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Process Algebra (2IF45) Basic Process Algebra (Soundness proof)

Process Algebra (2IF45) Basic Process Algebra (Soundness proof). Dr. Suzana Andova. Outline of today lecture. Soundness property of BPA(A) Example: cooking your own process algebra. BPA(A) Process Algebra – soundness property. Language: BPA(A) Signature: 0, 1, ( a._ ) a  A , +

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Process Algebra (2IF45) Basic Process Algebra (Soundness proof)

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  1. Process Algebra (2IF45)Basic Process Algebra (Soundness proof) Dr. Suzana Andova

  2. Outline of today lecture • Soundness property of BPA(A) • Example: cooking your own process algebra Process Algebra (2IF45)

  3. BPA(A) Process Algebra – soundness property 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? Process Algebra (2IF45)

  4. Soundness property of BPA(A) wrt to  Soundness property: If BPA(A) ├ t = r then t r, for any terms t and r in T(BPA(A)). Proof: All we need to show is that each axiom is sound, that is, t + r  r + t, for any terms t,r  T(BPA(A)) (t + r) + s  (t + r) + s, for any terms t, r, s  T(BPA(A)) t + t  t, for any term t  T(BPA(A)) t + 0  t, for any term t  T(BPA(A)). WHY IS THIS (considering only axioms) SUFFICIENT? Process Algebra (2IF45)

  5. BPA(A) Process Algebra – soundness property 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)

  6. Example: making a process algebra • Our first own BPA(A) extension • We extend BPA(A) basic process algebra with an unary operator D(_). • D(x) “doubles” every action that x can execute • Examples: • D(a.b.0) = a.a.b.b.0 • D(a.b.1 + c.0) = a.a.b.b.1 + c.c.0 • … • Construct the BPAD(A) process algebra! Process Algebra (2IF45)

  7. Example: recipe Follow the steps: • Define axioms for the new operator • Define SOS rules for the new operator • Mix the ingredients: check whether they all fit well! • Congruence of  with respect to the new operator • Soundness of the new axioms • Completeness q.e.d. Process Algebra (2IF45)

  8. Example: Our first own BPA(A) extension Solution: • Axioms of BPAD(A) are those of BPA(A) and: D(0) = 0 D(1) = 1 D(a.x) = a.a.D(x) D(x + y) = D(x) + D(y) • SOS rules for the term-deduction system of BPAD(A) includes the SOS rules of BPA(A) and the following two rules: • For the proofs of congruence of  with respect to the new D operator and the soundness proof for the new axioms you should use the proof strategies seen already. a x D(x)  x  x’ D(x)  a.D(x’) a Process Algebra (2IF45)

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