Inconsistency and underdefinedness in Z specifications

Abstract In software engineering, formal methods are meant to capture the requirements of software yet to be built using notations based on logic and mathematics. The formal language Z is such a notation. It has been found that in large projects inconsistencies are inevitable. It is also said, however, that consistency is required for Z specifications to have any useful meaning. Thus, it seems, Z is not suitable for large projects. Inconsistencies are a fact of life. We are constantly challenged by inconsistencies and we are able to manage them in a useful manner. Logicians recognised this fact and developed so called paraconsistent logics to continue useful, non-trivial, reasoning in the presence of inconsistencies. Quasi-classical logic is one representative of these logics. It has been designed such that the logical connectives behave in a classical manner and that standard inference rules are valid. As such, users of logic, like software engineers, should find it easy to work with QCL. The aim of this work is to investigate the support that can be given to reason about inconsistent Z specifications using quasi-classical logic. Some of the paraconsistent logics provide an extra truth value which we use to handle underdefinedness in Z. It has been observed that it is sometimes useful to combine the guarded and precondition approach to allow the representation of both refusals and underspecification. This work contributes to the development of quasi-classical logic by providing a notion of strong logical equivalence, a method to reason about equality in QCL and a tableau-based theorem prover. The use of QCL to analyse Z specifications resulted in a refined notion of operation applicability. This also led to a revised refinement condition for applicability. Furthermore, we showed that QCL allows fewer but more useful inferences in the presence of inconsistency. Our work on handling underdefinedness in Z led to an improved schema representation combining the precondition and the guarded interpretation in Z. Our inspiration comes from a non-standard three-valued interpretation of operation applicability. Based on this semantics, we developed a schema calculus. Furthermore, we provide refinement rules based on the concept that refinement means reduction of underdefinedness. We also show that the refinement conditions extend the standard rules for both the guarded and precondition approach in Z

Handling Inconsistency in Knowledge Bases

Real-world automated reasoning systems, based on classical logic, face logically inconsistent information, and they must cope with it. It is onerous to develop such systems because classical logic is explosive. Recently, progress has been made towards semantics that deal with logical inconsistency. However, such semantics was never analyzed in the aspect of inconsistency tolerant relational model. In our research work, we use an inconsistency and incompleteness tolerant relational model called Paraconsistent Relational Model. The paraconsistent relational model is an extension of the ordinary relational model that can store, not only positive information but also negative information. Therefore, a piece of information in the paraconsistent relational model has four truth values: true, false, both, and unknown. However, the paraconsistent relational model cannot represent disjunctive information (disjunctive tuples). We then introduce an extended paraconsistent relational model called disjunctive paraconsistent relational model. By using both the models, we handle inconsistency - similar to the notion of quasi-classic logic or four-valued logic -- in deductive databases (logic programs with no functional symbols). In addition to handling inconsistencies in extended databases, we also apply inconsistent tolerant reasoning technique in semantic web knowledge bases. Specifically, we handle inconsistency assosciated with closed predicates in semantic web. We use again the paraconsistent approach to handle inconsistency. We further extend the same idea to description logic programs (combination of semantic web and logic programs) and introduce dl-relation to represent inconsistency associated with description logic programs

Semantics of trace relations in requirements models for consistency checking and inferencing

Requirements traceability is the ability to relate requirements back to stakeholders and forward to corresponding design artifacts, code, and test cases. Although considerable research has been devoted to relating requirements in both forward and backward directions, less attention has been paid to relating requirements with other requirements. Relations between requirements influence a number of activities during software development such as consistency checking and change management. In most approaches and tools, there is a lack of precise definition of requirements relations. In this respect, deficient results may be produced. In this paper, we aim at formal definitions of the relation types in order to enable reasoning about requirements relations. We give a requirements metamodel with commonly used relation types. The semantics of the relations is provided with a formalization in first-order logic. We use the formalization for consistency checking of relations and for inferring new relations. A tool has been built to support both reasoning activities. We illustrate our approach in an example which shows that the formal semantics of relation types enables new relations to be inferred and contradicting relations in requirements documents to be determined. The application of requirements reasoning based on formal semantics resolves many of the deficiencies observed in other approaches. Our tool supports better understanding of dependencies between requirements

Opening address: Paraconsistent logic

I am honoured with and touched by the invitation of delivering the opening address of this Congress. Firstly, to see paraconsistent logic flourishing and growing, as we can readily see by simply glacing over the programme of this conference, is among one of my greatest joys. Secondly, and equally important, because this congress takes place in the University of Toruń.I am honoured for having lectured here, a most congenial and stimulating place, and could not think of a better place for a conference dedicated to the memory of Stanisław Jaśkowski. In particular, I am delighted for having had a correspondence with him, and although I was deprived of the pleasure of meeting him personally, I was fortunate enough for having collaborated with some of his disciples, such as L. Dubikajtis and T. Kotas. All and all, Toruń in particular and Poland in general are for me a second home, for all the kindness and care everyone has shown to me over several years, since my very first visit to this country

Probabilities and Quantum Reality: Are There Correlata?

