Mobile Calculi

Prof. Diletta Romana Cacciagrano

Some expressiveness results

Expressiveness

In 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:

A good notion of encoding

Compositional w.r.t. the operators

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

A good notion of encoding

Preserving some kind of semantics. Here there are several possibilities:

• Preserving observables

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

A 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.

Expressiveness: Process mobility as name mobility

Expressiveness: Process mobility as name mobility

Expressiveness: Polyadic as monadic

Expressiveness: Replication vs recursion

Expressiveness: Replication vs recursion

|

Theorem:

Expressiveness: Data (Boolean) as processes

|

||*

Expressiveness: Synchrony as asynchrony(Boudol’s 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.

Expressiveness: Synchrony as asynchrony(Boudol’s Encoding)

Honda-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 Tokoro’s Encoding)

Expressiveness: Synchrony as asynchrony(Honda and Tokoro’s Encoding)

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

Expressiveness: Synchrony as asynchrony(Honda and Tokoro’s Encoding)

Expressiveness: Synchrony as asynchrony and Testing semantics

We don’t 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

Expressiveness: Synchrony as asynchrony and Testing semantics

The encoding of Boudol and Honda-Tokoro preserve may and fair semantics, but not must semantics, i.e.

Expressiveness: Synchrony as asynchrony and Testing semantics

Theorem [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.

Expressiveness: Synchrony as asynchrony and Testing semantics

The 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.

The Pi-calculus hierarchy