Behavioral Equivalence Hossein Hojjat Formal Lab University of Tehran

Equivalence A concurrent system shouldn’t just be a single model (Petri net, process expression,…) Instead it is represented by an equivalence class of such objects The models give a representation that isn’t abstract enough For this purpose many equivalence notions have been proposed in the literature

Implementation Correctness Often equivalence relations are used to establish the correctness of implementations with respect to specifications of concurrent systems Suppose that P represents a specification Q represents an implementation P ~ Q states that the implementation is correct ~

Equivalence Notions Many different equivalence notions have been proposed in the literature Which aspects of system are crucial and which of them can be avoided We shall introduce bisimulation equivalence, which is an important semantic equivalence over CCS processes

Vending machine Consider a tea/coffee vending machine A = coin. (tea.A + coin.coffee.A) coin coffee tea A B C

Non-det vending machine Now consider a non deterministic vending machine A’ = coin. (tea.A’ + coin.coffee.A’) + coin.tea.A’ coin coffee tea B’ C’ A’ B0’B0’ tea coin

Equivalence coin coffee tea A B C coin coffee tea B’ C’ A’ B0’B0’ tea coin These two systems are language equivalent (why?)

Equivalence coin coffee tea A B C coin coffee tea B’ C’ A’ B0’B0’ tea coin These two systems are language equivalent (why?) But the machines are different When we supply a coin to the second machine, it can non- deterministically go to a state as before, or to a state that we can only obtain tea!

What’s wrong? The reactive behavior of the system is changed If every input and output seen as an interaction with the environment, they are not equivalent We need a different notion of equivalency

Strong simulation A Relation R on the states of an LTS is a strong simulation if p R q implies if p p’ then there exists q’ such that q q’ and p’ R q’ If such a relation exists, we say Q strongly simulates P α α pq p’q’ R R α α

Strong simulation in vending machine coin coffee tea A B C coin coffee tea B’ C’ A’ B0’B0’ tea coin We claim that the first system strongly simulates the second A’ R A B’ R B B 0 ’ R B C’ R C

Proof- Step1 coin coffee tea A B C coin coffee tea C’ A’ B0’B0’ tea coin B’ coin

Proof- Step2 coin coffee tea A B C coin coffee tea C’ A’ B0’B0’ tea coin B’ coin

Proof- Step3 coin coffee tea A B C coin coffee tea C’ A’ B0’B0’ tea coin B’ tea

Proof- Step4 coin coffee tea A B C coin coffee tea A’ B0’B0’ tea coin B’ C’ coin

Proof- Step5 coin coffee tea A B C coin coffee tea C’ A’ B0’B0’ tea coin B’ tea

Proof- Step6 coin coffee tea A B coin coffee tea A’ B0’B0’ tea coin B’ C’ C coffee

The opposite direction The second system also strongly simulates the first one A R A’ B R B’ C R C’ Intuitively it is correct Thee second machine can simulate every step the first machine can take

Problem It seems that we have defeated our original purpose The two machines should not be observationally equivalent But each one can strongly simulates the other We need a new idea

Strong bisimulation In 1981 David Park proposed a new approach to define the equivalence of automatons: bisimulation Strong Bisimulation: There is a single relation that both the relation and its converse are strong simulation Under this definition the two vending machines are not equivalent

Exercise Prove that these systems are bisimilar: A = a.A B = a.B + a.a.B def

τedges The original definition of bisimulation holds in LTS without τ Milner introduced a new kind of bisimulation Weak bisimulation equivalence It permits arbitrary sequences of τsteps to precede or follow corresponding atomic actions

Definition We define P P’ iff P … P’ τ*τ* τ τ P P’ iff P P 1 P 2 P’ τ*λ τ* τ*τ* λ τ*τ*

Weak simulation We say R is a weak simulation if the following two satisfied: 1. If P R Q and P P’ then there exists a Q’ such that Q Q’ and P’ R Q’ 2. If P R Q and P P’ then there exists a Q’ such that Q Q’ and P’ R Q’ P and Q are weakly bisimilar if there is a relation R such that both R and inverse are weak bisimulation τ τ*τ* λ τ*λ τ*

Weak simulation- graphically pq p’q’ R R τ pq p’q’ R R λ τ*λτ*τ*λτ* τ*τ*