Mobile CalculiProf. Diletta Romana Cacciagrano1Some expressiveness results2ExpressivenessIn general, in order to compare the expressive power of two languages, we look for the existence/non existence of an encoding with certain properties among these languages.

What is a good notion of encoding to be used as a basis to measure the relative expressive power?

In general we would be happy with an encoding L1 L2 being:

3A good notion of encodingCompositional w.r.t. the operators

(Preferably) homomorphic w.r.t. parallel (distribution-preserving)

4A good notion of encodingPreserving some kind of semantics. Here there are several possibilities:

Preserving observables

Preserving equivalence (This is one of the most popular requirements for an encoding)

5A good notion of encoding(Preferably) the encoding should not introduce divergences (tau-loops), in the sense that if in the original process all the computations converge, then the same holds for its translation.

Note that weak bisimulations are insensitive w.r.t. divergences. 6Expressiveness: Process mobility as name mobility

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Expressiveness: Process mobility as name mobility8Expressiveness: Polyadic as monadic

9Expressiveness: Replication vs recursion

10Expressiveness: Replication vs recursion

|Theorem:11Expressiveness: Data (Boolean) as processes

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*12Expressiveness: Synchrony as asynchrony(Boudols Encoding)Boudol (1992) provided the following encoding of Pi (without choice) into asynchronous Pi.

The idea is to force both partners to proceed only when it is sure that the communication can take place, by using a sort of rendez-vous protocol.

13Expressiveness: Synchrony as asynchrony(Boudols Encoding)

14Honda-Tokoro (1992) defined the following encoding of Pi (without choice) into asynchronous Pi, in which the communication protocol takes two steps instead than three.

The idea is to let the receiver take the initiative (instead than the sender).

Expressiveness: Synchrony as asynchrony(Honda and Tokoros Encoding)15Expressiveness: Synchrony as asynchrony(Honda and Tokoros Encoding)

16Honda proved this encoding sound and almost complete w.r.t. a certain logical semantics.

Expressiveness: Synchrony as asynchrony(Honda and Tokoros Encoding)17Expressiveness: Synchrony as asynchrony and Testing semanticsWe dont expect the encodings the encodings of output prefix to be correct w.r.t. testing semantics (why?) but we would like the encoding to satisfy at least the following property:

where can be

18Expressiveness: Synchrony as asynchrony and Testing semanticsThe encoding of Boudol and Honda-Tokoro preserve may and fair semantics, but not must semantics, i.e.

19Expressiveness: Synchrony as asynchrony and Testing semanticsTheorem [Cacciagrano, Corradini, Palamidessi, 2004]:

Let an encoding of Pi-calculus (without choice) into asynchronous Pi-calculus such that

compositional w.r.t. prefixes there exists P such that P diverges Then does not preserve must testing.

20Expressiveness: Synchrony as asynchrony and Testing semanticsThe problem, however, is uniquely a problem of fairness.

Theorem [Cacciagrano, Corradini, Palamidessi, 2004]:

The encodings of Boudol and of Honda-Tokoro

preserve must testing if we restrict to fair computations only. preserve a version of must testing called fair-must testing.

21The Pi-calculus hierarchy

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