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

Process Algebra (2IF45)

Basic Process Algebra (Soundness proof)

Dr. Suzana Andova

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Outline of today lecture

• Soundness property of BPA(A)

• Example: cooking your own process algebra

Process Algebra (2IF45)

4 Process Algebra (2IF45)

BPA(A) Process Algebra – soundness property

Language: BPA(A)

Signature: 0, 1, (a._ )aA, +

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’

aa

1

x (x + y)

a.x xa

y y’ x + y y’

aa

y (x + y)

⑥

Bisimilarity of LTSs Equality of terms

6

Soundness property of BPA(A) wrt to

Process Algebra (2IF45)

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, 1. t + r r + t, for any terms t,r T(BPA(A))2. (t + r) + s (t + r) + s, for any terms t, r, s T(BPA(A))3. t + t t, for any term t T(BPA(A))4. t + 0 t, for any term t T(BPA(A)).

WHY IS THIS (considering only axioms) SUFFICIENT?

9 Process Algebra (2IF45)

BPA(A) Process Algebra – soundness property

Language: BPA(A)

Signature: 0, 1, (a._ )aA, +

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’

aa

1

x (x + y)

a.x xa

y y’ x + y y’

aa

y (x + y)

⑥

Bisimilarity of LTSs Equality of terms

10

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)

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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)

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Example: Our first own BPA(A) extension

Process Algebra (2IF45)

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.

x x’ D(x) a.D(x’)

aa

x D(x)