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

2. 1 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. 2 BPA(A) Process Algebra – completeness property 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. 3 Process Algebra (2IF45) Ground completeness property: If t r then BPA(A) ├ t = r, for any closed terms t and r in C(BPA(A)). BPA(A) Process Algebra – ground-completeness property

5. 4 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. 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 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. 6 Lemma1: If p is a closed term in BPA(A) and p  then BPA(A) ├ p = 1 + p. 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. a

8. 7 Process Algebra (2IF45) Lemma3: If (p+q) + r r then p+r r and q + r r, for closed terms p,q, r  C(BPA(A)). Towards ground-completeness of BPA(A)

9. 8 Process Algebra (2IF45) Lemma4: If p and q are closed terms in BPA(A) and p+q q then BPA(A) ├ p+q = q. Towards ground-completeness of BPA(A) Lemma5: If p and q are closed terms in BPA(A) and p p+ q then BPA(A) ├ p = p +q.

10. 9 Process Algebra (2IF45) Ground completeness property: If t r then BPA(A) ├ t = r, for any closed terms t and r in C(BPA(A)). Ground-completeness of BPA(A)

11. 10 Process Algebra (2IF45) Lemma1: If p is a closed term in BPA(A) and p  then BPA(A) ├ p = 1 + p. 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