Logical Analysis and Verification of Cryptographic Protocols - Loria

2.1. PRELIMINARIES 27
Figure 2.2 Bachmair completion procedure
Deduce:
H,R
E∪{l · if 〈l, r〉 ∈ CR(R)
=r},R
Orient:
·
H∪{l =r},R
E,R∪{l→r} if l > r
Delete:
H∪{l · =l},R
E,R
Simplify-Identity:
H∪{l · =r},R
H∪{l ′ · =r},R
if l→Rl ′
R-Simplify-Rule
if r→Rl ′
H,R∪{l→r}
H,R∪{l→l ′ }
L-Simplify-Rule
H,R∪{l→r}
if l→Rl ′
H∪{l ′ · =r},R
Unfailing Knuth-Bendix procedure
The basic completion procedure described above will fail when an initial non trivial
equation is unorientable or when it generates an unorientable non trivial critical
pair or when it generates infinitely many new rules.
J. Hsiang and M. Rusinowitch [120] introduced an extension of the Knuth-
Bendix completion procedure, called the unfailing completion procedure or UKBprocedure
in short, which is a Knuth-Bendix type completion procedure that
does not fail. This procedure is described in Figure 2.3.
In [120], the authors make use of a complete simplification ordering >, i.e. >
is well-founded, monotone, stable, total over ground terms, and s[t] > t for any
terms s, t. They also make use of extended critical pairs defined as follow:
Definition 1 (extended critical pairs) Given two equations g · = d and l
integer p ∈ P os(g) such that
1. g|p �∈ X ,
2. g|p and l are unifiable with σ is their most general unifier,
3. rσ � lσ,
4. dσ � gσ.
·
= r, and an