Process Algebra (2IF45)
Basic notions of Equational theory
Dr. Suzana Andova
Deduction system for “Counting down”
Language (signature)
0
“zero”
s(_)
“successor function”
a(_, _) “addition”
m(_, _) “multiplication”
Deduction rules
1
s(x) 
x
0 
1
y
y’

1 a(x, y’)
a(x,y) 
x, y
a(x,y) 
Terms
s(s(0))
Terms
a(s(0),s(0))
s(s(0))


1
x
x’ , y
1
a(x,y) 
x’

a(s(0),s(0))
1
s(0)
1
0
1
Process Algebra (2IF45)
1
a(s(0),0)
1
0

Deduction system for “Counting down”
Language (signature)
0
“zero”
s(_)
“successor function”
a(_, _) “addition”
m(_, _) “multiplication”
Deduction rules
1
s(x) 
x
0 
1
y
y’

1 a(x, y’)
a(x,y) 
x, y
a(x,y) 
Terms
s(s(0))
Terms
a(s(0),s(0))
s(s(0))


1
x
x’ , y
1
a(x,y) 
x’

a(s(0),s(0))
1
s(0)
1
0
2
Process Algebra (2IF45)
1
a(s(0),0)
1
0

Deduction system vs. Equational theory
Language (signature)
0
s(_)
a(_, _)
Terms
“zero”
“successor function”
“addition”
e.g. a(s(0),s(0)), 0,. a(x,y)
Deduction rules
1 x
s(x) 
0
y 1 y’
1 a(x, y’)
a(x,y) 
1 x’ , y
x
1 x’
a(x,y) 
x, y
a(x,y) 
Equalities on terms
Equivalence relation

s(s(0))
a(s(0),s(0))
1
s(0)
a(s(0),0)
1
0
3
…├ a(s(0),s(0)) = s(s(0))
1
1
0
Process Algebra (2IF45)
Equational theory
reduction
on specification
componts’ specifications
reduction
on specification
the whole system specification
• simpler
• smaller
• in a particular form (basic)
•…
reduction
on LTSs
the state space
SOS rules
4
composition by axiom
Process Algebra (2IF45)
Example: Towards Equational theory for
Arithmetic of Natural numbers P(eano) A(rith.)
a(0,s(0))

s(0)
1
0
a(0,0)

0
1
0
So we need a set of axioms from which we can derive:
PA ├ a(0,s(0)) = …..= s(0) and also
PA ├ a(0,0) = …..= 0 and also
…..
1. Maybe axioms
2. Maybe axioms
(Ax11) a(0,s(0)) = s(0)
(Ax12) a(0,0) = 0
5
(Ax21) a(0,s(x)) = s(x)
(Ax22) a(0,0) = 0
Process Algebra (2IF45)
3. Maybe axioms
(Ax31) a(0,y) = y
Example: Towards Equational theory for
Arithmetic of Natural numbers PA (cont.)
a(0,s(0))

