1

Parallel composition, reduction

CS5270

2

Parallel Composition

• TTS = TTS1 || TTS2 || …… || TTSn

• Same principle as before:– Do common actions together– Take union of clock variables.– Take conjunction of the guards (state

invariants) !

3

An Example.

4

The Product Construction

• TTS1 = (S1, s01, Act1, X1, I1, →1)• TTS2 = (S2, s02, Act2, X2, I2, →2)• Assume X1 and X2 are disjoint (rename if

necessary).• TTS = TTS1 || TTS2 = (S, S0, Act, X, I, →)

– S = S1 S2

– (s01 , s02 )

– Act = Act1 Act2

– X = X1 X2

– I(s1, s2) = I1(s1) I2(s2)

5

The Product Construction

• TTSi = (Si, S0i, Acti, Xi, II, →i) i = 1, 2• TTS = TTS1 || TTS2 = (S, S0, Act, X, I, →)• → is the least subset of S Act (X) 2X S

satisfying:– Suppose (s1, a, 1, Y1, s1’) →1 and (s2, b, 2, Y2, s2’) →2. – Case1: a = b Act1 Act2

• Then ((s1, s2), a, 1 2, Y1 Y2, (s1’, s2’)) →.– Case2: a Act1 - Act2

• Then ((s1, s2), a, 1, Y1, (s1’, s2)) → .– Case3: b Act2 - Act1

• Then ((s1, s2), b, 2, Y2, (s1, s2’)) →.

6

The Gate-Train Example

7

What We Need to Do

• Problem: – We need to analyze the timed behavior of a TTS.– The timed behavior of TTS is given by TSTTS – But TSTTS is an infinite transition system!

• Solution:– Represent TSTTS as a finite transition system.– How?– By using the notion of regions, quotient TSTTS into a

finite transition system RTS.– Using regions we can compute RTS from TTS.– UPPAAL computes a refined version of RTS from

TTS.

8

The Reductions.

TSTTS

TATTS

RTS

Both the set of states and actions are infinite.

Time abstraction

Finite set of actions but infinite set of states.

Quotient via stable equivalence relation of finite index.

Both states and actions are finite sets.

TTSSemantics

Regions

9

The Reductions.

TSTTS

TATTS

RTS

Both the set of states and actions are infinite.

Finite set of actions but infinite set of states.

Both states and actions are finite sets.

RTS is computed directly from TTS (a finite object)

s is reachable in TTS iff the corresponding state is reachable in RTS.

TTSSemantics

Regions

10

The Reductions.

TSTTS

TATTS

RTS

Both the set of states and actions are infinite.

Finite set of actions but infinite set of states.

Both states and actions are finite sets.

TTSSemantics

Regions

11

Behaviors

• TTS = (S, sin, Act, X, I, )

• We associate a “normal” transition system with TTS while taking time into account:– TSTTS = (S, sin, Act R, )

– R, non-negative reals• S Act R S

• TSTTS is an infinite transition system!

12

The Reductions.

TSTTS

TATTS

RTS

Both the set of states and actions are infinite.

Finite set of actions but infinite set of states.

Both states and actions are finite sets.

TTSSemantics

Regions

13

Time Abstraction

• TTS = (S, S0, Act, X, I, ) s S

• TSTTS = (S, S0, Act R, ))

• TATTS = (S, S0, Act, ) where :

– (s, V) (s’, V’) iff there exists such that

– (s, V) (s, V+) in TS and

– (s, V+) (s’, V’) in TS.

a

a

14

Time Abstraction

• TTS = (S, S0, Act, X, I, ) s S

• TSTTS = (S, sin, Act R, ))

• TATTS = (S, sin, Act, )

• FACT: s is reachable in TTS (TS) iff s is reachable in TA.

• Infinite number of states but only a finite number of actions.

15

The Reductions.

TSTTS

TATTS

RTS

Both the set of states and actions are infinite.

Finite set of actions but infinite set of states.

Both states and actions are finite sets.

TTSSemantics

Regions

16

Quotienting

17

Quotienting

18

Quotienting

19

Quotienting

20

Quotienting …. ?

21

Bisimulation

• Finite index bisimulation relation – Used to quotient a big transition system into

small one.• big --- infinite• small ---- finite.

22

• TS = (S, sin, Act, )• S x S, an equivalence relation

– s s for every s in S (reflexive)– s s’ implies s’ s (symmetric)– s s’ and s’ s’’ implies s s’’ (transitive) – s t and s s’ implies there exists t’

such that t t’ and s’ t’.– s t and t t’ implies there exists s’

such that s s’ and s’ t’.

Bisimulation

a

a

a

a

23

Stable Equivalence Relation

s’

s t

a

s’

s t

a

t’

aImplies

24

Stable Equivalence Relation

s t

t’

a

s’

s t

a

t’

aImplies

25

Finite Index Bisimulation

• TS = (S, sin, Act, )

a bisimulation.

• s S

• [s]t – the equivalence class containing s.

– {s’ | s s’}

• is of finite index if {[s] | s S} is a finite set.

26

An Example

1 2 3 4 5 6a b a b a b

i j iff (i is odd and j is odd) OR (i is even and j is even).

is a bisimulation of finite index.

{1, 3, 5,….} = [5] {2, 4, 6, ..} = [8]

27

• TS = (S, sin, Act, )

• a bisimulation.

• QTS = (QS, qsin, Act, )

– The - quotient of TS.

– QS = { [s] | s S}

– qsin = [sin]

– [s] [s’] iff there exists s1 [s] and s1’ [s’] such that s1 s1’ in TS.

The Quotient Transition System

a

a

28

An Example

1 2 3 4 5 6a b a b a b

i j iff (i is odd and j is odd) OR (i is even and j is even).

is a stable equivalence relation of finite index.

{1, 3, 5,….} = [5] {2, 4, 6, ..} = [8]

[5] [8]

a

b

29

The Reductions.

TSTTS

TATTS

RTS

Both the set of states and actions are infinite.

Finite set of actions but infinite set of states.

Both states and actions are finite sets.

TTSSemantics

Regions

30

Equivalence based on Regions.

• TA = (S, S0, Act, )

• S x S , a bisimulation of finite index.

• (s, V) (s’, V’) iff– s = s’– V Reg V’

• V and V’ belong to the same clock region.