Logical Analysis and Verification of Cryptographic Protocols - Loria

164 CHAPTER 6. ON THE GROUND ENTAILMENT PROBLEMS
subsumption; tautology deletion deletes any clause of the form Γ, A → ∆, A, and
subsumption deletes any clause C ′ such that there exists another clause C with
Cσ ⊆ C ′ for a substitution σ. In [164], H. De Nivelle defined another variation
of the resolution with selection, called L-ordered resolution. He assumed a selection
function that, for each clause, selects the maximal atoms with respect to an
atom ordering ≺ that verifies the following property: for every atoms A, B, if
A � B then Aθ � Bθ for every substitution θ.
In this chapter, we make use of the resolution with selection introduced in
[146], and from now on, we consider only a subset of the valid selection functions,
namely the functions that select maximal atoms in clauses with respect
to an atom ordering. We remark that the selection functions we consider are
less general than in [146], and more general than in [164] since we consider an
arbitrary atom ordering.
Selected resolution is described by the following two inference rules:
Selected resolution
Γ → ∆, A A ′ , Γ ′ → ∆ ′
(Γ, Γ ′ → ∆, ∆ ′ )α
where α is the most general unifier of A and A ′ ,
and A, A ′ are selected in their respective premises.
Selected factoring
Γ → ∆, A, A ′
(Γ → ∆, A)α
where α is the most general unifier of A and A ′ ,
and A or A ′ is selected in the premise.
Selected resolution (respectively selected factoring) inference rule requires that
all atoms in the resolvent of two ground premises (respectively in the factor of
a ground premise) are strictly smaller than the resolved (respectively factored)
atom. We define a notion of redundancy that identifies clauses and inferences
that are not needed for establishing refutational completeness of selected resolution.
To this end, we define next the relations →RS and →R g over atoms, and
S
the notation A ↓S where A is a set of atoms and S a set of clauses.
Definition 55 Let S be a set of clauses. We define the relation →RS as follows: we have
A →RS B if there exists a clause C in S and A, B two atoms in C such that A≻aB;
and we define the relation → g
RS as follows: we have A →R g B if there exists two atoms
S
As , Bs such that As →RS Bs , Asσ = A and Bsσ = B for a substitution σ grounding
As , Bs .
Let the rewriting system RS (respectively R g
) be the set of rewriting rules
→RS (respectively → R g
S
S
). By definition, it is easy to see that rules in Rg
S are
ground, and that R g
S is set of ground instances of rewriting rules in RS. We also