An Efficient and Parallel Abstract Interpreter in Scala — Second Presentation

Universit´
e Libre de Bruxelles
Computer Science Department
MEMO-F524 Masters thesis
An Efficient and Parallel
Abstract Interpreter in Scala
— Second Presentation —
Olivier Pirson — [email protected]
orcid.org/0000-0001-6296-9659
March 20, 2018
(Little corrections: July 1st, 2018)
https://bitbucket.org/OPiMedia/efficient-parallel-abstract-interpreter-in-scala
Vrije Universiteit Brussel
Promotors Coen De Roover
Wolfgang De Meuter
Advisor Quentin Stievenart

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
1 The context: Abstract Interpretation for Static Analysis
2 The problematic: Too heavy for real programs
3 Parallel algorithms
4 Done and todo
5 References
An Eﬃcient and Parallel Abstract Interpreter in Scala — Second Presentation 2 / 21

View Slide

An Efficient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
The context: Abstract Interpretation for Static Analysis
We need tools to help us to build correct programs.
Figure:
First “flight” of
Ariane 5 in 1996.
Testing is not enough.
Static Analysis:
study the behaviour of programs without executing them.
Not trivial questions about the behaviour of a program
are undecidable (Rice’s theorem).
Abstract Interpretation:
approximation technique to perform static analysis.
Trade-off:
approximation must be enough precise to have an useful analysis,
and enough imprecise to make the problem decidable.

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
From Concrete Interpretation. . .
Trace: concrete interpretation with small-step semantics, for one instance.
e
s0 s1 s2 s3 s4 · · ·
injection
function
concrete transition function
Program is executed by interpreter,
described by an Abstract Machine (AM).
One execution is for one instance on this program.
e is for one expression, i.e. a program.
si
are the successive states during this execution.

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
From Concrete Interpretation to Abstract Interpretation
Trace: concrete interpretation with small-step semantics, for one instance.
e
s0 s1 s2 s3 s4 · · ·
s0 s1 s2 s3 s4
s3′
injection
function
injection
function
abstract transition function
abstraction
function α
Abstracting Abstract Machine (AAM).
2 over-approximations:
on addresses: by modulo; give a ﬁnite state space
on values: abstraction
Abstract transition function returns all directly reachable states.
State graph: abstract interpretation, for all instances.
“The abstract simulates the concrete” (Might).

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Approximation of values: abstraction
Based on mathematical notion of lattice
(partially ordered sets with additional properties).
Examples of abstraction for the set of integer values:
Figure: Ren´
e Magritte,
Le Calcul Mental. 1940.
the type integer
intervals
sign: {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}
abstracted by {⊥, −, 0, +, (− and 0), (0 and +), ⊤}
⊤ (top, no information)
(− and 0) (0 and +)
− 0 +
⊥ (bottom, no information yet)
Figure: Hasse diagram of the complete lattice of signs.
Properties of a lattice are such that it contains the join of 2 elements
and the succession of operation give a ﬁxed point.

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Example of a state graph on a very simple program
ev((letrec ((abs (lambda (x) (if (>= x 0) x (- x))))) (abs -42)))
0
ev((abs -42))
1
ev((if (>= x 0) x (- x)))
2
ev((>= x 0))
3
ko(Bool)
4
ev(x)
5
ev((- x))
6
ko(Int)
7
#nodes: 8
#edges: 8
graph density: 0,143
outdegree min: 1
outdegree max: 2
outdegree avg: 1,000
language: Scheme
machine: AAM
lattice: TypeSet
address: Classical
Figure: State graph of the abs program with a type lattice.
Algorithm 1: Absolute value of −42
( Ò ( × x )
( i f (>= x 0)
x
(− x ) ) )
( × −42)

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
4 Done and todo
5 References

