Process Algebra (2IF45) Some Extensions of Basic Process Algebra Dr. Suzana Andova
Process Algebra (2IF45) Some Extensions of Basic Process Algebra
Feb 06, 2016
Process Algebra (2IF45) Some Extensions of Basic Process Algebra. Dr. Suzana Andova. 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 - PowerPoint PPT Presentation
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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
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