Search

We have used the generic theorem prover Isabelle [181] to implement the non-classical logis we presented. The logical basis of Isabelle is a natural deduction presentation of minimal implicational predicate logic with universal quantification over all higher-types [179].1 We call this metalogic Meta, and to prevent object/meta confusion we use Λ to represent Meta’s universal quantifier and ⇒ for implication.

The process of information systems development seems lacking an information modelling stage, which requires a concrete expression of information. This paper suggests to use predicate logic to express items of information, and to use Barwise and Devlin’s situation theory to model the information aspect of a human purposeful activity as a remedy of the problem. The idea is to capture the information flow in a human purposeful activity from the point of view of the ‘actor’, an agent who performs the activity, by means of a set of formal concepts. This ‘information aspect’ will capture the information requirements of the actor for accomplishing the activity, and provides a basis for objects and data identification.

In this chapter, we present a brief overview of classical logical calculi and model theory. Our aim is twofold: first, we remind some of its main concepts and properties, and introduce notation. Second, our focus is directed especially to those properties which will be furthermore generalized due to the principle of the graded approach assumed in this book. In our opinion, this enables the reader to see the origins and way of the development of fuzzy logic. In endeavour for global point of view, we present general principles of formal logical systems (cf. A. Tarski [124]) in Section 3.3. They can be applied also to non-classical approaches and will be important for further explanation in Chapter 4. The interpretation, however, is there mostly a direct generalization of the classical approaches. A very abstract interpretation based on the category theory may have a special interest for fuzzy logic and it will be the topic of Chapter 7. A brief reminder of some fundamental notions of the model theory in categories is presented in Section 3.4. The reader who wishes to learn more about the topic of this chapter is referred to the special literature, for example [17, 121, 68, 76].

We propose an extension of classical predicate calculus, called Transaction Logic, which provides a logical foundation for the phenomenon of state changes in logic programs and databases. Transaction Logic comes with a natural model theory and a sound and complete proof theory. The proof theory not only verifies programs, but also executes them, which makes this logic an ideal tool for declarative programming of database transactions and state-modifying logic programs. The semantics of Transaction Logic leads naturally to features whose amalgamation in a single logic has proved elusive in the past. These features include hypothetical and committed updates, dynamic constraints on transaction execution, non-determinism, and bulk updates. Finally, Transaction Logic holds promise as a logical model of hitherto non-logical phenomena, including so-called procedural knowledge in AI, and the behavior of object-oriented databases, especially methods with side effects. This paper presents the semantics of Transaction Logic and a sound and complete SLD-style proof theory for a Horn-like subset of the logic.

How does one make inferences about the future, and how does one reason about the present? Reasoning requires that one relate, in some fashion, one event with another. Probability theory and predicate logic are the two most popular systems used to describe these relationships between states of nature. Probability theory can provide us with tools to compute the expected value of performance, the best guess as to the results that will occur.

Systems for helping users to select data sources in an environment, such as the Internet, must be expressive enough to allow a variety of data sources to be formally represented. We build upon and extend the concept language, description logic (DL), to propose a novel representation system to achieve that goal. We point out that there are technical barriers within description logic limiting the types of data sources that can be represented. Specifically, we show that (1) DL is awkward in representing sufficient conditions, and (2) DL can describe properties of a concept itself only in the case of existential quantification. To be concrete, whereas it is easy in DL to say ‘There are some objects in a concept C such that each of them holds the property P’, it is awkward, if not impossible, to say ‘All (Most, More-than-n, More-than-1/2) of those that hold the property P are in the concept C’ or ‘Most (More-than-n, More-than-1/2) of those that are in the concept C hold the property P’. These barriers cause us to extend DL with the notion of generalized quantifiers. We improve the previous results of generalized quantifiers to make them inter-operable with traditional logic. The proposed formalism integrates the nice features of generalized quantifiers into description logic, and hence achieves more expressive power than various representation systems based purely on description logic. It is also shown that our proposed language preserves those mathematical properties that traditional logic-based formalisms are known to hold.

Aspects of a draft version of the Aeronautical Telecommunications Network (ATN) Standards and Recommended Practices (SARPs) under development by ISO-compliant committees of the International Civil Aviation Organization (ICAO) have been mathematically modelled using a formal description technique. The ATN SARPs are a specification for a global telecommunications network for air traffic control systems. A version of Harel’s statecharts formalism embedded within a machine readable typed predicate logic has been used as a formal description technique to construct this mathematical model. Our model has been `typechecked’ to partially validate the internal consistency of the specification. The work described in this paper has already uncovered some problems in the draft SARPs, and will provide a basis for follow-on efforts to apply formal analysis methods such as model-checking and symbolic execution to aspects of the ATN SARPs. The success of this approach suggests that typed predicate logic is useful as a syntactic and semantic foundation for specialized Formal Description Techniques (FDTs).

