Online Quantitative Timed Pattern Matching with Semiring-Valued - - PowerPoint PPT Presentation

• M. Waga (NII)

Online Quantitative Timed Pattern Matching with Semiring-Valued Weighted Automata

1

Masaki Waga1,2,3 


National Institute of Informatics1, SOKENDAI2, JSPS Research Fellow3 27 August 2019, FORMATS 2019

This work is partially supported by JST ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), by JSPS Grants-in-Aid No. 15KT0012 & 18J22498

• M. Waga (NII)

Online Quantitative Timed Pattern Matching with Semiring-Valued Weighted Automata

1

Masaki Waga1,2,3 


National Institute of Informatics1, SOKENDAI2, JSPS Research Fellow3 27 August 2019, FORMATS 2019

This work is partially supported by JST ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), by JSPS Grants-in-Aid No. 15KT0012 & 18J22498

Online Quantitative Timed Pattern Matching with Semiring-Valued Weighted Automata

Monitoring

• M. Waga (NII)

Why Monitoring?

Exhaustive formal method
 (e.g. model checking, reachability analysis)

• The system is correct/incorrect for any execution
• We need system model (white box)
• Scalability is a big issue

Monitoring

• The system is correct/incorrect for the given execution
• data-driven analysis
• We do not need system model (black box is OK)
• Usually scalable

2

• M. Waga (NII)

Input

• Finite-valued signal σ
• System log
• e.g.,
• Real-time spec.
• Spec. to be monitored
• e.g., The velocity should not keep high for > 1 sec.


Output

• All the subsignals σ([t,t′)) of the log satisfies the spec.
• e.g., σ([4.0,8.0)), σ([6.0,8.0)), σ([6.0,7.5)), …

3

(Qualitative) timed pattern matching

[Ulus+, FORMATS’14]

t v vhigh vlow 4.0 8.0

W

discretized!!

• M. Waga (NII)

Input

• Finite-valued signal σ
• System log
• e.g.,
• Real-time spec.
• Spec. to be monitored
• e.g., The velocity should not keep high for > 1 sec.


Output

• All the subsignals σ([t,t′)) of the log satisfies the spec.
• e.g., σ([4.0,8.0)), σ([6.0,8.0)), σ([6.0,7.5)), …

3

(Qualitative) timed pattern matching

[Ulus+, FORMATS’14]

t v vhigh vlow 4.0 8.0

W

discretized!!

• M. Waga (NII)

Input

• Finite-valued signal σ
• System log
• e.g.,
• Real-time spec.
• Spec. to be monitored
• e.g., The velocity should not keep high for > 1 sec.


Output

• All the subsignals σ([t,t′)) of the log satisfies the spec.
• e.g., σ([4.0,8.0)), σ([6.0,8.0)), σ([6.0,7.5)), …

3

(Qualitative) timed pattern matching

[Ulus+, FORMATS’14]

t v vhigh vlow 4.0 8.0

W

discretized!! 4.0 8.0

8.0 4.0

• M. Waga (NII)

Input

• Finite-valued signal σ
• System log
• e.g.,
• Real-time spec.
• Spec. to be monitored
• e.g., The velocity should not keep high for > 1 sec.


Output

• All the subsignals σ([t,t′)) of the log satisfies the spec.
• e.g., σ([4.0,8.0)), σ([6.0,8.0)), σ([6.0,7.5)), …

3

(Qualitative) timed pattern matching

[Ulus+, FORMATS’14]

t v vhigh vlow 4.0 8.0

W

discretized!! 4.0 8.0 6.0

8.0 4.0 6.0 8.0

• M. Waga (NII)

Input

• Finite-valued signal σ
• System log
• e.g.,
• Real-time spec.
• Spec. to be monitored
• e.g., The velocity should not keep high for > 1 sec.


Output

• All the subsignals σ([t,t′)) of the log satisfies the spec.
• e.g., σ([4.0,8.0)), σ([6.0,8.0)), σ([6.0,7.5)), …

3

(Qualitative) timed pattern matching

[Ulus+, FORMATS’14]

t v vhigh vlow 4.0 8.0

W

discretized!! 4.0 8.0 6.07.5

8.0 4.0 6.0 8.0 6.0 7.5

• M. Waga (NII)

Input

• Finite-valued signal σ
• System log
• e.g.,
• Real-time spec.
• Spec. to be monitored
• e.g., The velocity should not keep high for > 1 sec.


