Logical Analysis and Verification of Cryptographic Protocols - Loria

150 CHAPTER 6. ON THE GROUND ENTAILMENT PROBLEMS
S with succedent A. Leaves are ground instances of positive unit clauses in S.
A direct proof of a ground clause A1, . . . , An → B from S is a direct proof of B
from S ∪ {A1, . . . , An}.
A term t occurs in either kind of proof if t occurs in a clause or in an atom
labelling the proof tree and an atom A occurs in either kind of proof if A occurs
in a clause labelling the proof tree. If π is a proof, by T erms(π) (respectively
µ(π)) we denote the set of terms (respectively set of atoms) occurring in π.
An inference system J is said to be complete if the empty clause can be derived
from any unsatisfiable set of clauses. An inference system J is said to be
sound if the empty clause can not derived from any satisfiable set of clauses. It
is easy to see that S |= C if and only if the ∅ clause is derived from S ∪ ¬C for
any complete and sound inference system J.
In the rest of this chapter, we will be interested only by the refutational
proofs, and for simplicity, we will write proofs instead of refutational
proofs. In addition, when the inference system is complete and sound,
we will abuse the notation and use the notation S |= C to mean that there
is a proof of S |= C.
6.1.2 Resolution
The resolution is one of the most successful methods for automated proof
search. It was developed in [176]. We introduce in this section some of the
well-known resolution inference strategies. Some of these strategies use orderings,
transitive and irreflexive binary relations, on atoms and terms. We denote
by ≻a the ordering on atoms and by ≻t the ordering on terms.
(Binary) Resolution
The (binary) resolution is described by the following two inference rules:
Resolution
Γ → ∆, A A ′ , Γ ′ → ∆ ′
(Γ, Γ ′ → ∆, ∆ ′ )α
where α is the most general unifier of A and A ′ .
The clause (Γ, Γ ′ → ∆, ∆ ′ )α is called a resolvent of the premises or a conclusion
of the inference, and the atom Aα is called the resolved atom.
F actoring
Γ → ∆, A, A ′
(Γ → ∆, A)α
where α is the most general unifier of A and A ′ .