s(a(0,0))
1
0
1
0
So we need a set of axioms from which we can derive:
PA ├ a(0,s(0)) = …..= s(a(0,0)) and also
PA ├ a(s(0),s(0)) = …..= s(s(0))
…..
1’. Maybe axioms
(Ax11) a(s(0),s(0)) = s(s(0))
(Ax12) a(0,s(0)) = s(0)
2’. Maybe axioms
3’. Maybe axioms
(Ax31) a(x,s(y)) = s(a(x,y))
(Ax32) a(s(x),y) = s(a(x,y))
(Ax21) a(s(x),s(y)) = s(s(a(x,y)))
6
Process Algebra (2IF45)
Example: Towards Equational theory for
Arithmetic of Natural numbers PA (cont.)
1. Maybe axioms
3. Maybe axioms
(Ax11) a(0,s(0)) = s(0)
(Ax12) a(0,0) = 0
(Ax31) a(0,y) = y
(Ax11) a(s(0),s(0)) = s(s(0))
(Ax12) a(0,s(0)) = s(0)
2. Maybe axioms
(Ax31) a(x,s(y)) = s(a(x,y))
(Ax32) a(s(x),y) = s(a(x,y))
(Ax21) a(0,s(x)) = s(x)
(Ax22) a(0,0) = 0
(Ax21) a(s(x),s(y)) = s(s(a(x,y)))
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Process Algebra (2IF45)
Example: Towards Equational theory for
Arithmetic of Natural numbers PA (cont.)
1. Maybe axioms
3. Maybe axioms
(Ax11) a(0,s(0)) = s(0)
(Ax12) a(0,0) = 0
(Ax31) a(0,y) = y
(Ax11) a(s(0),s(0)) = s(s(0))
(Ax12) a(0,s(0)) = s(0)
2. Maybe axioms
(Ax31) a(x,s(y)) = s(a(x,y))
(Ax32) a(s(x),y) = s(a(x,y))
(Ax21) a(0,s(x)) = s(x)
(Ax22) a(0,0) = 0
(Ax21) a(s(x),s(y)) = s(s(a(x,y)))
8
Process Algebra (2IF45)
Example: Towards Equational theory for
Arithmetic of Natural numbers PA (cont.)
1. Maybe axioms
3. Maybe axioms
(Ax11) a(0,s(0)) = s(0)
(Ax12) a(0,0) = 0
(Ax31) a(0,y) = y
(Ax11) a(s(0),s(0)) = s(s(0))
(Ax12) a(0,s(0)) = s(0)
(Ax31) a(x,s(y)) = s(a(x,y))
(Ax32) a(s(x),y) = s(a(x,y))
3. Maybe axioms
(Ax31) a(0,y) = y
(Ax21) a(s(x),s(y)) = s(s(a(x,y)))
? but can we derive here
a(s(x),0) = s(x) ?
9
Process Algebra (2IF45)
Example: Equational theory for Arithmetic of
Natural numbers PA (cont.)
3. Maybe axioms
(Ax31) a(0,y) = y
Axioms in PA
(PA1) a(x,0) = x
(PA2) a(x,s(y)) = s(a(x,y))
10
Process Algebra (2IF45)
(Ax31) a(x,s(y)) = s(a(x,y))
(Ax32) a(s(x),y) = s(a(x,y))
Equational Theory
Question:
• What did we have to take into account during our quest for the right
set of axioms?
Answer:
• Consistency between
bisimulation 
and
derivation in PA using the axioms
11
Process Algebra (2IF45)
Consistency
Questions:
• How do you understand “consistency” between the deduction
system (semantics) and the equational theory?
Answer:
• Consistency is two-directional
1. Everything that I am able to derive equal using the axioms
must be bisimilar, namely
if
PA ├ t = r then t  r
2. Everything that I can show bisimilar, I have to be able to
derive equial using the axioms, namely
if
12
t r
then PA ├ t = r
Process Algebra (2IF45)
Consistency
Questions:
• How do you understand “consistency” between the deduction
system (semantics) and the equational theory?
Answer:
• Consistency is two-directional
1. Everything that I am able to derive equal using the
axiomsmust be bisimilar, namely
if
PA ├ t = r then t  r
2. Every closed terms that I can show bisimilar, I have to be
able to derive equial using the axioms, namely
if
13
t r
then PA ├ t = r, for t and r closed terms
Process Algebra (2IF45)
Prerequisites for soundness of axioms
Questions:
• Are we on a safe ground now?
Soundness:
Everything that I am able to derive equal using the axioms must be
bisimilar, namely
if
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PA ├ t = r then t  r
Process Algebra (2IF45)
Derivation in equational theory
Questions:
• What is allowed (can be used) in derivation?
PA ├ t = r
1. Axioms (as ‘basic’ equalities)
2. Properties of = such as
• reflexivity PA ├ t = t
• symmetry if PA ├ t = r then PA ├ r = t
• transitivity if PA ├ t = r and PA ├ r = u then PA ├ t = u
3. Substitution (substituting variables by terms)
e.g. from PA ├ a(x,y) = a(y,x) follows PA ├ a(s(0),0) = a(0, s(0))
where s(0)/x and 0/y
4. Context rule
15
Process Algebra (2IF45)
Derivation in equational theory
Questions:
• What is allowed (can be used) in derivation?
PA ├ t = r
4. Context rule
e.g. if PA ├ t1 = r1 and PA ├ t2 = r2
then PA├ s(t1) = s(r1)
and also PA├ a(t1 ,t2) = a(r1, r2) …
‘Mapping’ the context rule back to semantics and 
e.g. if PA ├ t1  r1 and PA ├ t2  r2
then PA├ s(t1)  s(r1)
and also PA├ a(t1 ,t2)  a(r1, r2) …
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Process Algebra (2IF45)
Prerequisites for soundness
‘Mapping’ the context rule back to semantics and 
e.g. if t1  r1 and t2  r2
then s(t1)  s(r1)
and also a(t1 ,t2)  a(r1, r2) …
Since
a(0,s(0))

