Amílcar Sernadas

Structuring theories on consequence

Building on the work of Goguen and Burstall on institutions and on Tarskis notion of deductive system, a categorial framework for manipulating theories in an arbitrary logic is presented. Its main contribution is the formalisation of the semantics of theory-building operations on top of a consequence relation. For that purpose, the notion of π-institution is proposed as an alternative to the notion of institution, replacing the notions of model and satisfaction by a primitive consequence operator in the definition of a logic. The resulting approach to the semantics of specification languages is intrinsically different from the original one in the sense that the ultimate denotation of a specification is taken herein to be a class of theories (sets of formulae closed for the consequence relation) and not a class of models of that logic. Adopting this point of view, the semantics of Clear-like specification building operations is analysed.

Temporal aspects of logical procedure definition

Abstract This paper discusses the inclusion of time in a message-oriented relational model of information systems in order to achieve memory independent specifications. The concept of memory independence is reviewed and several systems languages are analysed from this point of view and also in other aspects of their temporal properties. The concept of temporal database is formally introduced and a special modal tense logic is developed to deal with it. This logic is extended with a transition imperative logic to allow the specification of processes within the information system at a very high level. The information system is seen as composed of theses concurrent autonomous processes that communicate through the temporal database divided in event and state assertions.

Fibring of logics as a categorial construction

Much attention has been given recently to the mechanism of fibring of logics, allowing free mixing of the connectives and using proof rules from both logics. Fibring seems to be a rather useful and general form of combination of logics that deserves detailed study. It is now well understood at the proof-theoretic level. However, the semantics of fibring is still insufficiently understood. Herein we provide a categorial definition of both proof-theoretic and model-theoretic fibring for logics without terms. To this end, we introduce the categories of Hilbert calculi, interpretation systems and logic system presentations. By choosing appropriate notions of morphism it is possible to obtain pure fibring as a coproduct. Fibring with shared symbols is then easily obtained by cocartesian lifting from the category of signatures. Soundness is shown to be preserved by these constructions. We illustrate the constructions within propositional modal logic.

Fibring: Completeness Preservation

A completeness theorem is established for logics with congruence endowed with general semantics (in the style of general frames). As a corollary, completeness is shown to be preserved by fibring logics with congruence provided that congruence is retained in the resulting logic. The class of logics with equivalence is shown to be closed under fibring and to be included in the class of logics with congruence. Thus, completeness is shown to be preserved by fibring logics with equivalence and general semantics. An example is provided showing that completeness is not always preserved by fibring ligics endowed with standard (non general) semantics. A categorial characterization of fibring is provided using coproducts and cocartesian liftings.

Weakly complete axiomatization of exogenous quantum propositional logic

A finitary axiomatization for EQPL (exogenous quantum propositional logic) is presented. The axiomatization is shown to be weakly complete relative to an oracle for analytical reasoning. The proof is carried out using a non-trivial extension of the Fagin-Halpern-Megiddo technique together with three Henkin style completions.

Abstract Object Types: A Temporal Perspective

The notion of abstract object type (AOT) tends to overlay the already classical concept of abstract data type (ADT) in several fields of application. Objects, although much more complex than data, have the advantage of dealing with states and processes. For that reason, they become useful, for instance, in the design of database applications and in software engineering. The difficulty lies in finding a suitable formalism for the abstract definition of objects, at least as effective as the equational formalism has been in the definition of abstract data types. The purpose of this paper is to present and discuss the main features of such a formalism. Concepts, tools and techniques are provided for the abstract definition of objects. A primitive language is presented allowing structured and rather independent definitions of object types. Each object is described as a temporal entity that evolves because of the events that happen during its life. The interaction between objects is reduced to event sharing. Both liveness and safety requirements can be stated and verified. Two case studies are presented for illustrating every aspect of the approach: the stack example which is very popular in the ADT area, thus allowing the comparison between the concepts of ADT and AOT, and the well known example of the eating philosophers which allows the discussion of the dynamic aspects.

Probabilistic Situation Calculus

In this article we propose a Probabilistic Situation Calculus logical language to represent and reason with knowledge about dynamic worlds in which actions have uncertain effects. Uncertain effects are modeled by dividing an action into two subparts: a deterministic (agent produced) input and a probabilistic reaction (produced by nature). We assume that the probabilities of the reactions have known distributions.Our logical language is an extension to Situation Calculae in the style proposed by Raymond Reiter. There are three aspects to this work. First, we extend the language in order to accommodate the necessary distinctions (e.g., the separation of actions into inputs and reactions). Second, we develop the notion of Randomly Reactive Automata in order to specify the semantics of our Probabilistic Situation Calculus. Finally, we develop a reasoning system in MATHEMATICA capable of performing temporal projection in the Probabilistic Situation Calculus.

Logics for specifying concurrent information systems

This chapter concentrates on a challenging problem of information system specification and design, namely how to cope on a high level of abstraction with concurrent behaviour and communication as implied by distribution. Since distributed information systems are reactive and open systems maintaining data bases and applications, it is crucial to develop high-level specification techniques that can cope with data and programs as well as with concurrent workflow and communication issues. Techniques from conceptual modeling, abstract data types, concurrent processes and communication protocols are relevant and have to be combined. In the approach presented here, temporal logic is used for specifying sequential object behaviour, and communication facilities are added for specifying interaction between concurrent objects. We study two distributed temporal logics dealing with communication in two different ways. D0 adds basic statements that can only express synchronous “calling” of predicates, while D1 adds much richer facilities for making local statements about other objects in their respective local logics. D0 is more operational and can be animated or implemented more easily, while D1 is intuitively more appealing and convenient for modeling and specification. We demonstrate by example how D1 can be effectively reduced to D0 in a sound and complete way.

Fibring Non-Truth-Functional Logics: Completeness Preservation

Fibring has been shown to be useful for combining logics endowed withtruth-functional semantics. However, the techniques used so far are unableto cope with fibring of logics endowed with non-truth-functional semanticsas, for example, paraconsistent logics. The first main contribution of thepaper is the development of a suitable abstract notion of logic, that mayalso encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that thisextended notion of fibring preserves completeness under certain reasonableconditions. This completeness transfer result, the second main contributionof the paper, generalizes the one established in Zanardo et al. (2001) butis obtained using new techniques that explore the properties of a suitablemeta-logic (conditional equational logic) where the (possibly)non-truth-functional valuations are specified. The modal paraconsistentlogic of da Costa and Carnielli (1988) is studied in the context of this novel notionof fibring and its completeness is so established.

Objects and their Specification

Object-oriented concepts and constructions are explained in an informal and language-independent way. Various algebraic approaches for dealing with objects and their specification are examined, ADT-based ones as well a~ process-based ones. The conclusion is that the process view of objects seems to be more appropriate than the data type view.