Proving that programs eventually do something good. Byron Cook
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1 Proving that programs eventually do something good Byron Cook 1
2 Collaborators Domagoj Babic, Josh Berdine, Aziem Chawdhary, Dino Distefano, Alexey Gotsman, Sumit Gulwani, Alan Hu, Samin Ishtiaq, Eric Koskinen, Tal Lev-Ami, Peter O Hearn, Matthew Parkinson, Andreas Podelski, Zvonimir Rakameric, Andrey Rybalchenko, Mooly Sagiv, Moshe Vardi, Viktor Vafeiadis, Hongseok Yang, & the East London Massive. 2
3 Collaborators Domagoj Babic, Josh Berdine, Aziem Chawdhary, Dino Distefano, Alexey Gotsman, Sumit Gulwani, Alan Hu, Samin Ishtiaq, Eric Koskinen, Tal Lev-Ami, Peter O Hearn, Matthew Parkinson, Andreas Podelski, Zvonimir Rakameric, Andrey Rybalchenko, Mooly Sagiv, Moshe Vardi, Viktor Vafeiadis, Hongseok Yang, & the East London Massive. 3
4 Formal verification 4
5 Formal verification 5
6 Automatic formal verification View artifact of interest as a mathematical system: Software Hardware Biological system etc Build tools that find proofs of correctness using mathematics and logic 100% testing coverage Faster and more scalable than brute force Allows for 100% coverage even for infinite-state systems 6
7 Example property The parallel port device driver s event-handling routine only calls KeReleaseSpinLock() when IRQL=PASSIVE 7
8 Example property The parallel port device driver s event-handling routine only calls KeReleaseSpinLock() when IRQL=PASSIVE 8
9 Example property The parallel port device driver s event-handling routine only calls KeReleaseSpinLock() when IRQL=PASSIVE 9
10 Example property The parallel port device driver s event-handling routine only calls KeReleaseSpinLock() when IRQL=PASSIVE 10
11 Example property The mouse device driver s event-handling routine always eventually terminates 11
12 Example property The mouse device driver s event-handling routine always eventually terminates 12
13 Example property The mouse device driver s event-handling routine always eventually terminates 13
14 Example property The mouse device driver s event-handling routine always eventually terminates 14
15 Example property The mouse device driver s event-handling routine always eventually terminates 15
16 Formal verification 16
17 Formal verification 17
18 Formal verification 18
19 Outline Introduction Termination basics New advances for program termination proving Proving termination argument validity Finding termination arguments Conclusion 19
20 Outline Introduction Termination basics New advances for program termination proving Proving termination argument validity Finding termination arguments Conclusion 20
21 Proving termination Traditional termination proving method originally proposed by the forefathers of computing E.g. Turing, Checking a large routine,
22 Proving termination Traditional termination proving method originally proposed by the forefathers of computing E.g. Turing, Checking a large routine,
23 Proving termination R 23
24 Proving termination R 24
25 Proving termination 25
26 Proving termination 26
27 Proving termination 27
28 Proving termination 28
29 Proving termination f 29
30 Proving termination f f f f f f 30
31 Proving termination 31
32 Proving termination 32
33 Proving termination f > R f 33
34 Proving termination f > R f 34
35 Proving termination 35
36 Proving termination f > R f 36
37 Proving termination 37
38 Outline Introduction Termination basics New advances for program termination proving Proving termination argument validity Finding termination arguments Conclusion 38
39 Outline Introduction Termination basics & history New advances for program termination proving Proving termination argument validity Finding termination arguments Conclusion 39
40 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 40
41 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 41
42 Automating the search for proofs Transition relations must be computed Technically, computing is undeciable, so we must find a sound over-approximation using available techniques: represents an infinite set of states, but has a compact expression 42
43 Automating the search for proofs Transition relations must be computed Technically, computing is undeciable, so we must find a sound over-approximation using available techniques: represents an infinite set of states, but has a compact expression 43
44 Automating the search for proofs Transition relations must be computed Technically, computing is undeciable, so we must find a sound over-approximation using available techniques: represents an infinite set of states, but has a compact expression 44
45 Automating the search for proofs Transition relations must be computed Technically, computing is undeciable, so we must find a sound over-approximation using available techniques: represents an infinite set of states, but has a compact expression 45
46 Automating the search for proofs We use an over-approximation of the transition relation Since, we can prove termination by showing 46 Meaning: there might be unrealistic transitions that we have to worry about R
47 Automating the search for proofs In practice, its extremely hard to find the right overapproximation Luckily: recent breakthroughs in safety proving now make this possible. In fact: the checking the validity of a termination argument can be directly encoded as a safety property Tools like SLAM can be used to prove validity 47
48 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 48
49 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 49
50 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 50
51 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 51
52 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 52
53 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 53
54 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 54
