0. Process Algebra (2IF45) Introduction From Labeled Transition Systems back to Equational Theory
Dr. Suzana Andova

1. Reactive systems – general
Computing systems which are supposed to offer some (well-defined) services to their users
These systems are large and usually consist of a number of components which interact with each other
Many of them are control crucial and/or safety critical
Process Algebra (2IF45)

2. Reactive systems - Correctness problem
It is important that a realization of these systems is just as intended
Intended behaviour can be validated
Solution: MODELING
abstract model of the system
unambiguous description
methods and tools for model analysis
Process Algebra (2IF45)

3. Model of Labeled Transition Systems (LTS)
in(x);
y:= x+1;
while (true) {
out(y); }. ?x
y:=x+1 !y
out(x);
in(y). !x ?y
x:= 1;
y:= x+1;
out(y). ?x
y:=x+1 !y
Process Algebra (2IF45)

4. Model of Labeled Transition Systems (LTS) Intermezzo
Process Algebra (2IF45)

5. Model of Labeled Transition Systems
VM1
?return
!tea
!coffee
VM2
?coin
?coin
?return
!tea
!coffee
error
VM3
!tea
!coffee
?coin
Using VM1
coin
coffee
!coin
User
?coffee
?coin
!coffee
!tea
Process Algebra (2IF45)

6. Model of Labeled transition systems
VM1
VM2’
VM3
?return
?return
error
?coin
!tea
?coin
!coffee
?coin
?coffee
?tea
!tea
!tea
!coffee
!coffee
User
!coin
!coffee
?coffee
Process Algebra (2IF45)

7. Observation
LTSs consist of states and transitions labeled with (action) labels
Initial state is indicated, final states are indicated
LTSs can interact according to predefined communications
Process Algebra (2IF45)

8. Questions What is a state?
How do we know drawing a transition from a state s to a state s’ is right? How do we know which label to assign to it?
How do we combine LTSs?
When modeling a system, is an LTS a model to start with or is it something to be obtained as a final or side product?
What ingredient do we need to have predefined, to be able to produce an LTS?
Process Algebra (2IF45)

9. Answers
What is a state?
A state is an expression in a specification language (signature)
How do we know whether drawing a transition from a state s to a state s’ is right? How do we know which label to assign to it?
A transition and its label are determined by a set of rules, defining the behaviour of any specification given in the language
How do we combine LTSs?
Any composition (operation) on LTSs must be predefined in the language and with the rules
Process Algebra (2IF45)

10. Answers
When modeling a system, is an LTS a model to start with or is it something to be obtained as a final or side product?
What ingredient do we need to have predefined, to be able to produce and work with LTSs?
Process Algebra (2IF45)

11. Answers offers flexibility
When modeling a system, is an LTS a model to start with or is it something to be obtained as a final or side product?
In (model checking) tools
manipulating
the state space (LTSs):
UPPAAL, Prism, MRMC
manipulating
the specification (language):
mCRL2, Chi, CADP, FDR, PEPA,
MRMC updated IMC
offers flexibility
reduction
on specification
components’ specifications
reduction
on specification …
the whole system specification
composition by axiom
reduction
on LTSs
No!
the state space
SS generation by the SOS rules
verification
model checking
property specification
Yes!
Process Algebra (2IF45)

12. Equational theory in place
In (model checking) tools
manipulating
the state space (LTSs):
UPPAAL, Prism, MRMC
manipulating
the specification (language):
mCRL2, Chi, CADP, FDR, PEPA,
MRMC updated IMC
reduction to basic forms
reduction
on specification
equiational theory (axioms)
reduction by equations
components’ specifications
reduction
on specification …
the whole system specification
Operational semantics
(SOS)
composition by axiom
reduction by equivalence relations (bisimulation)
reduction
on LTSs
No!
the state space
SS generation by the SOS rules
verification
model checking
property specification
Yes!
Process Algebra (2IF45)

13. Equational theory in place
In (model checking) tools
manipulating
the state space (LTSs):
UPPAAL, Prism, MRMC
manipulating
the specification (language):
mCRL2, Chi, CADP, FDR, PEPA,
MRMC updated IMC
reduction to basic forms
reduction
on specification
equiational theory (axioms)
reduction by equations
consistent
components’ specifications
reduction
on specification …
the whole system specification
Operational semantics
(SOS)
composition by axiom
reduction by equivalence relations (bisimulation)
reduction
on LTSs
No!
the state space
SS generation by the SOS rules
verification
model checking
property specification
Yes!
Process Algebra (2IF45)

14. Equational theory in place
In this course we will learn HOW to build a consistent
Process Algebra = specification language
+ axioms
+ SOS rules
+ reduction equivalence relations
so that the initial specification and the model checked LTS, they both describe the same system!
In (model checking) tools
manipulating
the state space (LTSs):
UPPAAL, Prism, MRMC
manipulating
the specification (language):
mCRL2, Chi, CADP, FDR, PEPA,
MRMC updated IMC
reduction to basic forms
reduction
on specification
equiational theory (axioms)
reduction by equations
consistent
components’ specifications
reduction
on specification …
the whole system specification
Operational semantics
(SOS)
composition by axiom
reduction by equivalence relations (bisimulation)
reduction
on LTSs
No!
the state space
SS generation by the SOS rules
verification
model checking
property specification
Yes!
Process Algebra (2IF45)

