Post on 06-Feb-2016
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Process Algebra (2IF45)
Some Extensions of Basic Process Algebra
Dr. Suzana Andova
2
Outline of today lecture
• Complete the proof of the Ground-completeness property of BPA(A) – the last lemma
• Extensions in process algebra• What are the main aspects to be taken care of
• Illustrate those by an example
Process Algebra (2IF45)
3 Process Algebra (2IF45)
Lemma1: If p is a closed term in BPA(A) and p then BPA(A) ├ p = 1 + p.
Results 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
4 Process Algebra (2IF45)
BPA(A) Process Algebra fully defined
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
5 Process Algebra (2IF45)
Extension of Equational theory
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 x a
y y’ x + y y’
aa
y (x + y)
⑥
Bisimilarity of LTSs Equality of terms
New Axiom:
(NA1) 0 + x = x
6 Process Algebra (2IF45)
Extension of Equational theory
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 x a
y y’ x + y y’
aa
y (x + y)
⑥
Bisimilarity of LTSs Equality of terms
New Axiom:
(NA1) 0 + x = x
New Axiom:
(NA2) 0 = 1
7 Process Algebra (2IF45)
Ground extension of T1 with T2: T1 = (1, E1) and T2 = (2, E2) are two equational theories. If 1. 2 contains 1 and
2. for any closed terms s and t in T1 it holds that T1 ├ s = t T2 ├ s = t
Extension of Equational theory
8 Process Algebra (2IF45)
Ground extension of T1 with T2: T1 = (1, E1) and T2 = (2, E2) are two equational theories. If 1. 2 contains 1 and
2. for any closed terms s and t in T1 it holds that T1 ├ s = t T2 ├ s = t
Extension of Equational theory
Axioms of BPA(A):
(A1) x+ y = y+x
(A2) (x+y) + z = x+ (y + z)
(A3) x + x = x
(A4) x+ 0 = x
New Axioms:
(NA1) 0 + x = x
E1
E2
9 Process Algebra (2IF45)
Ground extension of T1 with T2: T1 = (1, E1) and T2 = (2, E2) are two equational theories. If 1. 2 contains 1 and
2. for any closed terms s and t in T1 it holds that T1 ├ s = t T2 ├ s = t
Extension of Equational theory
Axioms of BPA(A):
(A1) x+ y = y+x
(A2) (x+y) + z = x+ (y + z)
(A3) x + x = x
(A4) x+ 0 = x
New Axioms:
(NA1) 0 + x = x
(NA2) 0 = 1
E1
E2
10 Process Algebra (2IF45)
Conservative Ground extension of T1 with T2: T1 = (1, E1) and T2 = (2, E2) are two equational theories. If 1. T2 ground extension of T1 and
2. for any closed terms s and t in T1 it holds that T2 ├ s = t T1 ├ s = t
Extension of Equational theory
11 Process Algebra (2IF45)
Conservative ground extension of T1 with T2: T1 = (1, E1) and T2 = (2, E2) are two equational theories. If 1. T2 ground extension of T1 and
2. for any closed terms s and t in T1 it holds that T2 ├ s = t T1 ├ s = t
Extension of Equational theory
Axioms of BPA(A):
(A1) x+ y = y+x
(A2) (x+y) + z = x+ (y + z)
(A3) x + x = x
(A4) x+ 0 = x
New Axioms:
(NA1) 0 + x = x
E1
E2
12 Process Algebra (2IF45)
Conservative ground extension of T1 with T2: T1 = (1, E1) and T2 = (2, E2) are two equational theories. If 1. T2 ground extension of T1 and
2. for any closed terms s and t in T1 it holds that T2 ├ s = t T1 ├ s = t
Extension of Equational theory
Axioms of BPA(A):
(A1) x+ y = y+x
(A2) (x+y) + z = x+ (y + z)
(A3) x + x = x
(A4) x+ 0 = x
New Axioms:
(NA1) 0 + x = x
(NA2) 0 = 1
E1
E2
13 Process Algebra (2IF45)
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
Extension of Equational theory
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
New Axioms in BPA+(A):…..New deduction rules for BPA+(A):
…..
14
Extension of BPA(A) with Projection operators
- Intuition what we want this operators to capture
Process Algebra (2IF45)
15
Extension of BPA(A) with Projection operators
- Intuition what we want this operators to capture
- OK! Now we can make axioms and later SOS rules
Process Algebra (2IF45)
16 Process Algebra (2IF45)
Language: BPAPR(A)
Signature: 0, 1, (a._ )aA, +
n(_), n 0
Language terms T(BPAPR(A))Axioms of BPAPR(A):
(A1) x+ y = y+x
(A2) (x+y) + z = x+ (y + z)
(A3) x + x = x
(A4) x+ 0 = x
(PR1) n(1) = 1
(PR2) n(0) = 0
(PR3) 0(a.x) = 0
(PR4) n+1(a.x) = a. n(x)
(PR5) n(x+y) = n(x) + n(y)
BPA(A)
BPAPR(A)
Extension of BPA(A) with Projection operators
17 Process Algebra (2IF45)
BPAPR(A) is a ground extension of BPA(A) (easy to conclude)
Extension of Equational theory
BPAPR(A) is a conservative ground extension of BPA(A)
18 Process Algebra (2IF45)
BPAPR(A) is a ground extension of BPA(A).
Extension of Equational theory
BPAPR(A) is a conservative ground extension of BPA(A).
Is BPAPR(A) more expressive than BPA(A)?
19 Process Algebra (2IF45)
If p is a closed terms in BPAPR(A), then there is a closed term q in BPA(A) such that BPAPR(A) ├ p = q.
Elimination theorem for BPAPR
20 Process Algebra (2IF45)
Operational semantics of BPAPR
21
New deduction rules for BPAPR(A):
Process Algebra (2IF45)
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
Extension of Equational theory
Language: BPAPR(A)
Signature: 0, 1, (a._ )aA, +, n(x), n 0 Language terms T(BPAPR(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
New Axioms in BPAPR(A):(PR1) n(1) = 1
(PR2) n(0) = 0
(PR3) 0(a.x) = 0
(PR4) n+1(a.x) = a. n(x)
(PR5) n(x+y) = n(x) + n(y)
x n (x)
x x’ n +1(x) n (x’)
aa