Prabhat Ppt

8/7/2019 Prabhat Ppt

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PRESENTATION

on

Axiomatizations

for probabilistic finite-state behavioursBy:Prabhat kumar(A3/64)

Under the guidence of:Mr. ArpitDepartment of CSE

IERT Allahabad


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CONTENTS

Introduction

AxiomatizationProbabilistic automata

Probabilistic models

Axiomatic systemBehavioral equivalences

Conclusion


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INTRODUCTION

The reason why we are interested in studying a model which expresses both nondeterministic and probabilistic behavior, and an equivalence sensitive to divergency, is that one of the long-term goals of this line of research is to

develop a theory which will allow us to reason about probabilistic algorithms used in distributed computing. In that domain it is important to ensure that an algorithm will work under any scheduler, and under other unknown oruncontrollable factors.


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Axiomatization

Axiomatization is the formulation of a system

ofstatements(i.e.axioms)thatrelateanumber

of primitive terms in order that

a consistent body of propositions may be

derived deductively from these statements.

Thereafter, theproofofanypropositionshouldbe,inprinciple,traceablebacktotheseaxioms.


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PROBABILISTIC AUTOMATA

Basically, a probabilistic automaton is just an ordinary automaton (also

called labeled transition system or state machine) with the only difference

thatthetargetofatransitionisaprobabilisticchoiceoverseveralnextstates.


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PROBABILISTIC MODELS

In 1995 van Glabbeek et al. classify probabilisticmodels into following three models:-

Reactive

Generative

Stratified

After that Segala pointed out that neitherreactive nor generative nor stratified models

capture real nondeterminism, an essentialnotion for modeling scheduling freedom. Hethen introduced two models:-

Probabilistic automata (PA)

Simple probabilistic automata (SPA)


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1) Inreactivemodels,eachlabeledtransitionisassociatedwithaprobability,andforeachstatethesumoftheprobabilitieswiththesame

labelis1.

2) Ingenerativemodelsforeachstatethesumoftheprobabilitiesofalltheoutgoingtransitionsis1.

3) Stratifiedmodelshavemorestructureandforeachstateeitherthereisexactlyoneoutgoinglabeledtransitionorthereareonly

unlabeledtransitionsandthesumoftheirprobabilitiesis1.

4) SPAis asimplifiedversionofPAcalledsimpleprobabilisticautomatawhicharelikeordinaryautomataexceptthatalabeled

transitionleadstoaprobabilisticdistributionoverasetofstatesinsteadofasinglestate.

5) InProbabilisticautomata(PA),bothprobabilityandnondeterminismaretakenintoaccount.Probabilisticchoiceisexpressedbythenotionoftransition,which,inPA,leadstoaprobabilisticdistributionoverpairs(action,state)anddeadlock.Nondeterministic

choice,ontheotherhand,isexpressedbythepossibilityofchoosingdifferenttransitions


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AXIOMATIC SYSTEM

Amathematicaltheoryconsistsofanaxiomaticsystemandallitsderivedtheorems.

Anaxiomatic systemthatiscompletelydescribedisaspecial kindofformal system; usuallythough theeffort towards complete formalisation brings diminishing returns in certainty, and a lack of readability

forhumans.Thereforediscussiono A axiomatic system is any set of axioms from which some or all axioms can be used in conjunction

tologicallyderivetheorems.

f axiomatic systems is normally only semi-formal. A formal theory typically means an axiomaticsystem,forexampleformulatedwithinmodeltheory.

Aformalproofisacompleterenditionofamathematicalproofwithinaformalsystem.


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BEHAVIORALEQUIVALENCES

Strong bisimulationStrong probabilistic bisimulation

Weak bisimulation

Weak probabilistic bisimulationDivergency-sensitive equivalence

Observational equivalence


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CONCLUSION

Here we have proposed a probabilistic process calculus which corresponds tosegalas probabilistic automata We have presented strong bisimulation, strongprobabilisticbisimulation.Wehaveaxiomatized divergency-sensitive equivalence

and observational equivalenceonlyforguardedexpressions.Weconjecturethatthe two behavioral equivalences are undecidable and therefore not finitelyaxiomatizable.In thefuture itmight beintresting toseehow to refineour processalgebratoallowforparallelcomposition.


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