An on-the-fly tableau construction for a real-time temporal logic

<strong>An</strong> on-the-fly tableau construction for a
real-time temporal logic
Marc Geilen
Information and Communication Systems Group
Faculty of Electrical Engineering
Eindhoven University of Technology
P.O.Box 513, 5600 MB Eindhoven
The Netherlands
E-mail: M.C.W.Geilen@tue.nl
November 29, 2001

Overview
1. Introduction
2. Modelling Timed Systems
3. Real-Time Temporal Logic
4. Efficient Tableau Constructions
5. Conclusions
A 1

1. Introduction: Real-time Systems
• Real-time systems must adhere to
quantitative timing constraints
• Embedded systems, control systems,
etcetera
• Hard real-time, soft real-time
A 2

1. Introduction: LTL Model-checking
System
S
Büchi Automaton
A S
Product Automaton
A S £A ¡
• Automata-theoretic approach
LTL Property
'
Büchi Automaton
A ¡
S satisfies '
iff
L(A S )\L(A :' )=∅
iff
L(A S £A ' )=∅
• Tableau construction relates temporal logic and automata
A 3

1. Introduction: Related Work
• This approach has not yet been applied to timed linear temporal
logics
• TCTL model-checking
• ‘proof of concept’ tableau construction by Alur
No implementation, not practical
• Tools DT-Spin/UPPAAL/Kronos use LTL model-checking or
reachability analysis only
A 4

1. Introduction: Goals
• Goal: an efficient tableau construction for a (dense) time linear
temporal logic
• Enabling the automata theoretic approach to timed temporal logic
model-checking
• (Enabling the monitoring of timed temporal logic properties during
simulations)
A 5

2. Modelling Timed Systems
A 6

2. Labelled Transition Systems
F G F G
F G
F G
System can be viewed as LTS (or Kripke structure)
Set Prop of observable (boolean) propositions (p, q)
F G
Observable states are subsets of Prop.
A 7
F G
F G
F G

2. Exploring a Labelled Transition System
F G
F G F G
F G
The (linear) view on behaviour of the system is a path through the LTS.
A trace ¯σ ∈ (2 Prop ) ω is an infinite sequence of states
A 8
F G
F G
F G
F G

2. Timed Transition Systems
F
G
F G
F G
N #
F G
N F G
For quantitative timing aspects, the LTS is augmented with timers to
express constraints on the timing of a run.
A
We use interval automata to model such systems
9
F G
N #
F G
F G

3. Real-Time Temporal Logic
A 10

3. Linear Temporal Logic
To specify properties of reactive systems, one often uses LTL formulas
(in positive form, p ∈ Prop):
ϕ ::= true | false | p | ¬p | ϕ1 ∨ ϕ2 | ϕ1 ∧ ϕ2 | ○ ϕ | ϕ1Uϕ2 | ϕ1Vϕ2.
LTL formulas are interpreted on infinite traces in the standard way
A 11

3. Real-time Temporal Logic
To deal with quantitative timing requirements, temporal logics are
extended.
The untimed Until operator ϕ1Uϕ2 states that sometime in the future, ϕ2
must hold, but cannot express hard constraints on that moment.
The extended operator ϕ1UIϕ2 adds an interval I that denotes that the
occurrence of ϕ2 must be within I.
A 12

3. Model-checking of Timed Systems
Timed System
S
Timed Automaton
A S
Product Automaton
A S £A ¡
MITL Property
'
Timed Automaton
A ¡
S satisfies '
iff
L(A S )\L(A :' )=∅
iff
L(A S £A ' )=∅
Tableau procedure for real-time temporal logic is required.
Must be efficient to make model-checking feasible in practice.
⇒ On-the-fly construction
A 13

3. Real-time Temporal Logic
To specify properties, we use MITL≤ formulas (in positive form)
(p ∈ Prop):
ϕ ::= true | false | p | ¬p | ϕ1 ∨ ϕ2 | ϕ1 ∧ ϕ2 | ○ ϕ | ϕ1UIϕ2 | ϕ1VIϕ2.
Let’s keep it simple!
I is an interval of the form [0, d] with d some integer.
MITL≤ formulas are interpreted on timed state sequences.
¯ρ = (I0I1I2 . . . ,σ0σ1σ2 . . . )
A 14

4. Efficient Tableau Constructions
A 15

4. On-the-fly tableau construction
• Separating now and the the future
• Disjunctive form mapped to nondeterministic
choice
• Normal form
ϕ = (Π1 ∧ ○Φ1) ∨ (Π2 ∧ ○Φ2) ∨ . . .
• Rewriting procedure NF
E.g. pUq ≡ q ∨ (p ∧ ○pUq)
ϕ1 ∧ (ϕ2 ∨ ϕ3) ≡ (ϕ1 ∧ ϕ2) ∨ (ϕ1 ∧
ϕ3)
A 16

4. On-the-fly tableau construction
pUq
q
p
°pUq
true
• E.g.
pUq ≡ q(∧ ○ true) ∨ (p ∧ ○pUq)
• true ≡ true ∧ ○true
• incremental / on-the-fly
A 17

4. Real-time On-the-fly Tableaux
To obtain a similar normal form for timed temporal logic we extend the
logic with timers.
ψ ::= ϕ | TS.ψ | x > 0 | x ≥ 0 | x < 0 | x ≤ 0 |
○ψ | ϕ1U≤xϕ2 | ϕ1U≤x+ɛϕ2 | ϕ1V≤xϕ2 | ϕ1V

