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Process Algebra (2IF45) Recursion in Process Algebra Suzana Andova http://fusionanomaly.net/recursion.jpg

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1 Language: BPA(A) Signature: 0, 1, (a._ ) a A, +, … Language terms T(BPA(A)) Closed terms C(BPA(A)) Process Algebra (2IF45) Deduction rules for BPA(A): x x’ x + y x’ a a 11 x (x + y) a.x x a y y’ x + y y’ a a 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

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2 Language: BPA(A) Signature: 0, 1, (a._ ) a A, +, … Language terms T(BPA(A)) Closed terms C(BPA(A)) Process Algebra (2IF45) Deduction rules for BPA(A): x x’ x + y x’ a a 11 x (x + y) a.x x a y y’ x + y y’ a a y (x + y) ⑥ Bisimilarity of LTSsEquality of terms 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.1 coin coffee coin coffee

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3 Recursive processes Process Algebra (2IF45) Socrates_thinks Socrates_eats getHungry goThinking thinking eating

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4 Recursive processes Process Algebra (2IF45) Socrates_thinks Socrates_eats getHungry goThinking Socrates_thinks = getHungry.Socrates_eats Socrates_eats = goThinking.Socrates_thinks thinking eating

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5 Process Algebra (2IF45) Deduction rules for BPA(A): x x’ x + y x’ a a 11 x (x + y) a.x x a y y’ x + y y’ a a y (x + y) ⑥ Recursive specifications and LTSs Language: BPA(A) Signature: 0, 1, (a._ ) a A, +, … 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

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6 Process Algebra (2IF45) Deduction rules for BPA(A): x x’ x + y x’ a a 11 x (x + y) a.x x a y y’ x + y y’ a a y (x + y) ⑥ Recursive specifications and LTSs Language: BPA(A) Signature: 0, 1, (a._ ) a A, +, … 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

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7 Process Algebra (2IF45) Recursive equations and specifications E = { X = a.0 } E 1 = { X = a.Y, Y = b.0 }

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8 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._ ) a A, +, … 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

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9 Process Algebra (2IF45) Solutions of recursive equations Example: 1. E = { X = a.0 } 2. E 1 = { X = a.Y, Y = b.0 }

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10 Process Algebra (2IF45) Solutions of recursive equations Example: 1. E = { X = X } 2. E 1 = { X = X + a.0 }

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11 Process Algebra (2IF45) Solutions of recursive equations Example: 1. E = { X = a.X } 2. E 1 = { X = a.(a.(X+1)) +1 }

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12 Process Algebra (2IF45) Solutions of recursive specifications E 1 = { 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. E1 Substitute it on the left-hand side and on the right-hand side and check bisimilarity.

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13 This is also recursion …. Process Algebra (2IF45)

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14 Process Algebra (2IF45) Deduction rules for BPA(A): x x’ x + y x’ a a 11 x (x + y) a.x x a y y’ x + y y’ a a y (x + y) ⑥ Semantics of Recursive specifications Language: BPA(A) Signature: 0, 1, (a._ ) a A, +, … 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

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15 Process Algebra (2IF45) Deduction rules for BPA(A): x x’ x + y x’ a a 11 x (x + y) a.x x a y y’ x + y y’ a a y (x + y) ⑥ Semantics of Recursive specifications Language: BPA rec (A) Signature: 0, 1, (a._ ) a A, +, 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

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16 Process Algebra (2IF45) Semantics of Recursive specifications

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17 Process Algebra (2IF45) Semantics of Recursive specifications t X,E w, X=t in E X w a a

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18 Process Algebra (2IF45) Deduction rules for BPA(A): x x’ x + y x’ a a 11 x (x + y) a.x x a y y’ x + y y’ a a y (x + y) ⑥ Semantics of Recursive specifications Language: BPA rec (A) Signature: 0, 1, (a._ ) a A, +, X E Language terms T(BPA rec (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 ….

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19 Process Algebra (2IF45) Deduction rules for BPA(A): x x’ x + y x’ a a 11 x (x + y) a.x x a y y’ x + y y’ a a y (x + y) ⑥ Semantics of Recursive specifications Language: BPA rec (A) Signature: 0, 1, (a._ ) a A, +, X E Language terms T(BPA rec (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 ….

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20 Term model T(BPA rec (A)) and BPA rec (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)

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21 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)

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22 Recursive Definition Principle (RDP) For every recursive specification there is a solution in the model Process Algebra (2IF45)

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23 Guarded recursions Does E = {X = a.Y, Y = Z, Z = b.X} has a solution and is this unique in T(BPA rec (A))? Process Algebra (2IF45)

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24 Recursive Specification Principle (RSP) Every guarded recursive specification has at most one solution. Process Algebra (2IF45)

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25 Restricted Recursive Definition Principle (RSP - ) Every guarded recursive specification has a solution. Process Algebra (2IF45)

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26 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)

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