Terminological Logic Research Articles

An epistemic operator for description logics

Description logics (also called terminological logics, or concept languages) are fragments of first-order logic that provide a formal account of the basic features of frame-based systems. However, there are aspects of frame-based systems—such as nonmonotonic reasoning and procedural rules—that cannot be characterized in a standard first-order framework. Such features are needed for real applications, and a clear understanding of the logic underlying them is necessary for principled implementations. We show how description logics enriched with an epistemic operator can formalize such aspects. The logic obtained is a fragment of a first-order nonmonotonic modal logic. We show that the epistemic operator formalizes procedural rules, as provided in many knowledge representation systems, and enables sophisticated query formulation, including various forms of closed-world reasoning. We provide an effective procedure for answering epistemic queries posed to a knowledge base expressed in a description logic and extend this procedure in order to deal with rules. We also address the computational complexity of reasoning with the epistemic operator, identifying cases in which an appropriate use of the epistemic operator can help in decreasing the complexity of reasoning.

Classic learning

Description logics, also called terminological logics, are commonly used in knowledge-based systems to describe objects and their relationships. We investigate the learnability of a typical description logic, Classic, and show that Classic sentences are learnable in polynomial time in the exact learning model using equivalence queries and membership queries (which are in essence, “subsumption queries”—we show a prediction hardness result for the more traditional membership queries that convey information about specific individuals). We show that membership queries alone are insufficient for polynomial time learning of Classic sentences. Combined with earlier negative results (Cohen & Hirsh, 1994a) showing that, given standard complexity theoretic assumptions, equivalence queries alone are insufficient (or random examples alone in the PAC setting are insufficient), this shows that both sources of information are necessary for efficient learning in that neither type alone is sufficient. In addition, we show that a modification of the algorithm deals robustly with persistent malicious two-sided classification noise in the membership queries with the probability of a misclassification bounded below 1/2. Other extensions are considered.

An alternative proof method for possibilistic logic and its application to terminological logics

Possibilistic logic, an extension of first-order logic, deals with uncertainty that can be estimated in terms of possibility and necessity measures. Syntactically, this means that a first-order formula is equipped with a possibility degree or a necessity degree that expresses to what extent the formula is possibly or necessarily true. Possibilistic resolution yields a calculus for possibilistic logic which respects the semantics developed for possibilistic logic. A drawback, which possibilistic resolution inherits from classical resolution, is that it may not terminate if applied to formulas belonging to decidable fragments of first-order logic. Therefore we propose an alternative proof method for possibilistic logic. The main feature of this method is that it completely abstracts from a concrete calculus but uses as basic operation a test for classical entailment. We then instantiate possibilistic logic with a terminological logic, which is a decidable subclass of first-order logic but nevertheless much more expressive than propositional logic. This yields an extension of terminological logics towards the representation of uncertain knowledge which is satisfactory from a semantic as well as algorithmic point of view.

The learnability of description logics with equality constraints

Although there is an increasing amount of experimental research on learning concepts expressed in first-order logic, there are still relatively few formal results on the polynomial learnability of first-order representations from examples. Most previous analyses in the pac-model have focused on subsets of Prolog, and only a few highly restricted subsets have been shown to be learnable. In this paper, we will study instead the learnability of the restricted first-order logics known as “description logics”, also sometimes called “terminological logics” or “KL-ONE-type languages”. Description logics are also subsets of predicate calculus, but are expressed using a different syntax, allowing a different set of syntactic restrictions to be explored. We first define a simple description logic, summarize some results on its expressive power, and then analyze its learnability. It is shown that the full logic cannot be tractably learned. However, syntactic restrictions exist that enable tractable learning from positive examples alone, independent of the size of the vocabulary used to describe examples. The learnable sublanguage appears to be incomparable in expressive power to any subset of first-order logic previously known to be learnable.

ON AUTOMATIC REASONING FOR SCHEMA INTEGRATION

Success in database schema integration depends on the ability to capture real world semantics of the schema objects, and to reason about the semantics. Earlier schema integration approaches mainly rely on heuristics and human reasoning. In this paper, we discuss an approach to automate a significant part of the schema integration process. Our approach consists of three phases. An attribute hierarchy is generated in the first phase. This involves identifying relationships (equality, disjointness and inclusion) among attributes. We discuss a strategy based on user-specified semantic clustering. In the second phase, a classification algorithm based on the semantics of class subsumption is applied to the class definitions and the attribute hierarchy to automatically generate a class taxonomy. This class taxonomy represents a partially integrated schema. In the third phase, the user may employ a set of well-defined comparison operators in conjunction with a set of restructuring operators, to further modify the schema. These operators as well as the automatic reasoning during the second phase are based on subsumption. The formal semantics and automatic reasoning utilized in the second phase is based on a terminological logic as adapted in the CANDIDE data model. Classes are completely defined in terms of attributes and constraints. Our observation is that the inability to completely define attributes and thus completely capture their real world semantics imposes a fundamental limitation on the possibility of automatically reasoning about attribute definitions. This necessitates human reasoning during the first phase of the integration approach.

Terminological logic users workshop Technical University of Berlin, October 24–25, 1991

Terminological logic users workshop Technical University of Berlin, October 24–25, 1991

The BACK system—an overview

The BACK project was begun at the Technical University Berlin in January 1985 as part of a larger project within the ESPRIT programme 1 . Our task in the project group was the specification, design and implementation of a knowledge representation system which we called BACK ("Berlin Advanced Computational Knowledge Representation System"). It is a based on a terminological logic (term description language) and supports complex representation of a domain terminology, description of domain objects using that terminology, and database access via a uniform interface language.

Undecidability of subsumption in NIKL

Subsumption—determining whether one concept is more general than another—is known to be NP-hard for all reasonably expressive terminological logics, but, up to now, the decidability of subsumption for terminological logics used in current knowledge representation systems such as NIKL remained unknown. This paper shows that subsumption in the terminological logic of NIKL is undecidable and thus that there are no complete algorithms for subsumption or classification in NIKL.

A four-valued semantics for terminological logics

An intuitive four-valued semantics can be used to develop expressively powerful terminological logics which have tractable subsumption. The subsumptions supported by the logic are a type of “structural” subsumption, where each structural component of one concept must have an analogue in the other concept. Structural subsumption captures an important set of subsumptions, similar to the subsumptions computed in kl-one and nikl. Thus the trade-off between expressive power and computational tractability which plagues terminological logics based on standard, two-valued semantics can be avoided while still retaining a useful and semantically supported set of subsumptions.