Improved Algorithms for Theory Revision with Queries

This paper presents an approach to belief revision in which revision is a function from a belief state and a finite set of formulas to a new belief state. In the interesting case, the set for revision S may be inconsistent but individual members of S are consistent. We argue that S will still contain interesting information regarding revision; in particular, maximum consistent subsets of S will determine candidate formulas for the revision process, and the agent's associated faithful ranking will determine the plausibility of such candidate formulas. Postulates and semantic conditions characterizing this approach are given, and representation results are provided. As a consequence of this approach, we argue that revision by a sequence of formulas, usually considered as a problem of iterated revision, is more appropriately regarded as revision by the possibly-inconsistent set of these formulas. Hence we suggest that revision by a sequence of formulas is foremost a problem of (uniterated) set revision.

Machine Learning, 2002

Artificial Intelligence, 2012

The area of belief revision studies how a rational agent may incorporate new information about a domain into its belief corpus. An agent is characterized by a belief state K, and receives a new item of information α which is to be included among its set of beliefs. Revision then is a function from a belief state and a formula to a new belief state. We propose here a more general framework for belief revision, in which revision is a function from a belief state and a finite set of formulas to a new belief state. In particular, we distinguish revision by the set {α, β} from the set {α ∧ β}. This seemingly innocuous change has significant ramifications with respect to iterated belief revision. A problem in approaches to iterated belief revision is that, after first revising by a formula and then by a formula that is inconsistent with the first formula, all information in the original formula is lost. This problem is avoided here in that, in revising by a set of formulas S, the resulting belief state contains not just the information that members of S are believed to be true, but also the counterfactual supposition that if some members of S were later believed to be false, then the remaining members would nonetheless still be believed to be true. Thus if some members of S were in fact later believed to be false, then the other elements of S would still be believed to be true. Hence, we provide a more nuanced approach to belief revision. The general approach, which we call parallel belief revision, is independent of extant approaches to iterated revision. We present first a basic approach to parallel belief revision. Following this we combine the basic approach with an approach due to Jin and Thielscher for iterated revision. Postulates and semantic conditions characterizing these approaches are given, and representation results provided. We conclude with a discussion of the possible ramifications of this approach in belief revision in general.

unpublished notes, 1998

The purpose of this note is to acknowledge a gap in a previous paper -"The Complexity of Revision", see [1] -and provide a corrected version of argument. The gap was originally pointed out by Francesco Orilia (personal communication and [4]), and the fix was developed in correspondence with Vann McGee.

Lecture Notes in Computer Science, 2004

A revision algorithm is a learning algorithm that identifies the target concept, starting from an initial concept. Such an algorithm is considered efficient if its complexity (in terms of the resource one is interested in) is polynomial in the syntactic distance between the initial and the target concept, but only polylogarithmic in the number of variables in the universe. We give efficient revision algorithms in the model of learning with equivalence and membership queries. The algorithms work in a general revision model where both deletion and addition type revision operators are allowed. In this model one of the main open problems is the efficient revision of Horn sentences. Two revision algorithms are presented for special cases of this problem: for depth-1 acyclic Horn sentences, and for definite Horn sentences with unique heads. We also present an efficient revision algorithm for threshold functions. Efficient learnability has been studied from many different angles in computational learning theory for the last two decades, for example, in both the PAC and query learning models, and by measuring complexity in terms of sample size, the number of queries or running time. Attribute-efficient learning algorithms are required to be efficient (polynomial) in the number of relevant variables, and "super-efficient" (polylogarithmic) in the total number of variables . It is argued that practical and biologically plausible learning algorithms need to be attribute efficient.

