A short overview of Hidden Logic

In this paper we review a hidden (sorted) generalization of k-deductive systems - hidden k-logics. They encompass deductive systems as well as hidden equational logics and inequational logics. The special case of hidden equational logics has been used to specify and to verify properties in program development of behavioral systems within the dichotomy visible vs. hidden data. We recall one of the main applications of this work - the study of behavioral equivalence. Related results are obtained through combinatorial properties of the Leibniz congruence relation. In addition we obtain a few new developments concerning hidden equational logic, namely we present a new characterization of the behavioral consequences of a theory

Behavioral institutions and refinements in generalized hidden logics

We investigate behavioral institutions and refinements in the context of the object oriented paradigm. The novelty of our approach is the application of generalized abstract algebraic logic theory of hidden heterogeneous deductive systems (called hidden k-logics) to the algebraic specification of object oriented programs. This is achieved through the Leibniz congruence relation and its combinatorial properties. We reformulate the notion of hidden k-logic as well as the behavioral logic of a hidden k-logic as institutions. We define refinements as hidden signature morphisms having the extra property of preserving logical consequence. A stricter class of refinements, the ones that preserve behavioral consequence, is studied. We establish sufficient conditions for an ordinary signature morphism to be a behavioral refinement. © J.UCS.FCT via UIM

Behavioral equivalence of hidden k-logics: an abstract algebraic approach

This work advances a research agenda which has as its main aim the application of Abstract Algebraic Logic (AAL) methods and tools to the specification and verification of software systems. It uses a generalization of the notion of an abstract deductive system to handle multi-sorted deductive systems which differentiate visible and hidden sorts. Two main results of the paper are obtained by generalizing properties of the Leibniz congruence — the central notion in AAL. In this paper we discuss a question we posed in [1] about the relationship between the behavioral equivalences of equivalent hidden logics. We also present a necessary and sufficient intrinsic condition for two hidden logics to be equivalent

Behavioural reasoning for conditional equations

Object-oriented (OO) programming techniques can be applied to equational specification logics by distinguishing visible data from hidden data (that is, by distinguishing the output of methods from the objects to which the methods apply), and then focusing on the behavioural equivalence of hidden data in the sense introduced by H. Reichel in 1984. Equational specification logics structured in this way are called hidden equational logics, HELs. The central problem is how to extend the specification of a given HEL to a specification of behavioural equivalence in a computationally effective way. S. Buss and G. Roşu showed in 2000 that this is not possible in general, but much work has been done on the partial specification of behavioural equivalence for a wide class of HELs. The OO connection suggests the use of coalgebraic methods, and J. Goguen and his collaborators have developed coinductive processes that depend on an appropriate choice of a cobasis, which is a special set of contexts that generates a subset of the behavioural equivalence relation. In this paper the theoretical aspects of coinduction are investigated, specifically its role as a supplement to standard equational logic for determining behavioural equivalence. Various forms of coinduction are explored. A simple characterisation is given of those HELs that are behaviourally specifiable. Those sets of conditional equations that constitute a complete, finite cobasis for a HEL are characterised in terms of the HEL's specification. Behavioural equivalence, in the form of logical equivalence, is also an important concept for single-sorted logics, for example, sentential logics such as the classical propositional logic. The paper is an application of the methods developed through the extensive work that has been done in this area on HELs, and to a broader class of logics that encompasses both sentential logics and HELs. © 2007 Cambridge University Press.FCT via UIM

Coiterative Morphisms: Interactive Equational Reasoning for Bisimulation, using Coalgebras

ter: SEN 3 Abstract: We study several techniques for interactive equational reasoning with the bisimulation equivalence. Our work is based on a modular library, formalised in Coq, that axiomatises weakly final coalgebras and bisimulation. As a theory we derive some coalgebraic schemes and an associated coinduction principle. This will help in interactive proofs by coinduction, modular derivation of congruence and co-fixed point equations and enables an extensional treatment of bisimulation. Finally we present a version of the lambda-coinduction proof principle in our framework

Equational and Rule-Based Programming: Visualization, Reliability, and Knowledge Base Generation

This document describes developing an environment for effective use of functional/equational programs and rule-based expert systems. There are significant advantages in using these paradigms for reliability, parallelism, and accumulation of expertise in knowledge bases. The environment will make it easier to understand and use these paradigms, construct more reliable systems, and automatically enrich rule-based knowledge bases with the expertise. It will consist of the following components: (1) Visualization: for composing systems using a graphical interface and for understanding of algorithms. (2) Consistency Checking: for an equational and a rule-based language in accordance with the semantics of the languages. (3) Knowledge Base Generation and Testing: a translator that extracts expertise from existing programs and accumulates it as rules in knowledge bases; the rules are tested to enhance reliability. (4) Verification: interactive heterogeneous reasoning that consists of equational reasoning based on visual and textual information. These tools will be integrated in the proposed environment. The environment will greatly reduce the costs and increase the reliability of functional/equational and rule-based systems

Algebraic specification : syntax, semantics, structure

Algebraic specification is the technique of using algebras to model properties of a system and using axioms to characterize such algebras. Algebraic specification comprises two aspects: the underlying logic used in the axioms and algebras, and the use of a small, general set of operators to build specifications in a structured manner. We describe these two aspects using the unifying notion of institutions. An institution is an abstraction of a logical system, describing the vocabulary, the kinds of axioms, the kinds of algebras, and the relation between them. Using institutions, one can define general structuring operators which are independent of the underlying logic. In this paper, we survey the different kind of logics, syntax, semantics, and structuring operators that have been used in algebraic specification

Higher Order Implementation of Kahn Networks in Maude: Alternating Bit Protocol

We implement Kahn networks in Maude system by using behavioural theory of streams and encoding higher order function types. As an example we implement the alternating bit protocol in our framework