TOWARDS A THEORY FOR ABSTRACT DATA TYPES

A rigorous framework for studying immutable data types having nondeterministic operations and operations exhibiting exceptional behavior is developed. The framework embodies the view of a data type taken in programming languages, and supports hierarchical and modular structure among data types. The central notion in this framework is the definition of a data type. An algebraic and behavioral approach for defining a data type is developed which focuses on the input-output behavior of a data type as observed through its operations. The definition of a data type abstracts from the representation structure of its values as well as from the multiple representations of the values for any representation structure. A hierarchical specification language for data types is proposed. The semantics of a specification is a set of related data types whose operations have the behavior captured by the specification. A clear distinction is made between a data type and its specification(s). The normal behavior and the exceptional behavior of the operations are specified separately. The specification language provides mechanisms to specify (i) a precondition for an operation thus stating its intended inputs, (ii) the exceptions which must be signalled by the operations, and (iii) the exceptions which the operations can optionally signal. Two properties of a specification, consistency and behavioral completeness, are defined. A consistent specification is guaranteed to specify at least one data type. A behaviorally complete specification ''completely'' specifies the observable behavior of the operations on their intended inputs. A deductive system based on first order multi-sorted predicate calculus with identity is developed for abstract data types. It embodies the general properties of data types, which are not explicitly stated in a specification. The theory of a data type, which consists of a subset of the first order properties of the data type, is constructed from its specification. The theory is used in verifying programs and designs expressed using the data type. Two properties of a specification, well definedness and completeness, are defined based on what can be proved from it using different fragments of the deductive system. The sufficient completeness property of Guttag and Horning is also formalized and related to the behavioral completeness property. The well definedness property is stronger than the consistency property, because the well definedness property not only requires that the specification specifies at least one data type, but also captures the intuition that it preserves other specifications used in it thus ensuring modular structure among specifications. The completeness property is stronger than the sufficient completeness property, since in addition to the requirement that the behavior of the observers can be deduced on any intended input by equational reasoning, it also requires that the equivalence of the observable effect of the construc