process algebra (2if45) extending process algebra: parallel composition more examples

Process Algebra (2IF45)

Extending Process Algebra:Parallel composition More examples

Dr. Suzana Andova

2

TCP language TCP(A, ) A is a pre-defied set of atomic actions is a pre-defined communication functionSignature: (constructs of the language)

constants 0,1 action prefix a._ non-deterministic choice _+_ sequential composition _ _

Process behaviour specification described by process equations, • recursive specifications• guarded recursive process variables and guarded process specifications• uniqueness of the process defined by a guarded Rspec• Additional mechanism for “equating” recursive specifications: RDP and RSP,

where “equating” = they define the same (up to bisimulation) process

parallel composition _ || _ communication composition. _ | _

encapsulation H(_), where H A

3 Process Algebra (2IF45)

components’ specifications

System specification manipulation (recap)

the whole system specification

the state space

reductionon specification


reductionon LTSs

composition by axiom

SOS rules

• simpler• smaller• in a particular form (basic)• …

4

Example: Specifying a Bag

• Assume a set of data elements D = {0,1}• Bn,m is the process that specify the current content of B• n is the number of 0s• m is the number of 1s

B0,0 = r1(0). B1,0 + r1(1). B0,1B0, m+1 = r1(0). B1,m+1 + r1(1). B0,m+2 + s2(1). B0,mBn+1,0 = r1(0). Bn+2,0 + r1(1). Bn+1,1 + s2(0). Bm,0Bn+1, m+1 = r1(0). Bn+2,m+1 + r1(1). Bn+1,m+2 + s2(0). Bn,m+1 + s2(1).Bn+1, m 1 2

5



B0,0 = r1(0). B1,0 + r1(1). B0,1B0, m+1 = r1(0). B1,m+1 + r1(1). B0,m+2 + s2(1). B0,mBn+1,0 = r1(0). Bn+2,0 + r1(1). Bn+1,1 + s2(0). Bn,0Bn+1, m+1 = r1(0). Bn+2,m+1 + r1(1). Bn+1,m+2 + s2(0). Bn,m+1 + s2(1).Bn+1, m

This is an infinite recursive specification!Can we do better?

1 2

6



Bag = r1(0). (Bag || s2(0)) + r1(1). (Bag || s2(1))

Do they define the same process, namely a bag process?TCP, RSP, RDP |- Bag = B0,0?

7


Bag = r1(0). (Bag || s2(0)) + r1(1). (Bag || s2(1))

1. Define new equations Dn,m using Bag such that Bag is exactly D0,0

2. and prove that TCP, RSP, RDP |- D0,0 = B0,0!


8


Bag = r1(0). (Bag || s2(0)) + r1(1). (Bag || s2(1))

1. Define new equations Dn,m using BagDn,m = Bag || (s2(0).s2(0)…. s2(0).1 )

|| (s2(1).s2(1).… s2(1).1 )

such that Bag is exactly D0,0 2. and prove that TCP, RSP, RDP |- D0,0 = B0,0?


n (n0)

m (m0)

9


Bag = r1(0). (Bag || s2(0)) + r1(1). (Bag || s2(1))

Dn,m = Bag || s2(0) n1 || s2(1) m1 for n,m 0

Our goal is to show that Dn,m = Bn,m, for any n and any m Namelly to show that I. D0,0 = B0,0 II. D0,m+1 = B, 0, m+1 (only this case on the next slides)III. Dn+1, 0 = Bn+1,0IV. Dn+1,m+1 = Bn+1,m+1

Observe thatD0,0 = Bag

?

10


Bag = r1(0). (Bag || s2(0)) + r1(1). (Bag || s2(1))


II. D0,m+1 = Bag || (s2(1) m+11) = Bag ||_ s2(1) m+11 + s2(1) m+11 ||_ Bag = (r1(0). (Bag || s2(0) ) + r1(1). (Bag || s2(1))) ||_ (s2(1) m+11)

+ (s2(1) m+11) ||_ Bag = (r1(0). (Bag || s2(0))) ||_ (s2(1) m+11) + (r1(1). (Bag || s2(1))) ||_ (s2(1) m+11) + (s2(1) m+11) ||_ Bag =

no communication


11


Bag = r1(0). (Bag || s2(0)) + r1(1). (Bag || s2(1))


II. D0,m+1 = Bag || s2(1) m+11 = Bag ||_ s2(1) m+11 + s2(1) m+11 ||_ Bag = (r1(0). (Bag || s2(0) ) + r1(1). (Bag || s2(1))) ||_ (s2(1) m+11)

+ (s2(1) m+11) ||_ Bag = (r1(0). (Bag || s2(0))) ||_ s2(1) m+11 + (r1(1). (Bag || s2(1))) ||_ s2(1) m+11 + s2(1) m+11 ||_ Bag = r1(0). ((Bag || s2(0)) || s2(1) m+11) + r1(1). ((Bag || s2(1)) || s2(1) m+11) + s2(1). (s2(1)m1|| Bag)


s2(1) m+11 ||_ Bag =

s2(1).s2(1) m1 ||_ Bag =

s2(1). (s2(1) m1|| Bag)

12


Bag = r1(0). (Bag || s2(0)) + r1(1). (Bag || s2(1))