15. Labeled transition systems – basic notions
Process Algebra (2IF45)

16. Labeled transition systems – basic notions
!tea
!coffee
VM1’
?coin
Given a set of labels L
An LTS consists of:
S is a set of states
  S x L x S
S0 S is the initial state
  S is the set of final states
nondeterministic choice
successful termination
deadlock
state
Process Algebra (2IF45)

17. Labeled transition systems - choice
VM1’
VM1’’
nondeterministic choice
?coin
?coin
?coin
nondeterministic choice
!coffee
!tea
!coffee
!tea
Using VM1’
coin
coffee 
Using VM1’’
coffee
coin
 or 
Process Algebra (2IF45)

18. Labeled transition systems - relations
VM1’
VM1’’
?coin
?coin
?coin
!coffee
!tea
!coffee
!tea
Similarities of the LTSs:
they both have the same traces, {?coin, ?coin !coffee, ?coin !tea}
Differences of the LTSs:
The moment a choice is made is different.
In VM1’ the choice is made before ?coin is executed.
In VM1’’ the choice is made after ?coin is executed.
Process Algebra (2IF45)

19. LTS Equivalence spectrum
Rob J. van Glabbeek “The Linear Time-Branching Time Spectrum”, CONCUR 1990
Process Algebra (2IF45)

20. Bisimulation on LTSs
!coffee
?coin
Bisimilar LTSs
Bisimulation relation: A binary relation R on the set of state S of an LTS is bisimulation relation iff the following transfer conditions hold:
for all states s, t, s’  S, whenever (s, t)  R and s –a-> s’ for some a  L, then there is a state t’  S such that t –a-> t’ and (s’, t’)  R;
vice versa, for all states s, t, s’  S, whenever (s, t)  R and t –a-> t’ for some a  L, then there is a state s’  S such that s –a-> s’ and (s’, t’)  R;
whenever (s, t)  R and s  then t ;
whenever (s, t)  R and t  then s ;
Two LTSs s and t are bisimilar, s  t, iff there is a bisimulation relation R such that (s, t)  R
Process Algebra (2IF45)

21. Bisimulation on LTSs
!coffee
?coin
Bisimilar LTSs
!tea
!coffee
?coin
Not bisimilar LTSs ??
Bisimulation relation: A binary relation R on the set of state S of an LTS is bisimulation relation iff the following transfer conditions hold:
for all states s, t, s’  S, whenever (s, t)  R and s –a-> s’ for some a  L, then there is a state t’  S such that t –a-> t’ and (s’, t’)  R;
vice versa, for all states s, t, s’  S, whenever (s, t)  R and t –a-> t’ for some a  L, then there is a state s’  S such that s –a-> s’ and (s’, t’)  R;
whenever (s, t)  R and s  then t ;
whenever (s, t)  R and t  then s ;
Two LTSs s and t are bisimilar, s  t, iff there is a bisimulation relation R such that (s, t)  R
Process Algebra (2IF45)

22. Structural Operational Semantics – general introduction
components’ specifications
the whole system specification
the state space …
Process Algebra (2IF45)

23. Structural Operational Semantics – general introduction
Ingredients
A set of labels L
Language  (signature/ syntax) : consists of symbols denoting constants, operators, variables, functions, additional symbols
All expressions (terms) in the language are build from the symbols in the signature, denoted C()
An expression corresponds to a state in a state space (LTS)
Example: Language of Natural numbers
0 “zero”
s(_) “successor function”
a(_, _) “addition”
m(_, _) “multiplication”
Terms in the language: s(s(0)), a(s(0),m(s(0),s(s(s(0))))), 0,
s(x) where x is a variable, …
Process Algebra (2IF45)

24. Structural Operational Semantics – general introduction
Ingredients (cont.)
Deduction (SOS) rules
Rules are in the form  
where
 is a set of formulas called premises; it can be an empty set
 is a formula called conclusion
Formula is either a transition s –a-> t or a termination s for some terms s and t in the language, s, t  C() and a label a  L
Deduction rules determine transitions in a LTS
A language and a set of rules defined over the language is called deduction system
Process Algebra (2IF45)

25. Example: Deduction system for “Counting down”
Example: Language of Natural numbers
0 “zero”
s(_) “successor function”
a(_, _) “addition”
m(_, _) “multiplication”
Question: How to define deduction rules that generate the following LTS
s(s(0)) 1
s(0) 1
Process Algebra (2IF45)

26. Example: Deduction system for “Counting down”
Example: Language of Natural numbers
0 “zero”
s(_) “successor function”
a(_, _) “addition”
m(_, _) “multiplication”
Question: How to define deduction rules that generate the following LTS
s(s(0))
s(x)  x 1 0   1 1
y  y’
a(x,y)  a(x, y’)
s(0) 1
x  x’ , y
a(x,y)  x’   1
x, y
a(x,y)  
Process Algebra (2IF45)

27. Example: Deduction system for “Counting down”1 1 1
a( s(s(0)), s(s(s(0))) )
a( s(s(0)), s(s(0)) )
a( s(s(0)), s(0) )
a( s(s(0)), 0 )    1 1  
s(0)  
s(x)  x 1 0   1
y  y’
a(x,y)  a(x, y’) 1
x  x’ , y
a(x,y)  x’  
x, y
a(x,y)  
Process Algebra (2IF45)

28. Example: Deduction system for “Counting down” --- Alternative rules
Exercise: Write an alternative rules for the
Counting down deduction system!