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

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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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