Process Algebra (2IF45)

Recursion in Process Algebra

Suzana Andova

http://fusionanomaly.net/recursion.jpg

2

Language: BPA(A) Signature: 0, 1, (a._ )aA, +, … Language terms T(BPA(A)) Closed terms C(BPA(A))

Process Algebra (2IF45)

Deduction rules for BPA(A):

x x’ x + y x’

aa

1

x (x + y)

a.x x a

y y’ x + y y’

aa

y (x + y)

⑥

Equational theory: terms and LTSs

Axioms of BPA(A):(A1) x+ y = y+x

(A2) (x+y) + z = x+ (y + z)(A3) x + x = x(A4) x+ 0 = x

coin.coffee.1

coin

coffee

one-t

o-one

3

Language: BPA(A) Signature: 0, 1, (a._ )aA, +, … Language terms T(BPA(A)) Closed terms C(BPA(A))

Process Algebra (2IF45)

Deduction rules for BPA(A):

x x’ x + y x’

aa

1

x (x + y)

a.x x a

y y’ x + y y’

aa

y (x + y)

⑥

Bisimilarity of LTSs Equality of termsSoundness

Completeness

Equational theory: terms and LTSs

Axioms of BPA(A):(A1) x+ y = y+x

(A2) (x+y) + z = x+ (y + z)(A3) x + x = x(A4) x+ 0 = x

coin.coffee.1

coin.coffee.1 + coin.coffee.1

coin

coffee

coin.coffee.1 = coin.coffee.1 + coin.coffee.1coin

coffee

coin

coffee

4

Recursive processes

Process Algebra (2IF45)

Socrates_thinks

Socrates_eats

getHungry

goThinking

thinking

eating

5

Recursive processes

Process Algebra (2IF45)

Socrates_thinks

Socrates_eats

getHungry

goThinking

Socrates_thinks = getHungry.Socrates_eats

Socrates_eats = goThinking.Socrates_thinks

thinking

eating

6 Process Algebra (2IF45)

Deduction rules for BPA(A):

x x’ x + y x’

aa

1

x (x + y)

a.x x a

y y’ x + y y’

aa

y (x + y)

⑥

Recursive specifications and LTSs Language: BPA(A) Signature: 0, 1, (a._ )aA, +, … Language terms T(BPA(A)) Closed terms C(BPA(A)) Axioms of BPA(A):

(A1) x+ y = y+x(A2) (x+y) + z = x+ (y + z)

(A3) x + x = x(A4) x+ 0 = x

goThinking getHungry

Socrates_thinks = getHungry.Socrates_eats

Socrates_eats = goThinking.Socrates_thinks

7 Process Algebra (2IF45)

Deduction rules for BPA(A):

x x’ x + y x’

aa

1

x (x + y)

a.x x a

y y’ x + y y’

aa

y (x + y)

⑥

Recursive specifications and LTSs Language: BPA(A) Signature: 0, 1, (a._ )aA, +, … Language terms T(BPA(A)) Closed terms C(BPA(A)) Axioms of BPA(A):

(A1) x+ y = y+x(A2) (x+y) + z = x+ (y + z)

(A3) x + x = x(A4) x+ 0 = x

GoThinking GetHungry

Socrates_thinks = getHungry.Socrates_eats

Socrates_eats = goThinking.Socrates_thinks

8 Process Algebra (2IF45)

Recursive equations and specifications

E = { X = a.0 }

E1 = { X = a.Y, Y = b.0 }

9 Process Algebra (2IF45)

Recursive Equations and Rec. Specification in Equational Theory

E = { X = a.Y + c.0,Y = b.X}

BPA(A), E ├ X = a.Y +c.0 = a.(b.X) +c.0 = a.(b.(a.Y + c.))) + c.0

Language: BPA(A) Signature: 0, 1, (a._ )aA, +, … Language terms T(BPA(A)) Closed terms C(BPA(A))

Axioms of BPA(A):(A1) x+ y = y+x

(A2) (x+y) + z = x+ (y + z)(A3) x + x = x(A4) x+ 0 = x

Recursive specification E

10 Process Algebra (2IF45)

Solutions of recursive equations

Example: 1. E = { X = a.0 }

2. E1 = { X = a.Y, Y = b.0 }

11 Process Algebra (2IF45)

Solutions of recursive equations

Example: 1. E = { X = X }

2. E1 = { X = X + a.0 }

12 Process Algebra (2IF45)

Solutions of recursive equations

Example: 1. E = { X = a.X }

2. E1 = { X = a.(a.(X+1)) +1 }

13 Process Algebra (2IF45)

Solutions of recursive specifications

E1 = { X = a.(a.(X+1)) +1 }

X

a

a.(X+1)

a a

X+1

This is a solution for X in the recursive spec. E1Substitute it on the left-hand side and on the right-hand side and check bisimilarity.

