## The Honda-Tokoro encoding for the output prefix

Honda and Tokoro have proposed in 1993 a calculus similar to what has been called later "the asynchronous pi-calculus", and have shown that the output-prefix construct of the pi-calculus can be encoded in their language. We give here the version of their encoding adapted to the syntax of the polyadic pi-calculus. "Polyadic" means that the input and output actions can have 0, 1, or more arguments. The arguments are passed all together at the moment of the communication.

Intuitively, the encoding is based on the idea of simulating synchronous communication via asynchronous communication and acknowledgement messages: the sender will first send its message (asynchronously), and then wait for an acknowledgement from the receiver. The encoding is compositional, i.e. the encoding of a compound process is defined in terms of the encoding of the components.

Formally, the encoding is a mapping [[.]] from the set of the pi-calculus processes (without "+") into the set of the polyadic asynchronous pi-calculus processes. It is defined compositionally as follows:

• [[ 0 ]] = 0
• [[ P|Q ]] = [[P]] | [[Q]]
• [[ C ]] = C and C =def= P becomes C =def= [[P]]
• [[ (x)P ]] = (x)[[P]]
• [[ [x=y]P ]] = [x=y][[P]]
• [[ x^y.P ]] = (a)(x^y a | a( ).[[P]])
• [[ x(z).Q ]] = x(z,w).( w^ | [[Q]] )
The interesting part of the encoding are the last two formulas. The translated processes behave as follows: the sender sends, together with the intended name y, also a private name "a" which will be used as the acknowledgement channel. The sender will not proceed until it receives the acknowledgement from the receiver, thus simulating the behaviour of output prefix. Correspondingly, the receiver has two parameters z and w; z is for the intended name (y), w is for the acknowledgement channel (a). After the receive action has taken place, the receiver will send an acknowledgement in form of an output action on w.

Note that in this encoding we use the restriction operator of the pi-calculus, which has the possibility of enlarging the scope during communication (scope extrusion). Thanks to this mechanism, the process ((a)( x^y a | a( ).[[P]] )) | x(z,w).( w^ | [[Q]] ) (which represents the encoding of [[ x^y.P | x(z).Q ]]) will become, after communication, (a) ( a( ).[[P]] | ( a^ | [[ Q[y/z] ]] )). Note that it is important that "a" (the acknowledgement channel) is private between the sender and the receiver, because we don't want interference with other processes possibly present in the environment and using the same name. Note that the same encoding could not be written in CCS (at least not compositionally) because of the impossibility of expressing scope-extrusion in CCS.