Propositional Knowledge Bases Research Articles

On measuring inconsistency in definite and indefinite databases with denial constraints

Real-world databases are often inconsistent. Although there has been an extensive body of work on handling inconsistency, little work has been done on measuring inconsistency in databases. In this paper, building on work done on measuring inconsistency in propositional knowledge bases, we explore inconsistency measures (IMs) for definite and indefinite databases with denial constraints. We first introduce database IMs that are inspired by well-established methods to quantify inconsistency in propositional knowledge bases, but are tailored to the relational database context where data is generally the reason for inconsistency, not the integrity constraints. Then, we analyze the compliance of the database IMs with rationality postulates for both definite and indefinite databases. Finally, we investigate the complexity of the inconsistency measurement problem as well as of the problems of deciding whether the inconsistency is lower than, greater than, or equal to a given threshold for both the definite and the indefinite cases.

General information spaces: measuring inconsistency, rationality postulates, and complexity

AI systems often need to deal with inconsistent information. For this reason, since the early 2000s, some AI researchers have developed ways to measure the amount of inconsistency in a knowledge base. By now there is a substantial amount of research about various aspects of inconsistency measuring. The problem is that most of this work applies only to knowledge bases formulated as sets of formulas in propositional logic. Hence this work is not really applicable to the way that information is actually stored. The purpose of this paper is to extend inconsistency measuring to real world information. We first define the concept of general information space which encompasses various types of databases and scenarios in AI systems. Then, we show how to transform any general information space to an inconsistency equivalent propositional knowledge base, and finally apply propositional inconsistency measures to find the inconsistency of the general information space. Our method allows for the direct comparison of the inconsistency of different information spaces, even though the data is presented in different ways. We demonstrate the transformation on four general information spaces: a relational database, a graph database, a spatio-temporal database, and a Blocks world scenario, where we apply several inconsistency measures after performing the transformation. Then we review so-called rationality postulates that have been developed for propositional knowledge bases as a way to judge the intuitive properties of these measures. We show that although general information spaces may be nonmonotonic, there is a way to transform the postulates so they can be applied to general information spaces and we show which of the measures satisfy which of the postulates. Finally, we discuss the complexity of inconsistency measures for general information spaces.

Measuring Inconsistency in Some Logics with Modal Operators

The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it.

On Limitations of Structured (Deterministic) DNNFs

The study of representations for propositional theories has been a central subject in knowledge compilation. Many known representations of propositional knowledge bases are restricted negation normal form circuits (NNFs) or binary decision diagrams. Sentential decision diagrams (SDDs) are one of them, by definition they are restricted structured deterministic DNNFs and more general than ordered binary decision diagrams (OBDDs). In this paper limitations of structured (deterministic) DNNFs and in particular SDDs are considered. The main result in the paper is a quasipolynomial simulation of structured (deterministic) DNNFs by equivalent (unambiguous) nondeterministic OBDDs. The result is tight in the sense that there cannot be a polynomial simulation due to a known separation result. One may ask whether there are simulations in smaller size for restricted structured (deterministic) DNNFs. Answering a question posed by Bova and Szeider (2017) in the negative it is shown that not every SDD with disjunctions of constant fan-in can be transformed into equivalent OBDDs of polynomial size.

On the measure of conflicts: an argumentation-based framework

An important issue in the management of knowledge-based systems is the handling of inconsistency. This problem has recently been attracting a lot of attention from Artificial Intelligence community. When inconsistency occurs in a knowledge base, there are mainly two ways to deal with it; we either resolve it or accept inconsistency and cope with it. This paper tackles the problem of evaluating the amount of contradiction in propositional knowledge bases, and provides a new measure of conflict based on deductive argumentation theory. Measuring the degree of conflict of a knowledge base can help us to deal with inconsistencies. Several semantic- and syntax-based approaches have been proposed separately. Given the pivotal role of argumentation in representing and handling inconsistency, in this paper, we use logical argumentation as a way to compute the inconsistency measure for propositional formulae. We show using the complete argumentation tree that our family of inconsistency measures is able to localise the conflict of a formula following its context and allows us to distinguish between formulae. We extend our measure to quantify the degree of inconsistency of a set of formulae and give a general formulation of the inconsistency using some logical properties. We also provide a general formulation of our method in order to quantify the conflict of the whole knowledge base. Finally, we address the problem of restoring consistency using inconsistency measures.

