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Process Algebra (2IF45) Some Extensions of Basic Process Algebra

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

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Process Algebra (2IF45) Some Extensions of Basic Process Algebra

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

  2. 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)

  3. Results towards ground-completeness of BPA(A) Lemma1: If p is a closed term in BPA(A) and p  then BPA(A) ├ p = 1 + p. 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)). Process Algebra (2IF45)

  4. BPA(A) Process Algebra fully defined Language: BPA(A) Signature: 0, 1, (a._ )aA, + Language terms T(BPA(A)) Deduction rules for BPA(A): 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 a x  x’ x + y  x’ a.x  x   a x (x + y)  1   a y  y’ x + y  y’  a y (x + y)  ⑥ Bisimilarity of LTSs Equality of terms Soundness Completeness Process Algebra (2IF45)

  5. Extension of Equational theory Language: BPA(A) Signature: 0, 1, (a._ )aA, + Language terms T(BPA(A)) Deduction rules for BPA(A): 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 a x  x’ x + y  x’ a.x x   a a y  y’ x + y  y’  1 a  x (x + y)   New Axiom: (NA1) 0 + x = x y (x + y)  ⑥ Bisimilarity of LTSs Equality of terms Soundness Completeness Process Algebra (2IF45)

  6. Extension of Equational theory Language: BPA(A) Signature: 0, 1, (a._ )aA, + Language terms T(BPA(A)) Deduction rules for BPA(A): 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 a x  x’ x + y  x’ a.x x   a a y  y’ x + y  y’  1 a  x (x + y)   New Axiom: (NA1) 0 + x = x y (x + y)  ⑥ New Axiom: (NA2) 0 = 1 Bisimilarity of LTSs Equality of terms Soundness Completeness Process Algebra (2IF45)

  7. Extension of Equational theory 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 Process Algebra (2IF45)

  8. Extension of Equational theory 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 Axioms of BPA(A): (A1) x+ y = y+x (A2) (x+y) + z = x+ (y + z) (A3) x + x = x (A4) x+ 0 = x E1 E2 New Axioms: (NA1) 0 + x = x Process Algebra (2IF45)

  9. Extension of Equational theory 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 Axioms of BPA(A): (A1) x+ y = y+x (A2) (x+y) + z = x+ (y + z) (A3) x + x = x (A4) x+ 0 = x E1 E2 New Axioms: (NA1) 0 + x = x (NA2) 0 = 1 Process Algebra (2IF45)

  10. Extension of Equational theory 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 Process Algebra (2IF45)

  11. Extension of Equational theory 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 Axioms of BPA(A): (A1) x+ y = y+x (A2) (x+y) + z = x+ (y + z) (A3) x + x = x (A4) x+ 0 = x E1 E2 New Axioms: (NA1) 0 + x = x Process Algebra (2IF45)

  12. Extension of Equational theory 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 Axioms of BPA(A): (A1) x+ y = y+x (A2) (x+y) + z = x+ (y + z) (A3) x + x = x (A4) x+ 0 = x E1 E2 New Axioms: (NA1) 0 + x = x (NA2) 0 = 1 Process Algebra (2IF45)

  13. Extension of Equational theory Language: BPA+(A) Signature: 0, 1, (a._ )aA, +, … Language terms T(BPA+(A)) Deduction rules for 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 a a x  x’ x + y  x’ a.x  x   a x (x + y)  1   New Axioms in BPA+(A):….. a y  y’ x + y  y’  a New deduction rules for BPA+(A): ….. y (x + y)  ⑥ Bisimilarity of LTSs Equality of terms Soundness Completeness Process Algebra (2IF45)

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

  15. 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)

  16. Extension of BPA(A) with Projection operators 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 BPA(A) 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) Process Algebra (2IF45)

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

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

  19. Elimination theorem for BPAPR 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. Process Algebra (2IF45)

  20. Operational semantics of BPAPR Process Algebra (2IF45)

  21. Extension of Equational theory Language: BPAPR(A) Signature: 0, 1, (a._ )aA, +, n(x), n  0 Language terms T(BPAPR(A)) Deduction rules for 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 a a a x  x’ n +1(x)  n (x’) x  x’ x + y  x’ a.x  x   a a 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 (x + y)  1   a y  y’ x + y  y’  a New deduction rules for BPAPR(A): x n (x)  y (x + y)  Soundness ⑥ Equality of terms Bisimilarity of LTSs Completeness Process Algebra (2IF45)

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