Abstract Often in Software Engineering, a modeling formalism has to support scenarios of inconsistency in which several requirements either reinforce or contradict each other. Paraconsistent transition systems are proposed in this paper as one such formalism: states evolve through two accessibility relations capturing weighted evidence of a transition or its absence, respectively. Their weights come, parametrically, from a residuated lattice. This paper explores both i) a category of these systems, and the corresponding compositional operators and ii) a modal logic to reason upon them. Furthermore, two notions of crisp and graded simulation and bisimulation are introduced in order to relate two paraconsistent transition systems. Finally, results of modal invariance, for specific subsets of formulas, are discussed.

The need for more flexible and robust models to reason about systems in the presence of conflicting information is becoming more and more relevant in different contexts. This has prompted the introduction of paraconsistent transition systems, where transitions are characterized by two pairs of weights: one representing the evidence that the transition effectively occurs and the other its absence. Such a pair of weights can express scenarios of vagueness and inconsistency. This paper establishes a foundation for a compositional and structured specification approach of paraconsistent transition systems, framed as paraconsistent institution. The proposed methodology follows the stepwise implementation process outlined by Sannella and Tarlecki.

Abstract This paper introduces a class of automata and associated languages, suitable to model a computational paradigm of fuzzy systems, in which both vagueness and simultaneity are taken as first-class citizens. This requires a weighted semantics for transitions and a precise notion of a synchronous product to enforce the simultaneous occurrence of actions. The usual relationships between automata and languages are revisited in this setting, including a specific Kleene theorem.

Fuzziness, as a way to express imprecision, or uncertainty, in computation is an important feature in a number of current application scenarios: from hybrid systems interfacing with sensor networks with error boundaries, to knowledge bases collecting data from often non-coincident human experts. Their abstraction in e.g. fuzzy transition systems led to a number of mathematical structures to model this sort of systems and reason about them. This paper adds two more elements to this family: two modal logics, framed as institutions, to reason about fuzzy transition systems and the corresponding processes. This paves the way to the development, in the second part of the paper, of an associated theory of structured specification for fuzzy computational systems.

Abstract We propose ε ↓ ( D → ) -logic as a formal foundation for the specification and development of event-based systems with data states. The framework is presented as an institution in the sense of Goguen and Burstall and the logic itself is parametrised by an underlying institution D → whose structures are used to model data states. ε ↓ ( D → ) -logic is intended to cover a broad range of abstraction levels from abstract requirements specifications up to constructive specifications. It uses modal diamond and box operators over complex actions adopted from dynamic logic. Atomic actions are pairs [inline-graphic not available: see fulltext] where e is an event and ψ a state transition predicate capturing the allowed reactions to the event. To write concrete specifications of recursive process structures we integrate (control) state variables and binders of hybrid logic. The semantic interpretation relies on event/data transition systems. For the presentation of constructive specifications we propose operational event/data specifications allowing for familiar, diagrammatic representations by state transition graphs. We show that ε ↓ ( D → ) -logic is powerful enough to characterise the semantics of an operational specification by a single ε ↓ ( D → ) -sentence. Thus the whole (formal) development process for event/data-based systems relies on ε ↓ ( D → ) -logic and its semantics as a common basis. It is supported by a variety of implementation constructors which can express, among others, event refinement and parallel composition. Due to the genericity of the approach, it is also possible to change a data state institution during system development when needed. All steps of our formal treatment are illustrated by a running example.

We propose $$\mathcal {E}^{\downarrow } $$ -logic as a formal foundation for the specification and development of event-based systems with local data states. The logic is intended to cover a broad range of abstraction levels from abstract requirements specifications up to constructive specifications. Our logic uses diamond and box modalities over structured actions adopted from dynamic logic. Atomic actions are pairs where e is an event and $$\psi $$ a state transition predicate capturing the allowed reactions to the event. To write concrete specifications of recursive process structures we integrate (control) state variables and binders of hybrid logic. The semantic interpretation relies on event/data transition systems; specification refinement is defined by model class inclusion. For the presentation of constructive specifications we propose operational event/data specifications allowing for familiar, diagrammatic representations by state transition graphs. We show that $$\mathcal {E}^{\downarrow } $$ -logic is powerful enough to characterise the semantics of an operational specification by a single $$\mathcal {E}^{\downarrow } $$ -sentence. Thus the whole development process can rely on $$\mathcal {E}^{\downarrow } $$ -logic and its semantics as a common basis. This includes also a variety of implementation constructors to support, among others, event refinement and parallel composition.

A ‘hybridization’ of a logic, referred to as the base logic, consists of developing the characteristic features of hybrid logic on top of the respective base logic, both at the level of syntax (i.e. modalities, nominals, etc.) and of the semantics (i.e. possible worlds). By ‘hybridized institutions’ we mean the result of this process when logics are treated abstractly as institutions (in the sense of the institution theory of Goguen and Burstall). This work develops encodings of hybridized institutions into (many-sorted) first-order logic (abbreviated $\mathcal{FOL}$ ) as a ‘hybridization’ process of abstract encodings of institutions into $\mathcal{FOL}$ , which may be seen as an abstraction of the well-known standard translation of modal logic into $\mathcal{FOL}$ . The concept of encoding employed by our work is that of comorphism from institution theory, which is a rather comprehensive concept of encoding as it features encodings both of the syntax and of the semantics of logics/institutions. Moreover, we consider the so-called theoroidal version of comorphisms that encode signatures to theories, a feature that accommodates a wide range of concrete applications. Our theory is also general enough to accommodate various constraints on the possible worlds semantics as well a wide variety of quantifications. We also provide pragmatic sufficient conditions for the conservativity of the encodings to be preserved through the hybridization process, which provides the possibility to shift a formal verification process from the hybridized institution to $\mathcal{FOL}$ .

Abstract Hybrid logics, which add to the modal description of transition structures the ability to refer to specific states, offer a generic framework to approach the specification and design of reconfigurable systems, i.e., systems with reconfiguration mechanisms governing the dynamic evolution of their execution configurations in response to both external stimuli or internal performance measures. A formal representation of such systems is through transition structures whose states correspond to the different configurations they may adopt. Therefore, each node is endowed with, for example, an algebra, or a first-order structure, to precisely characterise the semantics of the services provided in the corresponding configuration. This paper characterises equivalence and refinement for these sorts of models in a way which is independent of (or parametric on) whatever logic (propositional, equational, fuzzy, etc) is found appropriate to describe the local configurations. A Hennessy–Milner like theorem is proved for hybridised logics.

This paper extends the authors' previous work on a formal approach to the specification of reconfigurable systems, introduced in [7], in which configurations are taken as local states in a suitable transition structure. The novelty is the explicit consideration that not only the realisation of a service may change from a configuration to another, but also the set of services provided and even their functionality, may themselves vary. In other words, interfaces may evolve, as well.

Traditional notions of refinement of algebraic specifications, based on signature morphisms, are often too rigid to capture a number of relevant transformations in the context of software design, reuse and adaptation. This paper proposes an alternative notion of specification refinement, building on recent work on logic interpretation. The concept is discussed, its theory partially developed, its use illustrated through a number of examples.