Measuring inconsistency with constraints for propositional knowledge bases

Measuring inconsistency has been considered as a necessary starting point to understand the nature of inconsistency in a knowledge base better. For practical applications, however, we often have to face some constraints on resolving inconsistency. In this paper, we propose a graph-based approach to measuring the inconsistency for a propositional knowledge base with one or both of two typical types of constraints on modifying formulas. Here the first type of constraint, called the hard constraint, describes a pair of sets of formulas such that all the formulas in the first set should be protected from being modified on the condition that all the formulas in the second set must be modified in order to restore the consistency of that base, while the second type, called the soft constraint, describes a set of pairs of formulas that are not allowed to be modified together. At first, we use a bipartite graph to represent the relation between formulas and minimal inconsistent subsets of a knowledge base. Then we show that such a graph-based representation allows us to characterize the inconsistency with constraints in a concise way. Based on this characterization, we thus propose measures for the degree of inconsistency and for the responsibility of each formula for the inconsistency of a knowledge base with constraints, respectively. Finally, we show that these measures can be well explained based on Halpern and Pearl's causal model and Chockler and Halpern's notion of responsibility.

An algebraic approach to revising propositional rule-based knowledge bases

One of the important topics in knowledge base revision is to introduce an efficient implementation algorithm. Algebraic approaches have good characteristics and implementation method; they may be a choice to solve the problem. An algebraic approach is presented to revise propositional rule-based knowledge bases in this paper. A way is firstly introduced to transform a propositional rule-based knowledge base into a Petri net. A knowledge base is represented by a Petri net, and facts are represented by the initial marking. Thus, the consistency check of a knowledge base is equivalent to the reachability problem of Petri nets. The reachability of Petri nets can be decided by whether the state equation has a solution; hence the consistency check can also be implemented by algebraic approach. Furthermore, algorithms are introduced to revise a propositional rule-based knowledge base, as well as extended logic programming. Compared with related works, the algorithms presented in the paper are efficient, and the time complexities of these algorithms are polynomial.

What makes propositional abduction tractable

Abduction is a fundamental form of nonmonotonic reasoning that aims at finding explanations for observed manifestations. This process underlies many applications, from car configuration to medical diagnosis. We study here the computational complexity of deciding whether an explanation exists in the case when the application domain is described by a propositional knowledge base. Building on previous results, we classify the complexity for local restrictions on the knowledge base and under various restrictions on hypotheses and manifestations. In comparison to the many previous studies on the complexity of abduction we are able to give a much more detailed picture for the complexity of the basic problem of deciding the existence of an explanation. It turns out that depending on the restrictions, the problem in this framework is always polynomial-time solvable, NP-complete, coNP-complete, or Σ 2 P -complete. Based on these results, we give an a posteriori justification of what makes propositional abduction hard even for some classes of knowledge bases which allow for efficient satisfiability testing and deduction. This justification is very simple and intuitive, but it reveals that no nontrivial class of abduction problems is tractable. Indeed, tractability essentially requires that the language for knowledge bases is unable to express both causal links and conflicts between hypotheses. This generalizes a similar observation by Bylander et al. for set-covering abduction.

