Grey Box Data Refinement

We introduce the concepts of grey box and display box data types. These make explicit the idea that state variables in abstract data types are not always hidden. Programming languages have visibility rules which make representations observable and modifiable. Specifications in model-based notations may have implicit assumptions about visible state components, or are used in contexts where the representation does matter. Grey box data types are like the ``standard'' black box data types, except that they contain explicit subspaces of the state which are modifiable and observable. Display boxes indirectly observe the state by adding displays to a black box. Refinement rules for both these alternative data types are given, based on their interpretations as black boxes

Proof Theoretic Concepts for the Semantics of Types and Concurrency

We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can type-check in more than one way. Since interpretations follow the type-checking derivations, coherence theorems are required: that is, one must prove that the meaning of a program does not depend on the way it was type-checked. The proof of such theorems for our proposed interpretation are the basic technical results of this paper. Interestingly, proving coherence in the presence of recursive types, variants, and abstract types forced us to reexamine fundamental equational properties that arise in proof theory (in the form of commutative reductions) and domain theory (in the form of strict vs. non-strict functions)

Inheritance as Implicit Coercion

We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can type-check in more than one way. Since interpretations follow the type-checking derivations, coherence theorems are required: that is, one must prove that the meaning of a program does not depend on the way it was type-checked. The proof of such theorems for our proposed interpretation are the basic technical results of this paper. Interestingly, proving coherence in the presence of recursive types, variants, and abstract types forced us to reexamine fundamental equational properties that arise in proof theory (in the form of commutative reductions) and domain theory (in the form of strict vs. non-strict functions)

The theory of classification part 11: adding class types to object implementations

A model of strongly-typed object generators, in which the class-type information was attached to the object-information was developed. A class was defined as a family of objects that shared a similar implementation strategy, and a family of types shared a similar method interface. It provided a good foundation for developing further model interpretations of other object-oriented concepts such as class hierarchies, abstract classes, and interfaces. A new typed calculus was also used to present a model of strongly-typed inheritance, and a formal obligation to demonstrate that aspects of inheritance were type-sound were also fulfilled

Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms

In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negation-as-failure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly well-supported models, are generalizations of the notion of well-supported models of normal logic programs to the case of programs with c-atoms. As for the case of fixpoint-based semantics, the difference between these two definitions is rooted in the treatment of naf atoms. We prove that answer sets by reduct (resp. by complement) are equivalent to weakly (resp. strongly) well-supported models of a program, thus generalizing the theorem on the correspondence between stable models and well-supported models of a normal logic program to the class of programs with c-atoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone c-atoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with c-atoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature

Religious education and the unconscious: an investigation of children from seven to eleven years

This thesis examines recent research into the validity of Piaget's theories as to how children assimilate, think, and learn, and analyses in detail the research of Br. Ronald Goldman, who based his theories concerning Religious Education upon the psychology of Piaget. The thesis shows how, although having exerted influence over religious education in Britain for over twenty years, the theories of Dr. Goldman are unhelpful for effective teaching in the field of religious, education, as they fail to give recognition to the significance of environment and experience for a child's receptivity of teaching, concentrating as they do too* much upon chronological age. By the use of various enquiries, the current level of understanding of religion among children of Junior School age is investigated, and found to be lacking in any conceptual development from the Infant Stage. The problem, apparently, is that the children are not introduced to any other concepts of God than "Father", with the result that unhelpful anthropomorphic concepts form and receive reinforcement. The problem posed is how to make use of children's experiences in order to develop deeper thought concerning religious interpretations of life. Two basic types of experience are identified and examined, and related to the psychology of Dr. C. G. Jung: the personal experience and the collective experience. By practical classroom work, which is illustrated throughout, specific religious topics are investigated, drawing upon these two basic types of experience. It is shown that, contrary to Goldman's theory, junior children are capable of analysing quite sophisticated Biblical material, and further, show themselves capable of considerable thought about religious matters and symbolism: in general. The thesis concludes by listing various implications for the Primary School Curriculum. For religious education to be effective, it needs to have its foundations within both personal and collective experience. Thought needs to be drawn out from the children by the use of carefully planned schemes of work, and where possible the children should be given the opportunity of extending and deepening their thought by problem; solving activities related to the theme being followed. In contrast to Dr. Goldman's recommendations, the high potential of junior children for abstract thought and reasoning is illuminated throughout and illustrated, and the use of Biblical, and other religious writings and material is recommended for use in the classroom