Logical Analysis and Verification of Cryptographic Protocols - Loria

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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 ′ .
6.1. PRELIMINARIES 151 The clause (Γ → ∆, A)α is called a factor of the premise or a conclusion of the inference, and the atom Aα is called the factored atom. (Binary) resolution is a refutationally complete theorem proving method: the empty clause (i.e., a contradiction) can be derived from any unsatisfiable set of clauses. The search of a contradiction proceeds by saturating the given set of clauses, that is, systematically applying all inferences rules until no more new clauses can be added [176]. Ordered Resolution Ordered resolution is described by the following two inference rules: Ordered resolution Γ → ∆, A A ′ , Γ ′ → ∆ ′ (Γ, Γ ′ → ∆, ∆ ′ )α where α is the most general unifier of A and A ′ , Aα is strictly maximal with respect to Γα, ∆α, and Aα is maximal with respect to Γ ′ α, ∆ ′ α. Ordered factoring Γ → ∆, A, A ′ (Γ → ∆, A)α where α is the most general unifier of A and A ′ , Aα is strictly maximal with respect to Γα, and maximal with respect to ∆α. Ordered resolution (respectively ordered factoring) inference rule requires that there is no atom in the conclusion greater than the resolved (respectively factored) atom. Ordered resolution, i.e. ordered resolution inference rule together with ordered factoring rule, is refutationally complete and sound, and hence, for any set S of clauses, S is unsatisfiable if and only if empty clause can be derived from S [24]. We remark that not every ground instance of an inference by ordered resolution is an inference by ordered resolution. For example, let Inf be the following inference by ordered resolution I(y) → I ′ (x) I ′ (x ′ ), I(y ′ ) → I ′′ (y ′ ) I(y), I(y ′ ) → I ′′ (y ′ ) The most general unifier of I ′ (x), I ′ (x ′ ) is α = {x ′ ↦→ x}. By definition of ordered resolution inference, we have that I(yα) �≥ I ′ (xα), I(y ′ α) �> I ′ (xα), and I ′′ (y ′ α) �> I ′ (xα). Let the ground substitution σ be such that σ = αα ′ , I(yσ) > I(y ′ σ) > I ′′ (y ′ σ) > I ′ (xσ) and I ′ (xσ) = I ′ (x ′ σ). Infσ is a ground instance of Inf, and it is easy to see that Infσ is not an inference by ordered resolution.
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208 BIBLIOGRAPHY [83] H. Comon-Lund
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216 BIBLIOGRAPHY [187] H. Seidl and
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