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Example of a state graph on a other very simple program
ev((letrec ((fibonacci (lambda (n) (if (<= n 1) n (+ (fibonacci (- n 1)) (fibonacci (- n 2))))))) (fibonacci 10)))
0
ev((fibonacci 10))
1
ev((if (<= n 1) n (+ (fibonacci (- n 1)) (fibonacci (- n 2)))))
2
ev((<= n 1))
3
ko(Bool)
4
ev(n)
5
ev((+ (fibonacci (- n 1)) (fibonacci (- n 2))))
6
ko(Int)
7
ev((fibonacci (- n 1)))
8
ev((- n 1))
9
ko(Int)
10
11
ev((<= n 1))
12
ko(Bool)
13
ev(n)
14
15
ev(n)
16
17
ko(Int)
18
19
ko(Int)
20
21
ev((- n 1))
22
23
ev((- n 1))
24
ko(Int)
25
ev((- n 2))
26
ko(Int)
27
28
ko(Int)
29
30
31
ev((<= n 1))
32
ev((<= n 1))
33
ko(Bool)
34
ko(Bool)
35
ev(n)
36
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ev(n)
38
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ev(n)
42
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ev(n)
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ev(n)
45
ko(Int)
46
ko(Int)
47
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52
ko(Int)
48
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ko(Int)
50
ko(Int)
53
54
55
ev((- n 1))
56
ev((- n 1))
57
ev((- n 1))
58
59
ev((- n 2))
60
ev((- n 2))
61
ko(Int)
62
ko(Int)
63
ko(Int)
64
ev((- n 2))
65
ko(Int)
66
ko(Int)
67
68
69
70
ko(Int)
71
72
73
ev((<= n 1))
74
ev((<= n 1))
75
76
ev((<= n 1))
77
ko(Bool)
78
ko(Bool)
79
ko(Bool)
80
ev(n)
81
ev(n)
82
83
84
ev(n)
85
86
ev(n)
87
88
ev(n)
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ev(n)
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ev(n)
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ev(n)
96
97
ev(n)
98
ko(Int)
100
ko(Int)
99
101
ko(Int)
102
ko(Int)
103
ko(Int)
105
ko(Int)
104
ko(Int)
109
106
110
ko(Int)
108
ko(Int)
107
111
112
ev((- n 1))
113
114
115
ev((- n 1))
116
117
ev((- n 1))
118
ev((- n 2))
119
ev((- n 2))
120
ko(Int)
121
ev((- n 2))
122
ev((- n 2))
123
ko(Int)
124
ev((- n 2))
125
ko(Int)
126
ko(Int)
127
ko(Int)
128
129
ko(Int)
130
ko(Int)
131
132
ko(Int)
133
134
135
136
ev((<= n 1))
137
138
139
140
ev((<= n 1))
141
ev((<= n 1))
142
ko(Bool)
143
ev((<= n 1))
144
ko(Bool)
145
ko(Bool)
146
ev(n)
147
148
149
ev(n)
150
151
ev(n)
152
ko(Bool)
153
154
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ev(n)
156
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ev(n)
158
ev(n)
159
ev(n)
160
161
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ev(n)
163
ev(n)
164
165
ko(Int)
167
ko(Int)
168
ko(Int)
166
ev(n)
169
170
ev(n)
171
ev(n)
172
173
174
ko(Int)
177
ko(Int)
175
ko(Int)
176
ko(Int)
178
180
ko(Int)
181
ko(Int)
179
182
183
184
185
ko(Int)
190
187
ko(Int)
189
ko(Int)
188
186
191
192
ev((- n 1))
193
ev((- n 1))
194
ev((- n 2))
195
ev((- n 2))
196
ev((- n 2))
197
ev((- n 1))
198
ev((- n 1))
199
ev((- n 2))
200
ev((- n 2))
201
ko(Int)
202
ko(Int)
203
ko(Int)
204
ko(Int)
205
ko(Int)
206
ko(Int)
207
ko(Int)
208
ko(Int)
209
ko(Int)
210
211
212
213
214
215
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ev((<= n 1))
220
ev((<= n 1))
221
ev((<= n 1))
222
ko(Bool)
223
ko(Bool)
224
ko(Bool)
225
226
ev(n)
227
ev(n)
228
229
230
ev(n)
231
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ev(n)
233
ev(n)
234
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ev(n)
236
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ev(n)
239
240
241
ev(n)
242
ev(n)
243
ko(Int)
246
ko(Int)
244
ko(Int)
245
ko(Int)
248
ko(Int)
249
ko(Int)
247
251
ko(Int)
253
ko(Int)
250
ko(Int)
252
254
ev((- n 1))
255
ev((- n 2))
256
ko(Int)
257
ko(Int)
258
259
260
ev((<= n 1))
261
ko(Bool)
262
ev(n)
263
ev(n)
264
265
266
ev(n)
267
268
ko(Int)
271
ko(Int)
269
270
ko(Int)
272
ev((- n 1))
273
ko(Int)
274
275
#nodes: 276
#edges: 346
outdegree min: 1
outdegree max: 6
language: Scheme
machine: AAM
lattice: TypeSet
address: Classical
Figure: State graph of the Fibonacci program with a type lattice.
Algorithm 2: The 10th Fibonacci number.
; ; nth number of F i b o n a c c i ( by r e c u r s i v e p r o c e s s )
( Ò ( f i b o n a c c i n )
( i f (<= n 1)
n
(+ ( f i b o n a c c i (− n 1))
( f i b o n a c c i (− n 2 ) ) ) ) )
( f i b o n a c c i 10)

