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Process Algebra (2IF45) Abstraction in Process Algebra

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Process Algebra (2IF45)Abstraction in Process Algebra

Suzana Andova

- Our way of dealing with internal behaviour: branching bisimulation
- How we capture Abstraction in Process Algebra
- combining it with other concepts

Process Algebra (2IF45)

Abstraction is used to

- check the correctness of implementation against the system specification
- reduce and simplify the model to enable better, fasted and cleaner model analysis

Question: How do we chose to relate behaviours with internal steps?

Branching bisimulation

Process Algebra (2IF45)

a

b

a

is branching bisim to

b

a

“ related states must have the same potential

which does not change until an observable action is executed ”

Process Algebra (2IF45)

is branching bisim to

a

b

a

b

it is not branching bisim to

b

a

Process Algebra (2IF45)

s

s

s

t

t

t

t

a

t’

s’

s’

s’

t’’

a

t’’

- Branching Bisimulation relation: A binary relation R on the set of state S of an LTS is branching bisimulation relation iff the following transfer conditions hold:
- for all states s, t, s’ S, whenever (s, t) R and s → s’ for some a A, then there are
- states t’, t’’ S such that t t’ and t’ → t’’ and (s, t’), (s’,t’’) R;
- 2. vice versa, for all states s, t, s’ S, whenever (s, t) R and t→ t’ for some a A, then there are states s’,s’’ S such thats s’ and s’ → s’’ and (s’, t), (s’’,t’) R;
- 3. if (s, t) R and s then there is a state t’ such that t t’ , t’ and (s, t’) R
- 4. whenever (s, t) R and t then there is a state s’ such that s s’ , s’ and (s’, t) R
- Two LTSs s and t are branching bisimilar, s b t, iff there is a branching bisimulation relation R such that (s, t) R

a

a

a

a

Spectrum of behavioural relations

more

power of the observer

less

most powerful

a

b

c

d1

d2

d3

d4

b

a

b

c

d1

d2

d3

d4

b

a

b

c

d1

d2

d3

d4

Process Algebra (2IF45)

branching bisimilar!

b

a

a

+

+

b

b

branching bisimilar? NO!

a

a

branching bisimilar!

b

a

a

+

+

b

b

branching bisimilar? NO!

a

a

Painful conclusion: branching bisimilation is not compositional.

branching bisimilar

components!

Not branching bisimilar

compositions!

+

+

b

a

b

a

a

a

What to do? Two choices:

Make the relation weaker and relate the two compositions too!

Make the relation stronger and do not relate the two components

from the beginning!

Rooted branching bisimulationis strengthened variant of

branching bisimulation strict enough to obtain compositionality

s

t

s

s

t

t

a

a

a

a

a

a

s’

s’

s’

t’

t’

t

t’

b

q

p

p

p

q

b

(aA i.e. can be from A or can be )

r

- R is Rooted BB between state (s, t) R if R is Branching Bisimulation relation (as already defined) and the rootcondition:
- if s → s’ for a A, then there is a state t’S such that t → t’ and (s’, t’) R;
- if t → t’ for a A, then there is a state s’S such that s → s’ and (s’, t’) R;
- s if and only if t
- LTSss and t arerooted branching bisimilar, s rb t, iff there is a rooted branching bisimulation relation R such that (s, t) R

a

a

a

a

Language: BPA(A)

Signature: 0, 1, (a._ )aA, , +, •

Language terms T(BPA(A,))

Closed terms C(BPA(A))

Deduction rules for BPA(A) (a A):

x+ y = y+x

(x+y) + z = x+ (y + z)

x + x = x

x+ 0 = x

(x+ y) z = xz+yz

(xy) z = x(y z)

0 x = 0

x 1 = x

1 x = x

a.x y = a.(x y)

a

a

a.x x

x x’

x + y x’

a

a

x

(x + y)

y y’

x + y y’

a

1

a

a

x x’

x y x’

x y y’

x y y’

a

a

x y

(x y)

y

(x + y)

⑥

Soundness

Strong Bisimilarityon LTSs

Equality of terms

Completeness

Process Algebra (2IF45)

Language: BPA(A)

Signature: 0, 1, (a._ )aA, , +, •

Language terms T(BPA(A,))

Closed terms C(BPA(A))

Deduction rules for BPA(A) (a A):

x+ y = y+x

(x+y) + z = x+ (y + z)

x + x = x

x+ 0 = x

(x+ y) z = xz+yz

(xy) z = x(y z)

0 x = 0

x 1 = x

1 x = x

a.x y = a.(x y)

a

a

a.x x

x x’

x + y x’

a

a

x

(x + y)

y y’

x + y y’

a

1

a

a

x x’

x y x’ y

x y y’

x y y’

a

a

x y

(x y)

y

(x + y)

⑥

Soundness

Strong Bisimilarityon LTSs

Equality of terms

Rooted Branching

Completeness

Process Algebra (2IF45)

Axiomazing Rooted branching bisimulation

bb

y

x

x

+

+

y

x

+

Turned into equation looks like:

.(x+y) + x = x+y

Axiomazing Rooted branching bisimulation

…

rb

…

a

a

bb

y

x

x

+

+

y

x

+

Turned into equation looks like:

B axiom a.(.(x+y) + x) = a.(x+y)

Language: BPA(A)

Signature: 0, 1, (a._ )aA, , +, •

Language terms T(BPA(A,))

Closed terms C(BPA(A))

x+ y = y+x

(x+y) + z = x+ (y + z)

x + x = x

x+ 0 = x

(x+ y) z = xz+yz

(xy) z = x(y z)

