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By Rajeev Alur; T A Henzinger; Stanford University. Computer Science Department

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The following exercise presents a generalization of this remark. 4 {T2} [Stutter closure for passive modules] Consider a module P , and for each variable in XP , consider a sequence of k ≥ 1 values. Construct sequences of k +1 values by repeating the last value of each sequence of length k. (a) Prove that if P is passive and the sequences of length k form an initialized trajectory of P , then also the sequences of length k + 1 form an initialized trajectory of P . (b) In part (a), can you replace “passive” by “asynchronous” and “initialized trajectory” by “trace”?

Given an X-atom U , the renamed atom U [ρ] is the X[ρ]-atom with the set ctrXU [ρ] of controlled variables, the set readXU [ρ] of read variables, the set awaitXU [ρ] of awaited variables, the initial command initU [ρ], and the update command updateU [ρ]. Given a module P , and a renaming ρ for the set XP of module variables, the renamed module P [ρ] is the module with the set privXP [ρ] of private variables, the set intfXP [ρ] of interface variables, the set extlXP [ρ] of external variables, and the set {U [ρ] | U ∈ atomsP } of atoms.

Define two modules R1 and R2 for modeling the two processes of the protocol, encoding x1 = x2 by x = true and x1 = x2 by x = false. The resulting module module Pete2 is hide . . in R1 R2 Rx should have the same abstract type and the same traces as the module Pete. (c) How would you change the definition of Rx in part (a) if x is a nonnegativeinteger variable rather than a boolean variable? 30 uses many rounds for a single handshake. The two protocols have identical receiver agents. Every send-receive cycle of AsyncMsg consists of four phases —a message production phase, an agent coordination phase, a message transmission phase, and a message consumption phase— each consisting of any number of update rounds.

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