# Advanced Topics in SE Spring 2006 1 Process Algebra Hossein Hojjat Formal Methods Lab University of...

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Process AlgebraHossein HojjatFormal Methods LabUniversity of Tehran

IntroductionThe lambda calculus models the functional computation completelyLambda calculus is unique: There is only one way to observe a functional computation watch which output values yields when presented with different input valuesBut the concurrent computation is not neat as functionalThere are many aspects to concurrent programming

Introduction (cont.)Concurrent systems can be described in terms of many different constructsFor creating processes fork/waitcobegin/coend, etc.Exchanging information between processesshared memoryrendezvousmessage-passingdata flow, etc.

Introduction (cont.)For managing the shared resourcessemaphoresmonitorstransactionsThis variability has rise to a large class of formal systems: Petri nets, event structures, actor model,In the process algebra approach every thing is considered to be a process

Introduction (cont.)We need not distinguish betweenactive components like senders and receiverspassive components such as communication mediaAll of the processes interact via synchronous message passingThis is the only basic mode of interaction

Process AlgebraProcessA behavior of a systemA system is anything showing behaviorSoftware system, actions of a machine, etc.AlgebraWe take an algebraic/axiomatic approach in talking about processesBy using axioms, we can perform calculations with processes

Process Algebra: Automata Theory ExtensionAn automata has a number of states and a number of transitionsThere are some initial and final statesA behavior is a run: a path from initial state to final stateTwo automata are equivalent when their languages are equal

Process Algebra: Automata Theory Extension (cont.)The automata model lacks the notion of interactionDuring the execution from initial to final state, a system may interact with other systemsThis is needed in modeling the reactive systems

Well Known Process AlgebrasIn the early eighties, Milner introduced CCS, Hoare introduced CSP and Bergstra introduced ACP calculus: In 1989, Milner extended CCS to take full advantage of named channelsPEPA (Performance Evaluation Process Algebra): a stochastic process algebraAmbient calculus: a process calculus used to describe and theorize about concurrent systems that include mobility

CCSThe Calculus of Communicating Systems (or CCS) is a process calculus developed by Robin Milner

This lecture mainly focus on the CCS process algebra

The language CCSProcesses are called agents, built from a set of actionsActions can be:Observable, represented by letters a, b, etc.Unobservable (silent), marked by Observable actionsa, b, input actionsa, b, output actionsInput action a and output action a are complementary

The language CCS (cont.)If a means the reception of message a from the in port, the action a denotes the writing of the message to the out portThe following identity describes how the complementation of messages worksa = aThe most basic operator is 0 (read nill)Performs no action whatsoever

OperationsAgents may be constructed by employing process algebraic operations on other agentsThere are primarily five different methods of constructing agentsAction prefixingChoice operatorParallel compositionRestrictionRelabeling

Action PrefixingThe most basic process constructor in CCS is action prefixingIf a is an action and P is a process, then a.P is a processThe "." here is called "action prefixing" and denotes sequentializationP becomes active only after the action a has been performed

a.PP

ExampleMatch = strike.0defCM = coin.coffee.CMdefClock = tick.Clockdef

Choice OperatorIf P and Q are processes, then so is P + QThe process P + Q has the initial capabilities of both P and QChoosing to perform initially an action from P will preempt the further executions of actions from Q, and vice versa

Example

After having input a coin, the process CTM is willing to deliver either tea or coffeeIt depends upon the customers choiceteacoffee

Parallel CompositionLet Q and R be processes, (Q | R) is a processThe semantics of parallel composition:1. If M M, and N cannot perform , then M | N M | N2. If N N and M cannot perform , then M | N M | N3. If M M, and N N, then M | N M | N

ExampleCM = coin.coffee.CMdefCS = pub.coin.coffee.CSdefCS | CM

CommunicationThe computer scientist and coffee machine may communicate via the portsGiven two CCS expression P and Q, the process P | Q describes a system whichP and Q may proceed independentlyMay communicate via complementary ports

ExampleCMCScoincoffeecoffeecoinpubCM | CSCM|CS|CSCMCScoincoffeecoffeecoinpubCScoincoffeepub

RestrictionLet Q be a process and be a set of visible actions, . Then (Q) \ is a processIf Q Q and , , then ((Q) \ ) ((Q) \ )Process (Q) \ is like Q, but it cannot execute actions in the set or their complement actionsThe scope of port names in is restricted to Q

Example((CM|CS)\coin\coffee)|CSCMCSpubCScoincoffeepub

RelabelingLet Q be an agent and the set of its visible actions, Let be a set of actions and let m: U { } U { } such that m() = and : m()=m(). Then P ::= Q[m] is a processIf Q Q then Q[m] Q[m]Process P is like Q, but it actions have obtained by mapping the actions Q through m

ExampleVendingMachine = coin.item.VendingMachineCokeMachine = VendingMachine[coke/item]defCoffeeMachine = VendingMachine[coffee/item]def

Algebraic Operator Precedence1. Restriction and relabeling P\L P[f]2. Action prefixing .P3. Parallel composition P|Q4. Summation P+Q

Q: How is R+a.P|b.Q\L then to be read ?A: R+((a.P)|(b.(Q\L))) !tightest

Semantics1. Operational SemanticsSpecifies the behavior of a programming language by defining a simple abstract machine for it2. Denotational SemanticsMap programs to denotationsDenotations are abstract representations details3. Axiomatic SemanticsInstead of first defining the behaviors of programs and then deriving laws from this definition, take the laws themselves as the definition of the language

SOSStructural Operational SemanticsDeveloped by Gordon Plotkin in 1981Represents computation by means of deductive systemsThese systems turn the abstract machine into a system of logical interferencesWith SOS semantics definitions are given by inference rules

InferenceInference rule consists of a conclusion that follows from a set of premises, possibly under control of some conditionspremise1premise2premisenconclusioncondition

SOS of CCSThe prefix is the only axiom in the system

The parallel composition requires two symmetric rules:

So as choice operator:QQP + QQ

SOS of CCS (cont.)Communication may happen when compatible prefixes are allowed

Restriction and relabeling rulePPP | QP | QQQPPP \ LP \ L, LPPP [ f ]P [ f ]f()

Labeled transition systemSOS formally defines for each process term the labeled transition system that it stands forA labeled transition system (LTS) is a triple(S, Act, ) consisting ofa set S of statesa set Act of (action) labelsa transition relation S Act Sif (s,,s) we write s s

ExampleC0=is_0? . C0 + set_1 . C1 + set_0 . C0

C1=is_1? . C1 + set_0 . C0 + set_1 . C1C0C1set_1set_0set_0is_0?set_1is_1?

Next sessionsimulationbisimulationbranching bisimulation