Logical Analysis and Verification of Cryptographic Protocols - Loria

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152 CHAPTER 6. ON THE GROUND ENTAILMENT PROBLEMS Remarks. All these resolution inference rules have two premises. We implicitly suppose here that these premises do not share variables, which can be obtained by renaming the variables of one of the premises. Note that, by the definition of clauses, the same literal can not appear twice in a clause, for example the clause A1, A1, A2 → B1, B2 is not permitted. Indeed, we suppose that the resolution inference rules contain an implicit factorisation which immediately replaces A, A, Γ → ∆ (respectively Γ → B, B, ∆) by A, Γ → ∆ (respectively Γ → B, ∆). 6.1.3 Orderings We shall make use of various ordering relations on expressions. A (strict) ordering ≻ on a set of elements E is a transitive and irreflexive binary relation on E. The ordering ≻ is said to be: • well-founded if there is no infinite descending chain e ≻ e1 ≻ . . . for any element e in E • monotone if e ≻ e ′ then eσ ≻ e ′ σ for any elements e, e ′ in E and any substitution σ • stable if e ≻ e ′ then u[e] ≻ u[e ′ ] for any elements u, e and e ′ in E • subterm if e[e ′ ] ≻ e ′ for any elements e, e ′ in E • complete if it is total over ground elements of E Any ordering ≻ on a set E can be extended to an ordering ≻ set on finite sets over E as follows: if η1 and η2 are two finite sets over E, we have η1 ≻ set η2 if (i) η1 �= η2 and (ii) whenever for every e ∈ η2 \ η1 then there is e ′ ∈ η1 \ η2 such that e ′ ≻ e. Given a set, any smaller set is obtained by replacing an element by a (possibly empty) set of strictly smaller elements. We will call an element e maximal (respectively strictly maximal) with respect to a set η of elements, if for any element e ′ in η we have e ′ �≻ e (respectively e ′ �� e). Similarly, any ordering ≻ on a set E can be extended to an ordering ≻ mul on finite multisets over E as follows: if ξ1 and ξ2 are two finite multisets over E, we have ξ1 ≻ mul ξ2 if (i) ξ1 �= ξ2 and (ii) whenever ξ2(e) > ξ1(e) then ξ1(e ′ ) > ξ2(e ′ ), for some e ′ such that e ′ ≻ e; ξ(e) denotes the number of occurrences of e in the multiset ξ, and > denotes the standard “greater-than” relation on the natural numbers. Given a multiset, any smaller multiset is obtained by replacing an element by occurrences of smaller elements. We will call an element e maximal (respectively strictly maximal) with respect to a multiset ξ of elements, if for any element e ′ in ξ we have e ′ �≻ e (respectively e ′ �� e).
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.
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202 BIBLIOGRAPHY [12] M. Abadi and
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214 BIBLIOGRAPHY [160] J. C. Mitche
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216 BIBLIOGRAPHY [187] H. Seidl and
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