Output

• All the subsignals σ([t,t′)) of the log satisfies the spec.
• e.g., σ([4.0,8.0)), σ([6.0,8.0)), σ([6.0,7.5)), …

3

(Qualitative) timed pattern matching

[Ulus+, FORMATS’14]

t v vhigh vlow 4.0 8.0

W

discretized!!

We want to know how robustly the spec. is satisfied!!

4.0 8.0 6.07.5

8.0 4.0 6.0 8.0 6.0 7.5

• M. Waga (NII)

Input

• Real-valued piecewise-constant signal σ
• System log
• e.g., 

• Real-time spec. with signal constraints
• Spec. to be monitored
• e.g., The velocity should not keep > 80 for > 1 sec.

Output

• How robustly, each subsignals σ([t,t′)) of the log, satisfies the

spec.

• e.g., M(σ, W)(2.0,4.0) = -20, M(σ, W)(6.5,7.8) = 40, …

4

Quantitative timed pattern matching

[Bakhirkin+, FORMATS’17]

W

t v 120 60 45 4.0 8.010.0 6.2 100 satisfaction degree of for σ([2.0,4.0)) W

• M. Waga (NII)

Input

• Real-valued piecewise-constant signal σ
• System log
• e.g., 

• Real-time spec. with signal constraints
• Spec. to be monitored
• e.g., The velocity should not keep > 80 for > 1 sec.

Output

• How robustly, each subsignals σ([t,t′)) of the log, satisfies the

spec.

• e.g., M(σ, W)(2.0,4.0) = -20, M(σ, W)(6.5,7.8) = 40, …

4

Quantitative timed pattern matching

[Bakhirkin+, FORMATS’17]

W

t v 120 60 45 4.0 8.010.0 6.2 100

2 4 6 8 10 t 2 4 6 8 10

t

• 40
• 30
• 20
• 10

10 20 30 40

satisfaction degree of for σ([2.0,4.0)) W

• M. Waga (NII)

Input

• Real-valued piecewise-constant signal σ
• System log
• e.g., 

• Real-time spec. with signal constraints
• Spec. to be monitored
• e.g., The velocity should not keep > 80 for > 1 sec.

Output

• How robustly, each subsignals σ([t,t′)) of the log, satisfies the

spec.

• e.g., M(σ, W)(2.0,4.0) = -20, M(σ, W)(6.5,7.8) = 40, …

4

Quantitative timed pattern matching

[Bakhirkin+, FORMATS’17]

W

t v 120 60 45 4.0 8.010.0 6.2 100

2 4 6 8 10 t 2 4 6 8 10

t

• 40
• 30
• 20
• 10

10 20 30 40

satisfaction degree of for σ([2.0,4.0)) W

・

2.0 4.0

2.0 4.0

• M. Waga (NII)

Input

• Real-valued piecewise-constant signal σ
• System log
• e.g., 

• Real-time spec. with signal constraints
• Spec. to be monitored
• e.g., The velocity should not keep > 80 for > 1 sec.

Output

• How robustly, each subsignals σ([t,t′)) of the log, satisfies the

spec.

• e.g., M(σ, W)(2.0,4.0) = -20, M(σ, W)(6.5,7.8) = 40, …

4

Quantitative timed pattern matching

[Bakhirkin+, FORMATS’17]

W

t v 120 60 45 4.0 8.010.0 6.2 100

2 4 6 8 10 t 2 4 6 8 10

t

• 40
• 30
• 20
• 10

10 20 30 40

satisfaction degree of for σ([2.0,4.0)) W

・

2.0 4.0 6.5 7.8

2.0 4.0 6.5 7.8

• M. Waga (NII)

Online Pattern Matching

• After reading the prefix signal σ' of σ = σ'・σ'', we
• btain the partial result M(σ',W) of M(σ,W)
• Important in practice

