Logical Analysis and Verification of Cryptographic Protocols - Loria

5.5. DECIDABILITY RESULTS 129
However, the instance of x in the following reachability problem encodes a
word recognised by the automaton after a run encoded by the instance of y:
∅ ⊲ f(s(x, q0, ⊥, ⊥), y), g(f(s(x, q0, ⊥, ⊥), y)) ⊲ g(s(⊥, qf, ⊥, ⊥))
This example proves (with q0 ∈ QI and qf ∈ QF ) that the saturation can terminate
and yield a deduction system for which general reachability problems are
not decidable.
The undecidability comes from the fact that one can apply an unbounded
number of decreasing rules on a non-ground terms, and from the “lack of regularity”
on the terms obtained.
5.5.3 Decidability of the general reachability problems
We recall that the initial intruder system is given by I0 = 〈F, T0, H〉 while H is
generated by a convergent equational theory and has the finite variant property,
L0 = LI0 is the intruder deduction rules associated to I0. We recall also that
I ′ = 〈F, L ′ , ∅〉 is the saturated intruder system.
We give here a simple criterion that permits to ensure the termination of the
resolution of a constraint problem with a saturated deduction system. Let T
be a set of terms, T = {t1, . . . , tm}, we let ∆(T ) to be the set of strict maximal
subterms of T and we define:
δ(T ) =
� +∞ if T ⊆ X
|T \ X | − |V ar(T \ X ) \ (T ∩ X )| otherwise.
Now let us define µ(T ). We consider the image of the set of terms T by the
rewriting system U containing rules f(x1, . . . , xn) → x1, . . . , xn for every symbol
f in the signature of the deduction system. We define:
µ(T ) = min
T σ →∗ ′
U T
σ mgu of subterms of T
δ(T ′ )
We extend µ to rules as follows. Let L ′ be the set of deduction rules. We
recall that L ′ is partitioned into two disjoint sets of deduction rules, the set of
increasing rules L ′ inc and the set of decreasing rules L ′ dec . For every rule � l → r ∈
L ′ ,
µ( � l → r) =
�
µ(∆( � l \ X ∪ {r})) if � l → r is increasing,
µ(∆( � l \ X )) otherwise.
Definition 53 (Contracting deduction systems) A saturated deduction system I ′ =
〈F, L ′ , ∅〉 is contracting if for all rules � l → r in L ′ we have µ( � l → r) > 0.