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Example of a state graph on a other very simple program (bis)
ev((letrec ((fibonacci2 (lambda (n fn_1 fn) (if (= n 0) (cons fn_1 fn) (fibonacci2 (- n 1) fn (+ fn fn_1))))) (fibonacci (lambda (n) (cdr (fibonacci2 n 1 0))))) (fibonacci 10)))
0
ev((fibonacci 10))
1
ev((cdr (fibonacci2 n 1 0)))
2
ev((fibonacci2 n 1 0))
3
ev((if (= n 0) (cons fn_1 fn) (fibonacci2 (- n 1) fn (+ fn fn_1))))
4
ev((= n 0))
5
ko({#f,#t})
6
ev((cons fn_1 fn))
7
ev((fibonacci2 (- n 1) fn (+ fn fn_1)))
8
ko(Cons(@fn_1-Time(),@fn-Time()))
9
ev((- n 1))
10
ko(Int)
11
ko(Int)
12
ev((+ fn fn_1))
13
ko(Int)
14
ev((if (= n 0) (cons fn_1 fn) (fibonacci2 (- n 1) fn (+ fn fn_1))))
15
ev((= n 0))
16
ko({#f,#t})
17
ev((cons fn_1 fn))
18
ev((fibonacci2 (- n 1) fn (+ fn fn_1)))
19
ko(Cons(@fn_1-Time(),@fn-Time()))
20
ev((- n 1))
21
ko(Int)
22
ko(Int)
23
#nodes: 24
#edges: 24
outdegree min: 1
outdegree max: 2
language: Scheme
machine: AAM
lattice: TypeSet
address: Classical
Figure: State graph of the Fibonacci program with a type lattice.
Algorithm 3: The 10th Fibonacci number.
; ; nth number of F i b o n a c c i ( by i t e r a t i v e p r o c e s s )
( Ò ( f i b o n a c c i 2 n f n 1 fn )
( i f (= n 0)
( ÓÒ× f n 1 fn )
( f i b o n a c c i 2 (− n 1) fn (+ fn f n 1 ) ) ) )
( Ò ( f i b o n a c c i n )
( c dr ( f i b o n a c c i 2 n 1 0 ) ) )
( f i b o n a c c i 10)

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
The problematic: Too heavy for real programs
With an AAM the problem is became decidable.
But in general an AAM is too slow to analyse real programs.
One way to have a faster AAM is to
parallelize generation of the state graph.
Goal of this master thesis:
implement and compare several parallelizations of AAM
in the framework Scala-AM.
Parallel model for the implementation: actors with Akka.
Language analysed: Scheme.
Figure: Ferdinand Hodler, View to Inﬁnity. 1913.

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
4 Done and todo
5 References

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Sequential worklist strategy
s
s
s
s
s
worklist
Figure: K. Dewey, V. Kashyap, B. Hardekopf. A parallel abstract interpreter for JavaScript. 2015.

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Naive worklist parallel strategy
s
s
s
s
s
worklist
merge

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Naive worklist parallel strategy
Redundant computations.
Synchronization at the merge step.
Article test on few real JavaScript programs.
Results show that this adaptation of the sequential algorithm is not optimal.
Figure: L. Andersen, M. Might. Multi-core Parallelization of Abstracted. 2013.
Improvements with producer/consumer model.

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Better, per-context parallel strategy
Authors introduce a per-context parallel strategy.
The main idea is to separate these two parts:
state exploration
control of state space by some merging operations.
The intuitive idea is to parallelize “functions” instead basic “blocks”.

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
4 Done and todo
5 References

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Work done
Reading of theoretical background on lattice and AAM.
Redaction of an introduction to the subject and a lattice presentation
(preparatory work, last year).
Reading of some articles on parallelization.
Learning actors, Akka.
Learning use of some tools: sbt, Scala environments. . .
(discovery of GraalVM!)
Learning functioning of Scala-AM.
Removed some parts of Scala-AM (other machines or languages),
implementation of little features.

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
Work to be done
Implement the naive parallel algorithm, and experiment, really ASAP.
Implement better parallel algorithms.
Evaluate all of them,
identify advantages and disadvantages for each of them.
Reading of other articles on parallelization.
Have a great idea!
Final redaction.

View Slide

An Eﬃcient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
4 Done and todo
5 References

View Slide

An Efficient and
Parallel Abstract
Interpreter
in Scala
—
Second
Presentation
The context:
Abstract
Interpretation for
Static Analysis
The problematic:
Too heavy for
real programs
Parallel
algorithms
Done and todo
References
References
Thank you! Questions time. . .
L. Andersen, M. Might. Multi-core Parallelization of Abstracted
Abstract Machines. 2013.
Patrick Cousot. Abstract Interpretation in a Nutshell.
K. Dewey, V. Kashyap, B. Hardekopf. A parallel abstract
interpreter for JavaScript. 2015.
Matthew Might. Tutorial: Small-step CFA. 2011.
Quentin Sti´
evenart. Static Analysis of Concurrency Constructs
in Higher-Order Programs. 2014.
D. Van Horn, M. Might. Abstracting Abstract Machines. 2010.
Document, L
A
TEX sources, complete references and first presentation on
https:// Ø Ù ØºÓÖ »ÇÈ Å /efficient-parallel-abstract-interpreter-in-scala
Olivier Pirson. An Efficient and Parallel Abstract Interpreter in
Scala — Preparatory Work. 2017.

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An Efficient and Parallel Abstract Interpreter in Scala — Second Presentation

An Efficient and Parallel Abstract Interpreter in Scala — Second Presentation

🌳 Olivier Pirson — OPi 🇧🇪🇫🇷🇬🇧 🐧 👨‍💻 👨‍🔬

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