Logical Analysis and Verification of Cryptographic Protocols - Loria

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154 CHAPTER 6. ON THE GROUND ENTAILMENT PROBLEMS 6.2.1 McAllester’s work McAllester’s work [150] is based on Horn clauses. He defined a set of Horn clauses S to be subterm local if for every ground Horn clause C, we have S |= C if and only if C is entailed from a set of ground instances of clauses in S in which all terms are subterms of ground terms in S and C. He proved that if a set S of Horn clauses is finite and subterm local then its ground entailment problem is decidable. 6.2.2 Basin and Ganzinger work In their work, D. Basin and H. Ganzinger [28] generalised McAllester’s work by allowing arbitrary term ordering, any strict well-founded order over terms, and full (not Horn) clauses. We recall some definitions used by D. Basin and H. Ganzinger. A set of clauses S is said to be order local with respect to a term ordering ≻ if for every ground clause C, we have S |= C if and only if C is entailed from a set of ground instances of clauses in S in which each term is smaller than or equal (under the �) to some term in C. A term ordering ≻ is said to be of complexity f, g, whenever for each clause of size n (the size of a term is the number of nodes in its tree representation, and the size of a clause is the sum of sizes of its terms) there exists O(f(n)) terms that are smaller or equal (under �) to a term in the clause, and that may be enumerated in time g(n). It is easy to see that if ≻ is of complexity f, g, each ground term has finitely many smaller terms that may be enumerated in finite time. D. Basin and H. Ganzinger obtained the following results: 1. If S is a set of Horn clauses that is order local with respect to a term ordering ≻ of complexity f, g then the ground entailment problem for S is decidable. 2. If S is a set of (full) clauses that is order local with respect to a term ordering ≻ of complexity f, g then the disentailment ground problem for S is decidable. In [28], D. Basin and H. Ganzinger showed that the saturation of a set S of Horn clauses and hence (full) clauses is not sufficient to obtain the decidability of the ground entailment problem for S. To show this, they considered the following example. We recall that for arbitrary Horn clause sets, the satisfiability and ground entailment problems are undecidable. Example 22 Let S be an arbitrary set of Horn clauses, and let S ′ consist of the set of Horn clauses q(a), Γ → A such that Γ → A is in S. Choose an ordering ≺ such that a is the maximal term. For any compatible atom ordering, there is no ordered inferences from S ′ , and hence S ′ is saturated under ordered resolution. It is undecidable if S ′ |= ¬q(a) since this is equivalent to the inconsistency of S.
6.3. FROM CRYPTOGRAPHIC PROTOCOLS TO LOGIC OF CLAUSES 155 In [28], D. Basin and H. Ganzinger showed that restrictions on occurrences of variables can lead to decidability for full and hence also Horn clauses. This result is given as follows: If S is a set of (full) clauses saturated up to redundancy under ordered resolution with respect to a complete well-founded ordering over atoms, and if, for each clause in S, each of its maximal atoms contains all the variables of the clause, then the ground entailment problem for S is decidable. 6.3 From cryptographic protocols to logic of clauses Lots of researchers [84, 82, 180, 187, 205] have been interested by clauses and Horn clauses in order to analyse cryptographic protocols. In this line of research, many models have been given. In the most common models, intruder behaviour, protocol description and security properties are encoded as clauses and in particular as Horn clauses; and, the insecurity problem of the cryptographic protocol is reduced to the satisfiability problem for a class of Horn clauses. While the satisfiability problem for first order logic is undecidable in general, several classes have been proved to be decidable [60], and in Section 6.2, we mentioned some of them. In this section, we focus on the fragments of first order logic which are well-suited to modelling security protocols. In sections 6.3.1, 6.3.2 and 6.3.3, we present some of such fragments that are shown to be decidable, and in Section 6.4 we introduce our result. In what follows, a class of clauses means a set of sets of clauses. In Sections 6.3.1 and 6.3.2, we assume a unique unary predicate symbol I. 6.3.1 Comon-Lundh and Cortier work H. Comon-Lundh and V. Cortier [84] were focused on the secrecy property. In [82], they have showed that, for the secrecy property, it is sufficient to verify the correctness of the protocol for only k honest agents and one dishonest agent, k being the number of roles in the protocol. In [84], they represented the capacities of the intruder (respectively the description of the protocol and the intruder knowledge) by a set of Horn clauses, and the secrecy property by a negative unit Horn clause. They reduced the insecurity problem to the satisfiability problem of the set of clauses constructed from the intruder clauses and the clauses modelising the protocol, intruder knowledge and security goal (secrecy property). Assuming CI to be a class of Horn clauses containing at most one function symbol, and CP to be a class of Horn clauses containing at most one variable, CI represents the class of intruder clauses, and CP represents the modelisation of the protocol. As noticed in [84], the protocols with single blind
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Logical Analysis and Verification of Cryptographic Protocols - Loria

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