s(a(0,0))
1
0
then
s(a(0,s(0)))

s(s(a(0,0)))
1
1
0
a(0,s(0))
1
s(a(0,0))
1
0
17
Process Algebra (2IF45)
1
0
Prerequisites for soundness
Congruence relation
Congruence relation
1. it is equivalence relation
• reflexive
• symmetric
• transitive
2. it is preserved by any context C[ _ ]
if t1  r1 then C[t1]  C[r1]
Examples:
Bisimilarity is congruence on the set of all terms
18
Process Algebra (2IF45)
Prerequisites for soundness
Congruence relation
complete trace equivalent LTSs {?coin !coffee, ?coin !tea}
?coin
?coin
?coin
put in the context
“communicate with the
User”
!coin
!coffee
!tea
!coffee
coin
?coffee
!tea
coin
coin
coffee
coffee
NOT complete trace equivalent LTSs {coin coffee} vs. {coin coffee, coin}
19
Process Algebra (2IF45)
Towards ground completeness
Questions:
• Is the set of axioms in PA (PA1) a(x,0) = x
(PA2) a(x,s(y)) = s(a(x,y))
ground complete wrt the derivation rules and  ?
Every closed terms that I can show bisimilar, I have to be able to
derive equal using the axioms, namely
if
t r
then PA ├ t = r, for t and r closed terms
• What about PA ├ a(t,r) = a(r,t)? Can we derive this equality from the
two axioms, for closed terms t and r?
20
Process Algebra (2IF45)
Towards ground completeness
Working with closed terms:
1. Basic terms
• defined in a structural way, such as:
I. 0 is a basic term in PA
II. if t is a basic term in PA then s(t) is also a basic term in PA
• Thus, in PA basic terms are 0, s(0), s(s(0)), ….
2.
Elimination of ‘other’ operators from the signature
• In PA every closed term can be rewritten to a basic term, e.g.
PA ├ a(s(0),a(s(0),0)) = s(s(0))
•
the elimination proof goes by structural induction on the
structure on closed terms
3. Any property on closed terms can be proven by structural
induction on basic terms only instead on all closed terms
21
Process Algebra (2IF45)
Towards ground completeness
Property1: For any closed term t, PA ├ a(0, t) = t.
Proof: By structural induction on basic terms (knowing the
elimination property)
Basic step t  0. Directly from PA1
Inductive step t  s(t’) for some basic term t’. Derive
PA ├ a(0, t) = a(0, s(t’)) = s(a(0,t’)) = s(t’) = t
PA2
22
by
induction
Process Algebra (2IF45)
Towards ground completeness
Property2: For any closed term t and r, PA ├ a(s(t), r) = a(t, s(r)).
Proof: By structural induction on basic terms (knowing the
elimination property)
Basic step r  0.
PA ├ a(s(t), r) = a(s(t), 0) = s(t) by axiom PA1
PA ├ a(t, s(r)) = a(t, s(0)) = s(a(t, 0)) = s(t) by PA2 and PA1
Inductive step r  s(r’) for some basic term r’. Derive
PA ├ a(s(t), r) = a(s(t), s(r’)) = s( a(s(t), r’) ) = s( a(t, s(r’)) )
PA2
= a( t, s(s(r’)) ) = a(t, s(r))
PA2
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Process Algebra (2IF45)
by
induction
Towards ground completeness
Property3: For any closed term t and r, PA ├ a(t, r) = a(r, t).
Proof: Home Work
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Process Algebra (2IF45)
Process Algebra (2IF45)
Basic Process Algebra
Dr. Suzana Andova