Investigating the Non-termination of Affine Loops

125 Pages·2013·1.81 MB·English

Investigating the Non-termination of Affine Loops
a thesis presented in partial fulfilment
of the requirements for the degree of
master of science
at the university of stellenbosch
By
K. Durant
October 2012
Supervised by: Prof. W. Visser
Stellenbosch University http://scholar.sun.ac.za
Declaration
I the undersigned hereby declare that the work contained in this thesis is my own original work
and has not previously in its entirety or in part been submitted at any university for a degree.
Signature: .................. Date: ..................
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Summary
Thesearchfornon-terminatingpathswithinaprogramisacrucialpartofsoftwareverification,
as the detection of an infinite path is often the only manner of falsifying program termination
— the failure of a termination prover to verify termination does not necessarily imply that a
program is non-terminating. This document describes the development and implementation of
two focussed techniques for investigating the non-termination of affine loops. The developed
techniques depend on the known non-termination concepts of recurrent sets and Jordan matrix
decomposition respectively, and imply the decidability of single-variable and cyclic affine loops.
Furthermore, the techniques prove to be practically capable methods for both the location of
non-terminating paths, as well as the generation of preconditions for non-termination.
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Afrikaans summary
Sagtewareverifikasie vereis of die bewys van die be¨eindiging van ’n program, of die deteksie
van oneindige uitvoerings. In hierdie tesis ontwikkel en implementeer ons twee tegnieke om
oor die oneindige eienskap van affiene lusse te beslis. Die tegnieke wat ontwikkel word is
gebaseeropkonseptesoosJordanmatriksdekomposisieenherhaaldegroepewatalindieverlede
gebruik is om die be¨eindiging van lusse te ondersoek. Die tegnieke kan gebruik word om
die uitvoerbaarheid van beide een-veranderlike en sikliese affiene lusse te bepaal. Feitlik alle
nie-eindige affiene lusse kan ge¨ıdentifiseer word en die toestande waaronder hierdie oneindige
eienskap verskyn kan beskryf word.
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Acknowledgements
I would like to thank:
• Prof. Willem Visser, for providing me with both the opportunity and supervision to
perform this work; and
• Prof. Stephan Wagner, for his innate ability to produce apt counter-examples.
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Contents
1 Introduction 1
1.1 Document outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background 4
2.1 A brief review of software verification and falsification . . . . . . . . . . . . . . 4
2.1.1 Safety and liveness properties . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 State representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Termination verification . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.3 Conditional termination . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.4 Termination falsification . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Approach 11
3.1 Affine loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Non-termination via recurrent sets . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Single-variable affine loops . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 Cyclic affine loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.3 Termination verification via recurrent sets . . . . . . . . . . . . . . . . . 23
3.2.4 Non-linear loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Non-termination via Jordan decomposition . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Diagonalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
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3.3.2 Jordan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.3 Sums of exponential functions . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.4 Proving non-termination . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.5 Approximating the set of non-termination witnesses . . . . . . . . . . . 49
3.3.6 Constraining polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.7 Deciding termination for simple affine loops . . . . . . . . . . . . . . . . 64
3.3.8 Termination verification via sign permutations . . . . . . . . . . . . . . 65
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Implementation 68
4.1 Detecting loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.1 Detecting loop boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.2 Constructing affine loops . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Non-termination algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Non-termination via recurrent sets . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Non-termination via Jordan decomposition . . . . . . . . . . . . . . . . . . . . 80
4.5 Algorithm complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 Evaluation 86
5.1 Non-termination detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Conditional non-termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 Conclusion 98
6.1 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.1.1 Termination verification and conditional termination . . . . . . . . . . . 99
6.1.2 Complex loop forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1.3 Test case generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A Supporting Concepts 101
A.1 Simple program representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.2 Complex arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.3 Complex eigenvalues and Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . 102
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B Implementation Notes 105
B.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
B.2 Class structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
B.3 Example loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Bibliography 112
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List of Tables
3.1 The maximal periods K(n) of cyclic loops with few loop variables. . . . . . . . 23
3.2 Several values of the function in Figure 3.16 at positive integer intervals. . . . . 41
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List of Figures
3.1 An unnested program loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 The general form of an affine loop. . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 A periodically monotonic loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 A general single-variable loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 A non-terminating single-variable loop. . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 A loop which is not periodically monotonic. . . . . . . . . . . . . . . . . . . . . 21
3.7 A 2-cyclic non-terminating loop. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.8 A 2-cyclic terminating loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.9 A non-terminating, non-cyclic loop. . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.10 A loop for which termination can easily be verified. . . . . . . . . . . . . . . . . 24
3.11 A diagonalisable two-variable (excluding the auxiliary variable x(cid:48)) loop. . . . . 28
3.12 A loop which is not diagonalisable. . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.13 A sampling of exponential functions. . . . . . . . . . . . . . . . . . . . . . . . . 36
3.14 Exponential sums with two positive coefficients. . . . . . . . . . . . . . . . . . . 37
3.15 Exponential sums with mixed (sign) coefficients. . . . . . . . . . . . . . . . . . 38
3.16 An exponential sum in three parts. . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.17 An exponential sum of three terms, as a pair function and an exponential function. 40
3.18 A loop whose exponential sum is not dominated by the leading term. . . . . . . 46
3.19 A loop which engenders the eigenvalue zero. . . . . . . . . . . . . . . . . . . . . 53
3.20 A non-terminating loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.21 A non-terminating loop which is detected by the techniques in Section 3.3.5, but
not those of Section 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.22 A positive polynomial with complex roots. . . . . . . . . . . . . . . . . . . . . . 62
x