Logical Analysis and Verification of Cryptographic Protocols - Loria

2.1. PRELIMINARIES 25
Figure 2.1 Knuth-Bendix completion procedure
Input:
An equational theory H and a reduction order > over T (F, X ).
Output:
A finite convergent rewrite system R that is equivalent to H, if the procedure
terminates successfully;
or “fail”, if the procedure terminates unsuccessfully.
Initialisation:
If there exists an equation l
·
= r ∈ H such that l �= r, l ≯ r and r ≯ l then
terminates with output fail
�
otherwise, i = 0 and R0 = l → r such that l · = r ∈ H and
�
l > r
repeat Ri+1 = Ri;
for all 〈l, r〉 ∈ CP (Ri) do
1. reduce l, r to some Ri-normal form l ↓, r ↓;
2. if l ↓�= r ↓ and neither l ↓> r ↓ nor r ↓> l ↓, then terminate with output fail;
3. if l ↓> r ↓, then Ri+1 = Ri ∪ {l ↓→ r ↓};
4. if r ↓> l ↓, then Ri+1 = Ri ∪ {r ↓→ l ↓};
od
i=i+1;
until Ri = Ri−1;
output Ri;
Knuth-Bendix completion procedure
The Knuth-Bendix completion procedures [131] (also called basic completion procedure),
which is described in Figure 2.1, starts with an equational theory H and
tries to find a convergent rewrite system R that is equivalent to H. We assume
that the order > used in the procedure is a reduction order and it is given as an
input of the procedure. We recall that a reduction order > is any order which is
stable, well-founded, and monotone (see Section 2.1.5).
The Knuth-Bendix procedure removes automatically all trivial equations (re-
·
spectively trivial critical pairs) of the form l = l (respectively 〈l, l〉 or 〈l, r〉 with
l ↓= r ↓).
Thus, the basic completion procedure may show three different types of behaviour,
depending on the particular input H and >: