Process Algebra (2IF45) Basic Process Algebra (Soundness proof) Dr. Suzana Andova
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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)
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)
11
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)
12
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)
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