prove permutation property

I Prove method as-is

I Prove entire sort(int[]) method

I Prove entire sort method

Page 84: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Further Work

I Prove permutation property

I Prove method as-is

I Prove entire sort(int[]) method

I Prove entire sort method

Page 85: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Statistics – Single Pivot Partition

Method Nodes Branches Time [s] Rule Apps Interactive SMT

case right 14784 114 17,7 18919 0 0

split 17609 90 23,8 24189 0 0

sort(array, left, right) 18495 101 18,8 22839 0 0

sort(array) 654 7 0,4 1342 0 0

Total 51542 312 60.7 67289 0 0

Page 86: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Statistics – Swap Pivot Values Partition

Method Nodes Branches Time [s] Rule Apps Interactive SMT

move great left 1245 16 0,8 2346 0 0

move less right 2120 14 1,8 3224 0 0

swap values 123636 407 246,6 138039 0 0

Total 127001 437 249.2 143609 0 0

Page 87: Dual Pivot Quicksort: Verification and Proof using KeYi12Dual Pivot Quicksort: Veri cation and Proof using KeY Jonas Schi Karlsruher Institut fur Technologie July 27th, 2016 Introduction

Statistics – Dual Pivot Partition

Method Nodes Branches Time [s] Rule Apps Interactive SMT

calc indices 24533 8 49,6 24835 0 0

insertionsort indices 50816 365 137,4 73056 0 34

prepare indices 5332 28 6,4 7153 0 0

move great left 1650 15 1,1 2605 0 0

move great in loop 1580 18 1,1 2787 0 0

move less right 1928 14 1,4 2967 0 0

loop body 52134 287 57,3 56263 18 0

split 28751 98 109,6 51666 0 36

sort(int[],left,right) 51342 305 459,6 76973 114 116

sort(int[]) 611 5 0,4 1236 0 0

Total 218677 1143 823,9 299541 132 186

Entire Proof 297220 1892 1133,8 510439 132 186


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