5

• M. Waga (NII)

Timed symbolic weighted automata (TSWA)

• New formalism for spec.
• Automata structure is good for online monitoring
• Generality of semiring (same as the usual WFA)

6

Boolean sup-inf tropical

S

{True/False} R ∪ {±∞} R ∪ {+∞}

⊕ ∨ sup inf ⊗ ∧ inf +

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

r

• u, (a1a2 . . . am)
• =

inf

r n ^

(xi . /i di), (a)

• =

inf

/i di, (a)) where . /i2 {>, , , <} r(x d, (a)) = a(x) d where 2 {, >} r(x d, (a)) = d a(x) where 2 {, <}

• M. Waga (NII)

Contribution

• Introduced timed symbolic weighted automata (TSWA)
• TSWA: timed automata with signal constraints (TSA)



 + semiring-valued weight function


• Gave online algorithm for quantitative timed pattern

matching

• Implementation + experiments → Scalable!!

7

Automata structure Quantitative semantics

• M. Waga (NII)

Related Works

8

Qualitative Quantitative Offline

[Ulus+, FORMATS’14]


(TRE)

[Bakhirkin+, FORMATS’17]


(Signal RE)

Online

[Ulus+, TACAS’16], [Bakhirkin+, FORMATS’18]


(TRE & TA)

[Contribution]


(TSWA)

Timed automata Timed automata
 with 
 signal constraints Only “Robust” Semantics

[Fainekos & Pappas, TCS’09]

Any Semantics
 defined by semiring- valued weight function

• M. Waga (NII)

Outline

• Motivation + Introduction
• Technical Part
• Timed symbolic weighted automata (TSWA)
• TSWA: TA with signal constraints + weight function
• Quantitative monitoring/timed pattern matching algorithm
• Idea: zone construction with weight
• Experiments

9

• M. Waga (NII)

TSWA: TA with signal constraints + weight function

Timed Automaton (TA)

10

l0 start l1 l2 c < 5 /c := 0 c < 10

• M. Waga (NII)

TSWA: TA with signal constraints + weight function

Timed Symbolic Automaton (TSA)

11

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

• M. Waga (NII)

TSWA: TA with signal constraints + weight function

Timed Symbolic Weighted Automaton (TSWA)

12

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

+

r

• u, (a1a2 . . . am)
• =

inf

i∈{1,2,...,n} r(u, (ai))

r n ^

i=1

(xi . /i di), (a)

• =

inf

i∈{1,2,...,n} r(xi .

/i di, (a)) where . /i2 {>, , , <} r(x d, (a)) = a(x) d where 2 {, >} r(x d, (a)) = d a(x) where 2 {, <}

• M. Waga (NII)

TSWA: TA with signal constraints + weight function

Timed Symbolic Weighted Automaton (TSWA)

12

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

+

r

• u, (a1a2 . . . am)
• =

inf

i∈{1,2,...,n} r(u, (ai))

r n ^

i=1

(xi . /i di), (a)

• =

inf

i∈{1,2,...,n} r(xi .

/i di, (a)) where . /i2 {>, , , <} r(x d, (a)) = a(x) d where 2 {, >} r(x d, (a)) = d a(x) where 2 {, <}

Automata structure Quantitative semantics

• M. Waga (NII)

Weight function 
 κ: Φ(X, D) × (DX)⊛ → S

• κ(Λ(l),u): weight for the stay at l with signal values u
• Semiring: set S with accumulating operators ⊕ and ⊗
• We can use any complete and idempotent semiring

13

Constraints on signal values at the location Sequence of signal values at the location Semiring value

• M. Waga (NII)

continuous transitions

Semantics: Weighted TTS

14

→(l, ν, t, u) (l, ν + 3, t + 3, u ○ u') (l', ν', t +3, ε) 3.0 e

κ(Λ(l), u ○ u')

4.0 (l, ν + 4, t + 4, u ○ u") (l', ν", t + 4, ε) e

κ(Λ(l), u ○ u")

⋮

• l: location
• ν: clock valuation
• t: absolute time
• u: sequence of signal values after the latest discrete transition

discrete transitions

• M. Waga (NII)

