Behavioral Reasoning in Generalized Hidden Logics

Equational logic serves as the underlying logic in many formal approaches to program specification. The algebraic data types specified in this formal way can be viewed as abstract machines on which the programs are to be run. This is one way of giving a precise algebraic semantics for programs, against which the correctness of a program can be tested. Object oriented (OO) programs, however, present a special challenge for equational methods. This is due to properties inherent to the OO programs. A more appropriate model for the abstract machine in the case of an OO program is, arguably, a state transition system: like a state of such a system, a state of an OO program can be viewed as encapsulating all pertinent information about the abstract machine when it reaches the state during execution of the program. As a way of meeting the afore mentioned challenge the standard equality predicate can be augmented by behavioral equivalence; in this way many of the characteristic properties of state transition systems can be grafted onto equational logic. In our approach we use hidden k-logics as the underlying logic of program specification. They are a natural generalization of a k-deductive systems. They encompass deductive systems as well as hidden equational logics and inequational logics. Hidden k-logics are used to specify systems whose data may be heterogeneous, i.e., data are split in different kinds, usually called sorts. Moreover, in hidden k-logics we are also able to distinguish internal data (hidden data) and the real data (visible data). This advantage is central in the specification of OO systems. For some programs it is worth considering those kinds of encapsulated data representation either for security reasons or to simplify the process of updating and improving program implementations.