this paper, we propose to use clause sets to represent models. This idea is very simple and allows to combine in a well balanced way, expressive power with the \good" required properties of a reasonable model representation. Obviously, in order that the set of clauses specifying the model brings new information w.r.t. the initial one, a basic requirement is that these clause sets must have exactly one Herbrand model (on a given signature). Such clause sets are straightforward representations of their Herbrand models

. Finding computationally valuable representations of models of predicate logic formulas is an important issue in the field of automated theorem proving, e.g. for automated model building or semantic resolution. In this article we treat the problem of representing single models independently of building them and discuss the power of different mechanisms for this purpose. We start with investigating context-free languages for representing single Herbrand models. We show their computational feasibility and prove their expressive power to be exactly the finite models. We show an equivalence with "ground atoms and ground equations" concluding equal expressive power. Finally we indicate how various other well known techniques could be used for representing essentially infinite models (i.e. models of not finitely controllable formulas), thus motivating our interest in relating model properties with syntactical properties of corresponding Herbrand models and in investigating connections betwe...

In [8] we investigated representing Herbrand models via context-free grammars and found the representation power of this method to be exactly the finite models. Based on these observations we now present a clause set evaluation algorithm that operates directly on grammars, avoiding the exponential blow-up from the number of nonterminals in the grammar to the number of elements in the finite domain of the corresponding model, ending up with a not-so-obvious evaluation procedure for arbitrary clause sets over finite interpretations (specified via grammars).

Introduction Automated model building is now widely recognized as a crucial issue in the eld of Automated Deduction (see for example [1]). Since the early 90's, several dierent approaches have been proposed for building automatically models of rst-order formulae. In particular, methods for using the resolution calculus for generating (Herbrand) models of clause sets have been described [3, 4]. The basic idea is to compute the \deductive closure" of the clause set - i.e. to compute all clauses that can be deduced from the original one using (renements of) the resolution, factorization and paramodulation rules - and then to take advantage of the particular properties of these saturated clause sets in order to extract a model of the initial formula. One of the main advantages of this approach is that the additional computation cost required to nd the model is rather low, once the saturated clause set has been generated. The main drawback is, of course, that the method does n

We prove that for a natural class of first-order formulas the validity problem is \Pi p 2 -complete. 1 Section 1. Introduction Section 1 Introduction Traditionally, the decision problem for first-order logic is studied for classes of formulas characterized by the quantifier prefix and/or signature restrictions (see e.g. the recent book of Borger Gradel and Gurevich [1]). Kozen [4] considered the decision problem for so-called positive first-order formulas. Recently, in connection with model building, some decidability results have been obtained for classes of formulas characterized in terms of the clause form of the formula (e.g. Fermuller and Leitsch [2, 3]). In this paper we study the complexity of the validity problem for the class of ground-negative formulas which is a natural generalization of a class of formulas considered by Kozen [4]. We show that this problem is \Pi p 2 -complete both for logics with and without equality, which is a quite unusual complexity class for na...

s true in the given model), the equivalence check (i.e. to decide whether or not two representations represent the same model) and for clause evaluation (i.e. to decide whether a given clause is true in the given model). From a theoretical point of view it makes sense to focus on Herbrand models of skolemized formulas (or clause sets), as Herbrand models exist for all satisfiable formulas of this kind. But this is also justified from a practical viewpoint to utilize the intuitive requirement of understandability, because in Herbrand models the domain and the interpretation of the function symbols are clear, fixed and intuitive. However a Herbrand model over a fixed signature is fully specified by a description of its potentially infinite set of (true) ground atoms, i.e. a set of terms (or strings, depending on how we want to look at them). This point of view reveals the (to our opinion) most interesting aspect of our approach: We are lead to

. Well established fields in Automated Deduction focus on algorithms for the detection of unsatisfiability (or, equivalently, validity) of sets of clauses (first-order formulas). However, these algorithms are rather useless if the input clause set turns out to be satisfiable: Only if the system terminates, the user can recognize the existence of a counterexample to the tested specification. Which one, he/she can only guess. In this paper we present TILT 1 , an automated deduction system designed to overcome this lack of expressiveness. 1 Introduction Assume a statement or specification expressed as a first-order formula F . Well established fields in Automated Deduction focus on algorithms for the detection of unsatisfiability of clause sets C, sat-equivalent to :F . However, it is at least thinkable that F does not hold in all possible worlds. Then, usually, the only information one can extract from the output---if the system terminates at all--- is: There exists a counter-...

This paper investigates the problem of testing clause sets for membership in classes known from literature. In particular, we are interested in classes defined via renaming: Is it possible to rename the predicates in a way such that positive and negative literals satisfy certain conditions? We show that for classes like Horn or OCC1N [5] the existence of such renamings can be decided in polynomial time, whereas the same problem is NP-complete for class PVD [5]. The decision procedures are based on hyper-resolution; if a renaming exists, it can be extracted from the final saturated clause set. 1 Introduction Within the last decade the classical decision problem for predicate logic was revisited in the frame of clause logic. Instead of deciding the validity of first order formulas in prenex normal form---usually without function symbols---the aim was now to decide the satisfiability of clause sets with a rich functional structure. Many classes of clause sets--- with and without ...

The guarded fragment of rst order logic [1] was dened in order to explain the nice properties (in particular decidability) of modal logics. In fact, many modal logics can be translated into the guarded fragment. Guarded clauses, dened by de Nivelle in [5], are a generalization of guarded formulas in clausal form. In [5], it is also shown that the class of guarded clause sets is decidable by ordered resolution. In this work, we describe a method for the transformation of a given set of non-negative guarded clauses in which each clause has a positive greatest literal into a set of so-called primitive guarded Horn clauses. Additionally, the clauses in the given set must full the following two restrictions: no functional ground terms must occur in non-ground clauses, and all literals must have a depth of at most 1. The purpose of this transformation is automated model building. A set of primitive guarded Horn clauses can be seen as a representation of a Herbrand interpret...

The guarded fragment of rst order logic, dened in [1], is interesting because many modal logics can be translated into it. Guarded clauses are a generalisation of clausal forms of guarded formulas, and sets of such clauses are decidable by ordered resolution [9]. We show that it is possible to transform a set S of guarded clauses (without equality) into a set of general Horn clauses G which has exactly one well-supported model M, and such that M is a model for S. Then G can be seen as a representation of M, since it is possible to evaluate ground atoms in M using G. By techniques presented in [4], G can further be transformed into a set of so-called primitive guarded clauses which represents M. An interpretation represented in this way allows the evaluation of guarded clauses and should be easy to understand for human users. The principle of our method is the processing of a set which is saturated under ordered resolution and factorisation. In order to determine a greate...