One path in Weighted TTS

15

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

t v 12.0 7.0 7.0 3.5

→(l0, c=0, 0,ε)

here

• M. Waga (NII)

One path in Weighted TTS

15

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

t v 12.0 7.0 7.0 3.5

→(l0, c=0, 0,ε) (l0, c=2, 2, {v = 7}) 2.0

2.0 here

• M. Waga (NII)

One path in Weighted TTS

15

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

t v 12.0 7.0 7.0 3.5

→(l0, c=0, 0,ε) (l0, c=2, 2, {v = 7}) (l1, c=0, 2, ε) 2.0 e

κ(v < 15, {v=7})

2.0 here

• M. Waga (NII)

One path in Weighted TTS

15

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

t v 12.0 7.0 7.0 3.5

→(l0, c=0, 0,ε) (l0, c=2, 2, {v = 7}) (l1, c=0, 2, ε) (l1, c=5, 7, {v = 7}{v = 12}) 2.0 e

κ(v < 15, {v=7})

5.0

2.0 7.0 here

• M. Waga (NII)

One path in Weighted TTS

15

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

t v 12.0 7.0 7.0 3.5

→(l0, c=0, 0,ε) (l0, c=2, 2, {v = 7}) (l1, c=0, 2, ε) (l1, c=5, 7, {v = 7}{v = 12}) (l2, c=5, 7, ε) 2.0 e

κ(v < 15, {v=7})

5.0 e

κ(v > 5, {v=7}{v=12})

2.0 7.0 here

• M. Waga (NII)

One path in Weighted TTS

15

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

t v 12.0 7.0 7.0 3.5

→(l0, c=0, 0,ε) (l0, c=2, 2, {v = 7}) (l1, c=0, 2, ε) (l1, c=5, 7, {v = 7}{v = 12}) (l2, c=5, 7, ε) 2.0 e

κ(v < 15, {v=7})

5.0 e

κ(v > 5, {v=7}{v=12})

2.0 7.0 here

⊗

Boolean sup-inf tropical

S

{True/False} R ∪ {±∞} R ∪ {+∞}

⊕ ∨ sup inf ⊗ ∧ inf +

• M. Waga (NII)

Accumulating paths in Weighted TTS

16

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

t v 12.0 7.0 7.0 3.5

κ(v < 15, {v=7}) ⊗ κ(v > 5, {v=7}{v=12})

→(l0, c=0, 0,ε) (l0, c=2, 2, {v = 7}) (l1, c=0, 3, ε) (l1, c=5, 7, {v = 7}{v = 12}) (l2, c=5, 7, ε) 2.0 e 5.0 e

7.0

Boolean sup-inf tropical

S

{True/False} R ∪ {±∞} R ∪ {+∞}

⊕ ∨ sup inf ⊗ ∧ inf +

κ(v < 15, {v=7}{v=12}) ⊗ κ(v > 5,{v=12})

→(l0, c=0, 0,ε) (l0, c=4, 4, {v = 7}{v = 12}) (l1, c=0, 4, ε) (l1, c=3, 7, {v = 12}) (l2, c=3, 7, ε) 4.0 e 3.0 e

2.0

⋮

• M. Waga (NII)

Accumulating paths in Weighted TTS

16

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

t v 12.0 7.0 7.0 3.5

κ(v < 15, {v=7}) ⊗ κ(v > 5, {v=7}{v=12})

→(l0, c=0, 0,ε) (l0, c=2, 2, {v = 7}) (l1, c=0, 3, ε) (l1, c=5, 7, {v = 7}{v = 12}) (l2, c=5, 7, ε) 2.0 e 5.0 e

7.0

Boolean sup-inf tropical

S

{True/False} R ∪ {±∞} R ∪ {+∞}

⊕ ∨ sup inf ⊗ ∧ inf +

κ(v < 15, {v=7}{v=12}) ⊗ κ(v > 5,{v=12})

→(l0, c=0, 0,ε) (l0, c=4, 4, {v = 7}{v = 12}) (l1, c=0, 4, ε) (l1, c=3, 7, {v = 12}) (l2, c=3, 7, ε) 4.0 e 3.0 e