14

This is also recursion ….

Process Algebra (2IF45)

15 Process Algebra (2IF45)

Deduction rules for BPA(A):

x x’ x + y x’

aa

1

x (x + y)

a.x x a

y y’ x + y y’

aa

y (x + y)

⑥

Semantics of Recursive specifications Language: BPA(A) Signature: 0, 1, (a._ )aA, +, … Language terms T(BPA(A))

Axioms of BPA(A):(A1) x+ y = y+x

(A2) (x+y) + z = x+ (y + z)(A3) x + x = x(A4) x+ 0 = x

16 Process Algebra (2IF45)

Deduction rules for BPA(A):

x x’ x + y x’

aa

1

x (x + y)

a.x x a

y y’ x + y y’

aa

y (x + y)

⑥

Semantics of Recursive specifications Language: BPArec(A)

Signature: 0, 1, (a._ )aA, +, X Language terms T(BPA(A))

Axioms of BPA(A):(A1) x+ y = y+x

(A2) (x+y) + z = x+ (y + z)(A3) x + x = x(A4) x+ 0 = x

17 Process Algebra (2IF45)

Semantics of Recursive specifications

18 Process Algebra (2IF45)

Semantics of Recursive specifications

tX,E w, X=t in E X w a

a

19 Process Algebra (2IF45)

Deduction rules for BPA(A):

x x’ x + y x’

aa

1

x (x + y)

a.x x a

y y’ x + y y’

aa

y (x + y)

⑥

Semantics of Recursive specifications Language: BPArec(A)

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

Language terms T(BPArec(A)) Axioms of BPA(A):

(A1) x+ y = y+x(A2) (x+y) + z = x+ (y + z)

(A3) x + x = x(A4) x+ 0 = x

GoThinking GetHungry

Socrates_thinks = getHungry.Socrates_eats

Socrates_eats = goThinking.Socrates_thinks

a term

….

20 Process Algebra (2IF45)

Deduction rules for BPA(A):

x x’ x + y x’

aa

1

x (x + y)

a.x x a

y y’ x + y y’

aa

y (x + y)

⑥

Semantics of Recursive specifications Language: BPArec(A)

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

Language terms T(BPArec(A)) Axioms of BPA(A):

(A1) x+ y = y+x(A2) (x+y) + z = x+ (y + z)

(A3) x + x = x(A4) x+ 0 = x

GoThinking GetHungry

Socrates_thinks = getHungry.Socrates_eats

Socrates_eats = goThinking.Socrates_thinks

a term

….

one-

to-o

ne

21

Term model T(BPArec(A)) and BPArec(A)

• Bisimulation is congruence• Soundness holds (it is a model indeed) • Ground completeness does not hold

• Every recursive specification has a solution• Not every recursive specification has unique solution

Process Algebra (2IF45)

22

Equational theories with recursion

• Model vs. Solution of recursive specification• Two important points for “useful models”

• Every recursive specification has a solution• Every recursive specification has a unique solution

Process Algebra (2IF45)

23

Recursive Definition Principle (RDP)

For every recursive specification there is a solution in the model

Process Algebra (2IF45)

24

Guarded recursions

• Does E = {X = a.Y, Y = Z, Z = b.X} has a solution and is this unique in T(BPArec(A))?

Process Algebra (2IF45)

25

Recursive Specification Principle (RSP)

• Every guarded recursive specification has at most one solution.

Process Algebra (2IF45)

26

Restricted Recursive Definition Principle (RSP-)

• Every guarded recursive specification has a solution.

Process Algebra (2IF45)

27

Combining principles

• RDP- + RSP implies Every guarded recursive specification has a unique solution.

Example E = {X = a.X} E’ = {X’ = a.a.X’}

Process Algebra (2IF45)