On probabilistic inference by weighted model counting

A recent and effective approach to probabilistic inference calls for reducing the problem to one of weighted model counting (WMC) on a propositional knowledge base. Specifically, the approach calls for encoding the probabilistic model, typically a Bayesian network, as a propositional knowledge base in conjunctive normal form (CNF) with weights associated to each model according to the network parameters. Given this CNF, computing the probability of some evidence becomes a matter of summing the weights of all CNF models consistent with the evidence. A number of variations on this approach have appeared in the literature recently, that vary across three orthogonal dimensions. The first dimension concerns the specific encoding used to convert a Bayesian network into a CNF. The second dimensions relates to whether weighted model counting is performed using a search algorithm on the CNF, or by compiling the CNF into a structure that renders WMC a polytime operation in the size of the compiled structure. The third dimension deals with the specific properties of network parameters (local structure) which are captured in the CNF encoding. In this paper, we discuss recent work in this area across the above three dimensions, and demonstrate empirically its practical importance in significantly expanding the reach of exact probabilistic inference. We restrict our discussion to exact inference and model counting, even though other proposals have been extended for approximate inference and approximate model counting.

The Language of Search

This paper is concerned with a class of algorithms that perform exhaustive search on propositional knowledge bases. We show that each of these algorithms defines and generates a propositional language. Specifically, we show that the trace of a search can be interpreted as a combinational circuit, and a search algorithm then defines a propositional language consisting of circuits that are generated across all possible executions of the algorithm. In particular, we show that several versions of exhaustive DPLL search correspond to such well-known languages as FBDD, OBDD, and a precisely-defined subset of d-DNNF. By thus mapping search algorithms to propositional languages, we provide a uniform and practical framework in which successful search techniques can be harnessed for compilation of knowledge into various languages of interest, and a new methodology whereby the power and limitations of search algorithms can be understood by looking up the tractability and succinctness of the corresponding propositional languages.

Prime implicants of first order formulas via transversal clauses

Knowledge compilation techniques are usually applied for propositional knowledge bases. In this article, an extension of the notion of prime implicants of a knowledge base using first order formulas in Skolem conjunctive normal form is suggested. The method of transversal clauses used earlier for computing prime implicants of a propositional formula in conjuctive normal form is adopted to the first order case via substitutions. Partial correctness of the algorithm is proved. The algorithm is adopted heuristically for computing approximate prime implicants.

A stratified first order logic approach for access control

Modeling information security policies is an important problem in many domains. This is particularly true in the health care sector, where information systems often manage sensitive and critical data. This article proposes to use nonmonotonic reasoning systems to control access to sensitive data in accordance with a security policy. In the first part of the article, we propose an access control model that overcomes several limitations of existing systems. In particular, it allows us to deal with contexts and to represent the two main kinds of privileges: permissions and prohibitions. This model will then be formally encoded using stratified (or prioritized) first-order knowledge bases. In the second part of the article, we discuss the problem of conflicts due to the joint handling of permissions and prohibitions. We show that approaches proposed for solving conflicts in propositional knowledge bases are not appropriate for handling inconsistent first-order knowledge bases. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 817–836, 2004.

New Polynomial Classes for Logic-Based Abduction

We address the problem of propositional logic-based abduction, i.e., the problem of searching for a best explanation for a given propositional observation according to a given propositional knowledge base. We give a general algorithm, based on the notion of projection; then we study restrictions over the representations of the knowledge base and of the query, and find new polynomial classes of abduction problems.

Fast algorithms for revision of some special propositional knowledge bases

In this paper, the computational complexity of propositional clause set counter-factuals is discussed. It is shown that the computational complexity of propositional clause set counterfactuals is at the second level of the polynomial hierarchy, and that the computational complexity of propositional Horn clause set counterfactuals is at the first level of the polynomial hierarchy. Furthermore, some polynomial algorithms are presented for some special propositional clause set, such as the unique satisfiable clause set and the clause set of which only one subset is minimally inconsistent with the input clause whose inconsistency check can be solved in polynomial time.