⊕ ⊕

2.0

⋮

• M. Waga (NII)
• TSA: the automata structure


• Weight function (κ): the one-step semantics


(weight on each transition)

• Semiring operations (⊗,⊕): how to accumulate

weights
 One-step semantics → semantics for a path/TSWA

Timed symbolic weighted automata (TSWA)

17

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

• M. Waga (NII)

Outline

• Motivation + Introduction
• Technical Part
• Timed symbolic weighted automata (TSWA)
• TSWA: TA with signal constraints + weight function
• Quantitative monitoring/timed pattern matching algorithm
• Idea: zone construction with weight
• Experiments

18

• M. Waga (NII)

Review: Reachability by zones

19

continuous transitions

→(l, ν) (l, ν + 0.1) (l', ν') 0.1 e 0.2 (l, ν + 0.2) (l', ν") e

⋮

discrete transitions

0.21 (l, ν + 0.21) (l'', ν'") e'

Infinitely many reachable states!! → symbolic analysis by zones Infinitely many delays!!

• M. Waga (NII)

Review: Reachability by zones

• Zone Z symbolically represents infinitely many

clock valuations!!

20

continuous transitions

→(l, Z) (l, {ν + T | ν ∈ Z, T ∈ R+}) (l', Z') T e

discrete transitions Infinitely many reachable states!! → symbolic analysis by zones

(l'', Z'') e'

• M. Waga (NII)

Observation: reachability is shortest distance over Boolean semiring!

21

• M. Waga (NII)

Observation: reachability is shortest distance over Boolean semiring!

22

→(l, Z) (l, {ν + T | ν ∈ Z, T ∈ R+}) (l', Z') T e (l'', Z'') e'

• M. Waga (NII)

Observation: reachability is shortest distance over Boolean semiring!

22

→(l, Z) (l, {ν + T | ν ∈ Z, T ∈ R+}) (l', Z') T e (l'', Z'') e' Reachable at all the transitions (∧)

• M. Waga (NII)

Observation: reachability is shortest distance over Boolean semiring!

22

→(l, Z) (l, {ν + T | ν ∈ Z, T ∈ R+}) (l', Z') T e (l'', Z'') e' Reachable at all the transitions (∧) Reachable for one path (∨)

• M. Waga (NII)

Observation: reachability is shortest distance over Boolean semiring!

22

→(l, Z) (l, {ν + T | ν ∈ Z, T ∈ R+}) (l', Z') T e (l'', Z'') e' Reachable at all the transitions (∧) Reachable for one path (∨)

⋁

p∈Paths(⋀ i

(pi, pi+1) ∈ E)

Reachability

• M. Waga (NII)

Observation: reachability is shortest distance over Boolean semiring!

22

→(l, Z) (l, {ν + T | ν ∈ Z, T ∈ R+}) (l', Z') T e (l'', Z'') e' Reachable at all the transitions (∧) Reachable for one path (∨)

⋁

p∈Paths(⋀ i

(pi, pi+1) ∈ E)

⨁

p∈Paths (⨂ i

w(pi, pi+1))

Shortest Distance (for semiring)

Boolean sup-inf tropical

S

{True/False} R ∪ {±∞} R ∪ {+∞}

⊕ ∨ sup inf ⊗ ∧ inf +

Reachability

• M. Waga (NII)

Observation: reachability is shortest distance over Boolean semiring!

23

Qualitative [Bakhirkin+, FORMATS’18] Quantitative [Contribution]

TA zone graph
 (with Boolean weight) reachability
 (qualitative 
 semantics)

zone constr. reachability checking
 (Boolean shortest distance)

TSWA zone graph
 with semiring weight quantitative
 semantics

zone constr. semiring 
 shortest distance

• M. Waga (NII)

Zone construction with weight

24

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε)

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

24

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

24

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε)

κ(v < 15, {v=7})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

24

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

25

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε)

(l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

3.5

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

25

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε)

(l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

3.5

(l0, c = T = 3.5, {v = 7})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

25

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε)

(l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

3.5

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

25

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε)

