Horn Knowledge Bases Research Articles

Query–subquery nets for Horn knowledge bases in first-order logic

ABSTRACTWe formulate query–subquery nets and use them to create the first framework for developing algorithms for evaluating queries to Horn knowledge bases with the properties that: the approach is goal-directed; each subquery is processed only once and each supplement tuple, if desired, is transferred only once; operations are done set-at-a-time; and any control strategy can be used. Our intention is to increase efficiency of query processing by eliminating redundant computation, increasing adjustability (i.e. easiness in adopting advanced control strategies) and reducing the number of accesses to the secondary storage. For this purpose, we transform a logic program into an equivalent net structure and use it to determine which set of tuples or subqueries should be evaluated at each step, in an efficient way. The framework forms a generic evaluation method called QSQN, which is sound and complete and has polynomial time data complexity when the term-depth bound is fixed. The experimental results confirm the efficiency and usefulness of this method.

A Conceptual Framework for Secrecy-preserving Reasoning in Knowledge Bases

In many applications, Knowledge Bases (KBs) contain confidential or private information (secrets). The KB should be able to use this secret information in its reasoning process but in answering user queries care must be exercised so that secrets are not revealed to unauthorized users. We consider this problem under the Open World Assumption (OWA) in a setting with multiple querying agents M 1 ,…, M m that can pose queries against the KB K and selectively share answers that they receive from K with one or more other querying agents. We assume that for each M i , the KB has a prespecified set of secrets S i that need to be protected from M i . Communication between querying agents is modeled by a communication graph, a directed graph with self-loops. We introduce a general framework and propose an approach to secrecy-preserving query answering based on sound and complete proof systems. The idea is to hide the truthful answer from a querying agent M i by feigning ignorance without lying (i.e., to provide the answer ‘Unknown’ to a query q if it needs to be protected. Under the OWA, a querying agent cannot distinguish between the case that q is being protected (for reasons of secrecy) and the case that it cannot be inferred from K . In the pre-query stage we compute a set of envelopes E 1 , …, E m (restricted to a finite subset of the set of formulae that are entailed by K ) so that S i ⊆ E i , and a query α posed by agent M i can be answered truthfully whenever α ∉ E i and ¬ α ∉ E i . After the pre-query stage, the envelope is updated as needed. We illustrate this approach with two simple cases: the Propositional Horn KBs and the Description Logic AL KBs.

Horn Clause Contraction Functions

In classical, AGM-style belief change, it is assumed that the underlying logic contains classical propositional logic. This is clearly a limiting assumption, particularly in Artificial Intelligence. Consequently there has been recent interest in studying belief change in approaches where the full expressivity of classical propositional logic is not obtained. In this paper we investigate belief contraction in Horn knowledge bases. We point out that the obvious extension to the Horn case, involving Horn remainder sets as a starting point, is problematic. Not only do Horn remainder sets have undesirable properties, but also some desirable Horn contraction functions are not captured by this approach. For Horn belief set contraction, we develop an account in terms of a model-theoretic characterisation involving weak remainder sets. Maxichoice and partial meet Horn contraction is specified, and we show that the problems arising with earlier work are resolved by these approaches. As well, constructions of the specific operators and sets of postulates are provided, and representation results are obtained. We also examine Horn package contraction, or contraction by a set of formulas. Again, we give a construction and postulate set, linking them via a representation result. Last, we investigate the closely-related notion of forgetting in Horn clauses. This work is arguably interesting since Horn clauses have found widespread use in AI; as well, the results given here may potentially be extended to other areas which make use of Horn-like reasoning, such as logic programming, rule-based systems, and description logics. Finally, since Horn reasoning is weaker than classical reasoning, this work sheds light on the foundations of belief change

A Rational and Efficient Algorithm for View Revision in Databases

The dynamics of belief and knowledge is one of the major components of any autonomous system that should be able to incorporate new pieces of information. In this paper, we argue that to apply rationality result of belief dynamics theory to various practical problems, it should be generalized in two respects: first of all, it should allow a certain part of belief to be declared as immutable; and second, the belief state need not be deductively closed. Such a generalization of belief dynamics, referred to as base dynamics, is presented, along with the concept of a generalized revision algorithm for Horn knowledge bases. We show that Horn knowledge base dynamics has interesting connection with kernel change and abduction. Finally, we also show that both variants are rational in the sense that they satisfy certain rationality postulates stemming from philosophical works on belief dynamics.

