# Process Algebra (2IF45) Some Extensions of Basic Process Algebra Dr. Suzana Andova.

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Process Algebra (2IF45) Some Extensions of Basic Process Algebra Dr. Suzana Andova

1 Outline of today lecture Complete the proof of the Ground-completeness property of BPA(A) – the last lemma Extensions in process algebra What are the main aspects to be taken care of Illustrate those by an example Process Algebra (2IF45)

2 Lemma1: If p is a closed term in BPA(A) and p  then BPA(A) ├ p = 1 + p. Results towards ground-completeness of BPA(A) Lemma2: If p is a closed term in BPA(A) and p  p’ then BPA(A) ├ p = a.p’ + p. Lemma3: If (p+q) + r r then p+r r and q + r r, for closed terms p,q, r  C(BPA(A)). Lemma4: If p and q are closed terms in BPA(A) and p+q q then BPA(A) ├ p+q = q. Lemma5: If p and q are closed terms in BPA(A) and p p+ q then BPA(A) ├ p = p +q. Ground completeness property: If t r then BPA(A) ├ t = r, for any closed terms t and r in C(BPA(A)). a

3 Process Algebra (2IF45) BPA(A) Process Algebra fully defined 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 Deduction rules for BPA(A): x  x’ x + y  x’ a a  11  x  (x + y)   a.x  x  a  y  y’ x + y  y’ a a y  (x + y)  ⑥ Bisimilarity of LTSsEquality of terms

4 Process Algebra (2IF45) Extension of Equational theory 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 Deduction rules for BPA(A): x  x’ x + y  x’ a a  11  x  (x + y)   a.x  x  a  y  y’ x + y  y’ a a y  (x + y)  ⑥ Bisimilarity of LTSs Equality of terms New Axiom: (NA1) 0 + x = x

5 Process Algebra (2IF45) Extension of Equational theory 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 Deduction rules for BPA(A): x  x’ x + y  x’ a a  11  x  (x + y)   a.x  x  a  y  y’ x + y  y’ a a y  (x + y)  ⑥ Bisimilarity of LTSs Equality of terms New Axiom: (NA1) 0 + x = x New Axiom: (NA2) 0 = 1

6 Process Algebra (2IF45) Ground extension of T1 with T2: T1 = (  1, E1) and T2 = (  2, E2) are two equational theories. If 1.  2 contains  1 and 2. for any closed terms s and t in T1 it holds that T1 ├ s = t  T2 ├ s = t Extension of Equational theory

7 Process Algebra (2IF45) Ground extension of T1 with T2: T1 = (  1, E1) and T2 = (  2, E2) are two equational theories. If 1.  2 contains  1 and 2. for any closed terms s and t in T1 it holds that T1 ├ s = t  T2 ├ s = t Extension of Equational theory Axioms of BPA(A): (A1) x+ y = y+x (A2) (x+y) + z = x+ (y + z) (A3) x + x = x (A4) x+ 0 = x New Axioms: (NA1) 0 + x = x E1 E2

8 Process Algebra (2IF45) Ground extension of T1 with T2: T1 = (  1, E1) and T2 = (  2, E2) are two equational theories. If 1.  2 contains  1 and 2. for any closed terms s and t in T1 it holds that T1 ├ s = t  T2 ├ s = t Extension of Equational theory Axioms of BPA(A): (A1) x+ y = y+x (A2) (x+y) + z = x+ (y + z) (A3) x + x = x (A4) x+ 0 = x New Axioms: (NA1) 0 + x = x (NA2) 0 = 1 E1 E2

9 Process Algebra (2IF45) Conservative Ground extension of T1 with T2: T1 = (  1, E1) and T2 = (  2, E2) are two equational theories. If 1. T2 ground extension of T1 and 2. for any closed terms s and t in T1 it holds that T2 ├ s = t  T1 ├ s = t Extension of Equational theory