(l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

3.5

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

26

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

26

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

26

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), {v = 12})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

26

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), {v = 12}) (l1, 0 < c < T ∈ (3.5,7), 
 {v = 7}{v = 12})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

26

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), {v = 12}) (l1, 0 < c < T ∈ (3.5,7), 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), ε)

κ(v < 15, {v=7}{v=12})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

26

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), {v = 12}) (l1, 0 < c < T ∈ (3.5,7), 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), ε)

κ(v < 15, {v=7}{v=12})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

27

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), {v = 12}) (l1, 0 < c < T ∈ (3.5,7), 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), ε)

κ(v < 15, {v=7}{v=12})

7.0

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

27

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), {v = 12}) (l1, 0 < c < T ∈ (3.5,7), 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), ε)

κ(v < 15, {v=7}{v=12})

7.0

(l1, 0 < c < T =7, 
 {v = 12})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

27

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), {v = 12}) (l1, 0 < c < T ∈ (3.5,7), 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), ε)

κ(v < 15, {v=7}{v=12})

7.0

(l1, 0 < c < T =7, 
 {v = 12}) (l1, 0 < c < T = 7, 
 {v = 7}{v = 12})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

27

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), {v = 12}) (l1, 0 < c < T ∈ (3.5,7), 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), ε)

κ(v < 15, {v=7}{v=12})

7.0

(l1, 0 < c < T =7, 
 {v = 12}) (l1, 0 < c < T = 7, 
 {v = 7}{v = 12}) (l2, 0 < c < T = 7, ε)

κ(v > 5, {v=12})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Zone construction with weight

27

t v 12.0 7.0 7.0 3.5

→(l0,c = T = 0, ε) (l0, 0 < c = T < 3.5, 
 {v = 7}) (l1, 0 = c < T < 3.5, ε) (l1, 0 < c < T < 3.5, {v = 7})

κ(v < 15, {v=7})

(l0, c = T = 3.5, {v = 7}) (l1, 0 < c < T = 3.5, {v = 7}) (l1, c = 0 < T = 3.5, ε)

κ(v < 15,{v=7})

(l0, 3.5 < c = T = 7, 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), {v = 12}) (l1, 0 < c < T ∈ (3.5,7), 
 {v = 7}{v = 12}) (l1, 0 < c < T ∈ (3.5, 7), ε)

κ(v < 15, {v=7}{v=12})

7.0

(l1, 0 < c < T =7, 
 {v = 12}) (l1, 0 < c < T = 7, 
 {v = 7}{v = 12}) (l2, 0 < c < T = 7, ε)

κ(v > 5, {v=12}) κ(v > 5, {v=7}{v=12})

• T: absolute time
• Accepted ⇔ transit to acc. loc. at T = |σ| (= 7.0)

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

This is OK for monitoring

• M. Waga (NII)

Main Theorem: Correctness

28

Thm. The shortest distance in the zone graph with weight is same as the shortest distance in the weighted TTS for any complete and idempotent semiring.

Boolean sup-inf tropical

S

{True/False} R ∪ {±∞} R ∪ {+∞}

⊕ ∨ sup inf ⊗ ∧ inf +

All of them work!!

• M. Waga (NII)

Local Conclusion: Zone Construction with Weight

• The construction is basically same as the usual zone

construction

• Weights are same as weighted TTS
• The state space is finite thanks to zones and finite

horizon of the input signal

29

• M. Waga (NII)

Matching Automata for Pattern Matching

30

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

linit, > start l0, v < 15 l1, v > 5 l2, > >/T 0 := 0, c := 0 c < 5/c := 0 c < 10

• Add linit to wait for the beginning of the

matching

• Add clock variable T’ for the beginning
• f the matching

[Bakhirkin+, FORMATS’18]

• M. Waga (NII)

Outline

• Motivation + Introduction
• Technical Part
• Timed symbolic weighted automata (TSWA)
• TSWA: TA with signal constraints + weight function
• Quantitative monitoring/timed pattern matching algorithm
• Idea: zone construction with weight
• Experiments

