Logical Analysis and Verification of Cryptographic Protocols - Loria

5.5. DECIDABILITY RESULTS 131
• if the algorithm terminates with failure, U has no unifier, and if the algorithm
terminates with success, U has been transformed into an equivalent
∅-unification system in solved form.
We remark that Martelli-Montanari ∅-unification algorithm provides a widely
general version from which most unification algorithms [35, 55, 148, 167, 177,
176, 197, 198] can be derived.
Lemma 49 Let S = {s1, . . . , sn} and T = � {t1, . . . , tn} be two sets � of terms and
let σ be the most general unifier of V =
. If µ(T ) > 0
s1
?
?
=∅ t1, . . . , sn =∅ tn
then either nbv(s1, . . . , sn) > nbv((s1, . . . , sn, t1, . . . , tn)σ) or nbv(s1, . . . , sn) =
nbv((s1, . . . , sn, t1, . . . , tn)σ), S = Sσ and for all x ∈ V ar(T ) there is i ∈ {1, . . . , n}
such that σ(x) � si.
PROOF.
Let V =
�
s1
?
=∅ t1, . . . , sn
�
?
=∅ tn . In order to solve V , we apply the first
step of the unification algorithm of Martelli-Montanari [147] (Figure 5.4). We
reduce V to V ′ �
�
=
such that
x1
?
=∅ u1, . . . , xk
?
=∅ uk, xk+1
?
=∅ uk+1, . . . , xm
?
=∅ um
for every equation x ? =∅ u ∈ V ′ , we have either x ∈ V ar(S) and u ∈ Sub(T ) or
x ∈ V ar(T ) and u ∈ Sub(S). We suppose that xj ∈ V ar(T ) for j ∈ {1, . . . , k}.
• If k = m then we have xj ∈ V ar(T ) for j ∈ {1, . . . , m}. We suppose
that xi �= xj for all i, j ∈ {1, . . . , m} and i �= j. This implies that
Sσ = S and V ar(T ) are instantiated by subterms of S, that is V ar(T )σ are
smaller or equal than terms in S. We conclude also that nbv(s1, . . . , sn) =
nbv((s1, . . . , sn, t1, . . . , tn)σ).
• If k �= m assume {uk+1, . . . , um} /∈ V ar(T ), we have different cases:
– If for all different i, j ∈ {1, . . . , m} we have xi �= xj then m−k variables
of S, xk+1, . . . , xm, are instantiated by subterms of T , uk+1, . . . , um. This
implies that when we apply σ to U, new variables, V ar(uk+1, . . . , um)\
{x1, . . . , xk} will appear in Sσ. There exists a set T ′ �⊆ X such that
T → ∗ U T ′ and T ′ = {x1, . . . , xk, uk+1, . . . , um}. Since µ(T ) > 0, we have
|T ′ \X | > |V ar(T ′ \X )\(x1, . . . , xk)|. This implies that nbv(s1, . . . , sn) >
nbv((s1, . . . , sn, t1, . . . , tn)σ).
– If there is different i, j ∈ {1, . . . , m} such that xi = xj:
∗ If i, j ≤ k then we have to unify two subterms of S. Let ui and uj
be these two subterms and α be their most general unifier.
Let us apply α on V and
�
to solve V we have to solve V α =
. To solve V α we reduce it to another
�
s1α ? =∅ t1, . . . , snα ? =∅ tn