The lexicographic closure as a revision process

The connections between nonmonotonic reasoning and belief revision are well-known. A central problem in the area of nonmonotonic reasoning is the problem of default entailment, i.e., when should an item of default information representing “if θ is true then, normally, φ is true” be said to follow from a given set of items of such information. Many answers to this question have been proposed but, surprisingly, virtually none have attempted any explicit connection to belief revision. The aim of this paper is to give an example of how such a connection can be made by showing how the lexicographic closure of a set of defaults may be conceptualised as a process of iterated revision by sets of sentences. Specifically we use the revision method of Nayak. Introduction and Preliminaries The methodological connections between the areas of nonmonotonic reasoning, i.e., the process by which an agent may, possibly, withdraw previously derived conclusions upon enlarging her set of hypotheses (Makinson 1994), and belief revision, i.e., the process by which an agent changes her beliefs upon discovering some new information (Alchourŕon, Gärdenfors, & Makinson 1985; Gärdenfors 1988), are well-known (see, for example, (Gärdenfors & Makinson 1994; Gärdenfors & Rott 1995; Makinson & Gärdenfors 1991; Rott 1996)). As a consequence, it is possible to translate particular problems in one area into problems in the other. One particular problem in nonmonotonic reasoning is the question of default entailment, i.e., when should we regard one item of so-called “default knowledge” (hereafter just “default”), i.e., an expression of the form θ ⇒ φ standing for “if θ then normally (or usually, or typically) φ”, as “following from” a given set of defaults. Several answers to this question have been proposed in the literature (such as in (Benferhat et al. 1993; Benferhat, Saffiotti, & Smets 1995; Goldszmidt, Morris, & Pearl 1993; Lehmann 1995; Lehmann & Magidor 1992; Pearl 1990; Weydert 1996), to name but a few) but none of them (with the exception of the last named) seem to attempt any explicit connection with belief revision. The aim of this paper is to make a start on such a connection by showing how one particular method of default entailment, namely the lexicographic closure construction (Benferhat et al. 1993; Lehmann 1995) can be given a formulation in terms of a certain method of belief revision which was first given in (Nayak 1994) and studied further in (Nayak, Nelson, & Polansky 1996). In the process, we uncover one or two interesting avenues for further research on both sides. The plan of this paper is as follows. Firstly, in the next section we formally pose the basic question of default entailment outlined above and describe the lexicographic closure. The set of defaults defined by the lexicographic closure, considered as a binary relation, forms a rational consequence relation (in the sense of (Kraus, Lehmann, & Magidor 1990)). The section following this introduces the theory of belief revision and the important notion of epistemic entrenchment relation (E-relation for short) which it utilises. Also in this section we describe the correspondence between Erelations and rational consequence relations. Next, with the aid of this correspondence, we describe Nayak’s operation of revision. Nayak proposes to model revision of an epistemic state (represented as an E-relation) by an arbitrary set of sentences by first converting this set into an E-relation and then revising by this relation. We present one particular method for generating an E-relation from a set of sentences and show our main result: that, given this method, the E-relation corresponding to the lexicographic closure can be obtained by revising the initial epistemic state (which we take to be the E-relation in which the only sentences believed are the tautologies) firstly by the set of (the material counterparts of) those defaults which are the least specific, then those defaults which are the next-least specific and so on up to the set of the most specific defaults. After this we give our ideas for possible further study before offering some short concluding remarks. Before we get started, let us fix our notation. Throughout this paper, L is an arbitrary but fixed propositional language built up from a finite set of propositional variables using the usual connectives ¬,∧,∨,→,⊤ and ⊥. Semantics is provided by the (finite) set W of propositional worlds. For θ ∈ L we set Sθ = {w ∈ W | w |= θ}, i.e., Sθ is the set of worlds which satisfy θ. Given E ∪ {φ} ⊆ L we write E |= φ whenever ⋂ θ∈E Sθ ⊆ Sφ and let Cn(E) denote the set {φ | E |= φ}. As usual we write θ |= φ rather than {θ} |= φ etc. while, for any w ∈ W and E ⊆ L we set sentE(w) = {θ ∈ E | w |= θ}. Finally, for an arbitrary set X we use |X | to denote the cardinality of X . The Lexicographic Closure of a Set of