Compilability and compact representations of revision of Horn knowledge bases

Several methods have been proposed as an attempt to deal with dynamically changing scenarios. From a computational point of view, different formalisms have different computational properties. In this article we consider knowledge bases represented as sets of Horn clauses. The importance of this case is twofold: first, inference is polynomial, thus tractable; second, Horn clauses represents causal relations between facts, thus they are of great practical importance, although not all propositional knowledge bases can be represented in Horn form. The complexity of Horn revision is still high, and in some cases coincides with the complexity of the general (non-Horn) case. We analyze the complexity of belief revision from the point of view of the compilation [Cadoli et al. 1999]: we study the possibility of reducing the complexity by allowing a (possibly expensive) preprocessing of part of the input of the problem. Extending the work of Cadoli et al.[1996], we consider the problem of compact representation of revision in the Horn case, i.e., given a knowledge base T and an update P (both represented by Horn clauses) decide whether T * P , the result of the revision, can be represented with a propositional formula whose size is polynomial in the size of T and P . We give this representation for all formalisms for which it exists, and we show that the existence of a compact representation is related to the possibility of decreasing the complexity of a formalism via a proprocessing.

Semantical and computational aspects of Horn approximations

Selman and Kautz proposed a method, called Horn approximation, for speeding up inference in propositional Knowledge Bases. Their technique is based on the compilation of a propositional formula into a pair of Horn formulae: a Horn Greatest Lower Bound (GLB) and a Horn Least Upper Bound (LUB). In this paper we focus on GLBs and address two questions that have been only marginally addressed so far: 1. what is the semantics of the Horn GLBs? 2. what is the exact complexity of finding them? We obtain semantical as well as computational results. The major semantical result is: The set of minimal models of a propositional formula and the set of minimum models of its Horn GLBs are the same. The major computational result is: Finding a Horn GLB of a propositional formula in CNF is NP -equivalent.

PRACTICAL INCONSISTENCY MANAGEMENT FOR CRITICAL-TASKS DECISION-SUPPORT SYSTEMS

In this paper, the problem of logical consistency in critical-tasks decision systems is addressed. It is shown how local search techniques can be used to check the (in)consistency of very large propositional knowledge bases (KBs). Using new heuristic findings. we can now implement efficient forms of diagnostic reasoning in the context of very large ICBs modeling deep models of complex devices and including patterns of default reasoning.

A general-purpose natural language interface to an operating system

One of the most difficult tasks facing new computer users is the management of data and programs through an operating system (OS) interface. Both command languages and graphical interfaces can impede a user unfamiliar with the terminology and organizational techniques of modern file management systems. Various attempts have been made in recent years to develop systems that will enable users to interact with the computer in natural language. By simplifying the form of user input to an OS, such interfaces allow the user to concentrate on file management tasks rather than on the means of expressing OS-specific instructions. The Informational Network for a Natural Talking (INFANT) System was originally developed as a conversational system for communication at the level of a small child. It has since been adapted for use as a general-purpose English language interface for standard OS file management operations. The INFANT System reduces an input sentence to a hierarchy of standardized propositional forms, which are mapped through a propositional knowledge base and a compositional semantics process into a high-level representation of meaning. Its effectiveness as a working natural language interface has been verified in a pair of experiments performed in a college computer lab. © 1996 John Wiley & Sons, Inc.

Inference guiding in propositional knowledge bases

When initial data are not sufficient to accomplish inference in a knowledgedbased system, more information may be needed. The task of inference guiding or questiondasking in that situation is to select missing information to confirm. We present an inference guiding method for propositional knowledgedbased systems. Our computational experiments show that the new method is significantly better than the methods currently used in many knowledgedbased systems.

A practical propositional knowledge base revision algorithm

This paper gives an outline of knowledge base revision and some recently presented complexity results about propositional knowledge base revision. Different methods for revising propositional knowledge base have been proposed recently by several researchers, but all methods are intractable in the general case. For practical application, this paper presents a revision method for special case, and gives its corresponding polynomial algorithm.