10 Process Algebra (2IF45) Conservative ground extension of T1 with T2: T1 = (  1, E1) and T2 = (  2, E2) are two equational theories. If 1. T2 ground extension of T1 and 2. for any closed terms s and t in T1 it holds that T2 ├ s = t  T1 ├ s = t Extension of Equational theory Axioms of BPA(A): (A1) x+ y = y+x (A2) (x+y) + z = x+ (y + z) (A3) x + x = x (A4) x+ 0 = x New Axioms: (NA1) 0 + x = x E1 E2

11 Process Algebra (2IF45) Conservative ground extension of T1 with T2: T1 = (  1, E1) and T2 = (  2, E2) are two equational theories. If 1. T2 ground extension of T1 and 2. for any closed terms s and t in T1 it holds that T2 ├ s = t  T1 ├ s = t Extension of Equational theory Axioms of BPA(A): (A1) x+ y = y+x (A2) (x+y) + z = x+ (y + z) (A3) x + x = x (A4) x+ 0 = x New Axioms: (NA1) 0 + x = x (NA2) 0 = 1 E1 E2

12 Process Algebra (2IF45) Deduction rules for BPA(A): x  x’ x + y  x’ a a  11  x  (x + y)   a.x  x  a  y  y’ x + y  y’ a a y  (x + y)  ⑥ Bisimilarity of LTSsEquality of terms Extension of Equational theory 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 New Axioms in BPA + (A) :….. New deduction rules for BPA+(A): …..

13 Extension of BPA(A) with Projection operators - Intuition what we want this operators to capture Process Algebra (2IF45)

14 Extension of BPA(A) with Projection operators -Intuition what we want this operators to capture -OK! Now we can make axioms and later SOS rules Process Algebra (2IF45)

15 Process Algebra (2IF45) Language: BPAPR(A) Signature: 0, 1, (a._ ) a  A, +  n (_), n  0 Language terms T(BPAPR(A)) Axioms of BPAPR(A): (A1) x+ y = y+x (A2) (x+y) + z = x+ (y + z) (A3) x + x = x (A4) x+ 0 = x (PR1)  n (1) = 1 (PR2)  n (0) = 0 (PR3)  0 (a.x) = 0 (PR4)  n+1 (a.x) = a.  n (x) (PR5)  n (x+y) =  n (x) +  n (y) BPA(A) BPAPR(A) Extension of BPA(A) with Projection operators

16 Process Algebra (2IF45) BPAPR(A) is a ground extension of BPA(A) (easy to conclude) Extension of Equational theory BPAPR(A) is a conservative ground extension of BPA(A)

17 Process Algebra (2IF45) BPAPR(A) is a ground extension of BPA(A). Extension of Equational theory BPAPR(A) is a conservative ground extension of BPA(A). Is BPAPR(A) more expressive than BPA(A)?

18 Process Algebra (2IF45) If p is a closed terms in BPAPR(A), then there is a closed term q in BPA(A) such that BPAPR(A) ├ p = q. Elimination theorem for BPAPR

19 Process Algebra (2IF45) Operational semantics of BPAPR

20 New deduction rules for BPAPR(A): Process Algebra (2IF45) Deduction rules for BPA(A): x  x’ x + y  x’ a a  11  x  (x + y)   a.x  x  a  y  y’ x + y  y’ a a y  (x + y)  ⑥ Bisimilarity of LTSs Equality of terms Extension of Equational theory Language: BPAPR(A) Signature: 0, 1, (a._ ) a  A, +,  n (x), n  0 Language terms T(BPAPR(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 New Axioms in BPAPR(A) : (PR1)  n (1) = 1 (PR2)  n (0) = 0 (PR3)  0 (a.x) = 0 (PR4)  n+1 (a.x) = a.  n (x) (PR5)  n (x+y) =  n (x) +  n (y) x   n (x)  x  x’  n +1 (x)   n (x’) a a

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