Logical Analysis and Verification of Cryptographic Protocols - Loria

Chapter 6
Decidability of ground entailment
problems for saturated sets of
clauses
In this chapter, we study the ground entailment problems in the
first order logic, and more precisely the ground entailment problems
for a set of first order clauses. For clauses and Horn clauses sets,
the satisfiability and ground entailment problems are undecidable.
Many decidability results have been obtained for several fragments
of first order logic [150, 28, 84, 180, 205]. In this chapter, we introduce
a new fragment of first order logic and we prove the decidability of
its ground entailment problem using the selected resolution, and a
special ordering over atoms and terms. We then show how to use this
result in order to decide the insecurity problem for cryptographic
protocols in the case of bounded number of sessions.
Outline of the chapter We introduce in Section 6.1 the basic
notions of this chapter including clauses, ground entailment problems,
and resolution methods. In Section 6.2, we mention some
obtained decidability results for the ground entailment problem,
and in Section 6.3 we show that the ground entailment problem
and satisfiability problem in first order logic can be used to analyse
cryptographic protocols and we mention some obtained results in
that field. In Section 6.4, we introduce our modelisation of cryptographic
protocols using first order clauses and we show how to
reduce insecurity problem for cryptographic protocols to ground
entailment problem in the first order logic. In Section 6.5, we present
selected resolution for which we show soundness and completeness,
we introduce a new fragment of first order logic and we show the
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