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In this paper, we investigate rule semantics for deductive databases in the context of four-valued logic. In our approach, a database is a pair $\varDelta =(E,R)$, where $E$ is a set of pairs, each pair associating a ground fact with a truth value (thus, allowing to store true, false or inconsistent facts) and $R$ is a set of rules generalizing standard Datalog rules in the following sense: (i) the head of a rule can be a positive or a negative atom and (ii) the body can involve any among the connectors of four-valued logic. We define the database semantics as the least fixed point of a monotonic operator and we compare this semantics with that of k-existential programs defined by Fitting and paraconsistent extended logic programs defined by Arieli. Our main contribution is to show that, if we consider rules as implications (i.e. if we view the database as a set of formulas) then the semantics of the database is the unique minimal model of the set of database formulas. Here, minimality is understood with respect to the knowledge ordering of four-valued logic satisfying a monotonicity property whereby the truth value of the head of an instantiated rule is greater than that of the body. Moreover, we characterize databases having finite semantics and then, we address the issue of database updating. We argue that our approach allows for a new kind of updates, in which the update result depends not only on the fact involved in the update but also on its current truth value in the database.

Rough set theory is studied to manage uncertain and inconsistent information. Because Pawlak’s decision logic for rough sets is based on the classical two-valued logic, it is inconvenient for handling inconsistent information. We propose a bilattice logic as the deduction basis for the decision logic of rough sets to address inconsistent and ambiguous information. To enhance the decision logic to bilattice semantics, we introduce Variable Precision Rough Set (VPRS). As a deductive basis for bilattice decision logic, we define a consequence relation for Belnap’s four-valued semantics and provide a bilattice semantic tableau TB4 for a deduction system. We demonstrate the soundness and completeness of TB4 and enhance it with weak negation.

We apply a non-classical four-valued logic in the process of reasoning regarding strategies for cops in a modified game of “Cops and Robber” played on a graph. We extend the game by introducing uncertainty in a form of random failures of detecting devices. This is realized by allowing that a robber can be detected in a node only with the given probability PA. Additionally, with the probability PF, cops can be given a false-positive, i.e., they are informed that the robber is located at some node, whereas it is located somewhere else. Consequently, non-zero PFintroduces a measurement noise into the system. All the cops have access to information provided by the detectors and can communicate with each other, so they can coordinate the search. By adjusting the number of detectors, PA, and PFwe can achieve a smooth transition between the two well-known variants of the game: “with fully visible robber” and “with invisible robber”. We compare a simple probabilistic strategy for cops with the non-parametric strategy based on reasoning with a four-valued paraconsistent logic. It is shown that this novel approach leads to a good performance, as measured by the required mean catch-time. We conclude that this type of reasoning can be applied in real-world applications where there is no knowledge about the underlying source of errors which is particularly useful in robotics.

Four-valued semantics is applied to a deduction basis for granular reasoning with zooming operation. This enables the granular reasoning useful for the foundation of reasoning for incomplete and inconsistent information. We interpret the semantics of zooming operation with four-valued semantics based on partial interpretation and construct a deductive system with Belnap's four-valued logic. We also discuss an application with non-monotonic reasoning and paraconsistent logic using the zooming technique for frame problem.

There were two methods of doing modular transformations from Entity Relationship Diagrams to Class Diagrams according to international researchers. In order to establish which method would best suit software engineers, we conducted a survey by giving a group of students in the computer science field, whom we considered potential future software engineers. The results we got were valid, but did not match those of any of the international researchers. We found that this situation could only be explained using Eastern Four-Valued logic, also known by such names as Catuskoti and Tetralemma.

We propose a four-valued logic with intuitive semantics by the connectives that is useful for understanding the contradictions in knowledge representation. The intuitive semantics reflects that any assertion has dual character by whose information for or against the judgment. The four-valued logic is weakly paraconsistent and has the weak consistency to capture whether or not the contradictions are reconcilable with information. The four-valued logic is a normal extension of classical logic in a sense that it contains the schemata of classical axioms. We then propose an axiomatization of the four-valued logic. The soundness and completeness of the axiomatization with respect to the semantics are proved. The usefulness of the four-valued logic is discussed.

