This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming. The complexity of the unification problem is also addressed.

. In [9] we have shown how to construct a 3 {layer recurrent neural network (RNN) that computes the iteration of the meaning function TP of a given propositional logic program, what corresponds to the computation of the semantics of the program. In this paper we dene a notion of approximation for interpretations and prove that there exists a feed forward neural network (FNN) that approximates the calculation of TP for a given (rst order) acceptable logic program with an injective level mapping arbitrarily well. By extending the FNN by recurrent connections we get a RNN whose iteration approximates the xed point of TP . The proof is found by taking advantage of the fact that for acceptable logic programs, TP is a contraction mapping on the complete metric space of the interpretations for the program. Mapping this metric space to the metric space IR the real valued function fP corresponding to TP turns out to be continuous and a contraction and for this reason can be approximated by an indicated class of FNN. 1

First-order logic models of security for cryptographic protocols, based on variants of the Dolev-Yao model, are now well-established tools. Given that we have checked a given security protocol π using a given first-order prover, how hard is it to extract a formally checkable proof of it, as required in, e.g., common criteria at evaluation level 7? We demonstrate that this is surprisingly hard: the problem is non-recursive in general. On the practical side, we show how we can extract finite models M from a set S of clauses representing π, automatically, in two ways. We then define a model-checker testing M| = S, and show how we can instrument it to output a formally checkable proof, e.g., in Coq. This was implemented in the h1 tool suite. Experience on a number of protocols shows that this is practical.

. This paper surveys various complexity and expressiveness results on different forms of

Abstract. Clever algorithm design is sometimes superseded by simple encodings into logic. We apply this motto to a few case studies in the formal verification of security properties. In particular, we examine confidentiality objectives in hardware circuit descriptions written in VHDL. 1

Distinguished variables are variables present in the heads of the rules of a DATALOG (or PROLOG) program. Except for their relevance to a query, they are as undistinguished by previous works as others, introduced by resolution steps. What if the selection rule first selects literals with at least one distinguished variable ? This paper is an attempt to answer this question. To do this, we first define generating goals, which are the bodies of the resultants of a partial evaluation of a rule head in the program, the unfolding rule being "Unfold, as much as possible, atoms containing distinguished variables". Next the properties of generating goals are discussed, and an adaptation of SLD-resolution taking into account these properties is presented. The question of the finiteness of the set of generating goals is then raised. Although decidability of this property is still an open question in the general case, decidability is shown for significant classes of Datalog programs, including bo...