55 Modular termination arguments 55
56 Modular termination arguments 56
57 Modular termination arguments 57
58 Modular termination arguments 58
59 Modular termination arguments f f 59
60 Modular termination arguments f f or g or h h g 60
61 Modular termination arguments f f or g or h h g 61
62 Modular termination arguments Modularity gives us freedom when looking for valid arguments Strategy: refinement based on failed attempts Start with empty termination argument Check inclusion If inclusion check succeeds, termination has been proved If it fails, synthesize a new ranking function from a counterexample and add it in Go to start 62
63 Modular termination arguments 63
64 Modular termination arguments X 64
65 Modular termination arguments X 65
66 Modular termination arguments X f f 66
67 Modular termination arguments X f f 67
68 Modular termination arguments X f f 68
69 Modular termination arguments X f f 69
70 Modular termination arguments X 70
71 Modular termination arguments X X 71
72 Modular termination arguments X X 72
73 Modular termination arguments X g g 73
74 Modular termination arguments X X g g 74
75 Modular termination arguments X g g 75
76 Modular termination arguments X 76
77 Modular termination arguments X 77
78 Modular termination arguments X h h 78
79 Modular termination arguments X h h 79
80 Modular termination arguments 80
81 Modular termination arguments 81
82 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 82
83 Automating the search for proofs Difficulties: Proving the inclusion undecidable in theory) is hard in practice (and Finding an such that is even harder in practice (and undecidable in theory) 83
84 Terminator copied = 0;... while(x<y) { if x = (!copied) f(x,y); { g(&y,x); if (*) { } H[x] = x; H[y] = y; copied = 1; } } else { assert(t1 T2 T3); } 84 copied = 0;
85 85
86 86
87 87
88 88
89 89
90 Examples 90
91 Examples 91
92 Examples 92
93 Examples 93
94 Examples 94
95 Examples 95
96 Examples 96
97 Examples 97
98 Examples 98
99 Misunderstanding the halting problem 99
100 Misunderstanding the halting problem Terminator
101 Misunderstanding the halting problem Terminator
102 Misunderstanding the halting problem Terminator 2006 X X X X X X X 102
103 Misunderstanding the halting problem Terminator 2006 X X X X X X X 103
104 Misunderstanding the halting problem Terminator 2006 X X X X X X X 104
105 Misunderstanding the halting problem Terminator 2006 X X X X X X X 105
106 Misunderstanding the halting problem Terminator 2006 X X X X X X X 106
107 Misunderstanding the halting problem 107
108 Misunderstanding the halting problem 108
109 Misunderstanding the halting problem 109
110 Misunderstanding the halting problem 110
111 Misunderstanding the halting problem 111
112 Misunderstanding the halting problem 112
113 Misunderstanding the halting problem 113
114 Misunderstanding the halting problem 114
115 Misunderstanding the halting problem 115
116 Misunderstanding the halting problem 116
117 Misunderstanding the halting problem 117
118 Misunderstanding the halting problem? 118
119 Misunderstanding the halting problem? 119
120 Misunderstanding the halting problem Automatic searches for proofs of program termination don t make for exciting demos Termination bugs found from failed proof attempts are usually more entertaining 120
121 Misunderstanding the halting problem 121
122 Misunderstanding the halting problem 122
123 Misunderstanding the halting problem 123
124 Misunderstanding the halting problem 124
125 Misunderstanding the halting problem 125
126 Misunderstanding the halting problem 126
127 Misunderstanding the halting problem 127
128 Misunderstanding the halting problem 128
129 Misunderstanding the halting problem 129
130 Misunderstanding the halting problem 130
131 Misunderstanding the halting problem 131
132 Misunderstanding the halting problem 132
133 Misunderstanding the halting problem 133
134 Misunderstanding the halting problem 134
135 Misunderstanding the halting problem 135
136 Misunderstanding the halting problem 136
137 Outline Introduction Termination basics & history New advances for program termination proving Proving termination argument validity Finding termination arguments Conclusion 137
138 Outline Introduction Termination basics & history New advances for program termination proving Proving termination argument validity Finding termination arguments Conclusion 138
139 Future work Previous wisdom: proving termination for industrial systems code is impossible Now people are beginning to think that it s effectively solved. Much left to do, including Complex data structures (safety) Infinite-state systems w/ bit vectors (safety) Binaries (safety) Non-linear systems (liveness and safety) Better support for concurrent programs Modern programming features (e.g. closures) Finding preconditions to termination Scalability, performance, precision 139
140 Future work Termination proving is at the heart of many undecidable problems (e.g. Wang s tiling problem) Modern termination proving techniques could potentially be used to building working tools Challenge: black-box solutions to undecidable problems die in the most unpredictable ways 140
141 Conclusion Conventional wisdom about termination overturned Undecidable does not mean we cannot soundly approximate a solution Terminator shows that automatic termination proving is not hopeless for industrial systems code Current state-of-the-art solutions based on Abstraction search for safety property verification (e.g. SLAM) Farkas-based linear rank function synthesis Ramsey-based modular termination arguments Separation Logic based data structure analysis 141
142 For more information Research papers Recorded technical lectures Contact details CACM review article 142
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