We have recently developed a package in Maple that allows to perform logical computations in any existing or proposed many-valued logic (that can be defined using truth tables). It has applications in logic engineering (e.g. when creating a new logic adapted to some requirements), in theoretical logic (e.g. checking if two axiomatizations of a logic are equivalent), or simply in checking the properties of a well known logic. Now, such a package has been tuned so that it can simultaneously deal with the logic proposed by the end user plus the standard Boolean logic, in order to conveniently check properties. This new approach is used to easily develop some disquisitions on why Belnap's four-valued logic is defined the way it is. Finally, it is shown how a new four valued logic that mixes ideas from Łukasiewicz's and Kleene's logics behaves similarly to Belnap's logic.

In answer set programming, an extended logic program may have no answer set, or only one trivial answer set. In this paper, we propose a new stable model semantics based on the restricted four-valued logic to overcome both inconsistences and incoherences in answer set programming. Our stable models coincide with classical answer sets when reasoning on consistent and coherent logic programs, and can be solved by transformation in existing ASP solvers. We also show the connection between our stable models and the extensions of default logic.

The aim of this paper is to present application of non-standard logic in the process of reasoning regarding the state of environment in a multi-robotic system. We have investigated the possibility that the usage of four-valued logic can serve as an alternative to the traditional probabilistic approach when handling the uncertainties. As a proof of concept, we have considered a typical search-and-rescue scenario in which a group of n robots performs a search for k victims localized in random positions in an initially unknown building. The building is represented in the form of a graph while the uncertainties are modeled by parameters denoting probabilities of victim observation and random blocking of a robot transition between neighboring nodes (which precludes the possibility for the robot to know anything about the actual graph structure with full confidence). The simulation results show that the unique features of applied reasoning language give significant improvement over typical approaches based on the two-valued logic. The results have a preliminary character, in the sense that up to now, only numerical experiments have been performed—applying the idea in a real multi-robotic system is currently underway.

One of the interesting types of information in information systems is contingent information. Information systems should be able to deal with such information. It is interesting to formalize contingent information, and the task is closely related to the so-called future contingents. We try to model contingent information by means of Łukasiewicz’s four-valued logic. Although modalities in his logic are non-standard, the non-modal fragment is useful to express contingent information in information systems.

Positive extended disjunctive deductive databases are those that contain explicit negation both in the head and body of the clauses. For such databases, paraconsistent minimal models (p-minimal models) have been proposed based on multi-valued logic (four-valued logic). Moreover, the paraconsistent relational model is also based on four-valued logic. In this paper, we propose an algorithm, which converts clauses to equations and solves it, to find p-minimal models using the paraconsistent relational model. In order to accomplish that, we use disjunctive paraconsistent relation model.

We propose a combination of a fragment of four-valued logic and process algebra. We present an operational semantics in SOS-style, and a completeness result for ACP with conditionals and four-valued logic.

The paper presents some results obtained in searching for a new axiomatic foundation for partial comparability (PC) in the frame of non-conventional preference modeling. The basic idea is to define an extended preference structure able to represent lack of information, uncertainty, ambiguity, multidimensional and conflicting preferences, using formal logic as the basic formalism. A four-valued paraconsistent logic is therefore described in the paper as a more suitable language for the purposes of the research. The concepts of partition, general binary relations properties, fundamental relational system of preferences (f.r.s.p.), maximal f.r.s.p. and well founded f.r.s.p. are then introduced and some theorems are demonstrated in order to provide the axiomatic foundation of PC. The main result obtained is a preference structure that is a maximal well founded f.r.s.p. This preference structure facilitates a more flexible, reliable and robust preference modeling. Moreover it can be viewed as a generalization of the conventional approach, so that all the results obtained until now can be used under it. Two examples are provided at the end of the paper in order to give an account of the operational potentialities of the new theory, mainly in the area of multicriteria decision aid and social choice theory. Further research directions conclude the paper.