0 x = 0

x 1 = x

1 x = x

a.x y = a.(x y)

a.(.(x+y) + x) = a.(x+y)

Deduction rules for BPA(A) (a A):

a

a

a.x x

x x’

x + y x’

a

a

x

(x + y)

y y’

x + y y’

a

1

a

a

x x’

x y x’ y

x y y’

x y y’

a

a

x y

(x y)

y

(x + y)

⑥

Soundness

Strong Bisimilarityon LTSs

Equality of terms

Rooted Branching

Completeness

Process Algebra (2IF45)

- Prove soundness of B axiom wrt rooted BB
- Read the proof of ground completeness

Process Algebra (2IF45)

Language: BPA(A)

Signature: 0, 1, (a._ )aA, ,+, •

Language terms T(BPA(A,))

Closed terms C(BPA(A))

Axioms

Deduction rules

Process Algebra (2IF45)

Language: BPA(A)

Signature: 0, 1, (a._ )aA, ,+, •, I(I A)

Language terms T(BPA(A,))

Closed terms C(BPA(A))

turns external actions into internal steps

Axioms for I

Deduction rules for I

Process Algebra (2IF45)

Languagewith

Signature: 0, 1, (a._ )aA, ,+, H(H A)

blocks actions

Process Algebra (2IF45)

Language: TCP(A)

Signature: 0, 1, (a._ )aA, ,+, •, I(I A), ||, |, ╙, H,

Language terms T(BPA(A, ))

Closed terms C(BPA(A, ))

Axioms for parallel composition with silent step:

x ╙.y = x ╙ y

x |.y = 0

Process Algebra (2IF45)

- see distributed copies

Process Algebra (2IF45)

Guardedness and silent steps: cannot be a guard of a variable

X = .X has solutions ..a.1 but also ..b.1

Guardedness and hiding operator: I cannot appear in tX in X = tX

X = i.I(X), where i I has solutions i.i.a.1 but also i.i.b.1

Process Algebra (2IF45)

- Observation:
- they are rooted bb bisimilar
- implicitly internal loop is left eventually
- = fairness

Z

X

U

Y

a

a

0

0

Process Algebra (2IF45)

- Observation on LTSs:
- they are rooted bb bisimilar
- implicitly internal loop is left eventually
- = fairness

Z

X

U

Y

a

a

0

0

As recursive specifications:

X = .Y

Y = .Y + a.0

Z = .U

U = a.0

RSP+RDP?

X = Z

Process Algebra (2IF45)

- Observation on LTSs:
- they are rooted bb bisimilar
- implicitly internal loop is left eventually
- = fairness

Z

X

U

Y

a

a

0

0

As recursive specifications:

X = .Y

Y = .Y + a.0

Z = .U

U = a.0

RSP+RDP?

X = Z

At least two problems:

Those are not guarder recursive specifications!

Even if they are somehow made guarded, B axiom is not sufficient

to rewrite one spec into another

Process Algebra (2IF45)

Abstraction and Recursion and Fairness:

problem 1. dealing with guardedness

for some action i

to be turned internal “soon”

by applying I for I = {i}

X’ = i.Y’

Y’ = i.Y’ + a.0

represents

X = .Y

Y = .Y + a.0

applying {i}

X

X’

i

i

Y

Y’

a

a

0

0

Process Algebra (2IF45)

Abstraction and Recursion and Fairness:

problem 1. dealing with guardedness

for some action i

to be turned internal “soon”

by applying I for I = {i}

Z’ = i.U’

U’ = a.0

X’ = i.Y’

Y’ = i.Y’ + a.0

represents

represents

Z = .U

U = a.0

X = .Y

Y = .Y + a.0

applying {i}

Z

Z’

i

applying {i}

U’

U

X

X’

i

a

a

i

0

0

Y

Y’

a

a

0

0

Process Algebra (2IF45)

Abstraction and Recursion and Fairness:

problem 1. dealing with guardedness

for some action i

to be turned internal “soon”

by applying I for I = {i}

Z’ = i.U’

U’ = a.0

X’ = i.Y’

Y’ = i.Y’ + a.0

represents

represents

Z = .U

U = a.0

X = .Y

Y = .Y + a.0

applying {i}

applying {i}

Z

Z’

X

X’

i

i

How to

connect them

?

i

U’

U

Y

Y’

a

a

a

a

0

0

0

0

OK!

OK!

Process Algebra (2IF45)

Abstraction and Recursion and Fairness:

problem 2. derivation rules

We want to derive that I(X’) = I(Z’)! We need new rules for this!

X’ = i.Y’

Y’ = i.Y’ + a.0

Something like this shall help:

Y’ = i.Y’ + a.0

.I(Y’) =. I(a.0)

Process Algebra (2IF45)

Abstraction and Recursion and Fairness: Fairness rule KFAR1b

a bit more general rule:

x1 = i1.x1 + y1, i1 I

.I(x1) =. I(y1)

Process Algebra (2IF45)

Abstraction and Recursion and Fairness: Fairness rule KFARnb

General KFAR rule is:

x1 = i1.x2 + y1,

x2 = i2.x3 + y2,

…

xn = in.x1 + yn, i1, … in I , there is ik

.I(x1) =. (I(y1) + … + I(yn))

Process Algebra (2IF45)

Abstraction and Recursion and Fairness:

Example of tossing a coin

Process Algebra (2IF45)

- Study the Coin tossing example
- Study the complete proof for ABP, derivation up to abstraction and derivation by means of fairness derivation rules.

Process Algebra (2IF45)