Logical Analysis and Verification of Cryptographic Protocols - Loria

130 CHAPTER 5. SATURATED DEDUCTION SYSTEMS
Figure 5.4 Martelli-Montanari ∅-unification algorithm
Given an ∅-unification system U, repeatedly perform any of the following transformations.
If no transformation applies, stop with success.
• Select any equation of the form t ? = ∅ x where t is not a variable and x is a variable,
and rewrite it as x ? = ∅ t.
• Select any equation of the form x ? = ∅ x where x is variable, and erase it.
• Select any equation of the form t ′ ? =∅ t” where t ′ and t” are not variables. If
the two root function symbols are different, stop with failure; otherwise, assume
t ′ = f(t1, . . . , tn) and t” = f(s1, . . . , sn) with f a function symbol with arity n and
apply the following:
If n = 0, then f is a constant symbol, and the equation is simply erased;
otherwise, replace f(t1, . . . , tn) ? = ∅ f(s1, . . . , sn) with the following equations:
t1 ? = ∅ s1, . . . , tn ? = ∅ sn.
• Select any equation of the form x ? = ∅ t where x is a variable which occurs somewhere
else in the unification system and where t �= x. If x occurs in t, then stop
with failure; otherwise, apply the substitution σ = {x ↦→ t} to both terms of all
other equations in the unification system (without erasing x ? = ∅ t).
Martelli-Montanari unification algorithm. In this paragraph, we recall the unification
algorithm due to Martelli-Montanari [147], which is used in the proof
of the next lemma. In [147], A. Martelli and U. Montanari gave a ∅-unification
algorithm based on the transformation of a given ∅-unification system U into an
equivalent and simpler unification system. An unification system U is said to
be in solved form if and only if it satisfies the following conditions:
• every equation in U is of the form x ? =∅ t;
• every variable which is the left member of some equation occurs only
there.
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An ∅-unification system U = x1 =∅ t1, . . . , xn =∅ tn in solved form has the
obvious unifier σ = {x1 ↦→ t1, . . . , xn ↦→ tn}, which is its most general ∅-unifier.
We give in Figure 5.4 Martelli-Montanari ∅-unification algorithm. In [147],
A. Martelli and U. Montanari proved that for any ∅-unification system U
• their algorithm always terminates, no matter which choices are made,