We describe an extension of the BeepBeep stream processing library for the offline verification of arbitrary expressions of Linear Temporal Logic using bitmap manipulations. Experimental results show that, for complex LTL formulæ containing up to 20 operators, event traces can be evaluated at a throughput of millions of events per second and provide a considerable speed-up compared to the current implementation of the tool.

1. K. Kie, S. Hallé
Kun Xie and Sylvain Hallé
Université du Québec à Chicoutimi
CANADA
Offline Monitoring of LTL
with Bit Vectors
CRSNG
NSERC
ACM Symposium on Applied Computing, March 24th, 2021

2. K. Kie, S. Hallé
What is runtime monitoring?

3. K. Kie, S. Hallé
What is runtime monitoring?
System

4. K. Kie, S. Hallé
What is runtime monitoring?
System

5. K. Kie, S. Hallé
What is runtime monitoring?
System
Instrumentation

6. K. Kie, S. Hallé
What is runtime monitoring?
System
Instrumentation

7. K. Kie, S. Hallé
What is runtime monitoring?
System
Instrumentation
Trace

8. K. Kie, S. Hallé
What is runtime monitoring?
System
Instrumentation
Trace
Events

9. K. Kie, S. Hallé
What is runtime monitoring?
System
Instrumentation
Trace
Events

10. K. Kie, S. Hallé
What is runtime monitoring?
System
Instrumentation
Trace
Events
Monitoring

11. K. Kie, S. Hallé
What is runtime monitoring?
Runtime monitoring is the process of observing an
actual run of a system and checking constraints on
its execution
Monitor
Verifies that
sequence of
events follows
specification
System
Event
Event
. . .
}
The execution of
the system
produces events
Specification
Gives conditions on
events and
sequences of events
allowed to happen

12. K. Kie, S. Hallé
p
p
now
p
p
now
q
G p
F p
X p
p U q
now
now
Semantics of LTL operators
"Globally"
"Eventually"
"Next"
"Until"

13. K. Kie, S. Hallé
Semantics of LTL operators
A call to next must be followed by a call
to hasNext
No CartCreate request can occur
before a LoginResponse message
A received order must eventually
be shipped
Three successive login attempts should
trigger an alarm
G (next → X hasNext)
¬ CartCreate U hasNext
G (receive → F ship)
G ¬(fail ∧ (X (fail ∧ X fail)))

14. K. Kie, S. Hallé
M
Classical LTL monitoring

15. K. Kie, S. Hallé
M
Classical LTL monitoring

16. K. Kie, S. Hallé
M
Classical LTL monitoring
A
e

17. K. Kie, S. Hallé
M
Classical LTL monitoring
A
e
S
s
X

18. K. Kie, S. Hallé
M
Classical LTL monitoring
A
e
S
s
X
s
’
s
’

19. K. Kie, S. Hallé
M
Classical LTL monitoring
A
e
S
s
X
s
’
s
’

20. K. Kie, S. Hallé
M
Classical LTL monitoring
A
e
S
s
X
s
’
s
’
2
1
3
4
5
The trace must be read linearly

21. K. Kie, S. Hallé
Each event is encoded by a set of Boolean
variables; each variable can be either true or false
in each event.
Event sequences as bit vectors
We create one bit vector vx for each Boolean
variable x, with the property that vx[i] = 1 if and
only if x is true in event i.
101001...
va =
001100...
vb =
.
.
.

22. K. Kie, S. Hallé
Bit vectors for LTL formulas are built by combining
and manipulating bit vectors of simpler formulas.
Bit vector manipulations
101001...
va =
001100...
vb =
va∨b = va ⊕ vb
⊕
=
101101...
va∨b = Boolean
operators
are direct!
Example:

23. K. Kie, S. Hallé
For a formula φ, vφ[i] = 1 if and only if the
sequence starting at position i satisfies φ.
Intuition
How to deal with temporal operators?

24. K. Kie, S. Hallé
Bitmap manipulations for LTL operators

25. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ

26. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END

27. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START

28. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START
END
0

29. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START
END
0
φ
F φ

30. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START
END
0
φ
F φ
1 0
1
START
END
START

31. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START
END
0
φ
F φ
1 0
1
START
END
START
0
END

32. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START
END
0
φ
F φ
1 0
1
START
END
START
0
END
φ
G φ
START
END
START
END

33. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START
END
0
φ
F φ
1 0
1
START
END
START
0
END
φ
G φ
START
END
START
END
0 1
0 1

34. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START
END
0
φ
F φ
1 0
1
START
END
START
0
END
φ
G φ
START
END
START
END
0 1
0 1
φ
φ U ψ
ψ

35. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START
END
0
φ
F φ
1 0
1
START
END
START
0
END
φ
G φ
START
END
START
END
0 1
0 1
φ
φ U ψ
ψ
1
1

36. K. Kie, S. Hallé
Bitmap manipulations for LTL operators
φ
X φ
START
END
START
END
0
φ
F φ
1 0
1
START
END
START
0
END
φ
G φ
START
END
START
END
0 1
0 1
φ
φ U ψ
ψ
1
1
1