Reasoning with ordered binary decision diagrams

We consider problems of reasoning with a knowledge-base, which is represented by an ordered binary decision diagram, for two cases of general and Horn knowledge-bases. Our main results say that both finding a model of a knowledge-base and deducing from a knowledge-base can be done in linear time for a general knowledge-base, but that abduction is NP-complete even for a Horn knowledge-base. Then, we consider abduction when its assumption set consists of all propositional literals (i.e., an answer for a given query is allowed to include any positive literals), and show that it can be done in polynomial time if the knowledge-base is Horn, while it remains NP-complete for the general case. Some other solvable cases are also discussed.

KNOWLEDGE BASE REFORMATION: PREPARING FIRST-ORDER THEORIES FOR EFFICIENT PROPOSITIONAL REASONING

We present an approach to knowledge compilation that transforms a function-free first-order Horn knowledge base to propositional logic. This form of compilation is important since the most efficient reasoning methods are defined for propositional logic, while knowledge is most conveniently expressed within a first-order language. To obtain compact propositional representations, we employ techniques from (ir)relevance reasoning as well as theory transformation via unfold/fold transformations. Application areas include diagnosis, planning, and vision. Preliminary experiments with a hypothetical reasoner indicate that our method may yield significant speed-ups.

Essential and redundant rules in Horn knowledge bases

A production rule of a knowledge base is called essential if it is present in any prime knowledge base which is logically equivalent to the given one. Identification of essential rules constitutes a crucial part of the structural analysis of any knowledge base. It specifies the degree of freedom we have in constructing logically equivalent transformations of the base, by specifying the set of rules which must remain in place, and implicitly showing which rules could be replaced by other ones. A prime rule is called redundant if it is not present in any irredundant prime knowledge base which is logically equivalent to the given one. Redundant rules can be viewed as the “least important” prime rules of a knowledge base. The recognition of redundancy of a particular prime rule in a knowledge base will eliminate such rule from consideration in any future simplifications of the knowledge base. This paper provides combinatorial characterization of essential and redundant rules of propositional Horn knowledge bases, and develops, whenever possible, efficient computational procedures for recognizing essentiality and redundancy of prime rules of Horn knowledge bases.

Quasi-acyclic propositional Horn knowledge bases: optimal compression

Horn knowledge bases are widely used in many applications. The paper is concerned with the optimal compression of propositional Horn production rule bases-one of the most important knowledge bases used in practice. The problem of knowledge compression is interpreted as a problem of Boolean function minimization. It was proved by P.L. Hammer and A. Kogan (1993) that the minimization of Horn functions, i.e., Boolean functions associated with Horn knowledge bases, is NP complete. The paper deals with the minimization of quasi acyclic Horn functions, the class of which properly includes the two practically significant classes of quadratic and of acyclic functions. A procedure is developed for recognizing in quadratic time the quasi acyclicity of a function given by a Horn CNF, and a graph based algorithm is proposed for the quadratic time minimization of quasi acyclic Horn functions. >

Optimal compression of propositional Horn knowledge bases: complexity and approximation

Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge bases. The standard approach to this problem, consisting in the removal of redundant rules from a knowledge base, leads to an “irredundant” but not necessarily optimal knowledge base. We prove here that the number of rules in any irredundant Horn knowledge base involving n propositional variables is at most n − 1 times the minimum possible number of rules. Therefore, the quadratic time transformation of an arbitrary Horn production rule base to an equivalent irredundant and prime one (presented in [9]) provides a reasonable approximation algorithm. In order to formalize the optimal compression problem, we define a Boolean function of a knowledge base as being the function whose set of true points is the set of models of the knowledge base. In this way the optimal compression of production rule knowledge bases becomes a problem of Boolean function minimization. In this paper we prove that the minimization of Horn functions (i.e. Boolean functions associated to Horn knowledge bases) is NP-complete.