Abstract

This paper gives an example of such an inductive proof for a combinatorial problem. While there exist other constructive proofs of Higman's lemma (see for instance [10, 14]), the present argument has been recorded for its extreme formal simplicity. This simplicity allows us to give a complete description of the computational content of the proof, first in term of a functional program, which follows closely the structure of the proof, and then in term of a program with state. The second program has an intuitive algorithmic meaning. In order to show that these two programs are equivalent, we introduce an intermediary program, which is a first-order operational interpretation of the functional program. The relation between this program and the program with state is simple to establish. We can thus claim that we understand completely the computational behaviour of the proof. It is possible to give still another description of this algorithm, in term of process computing in parallel. In this form, the connection with Nash-Williams non constructive argument is quite clear (though this algorithm was found first only as an alternative description of the computational content of the inductive proof). This inductive proof was actually found from the usual non constructive argument by using the technique described in [3]. These two facts give strong indication that this algorithm can be considered as the computational content of the Nash-Williams argument.

Citations