FUNCTIONAL DEPENDENCIES AND INCOMPLETE INFORMATION

Functional dependencies play an important role in relational database design. They are defined in the context of a single relation which at all times must contain tuples with non-null entries. In this paper we examine an extension of the functional dependency interpretation to handle null values, that is, entries in tuples that represent incomplete information in a relational database. A complete axiomatization of inference rules for extended functional dependencies is also presented. Only after having such results is it possible to talk about decompositions and normalization theory in a context of incomplete information. Finally, we show that there are several practical advantages in using nulls and a weaker notion of constraint satisfiability.Information Systems Working Papers Serie

A logic for unknown outcomes

In computer question answering and problem solving programs many of the questions of modal and tense logics appear as practical design problems. One problem of particular interest appears when we allow events to have the truth value "unknown", a natural value to assign to some events which occur at other times than the present. However, allowing a third value is not as simple as it seems.published or submitted for publicationis peer reviewe

The logical structure underlying temporal references in natural language

textTemporal references in natural language include tenses and other time relations, references to specific times, and a variety of phrases such as "present", "later", "when", "how often", and "never". Their high frequency of occurrence reflects the importance of time to the users of natural language. Although the structure underlying temporal references may appear complicated, it is a working assumption of this thesis that a sound logical explanation of its characteristics can be made. The frequent use of temporal references makes a correct exhibition of their underlying structure vital to a full understanding of natural language. Such an understanding is important in teaching and translating, indeed in all uses of natural language. In addition, understanding language better should aid in the design of computer programs which process natural languages. Chapter 2 of this thesis surveys some relevant work on temporal references, both to show what has been done and to show the scope of the problem. Despite the divergence in terminology and viewpoint, a unified theory can be derived which relates and extends the previous work. The new theory is presented in Chapter 3. It is a formal system which models the intuitive meaning of tenses, time relations, and other references to the time of events. The system precisely defines and shows the interrelationships of concepts which are often only vaguely defined. By its generality and its logical foundation, the system is able to serve as a skeleton for further studies of time in language. To illustrate some of the features of the system a question answering computer program, called Chronos, was written which accepts information in the form of tensed sentences and answers questions about the time of events. This program is discussed in Chapter 4. Chapter 5 discusses a problem which arises when we consider assigning truth values to statements about events occurring at times other than the present. The problem is to define a logic for unknown outcomes which retains the two valued tautologies. A logic is presented which has two kinds of implication: a material implication for which all the classical tautologies hold, and a strict implication defined in terms of logical necessity. The strict implication fragment of this logic is shown to be slightly stronger than the Lewis (1959) system S5, although it avoids many of the so-called paradoxes of material implication. The logic of Chapter 5 is a useful extension of the system for tenses (Chapter 3) to situations in which future (and perhaps past) events may have the truth value "unknown". Chapter 6 is a discussion section which evaluates the tense system, the logic for unknown outcomes, and the program Chronos. Several possibilities for extending the thesis are discussed.Computer Science