37. K. Kie, S. Hallé
Experimental evaluation
Goal: compare processing speed and memory
consumption between LTL bitmap evaluation and a
baseline
Sample of 50+ LTL formulas
Between 2 and 20 operators
Nesting depth between 2 and 11
Evaluated on traces of 1M events
Reference:
Event stream processing library
Front-end accepting LTL formulas (Polyglot)
Hallé & Khoury, RV 2018
10.1007/978-3-030-03769-7_27

38. K. Kie, S. Hallé
LTL in BeepBeep
f
→
=?
a
1
2
f
∧
=?
b
1
2
¬
X
f
=?
c
1
U
f
=?
d
1
a → (¬ (b ∧ X c) U d)
S. Hallé. (2018). Event Stream Processing with BeepBeep 3, chapter 5.
ISBN 978-2-7605-5102-2

39. K. Kie, S. Hallé
1.6x106
1.8x106
2x106
2.2x106
2.4x106
2.6x106
2.8x106
3x106
3.2x106
3.4x106
3.6x106
0 5 10 15 20 25
Throughput
(Hz)
Formula size
0
500000
1x106
1.5x106
2x106
2.5x106
3x106
0 5 10 15 20 25
Throughput
(Hz)
Formula size
Raw bit vectors BeepBeep 3
Throughput comparison

40. K. Kie, S. Hallé
1.6x106
1.8x106
2x106
2.2x106
2.4x106
2.6x106
2.8x106
3x106
3.2x106
3.4x106
3.6x106
0 5 10 15 20 25
Throughput
(Hz)
Formula size
0
500000
1x106
1.5x106
2x106
2.5x106
3x106
0 5 10 15 20 25
Throughput
(Hz)
Formula size
Raw bit vectors BeepBeep 3
Throughput comparison

41. K. Kie, S. Hallé
0 5 10 15 20 25
Throughput
(Hz)
Formula size
0
500000
1x106
1.5x106
2x106
2.5x106
3x106
0 5 10 15 20 25
Throughput
(Hz)
Formula size
Raw bit vectors BeepBeep 3
500000
1x106
1.5x106
2x106
2.5x106
3x106
Throughput comparison

42. K. Kie, S. Hallé
Formula S01 Formula S02
Memory consumption
0
200000
400000
600000
800000
1x106
1.2x106
1.4x106
1.6x106
1.8x106
200000 300000 400000 500000 600000 700000 800000 900000 1x106
Memory
(B)
Trace length
EWAH64
Roaring
Concise
Raw
BeepBeep
EWAH32
WAH
0
2x106
4x106
6x106
8x106
1x107
1.2x107
1.4x107
1.6x107
1.8x107
2x107
2.2x107
200000 300000 400000 500000 600000 700000 800000 900000 1x106
Memory
(B)
Trace length
EWAH64
Roaring
Concise
Raw
BeepBeep
EWAH32
WAH

43. K. Kie, S. Hallé
Formula S01 Formula S02
Memory consumption
0
200000
400000
600000
800000
1x106
1.2x106
1.4x106
1.6x106
1.8x106
200000 300000 400000 500000 600000 700000 800000 900000 1x106
Memory
(B)
Trace length
EWAH64
Roaring
Concise
Raw
BeepBeep
EWAH32
WAH
0
2x106
4x106
6x106
8x106
1x107
1.2x107
1.4x107
1.6x107
1.8x107
2x107
2.2x107
200000 300000 400000 500000 600000 700000 800000 900000 1x106
Memory
(B)
Trace length
EWAH64
Roaring
Concise
Raw
BeepBeep
EWAH32
WAH
(¬(X (X ((G f) ∧ (X ((!a) ∨ d)))))) → ((X i) ∧ (X (i → e)))

44. K. Kie, S. Hallé
Formula S01 Formula S02
Memory consumption
0
200000
400000
600000
800000
1x106
1.2x106
1.4x106
1.6x106
1.8x106
200000 300000 400000 500000 600000 700000 800000 900000 1x106
Memory
(B)
Trace length
EWAH64
Roaring
Concise
Raw
BeepBeep
EWAH32
WAH
0
2x106
4x106
6x106
8x106
1x107
1.2x107
1.4x107
1.6x107
1.8x107
2x107
2.2x107
200000 300000 400000 500000 600000 700000 800000 900000 1x106
Memory
(B)
Trace length
EWAH64
Roaring
Concise
Raw
BeepBeep
EWAH32
WAH
(¬(X (X ((G f) ∧ (X ((!a) ∨ d)))))) → ((X i) ∧ (X (i → e)))
((X (g → b)) ∧ ((¬g) → ((¬(X (X (G (F (b ∧ c)))))) → f)))
∨ (F ((G (F (X g))) → ((¬(c ∨ g)) → (e ∨ (G (¬a))))))

45. K. Kie, S. Hallé
Upsides:
Take-home points
Bit vector manipulations can be done on
multiple bits at a time (32 or 64): parallelism
"for free"
Leverages quasi-random access to vector
elements
Implementation incurs a speed-up over linear
BeepBeep of up to 10x
Future work:
Only works offline
How to deal with more expressive logics (e.g.
LTL-FO+)?