This paper presents the design of a knowledge acquisition user interface for a compositional modeling system. To facilitate collaborative development of knowledge bases and their reuse, knowledge of the domain is represented in a formal language based on the first-order predicate logic. The vocabulary used is also formally defined explicitly in the same language. However, use of a formal representation language does not make the process of expressing knowledge about the domain easy or efficient for all users. We have designed the knowledge acquisition interface with multiple layers, each intended for a different type of users, such as programmers, knowledge engineers, and domain experts. Though the underlying modeling system employs one representation formalism, the layered interface allows each type of users to interact with the system in terms that are appropriate and efficient for their purposes.

Prolog is generally not the language of choice of business application developers. This, despite the fact that Prolog presents clear advantages in the declarative representation of business logic. One of the primary reasons for this is that the representational advantages of Prolog are perceived as being outweighed by the disadvantages of implementing integrated user interfaces and database access. With the advent of three-tiered client/server architectures the disadvantages faced by Prolog have been removed, and Prolog now presents a highly attractive platform for the implementation of business logic. This paper addresses two issues related to the use of Prolog Business Logic Objects. First is the integration of Prolog-based Business Logic Objects within the three-tiers. Second is the structure of the individual Business Logic Object in such a way that is can take advantage of the meta-level characteristics offered by Prolog.

After defining the problem in Chapter 4, designing the knowledge model and user interface model in Chapter 6, we can turn our attention to implementing the knowledge model (Figure 8–1). This model is detailed in section 6.1.1 (Figure 8–2). Having decided to implement the knowledge model in PROLOG, we must still choose between two major paradigms. On the one hand, we can implement the knowledge model using native PROLOG as the inference mechanism and predicate logic as the knowledge representation method. This approach is conceptually the easiest and has resulted in many useful expert systems (Roach et al. 1985, Rauscher et al. 1990, Thomson 1989). The other approach, is to use meta-programming in PROLOG. Meta-programming in PROLOG can be used to create a toolkit of PROLOG modules; each module implements some KBS feature. Unfortunately, the toolkit approach is more complex than native PROLOG programming and therefore requires a higher level of expertise from the developers.

In this chapter, a technique that is based on the idea of analyzing programs by decomposition 11, 15, 16 is introduced. It annotates loops with predicate logic assertions in a step by step process as depicted in Figure 3.1. The analysis of a loop starts by decomposing it into fragments, called events. Each event encapsulates the loop parts which are interdependent, with respect to data flow, and separates them from the rest of the loop. The resulting events are then analyzed, using plans stored in a knowledge base, to deduce their individual predicate logic annotations. Finally, the annotation of the whole loop is synthesized from the annotations of its events 60.

The era of employing a grand monolithic approach to develop data processing systems is coming to an end. Systems developers are now realizing that there are many different ways to specify and program systems. While the traditional IF… THEN, DO WHILE approach is popular in many situations, it is far from being the best or only way. For instance, a medical diagnosis system might be implemented best with rules and an inference engine. In other situations, conditioning a neural network or genetic algorithm could produce the desired results much more efficiently. Elsewhere, a predicate logic or functional expression would be the best way to specify an application. In other words, if a system can be specified in many ways, why not choose the mix of approaches best for the problem at hand? Why be bound to only one way of doing things, when a combination of approaches might produce a result that is more economical, accurate, understandable, and elegant?

In this paper we show that there are many examples of adequate models for intuitionistic predicate logic (IPL) which appear in algebra and logic. We present a class of lattices, which are complete Heyting algebras, such that every lattice from this class is adequate for IPL in the sense that for every formula which cannot be derived in IPL and for every lattice L from the class there is a model of IPL over L in which the formula is not valid.

We describe the early history of intuitonistic logic, its formalization and the genesis of the so-called Brouwer-Heyting-Kolmogorov interpretation. In particular we discuss at some length whether Heyting’s papers contain an anticipation of logic with existence predicate. Finally we publish some source material, in particular letters of Bemays, Glivenko and Kolmogorov to Heyting.

The design of the experimental logic programming language PLOP (predicative language for office procedure automation) and an associated application-oriented programming environment aims at a proper integration of office databases and office procedure programming in a distributed environment, without being principally confined to office applications.

We study the problem of null values. By this we mean that an attribute is applicable but its value at present is unknown and also that an attribute is applicable but its value is arbitrary. We adopt the view that tuples denote statements of predicate logic about database relations. Then, a null value of the first kind, respectively second kind, corresponds to an existentially quantified variable, respectively universally quantified variable. For instance if r is a database relation without null values and X is a range declaration for r then the tuple (a, ∀,b, ∃) ∈ R is intended to mean “there exists an x ∈ X such that for all y ∈ X: (a,y,b,x) ∈ r”. We extend basic operations of the well-known relational algebra to relations with null values. Using formal notions of correctness and completeness (adapted from predicate logic) we show that our extensions are meaningful and natural. Furthermore we reexamine the generalized join within our framework. Finally we investigate the algebraic structure of the class of relations with null values under a partial ordering which can be interpreted as a kind of logical implication.