4. Real-time On-the-fly Tableaux
• Unfolding
ϕ1U≤xϕ2 ≡ ϕ2 ∨ (x > 0 ∧ ϕ1 ∧ ○ϕ1U≤xϕ2
ϕ1V≤yϕ2 ≡ y ≤ 0 ∨ (ϕ2 ∧ (ϕ1 ∨ ○ϕ1V≤yϕ2))
(but what does ○ mean in dense time?)
• We cannot create a new timer every time we must check a timed
subformula
ϕ1U≤dϕ2 ∧ ϕ1U≤xϕ2 ≡ ϕ1U≤xϕ2 (if x ≤ d)
A 19

4. Real-time On-the-fly Tableaux
x:=5
pU ≤5 q
q
p
x>0
pU≤5q ≡ q ∨ {x := 5}.(p ∧ x > 0∧
○pU≤xq)
A 20

4. Real-time On-the-fly Tableaux
x:=5
pU ≤5 q
pU ≤x q
q
p
x>0
pU≤5q ≡ q ∨ {x := 5}.(p ∧ x > 0∧
○pU≤xq)
pU≤xq ≡ q ∨ (p ∧ x > 0 ∧ ○pU≤xq)
A 21

4. Real-time On-the-fly Tableaux
TS 1
TS 2
TS 3
'
¦ 1
¥ 1
¦ 2
¥ 2
¦ k
¥ k
© 2
• Normal form
ϕ = (Π1 ∧ Ξ1 ∧ ○Φ1)∨
(Π2 ∧ Ξ2 ∧ ○Φ2) ∨ . . .
• Propositional part Πi
• Timer Conditions Ξi
• Future part Φi
A 22

4. Real-time On-the-fly Tableaux
Two similar timed state sequences . . .
[
0 2
4
[
:p
:p
:p
0 2
4
A 23
:p
](
)[
p
p

4. Real-time On-the-fly Tableaux
Looking at the formula ♦≤2p . . .
[
0 2
4
[
: ≤ p
:p
: ≤ p
:p
](
)[
≤ p
:p
0 2
4
A 24
≤ p
:p
](
)[
p
p

4. Real-time On-the-fly Tableaux
Checking the formula using timers . . .
[
0 2
4
[
: ≤£ p
:p
: ≤ ¡ p
:p
¤ :=¥
](
¢ :=£
)[
≤ ¡ ¢ p
≤£ p
:p
≤ p
≤ ¡ p
:p
0 2
4
♦≤2p s ⇒ {x := 2}.♦≤xp ♦≤2p o ⇒ {x := 2}.♦≤x+ɛp
A 25
](
)[
p
p

4. Real-time On-the-fly Tableaux
As the type of interval transition influences the constraints to be verified.
It is controlled by enforcing alternation of open and singular intervals.
x:=0 x:=0
x=0
t 0 ) {t 0 } (t 0 ,
t 0
open
singular
x=0
t 1 ) {t 1 } (t 1
A 26
t 1

4. Real-time On-the-fly Tableaux
⇒ Two sets of normal form rewrite rules!
Normal form procedure NFs
Normal form procedure NFo
¯ρ |=ν NFs(ψ) iff ¯ρ |=ν ψ
¯ρ t |=ν NFo(ψ) iff ¯ρ t |=ν ψ for all t in the first open interval
A 27

4. Real-time On-the-fly Tableaux
TS 1
TS 2
TS 3
singular
¢
¢
¢
£
£
£
1
1
2
2
k
k
TS a
TS b
§ ¨©
open
¢
¡
¢
£
£
¢¥¤
£¦¤
2
a
a
b
b
singular
A 28
¡
b

4. The Tableau Algorithm
L0 := {((s, Now, Next), TS(¯0)) | (TS, Now, Next) ∈ NFs({ϕ})}
LNew := {(s, Now, Next) | ((s, Now, Next), TS) ∈ L0 }
L := ∅, E := ∅
while LNew = ∅ do
Let (it, Now, Next) ∈ LNew
LNew := LNew\{(it, Now, Next)}
L := L ∪ {(it, Now, Next)}
for every (TS ′ , Now ′ , Next ′ ) ∈ NF it (Next) do
E := E ∪ {((it, Now, Next), TS ′ , (it, Now ′ , Next ′ ))}
if (it, Now ′ , Next ′ ) /∈ L then
LNew := LNew ∪ {(it, Now ′ , Next ′ )}
od
od
A 29

4. Example
Tableau automaton of the formula ≤100♦≤5p
©
¥
¦ § ¡¢¤¨ ¥ ¥ ¡¢¤£
0 §
¥ §
¥
¥ §
¥ §
¥
§ ¥
¥ ¦
§ ¥
¥ §
¥ §
singular
open
sing.+open
A 30

5. Other Issues and Conclusions
A 31

5. Constructing Monitors for Runtime
Verification
• Deterministic and incremental
• On-the-fly determinisation
• Localise updates (stutter-closed formulas)
• Instrumenting programs and models
• Specification language for properties and
atomic propositions
A 32

5. Optimising the tableaux
Results can still be optimised
• Joining locations for open and singular intervals
• Using techniques applied to LTL tableaux
– Rewriting formulas before construction
– Optimising automata after construction
– Identify equivalent locations
A 33

5. Conclusions and Future Work
• On-the-fly tableau construction for a linear real-time temporal logic
• Weakly monotone time
• Observers for simulations
• Implementation
A 34