Any attempt to introduce probabilities into quantum mechanics faces difficulties due to the mathematical structure of Hilbert space, as reflected in Birkhoff and von Neumann's proposal for a quantum logic. The (consistent or decoherent) histories solution is provided by its single framework rule, an approach that includes conventional (Copenhagen) quantum theory as a special case. Mermin's Ithaca interpretation addresses the same problem by defining probabilities which make no reference to a sample space or event algebra (``correlations without correlata''). But this leads to severe conceptual difficulties, which almost inevitably couple quantum theory to unresolved problems of human consciousness. Using histories allows a sharper quantum description than is possible with a density matrix, suggesting that the latter provides an ensemble rather than an irreducible single-system description as claimed by Mermin. The histories approach satisfies the first five of Mermin's desiderata for a good interpretation of quantum mechanics, including Einstein locality, but the Ithaca interpretation seems to have difficulty with the first (independence of observers) and the third (describing individual systems).Comment: Latex 31 pages, 3 figures in text using PSTrick

An Automated Method for Identifying Inconsistencies within Diagrammatic Software Requirements Specifications

The development of large-scale, composite software in a geographically distributed environment is an evolutionary process. Often, in such evolving systems, striving for consistency is complicated by many factors, because development participants have various locations, skills, responsibilities, roles, opinions, languages, terminology and different degrees of abstraction they employ. This naturally leads to many partial specifications or viewpoints. These multiple views on the system being developed usually overlap. From another aspect, these multiple views give rise to the potential for inconsistency. Existing CASE tools do not efficiently manage inconsistencies in distributed development environment for a large-scale project. Based on the ViewPoints framework the WHERE (Web-Based Hypertext Environment for requirements Evolution) toolkit aims to tackle inconsistency management issues within geographically distributed software development projects. Consequently, WHERE project helps make more robust software and support software assurance process. The long term goal of WHERE tools aims to the inconsistency analysis and management in requirements specifications. A framework based on Graph Grammar theory and TCMJAVA toolkit is proposed to detect inconsistencies among viewpoints. This systematic approach uses three basic operations (UNION, DIFFERENCE, INTERSECTION) to study the static behaviors of graphic and tabular notations. From these operations, subgraphs Query, Selection, Merge, Replacement operations can be derived. This approach uses graph PRODUCTIONS (rewriting rules) to study the dynamic transformations of graphs. We discuss the feasibility of implementation these operations. Also, We present the process of porting original TCM (Toolkit for Conceptual Modeling) project from C++ to Java programming language in this thesis. A scenario based on NASA International Space Station Specification is discussed to show the applicability of our approach. Finally, conclusion and future work about inconsistency management issues in WHERE project will be summarized

On the Satisfiability of Quasi-Classical Description Logics

Though quasi-classical description logic (QCDL) can tolerate the inconsistency of description logic in reasoning, a knowledge base in QCDL possibly has no model. In this paper, we investigate the satisfiability of QCDL, namely, QC-coherency and QC-consistency and develop a tableau calculus, as a formal proof, to determine whether a knowledge base in QCDL is QC-consistent. To do so, we repair the standard tableau for DL by introducing several new expansion rules and defining a new closeness condition. Finally, we prove that this calculus is sound and complete. Based on this calculus, we implement an OWL paraconsistent reasoner called QC-OWL. Preliminary experiments show that QC-OWL is highly efficient in checking QC-consistency

Paraconsistência em lógica híbrida

Mestrado em Matemática e AplicaçõesThe use of hybrid logics allows the description of relational structures, at the same time that allows establishing accessibility relations between states and, furthermore, nominating and making mention to what happens at speci c states. However, the information we collect is subject to inconsistencies, namely, the search for di erent information sources can lead us to pick up contradictions. Nowadays, by having so many means of dissemination available, that happens frequently. The aim of this work is to develop tools capable of dealing with contradictory information that can be described as hybrid logics' formulas. To build models, to compare inconsistency in di erent databases, and to see the applicability of this method in day-to-day life are the basis for the development of this dissertation.O uso de lógicas híbridas permite a descrição de estruturas relacionais, ao mesmo tempo que permite estabelecer relações de acessibilidade entre estados, e, para além disso, nomear e fazer referência ao que acontece em estados específicos. No entanto, a informação que recolhemos está sujeita a inconsistências, isto é, a procura de diferentes fontes de informação pode levar a recolha de contradições. O que nos dias de hoje, com tantos meios de divulgação disponíveis, acontece frequentemente. O objetivo deste trabalho e desenvolver ferramentas capazes de lidar com informação contraditória que possa ser descrita através de fórmulas de lógicas híbridas. Construir modelos e comparar a inconsistência de diferentes bases de dados e ver a aplicabilidade deste método no dia-a-dia são a base para o desenvolvimento desta dissertação