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

Process Algebra (2IF45) Basic Process Algebra (Completeness proof). Dr. Suzana Andova. Outline of today lecture. Ground-completeness property of BPA(A) Proving techniques (e.g. mathematical induction) Dividing the proof in small steps (5+1). BPA(A) Process Algebra – completeness property.

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

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

  2. Outline of today lecture • Ground-completeness property of BPA(A) • Proving techniques (e.g. mathematical induction) • Dividing the proof in small steps (5+1) Process Algebra (2IF45)

  3. BPA(A) Process Algebra – completeness 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)

  4. BPA(A) Process Algebra – ground-completeness property 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)

  5. Structure of the closed terms of BPA(A) • The definition of closed terms is inductive: • 0 and 1 are closed terms • if p is a closed term and a  A then a.p is a closed term too • if p and q are closed terms then p+q is a closed term too Process Algebra (2IF45)

  6. Structure of the Closed terms of BPA(A) • The definition of closed terms is inductive: • 0 and 1 are closed terms • if p is a closed term and a  A then a.p is a closed term too • if p and q are closed terms then p+q is a closed term too • Proofs are easy by Structural induction • Prove a property for closed BPA(A) terms Basic case: prove the property for the constants 0 and 1 Inductive step: Step1. prove the property for the construct a._ Step2. prove the property for the construct _+_ • … Process Algebra (2IF45)

  7. 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. Process Algebra (2IF45)

  8. Towards ground-completeness of BPA(A) Lemma3: If (p+q) + r  r then p+r  r and q + r  r, for closed terms p,q, r  C(BPA(A)). Process Algebra (2IF45)

  9. Towards ground-completeness of 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. Process Algebra (2IF45)

  10. Ground-completeness of BPA(A) 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)

  11. 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)

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