Logical Analysis and Verification of Cryptographic Protocols - Loria

6.2. DECIDABLE FRAGMENTS OF FIRST ORDER LOGIC 153
If the ordering ≻ is total (respectively, well-founded), so is its multiset extension
[1]. It is easy to see that if the ordering ≻ is total (respectively, wellfounded),
so is its set extension: actually, by definition of multisets, a set is a
multiset; since the multiset extension ≻ mul of a total (respectively well-founded)
order ≻ is also total (respectively well-founded) [1], we deduce that the set extension
≻ set of a total (respectively well-founded) order ≻ is also total (respectively
well-founded).
Clause and proof orderings. By an atom ordering (respectively term ordering) we
mean an arbitrary ordering on atoms (respectively on terms). To extend an atom
ordering ≻a to an ordering on clauses, we identify a (positive or negative) literal
A with a set {A}, and a clause with the union of its literals, or more precisely
with the union of sets of atoms identifying its literals. For example, the clause
A1, A2 → B is identified with the following union of literals ¬A1 ∪ ¬A2 ∪ B, that
is with the following union of sets of atoms {A1} ∪ {A2} ∪ {B} which is equal to
the set of atoms {A1, A2, B}. From now on, we denote by µ(C) the set of atoms
representing the clause C, that is the set of atoms equal to the union of the set of
atoms identifying its literals. For example µ(A1, A2 → B) = {A1, A2, B}. Then,
for clauses C and C ′ , we define C≻cC ′ if and only if the set of atoms representing
C is strictly bigger than the set of atoms representing C ′ for the set extension of
≻a, that is µ(C)≻a set µ(C ′ ). Clearly, if the ordering ≻a is well-founded and total
on ground atoms, so is its extension to ground clauses. We extend the definition
of µ from a clause to a set of clauses, let S be a set of clauses, S = {C1, . . . , Cn},
µ(S) = µ(C1) ∪ . . . ∪ µ(Cn) = ∪ n i=1µ(Ci). By definition of proofs, each atom that
appears in a proof belongs to a clause labelling one of its leaves. We extend next
the atom ordering to an ordering on proofs. If π is a proof, the set leaves(π)
denotes the set of clauses labelling its leaves, and µ(π) = µ(leaves(π)). More
precisely µ(π) is the union of set of atoms identifying clauses labelling its leaves.
For example, let the proof π given as below:
∅→C
C→A,B B→∅
C→A
∅→A
∅
A→B B→∅
A→∅
leaves(π) is equal to the following set of clauses {∅ → C; C → A, B; A → B; B →
∅}, and µ(π) = {A, B, C}. Let π, π ′ be two proofs, we define π≻pπ ′ if and only if
µ(π)≻a set µ(π ′ ).
6.2 Decidable fragments of first order logic
It is known that the ground entailment problem for Horn clauses and full
clauses sets is undecidable. Here, we mention some obtained decidability results
under some restrictions.