Dn,m = Bag || s2(0) n1 || s2(1) m+11 for n,m 0

II. D0,m+1 = Bag || s2(1) m+11 = Bag ||_ s2(1) m+11 + s2(1) m+11 ||_ Bag = (r1(0). (Bag || s2(0) ) + r1(1). (Bag || s2(1))) ||_ (s2(1) m+11)

+ (s2(1) m+11) ||_ Bag = (r1(0). (Bag || s2(0))) ||_ s2(1) m+11 + (r1(1). (Bag || s2(1))) ||_ s2(1) m+11 + s2(1) m+11 ||_ Bag = r1(0). ((Bag || s2(0)) || s2(1) m+11) + r1(1). ((Bag || s2(1)) || s2(1) m+11) + s2(1). (s2(1)m1|| Bag)= r1(0). (D1,m+1) + r1(1). (D0,m+2) + s2(1). (D0,m)


s2(1).1 || s2(1) m+11 =s2(1) m+11

13

Example: Specifying a Bag - Conclusion

Bag = r1(0). (Bag || s2(0)) + r1(1). (Bag || s2(1))

Dn,m = Bag || s2(0) n1 || s2(1) m+11

From RDP and RSP it follows that I. D0,0 = B0,0II. D0,m+1 = B0,m+1III. Dn+1,0= Bn+1,0IV. Dn+1,m+1 = Bn+1,m+1

Final conclusion: B0,0 = Bag


14

Example: Buffer - Revision

Assume a set of data elements D = {0,1}

II. Two place buffer

BufTwo = r1(0).B0 + r1(1).B1 B0 = s2(0).BufTwo + r1(0).s2(0).B0 + r1(1).s2(0).B1 B1 = s2(1).BufTwo + r1(0).s2(1).B0 + r1(1).s2(1).B1


1 2BufTwo

15

Example: Buffer

Assume a set of data elements D = {0,1}III. Implementing a Two place buffer with Two One place buffers


21 3

BufOne13 = r1(0).s3(0).BufOne13 + r1(1).s3(1).BufOne13BufOne32 = r3(0).s2(0).BufOne32 + r3(1).s2(1).BufOne32communication: (r3(x), s3(x)) = c3(x), for x D blocking: H = {r3(x), s3(x) | x D}

BufTwoInOne = ∂H( BufOne13 || BufOne32 )

16

Example: BufTwoInOne Recursive Specification

III. Implementing a Two place buffer with Two One place buffersBufOne13 = r1(0).s3(0).BufOne13 + r1(1).s3(1).BufOne13BufOne32 = r3(0).s2(0).BufOne32 + r3(1).s2(1).BufOne32

• communication: (r3(x), s3(x)) = c3(x), for x D • blocking: H = {r3(x), s3(x) | x D}

BufTwoInOne = r1(0).c3(0).X0 + r1(1).c3(1).X1 X0 = r1(0).s2(0).c3(0).X0 + r1(1).s2(0). c3(1). X1 + s2(0). BufTwoInOne X1 = r1(0).s2(1).c3(0).X0 + r1(1).s2(1).c3(1). X1 + s2(1). BufTwoInOne

II. Back to our original specification: Two place buffer BufTwo = r1(0).B0 + r1(1).B1 B0 = s2(0).BufTwo + r1(0).s2(0).B0 + r1(1).s2(0).B1 B1 = s2(0).BufTwo + r1(0).s2(0).B0 + r1(1).s2(0).B1

NOT YET!


Extending Process Algebra:Abstraction

Dr. Suzana Andova


components’ specifications

System specification manipulation (recap)

the whole system specification

the state space



reductionon LTSs

composition by axiom

SOS rules

• simpler• smaller• in a particular form (basic)• …

19

TCP language extended with hiding feature TCP(A, ) A is a pre-defied set of atomic actions internal (silent) action , A is a pre-defined communication functionSignature: (constructs of the language)

constants 0,1 action prefix a._ non-deterministic choice _+_ sequential composition _ _

hiding operator I for I A Process behaviour specification described by process equations,

• guarded recursive process variables and guarded process specifications• equivalence relation that treats differently

parallel composition _ || _ communication composition. _ | _

encapsulation H(_), where H A


Towards equivalence relation(s)

in1euro

!coffee !coffee

in50c

in50c

!coffee !coffee

hiding reducing

!tea !coffee

40c 50c

!tea !coffee

hiding reducing

?

?

Think about different ways to reduce these processes?

Which reduced process preserves “the same moment of choice” as in the original process with s?


reducing insert

40c

hiding ? coffee

insert

coffee

reducing insert

card

hiding ?

coffee

coin

insert

coffee

Towards equivalence relation(s)

22

Towards our equivalence relation -Conclusions

• Some internal steps cannot be removed. • Some internal steps can be removed. They are called inert!• Inert internal steps occur in following situations:

a

P Q

P+

+

…

a

P Q+

…

reduces to

23

Towards our equivalence relation -Conclusions

• Some internal steps cannot be removed. • Some internal steps can be removed. They are called inert!• Inert internal steps occur in following situations:• Relation to be established is:

a

P Q

P+

+

…

a

P Q+

…

reduces to

process algebra (2if45) extending process algebra: parallel composition more examples

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