31

• M. Waga (NII)
• Semirings: sup-inf (R ∪ {± ∞}, sup, inf) and tropical (R ∪ {+ ∞}, inf, +)
• Used 3 original benchmarks (automotive):
• Inspired by ST-Lib [Kapinski+, SAE Technical Paper’16]
• Overshoot: |vref - v| gets large after vref changed
• Only matches the sub-signals of length < 150 time units
• Ringing: v(t) - v(t-10) gets positive and negative repeatedly
• Only matches the sub-signals of length < 80 time units
• Overshoot (unbounded): |vref - v| gets large after vref changed
• No such bounded
• Amazon EC2 c4.large instance / Ubuntu 18.04 LTS (64 bit)
• 2.9 GHz Intel Xeon E5-2666 v3, 2 vCPUs, 3.75 GiB RAM

Environment of Experiments

32

• M. Waga (NII)

Execution Time

33

• Execution time is linear for the bounded spec.
• 1,000 entries / 1 or 2 sec.
• Execution time explodes for the unbounded spec.

20 40 60 80 100 120 140 1 2 3 4 5 6 Execution time [s] Number of entries of the signal [×10, 000] Overshoot, sup-inf Ringing sup-inf Overshoot, tropical Ringing tropical 20 40 60 80 100 120 140 1 2 3 4 5 6 7 8 9 10 Execution time [s] Number of entries of the signal [×100]

Overshoot (Unbounded), sup-inf Overshoot (Unbounded), tropical

Bounded Unbounded

• M. Waga (NII)

Conclusion

• Introduced timed symbolic weighted automata (TSWA)
• TSWA: TA with signal constraints + weight function
• Gave quantitative monitoring/timed pattern matching

algorithm

• Idea: zone construction with weight
• Implementation + experiments
• scalable for bounded specifications

34

• M. Waga (NII)

Appendix

35

• M. Waga (NII)

Example: “Robust” Semantics

36

r

• u, (a1a2 . . . am)
• =

inf

i∈{1,2,...,n} r(u, (ai))

r n ^

i=1

(xi . /i di), (a)

• =

inf

i∈{1,2,...,n} r(xi .

/i di, (a)) where . /i2 {>, , , <} r(x d, (a)) = a(x) d where 2 {, >} r(x d, (a)) = d a(x) where 2 {, <}

Weight Function: minimum distance from the threshold Semiring: sup-inf semiring

Robustness

• M. Waga (NII)

Example: “Robust” Semantics

36

r

• u, (a1a2 . . . am)
• =

inf

i∈{1,2,...,n} r(u, (ai))

r n ^

i=1

(xi . /i di), (a)

• =

inf

i∈{1,2,...,n} r(xi .

/i di, (a)) where . /i2 {>, , , <} r(x d, (a)) = a(x) d where 2 {, >} r(x d, (a)) = d a(x) where 2 {, <}

Weight Function: minimum distance from the threshold Semiring: sup-inf semiring

Boolean sup-inf tropical

S

{True/False} R ∪ {±∞} R ∪ {+∞}

⊕ ∨ sup inf ⊗ ∧ inf +

Robustness

• M. Waga (NII)

Insights: Zone Construction with Weight

• The construction is basically same as the usual zone

construction

• The state space is finite thanks to zones and finite

horizon of the input signal

• The weight is constant because the signal is

piecewise-constant

37

• M. Waga (NII)

Comparison of the semiring

38

t v 13.0 7.0

l0, v < 15 start l1, v > 5 l2, > c < 5 /c := 0 c < 10

Boolean sup-inf tropical

S

{True/False} R ∪ {±∞} R ∪ {+∞}

⊕ ∨ sup inf ⊗ ∧ inf +

κ(v < 15, {v=13}) = 2

→(l0, c=0, 0,ε) (l0, c=2, 2, {v = 7}) (l1, c=0, 3, ε) (l1, c=5, 7, {v = 7}{v = 12}) (l2, c=5, 7, ε)

κ(v > 5, {v=13}) = 8

⊗

Sup-inf semiring Tropical semiring

2 ⊗ 8 = inf (2, 8) = 2 2 ⊗ 8 = 2 + 8 = 10