2004

The aim of this article is to revisit Dalal's operator for belief revision. Dalal has proposed a technique for revising belief bases based on the minimization of a distance between interpretations. The result is a concrete operator that can be considered either from a semantical point of view (distance between interpretations) or from a syntactical point of view (number of atoms that have their truth values changed). Dalal has shown that the so-called Alchourrón, Gärdenfors and Makinson (AGM) postulates are satisfied by its operator. The AGM postulates constrain the revision process so that minimal changes occur in the belief set. In this article, our contribution is twofold: first, we improve Dalal's algorithm by avoiding multiple satisfiability checking, which are NP-complete tasks. Our algorithm requires only one NP-stage if beliefs are expressed in a specific syntax, namely the prime implicates and prime implicants. Second, we propose a new distance based on the number of prime implicates in contradiction with the incoming new information. We argue that in some cases changing a minimal set of propositional symbols do not necessarily entail minimal changes.

1995

In this paper we address a specific computational aspect of belief revision: the size of the propositional formula obtained by means of the revision of a formula with a new one. In particular, we focus on the size of the smallest formula which is logically equivalent to the revised knowledge base. The main result of this paper is that not all formalizations of belief revision are equal from this point of view. For some of them we show that the revised knowledge base can be represented by a polynomial-size formula (we call these results "compactability" results). On the other hand, for other ones the revised knowledge base does not always admit a polynomial-space representation, unless the polynomial hierarchy collapses at a sufficiently low level ("non-compactability" results). We also show that the time complexity of query answering for the revised knowledge base has definitely an impact on being able to represent the result of the revision compactly. Nevertheless, formalisms with the same complexity may have different compactability properties. We also study compactability properties for a weaker form of equivalence, called query equivalence, which allows to introduce new propositional symbols. Moreover, we extend our analysis to the special case in which the new formula has constant size and to the case of sequences of revisions (i.e., iterated belief revision). A complete analysis along these four coordinates is shown. * An extended abstract of this paper appeared as [5].

1998

Approximating a general formula from above and below by Horn formulas (its Horn envelope and Horn core, respectively) was proposed by Kautz (1991, 1996) as a form of \knowledge compilation," supporting rapid approximate reasoning; on the negative side, this scheme is static in that it supports no updates, and has certain complexity drawbacks pointed out by . On the other hand, the many frameworks and schemes proposed in the literature for theory update and revision are plagued by serious complexity-theoretic impediments, even in the Horn case, as was pointed out by , and is further demonstrated in the present paper. More fundamentally, these schemes are not inductive, in that they may lose in a single update any positive properties of the represented sets of formulas (small size, Horn structure, etc.). In this paper we propose a new scheme, incremental recompilation, which combines Horn approximation and model-based updates; this scheme is inductive and very e cient, free of the problems facing its constituents. A set of formulas is represented by an upper and lower Horn approximation. To update, we replace the upper Horn formula by the Horn envelope of its minimum-change update, and similarly the lower one by the Horn core of its update; the key fact which enables this scheme is that Horn envelopes and cores are easy to compute when the underlying formula is the result of a minimum-change update of a Horn formula by a clause. We conjecture that e cient algorithms are possible for more complex updates.

A revision algorithm is a learning algorithm that identifies the target concept, starting from an initial concept. Such an algorithm is considered efficient if its complexity (in terms of the measured resource) is polynomial in the syntactic distance between the initial and the target concept, but only polylogarithmic in the number of variables in the universe. We give efficient revision algorithms in the model of learning with equivalence and membership queries. The algorithms work in a general revision model where both deletion and addition revision operators are allowed. In this model one of the main open problems is the efficient revision of Horn formulas. Two revision algorithms are presented for special cases of this problem: for depth-1 acyclic Horn formulas, and for definite Horn formulas with unique heads.

arXiv (Cornell University), 2016

Belief revision has been studied mainly with respect to background logics that are monotonic in character. In this paper we study belief revision when the underlying logic is nonmonotonic instead-an inherently interesting problem that is under explored. In particular, we will focus on the revision of a body of beliefs that is represented as a logic program under the answer set semantics, while the new information is also similarly represented as a logic program. Our approach is driven by the observation that unlike in a monotonic setting where, when necessary, consistency in a revised body of beliefs is maintained by jettisoning some old beliefs, in a nonmonotonic setting consistency can be restored by adding new beliefs as well. We will define a syntactic revision function and subsequently provide representation theorem for characterising it.