Process Algebra (2IF45)

Introduction From Labeled Transition Systems back to Equational Theory

Dr. Suzana Andova

2 Process Algebra (2IF45)

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

3 Process Algebra (2IF45)

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

4 Process Algebra (2IF45)

Model of Labeled Transition Systems (LTS)

x:= 1;y:= x+1;out(y).

?x

y:=x+1

!y

in(x);y:= x+1;while (true) { out(y);}.

?x

y:=x+1

!y

out(x);in(y).

!x

?y

5 Process Algebra (2IF45)

Model of Labeled Transition Systems (LTS)Intermezzo

6 Process Algebra (2IF45)

Model of Labeled Transition Systems

!tea ?coin !coffee

VM1

?coin

?return

!tea !coffee

error

VM3

?return

!tea !coffee

VM2

?coin

!coin

User

?coffee

Using VM1

coin

coffee

?coin !coffee !tea

7 Process Algebra (2IF45)

Model of Labeled transition systems

!tea ?coin !coffee

!coin

?return

!tea !coffee

?coin

?return

!tea !coffee

error

VM1 VM2’ VM3

User

?coin

!coffee

?coffee

?tea

?coffee

8 Process Algebra (2IF45)

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

9 Process Algebra (2IF45)

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?

10 Process Algebra (2IF45)

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

11 Process Algebra (2IF45)

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?

12 Process Algebra (2IF45)

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?

In (model checking) tools

manipulating the state space (LTSs):UPPAAL, Prism, MRMC

manipulating the specification (language):

mCRL2, Chi, CADP, FDR, PEPA, MRMC updated IMCoffers flexibility

components’ specifications

the whole system specification

the state space

verificationmodel checking

reductionon specification

reductionon specification

reductionon LTSs

composition by axiom

SS generation by the SOS rules

property specification

Yes!

No!

…

13 Process Algebra (2IF45)

components’ specifications

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

the whole system specification

the state space

verificationmodel checking

reductionon specification

reductionon specification

reductionon LTSs

composition by axiom

SS generation by the SOS rules

property specification

Yes!

No!

… equiational theory (a

xioms)

Operational semantics

(SOS)

reduction to

basic forms

reduction by

equations

reduction by

equivalence

relations

(bisimulation)

14 Process Algebra (2IF45)

components’ specifications

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

the whole system specification

the state space

verificationmodel checking

reductionon specification

reductionon specification

reductionon LTSs

composition by axiom

SS generation by the SOS rules

property specification

Yes!

No!

… equiational theory (a

xioms)

Operational semantics

(SOS)

reduction to

basic forms

reduction by

equations

reduction by

equivalence

relations

(bisimulation)

consistent

15 Process Algebra (2IF45)

components’ specifications

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

the whole system specification

the state space

verificationmodel checking

reductionon specification

reductionon specification

reductionon LTSs

composition by axiom

SS generation by the SOS rules

property specification

Yes!

No!

… equiational theory (a

xioms)

Operational semantics

(SOS)

reduction to

basic forms

reduction by

equations

reduction by

equivalence

relations

(bisimulation)

consistent

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!

16 Process Algebra (2IF45)

Labeled transition systems – basic notions

17 Process Algebra (2IF45)

Labeled transition systems – basic notions

• 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

!tea !coffee

VM1’

?coin

deadlock state

successful termination

nondeterministic choice

18 Process Algebra (2IF45)

Labeled transition systems - choice

!tea !coffee

VM1’

?coin

nondeterministic choice

!tea !coffee

VM1’’

?coin ?coin

nondeterministic choice

Using VM1’

coin

coffee

Using VM1’’

coffee

coin coin

or

19 Process Algebra (2IF45)

Labeled transition systems - relations

!tea !coffee

VM1’

?coin

!tea !coffee

VM1’’

?coin ?coin

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.

20 Process Algebra (2IF45)

LTS Equivalence spectrum

Rob J. van Glabbeek “The Linear Time-Branching Time Spectrum”, CONCUR 1990

21 Process Algebra (2IF45)

Bisimulation on LTSs

!coffee !coffee

?coin

!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:

1. 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;

2. 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;

3. whenever (s, t) R and s then t ; 4. 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

22

Bisimulation relation: A binary relation R on the set of state S of an LTS is bisimulation relation iff the following transfer conditions hold:

1. 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;

2. 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;

3. whenever (s, t) R and s then t ; 4. 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)

Bisimulation on LTSs

!coffee !coffee

?coin

!coffee

?coin

Bisimilar LTSs

!tea !tea !coffee

?coin ?coin

!coffee

?coin

Not bisimilar LTSs

??

23 Process Algebra (2IF45)

Structural Operational Semantics – general introduction

components’ specifications

the whole system specification

the state space

…

24 Process Algebra (2IF45)

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, …

25 Process Algebra (2IF45)

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

26

Example: Deduction system for “Counting down”

Process Algebra (2IF45)

• 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(0)

0

1

1

27

Example: Deduction system for “Counting down”

Process Algebra (2IF45)

• 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(0)

0

1

1

0

1

1 y y’

a(x,y) a(x, y’)

s(x) x1

1

1 x x’ , y a(x,y) x’

x, y a(x,y)

28

Example: Deduction system for “Counting down”

Process Algebra (2IF45)

1

0

1

1 y y’

a(x,y) a(x, y’)

s(x) x1

1

1 x x’ , y a(x,y) x’

x, y a(x,y)

a( s(s(0)), s(s(s(0))) ) a( s(s(0)), s(s(0)) )1

a( s(s(0)), s(0) )

1a( s(s(0)), 0 )

s(0)

1

01

29

Example: Deduction system for “Counting

down” --- Alternative rules

Exercise: Write an alternative rules for the Counting down deduction system!