When monitoring a system w.r.t. a property defined in a temporal logic such as LTL, a major concern is to settle with an adequate interpretation of observable system events; that is, models of temporal logic formulae are usually infinite words of events, whereas at runtime only finite but incrementally expanding prefixes are available.In this work, we review LTL-derived logics for finite traces from a runtime-verification perspective. In doing so, we establish four maxims to be satisfied by any LTL-derived logic aimed at runtime verification. As no pre-existing logic readily satisfies all of them, we introduce a new four-valued logic Runtime Verification Linear Temporal Logic RV-LTL in accordance to these maxims. The semantics of Runtime Verification Linear Temporal Logic (RV-LTL) indicates whether a finite word describes a system behaviour which either (i) satisfies the monitored property, (ii) violates the property, (iii) will presumably violate the property, or (iv) will presumably conform to the property in the future, once the system has stabilized. Notably, (i) and (ii) correspond to the classical semantics of LTL, whereas (iii) and (iv) are chosen whenever an observed system behaviour has not yet lead to a violation or acceptance of the monitored property.Moreover, we present a monitor construction for RV-LTL properties in terms of Moore machines signalizing the semantics of the so far obtained execution trace w.r.t. the monitored property. © The Author, 2010. Published by Oxford University Press. All rights reserved.

This paper proposes a specific four-valued logic (4VL) as a means of handling missing data in Relational Data Base Management systems. The proposed 4VL is a minor variant of the standard 4VL in which the “least true” condition is interpreted as “inapplicable” rather than “false”. The use of this 4VL is defended on the grounds that by defining several additional unary operators, the 4VL can be rendered intuitively manageable. The proposed unary operators contribute to the conceptual utility of the 4VL by providing an explicit way for users to relate familiar results of two-valued logic to analogous results in 4VL.

This report deals with the application of multi-valued logic in cellular automata. A four valued logic system with dibit representation has been considered in this case. The general properties and their relations to build such logical systems are also investigated and the question of implementation of this logical system in cellular automata environment has also been studied. It is shown, that chaotic fractals i.e, fractals as function of initial conditions are formed in such cases. It is interesting to note also that fractals so formed are multifractals and thus may have applications in analyzing natural fractal formation. Subject. terms: Multivalued Logic, Cellular Automata, Fractals.

This paper extends the basic rough set formalism introduced by Pawlak [1] to a rule-based knowledge representation language, called Rough Datalog, where rough sets are represented by predicates and described by finite sets of rules. The rules allow us to express background knowledge involving rough concepts and to reason in such a knowledge base. The semantics of the new language is based on a four-valued logic, where in addition to the usual values TRUE and FALSE, we also have the values BOUNDARY, representing uncertainty, and UNKNOWN corresponding to the lack of information. The semantics of our language is based on a truth ordering different from the one used in the well-known Belnap logic [2, 3] and we show why Belnap logic does not properly reflect natural intuitions related to our approach. The declarative semantics and operational semantics of the language are described. Finally, the paper outlines a query language for reasoning about rough concepts.

In this paper a four-valued logic on B(4) called Hyawic Form UniLogic is presented. Examples of this UniLogic are given which model problems in the area of human interaction, dialog and communication, i.e., human speech. The four values, or states, are isolated, conflicting, coexisting and unifying, forming a square lattice of B(4), Post Algebra of order 2 with 4 elements, inside a Post Algebra on E(9) of order 3 with 9=3/sup */3 elements. This logic has been used to represent human speech problems and their solutions in college classrooms. This logic is used in different universities, countries and languages including France, Algeria, Egypt and USA. It is used for policy analysis and planning, as well as for academic research in inter-communications analysis. Software packages using this logic are available from the developer Dr. Raiek Alnakari.

We address the problem of integrating information coming from different sources. The information consists of facts that a central server collects and tries to combine using (a) a set of logical rules, i.e. a logic program, and (b) a hypothesis representing the server's own estimates. In such a setting incomplete information from a source or contradictory information from different sources necessitate the use of many-valued logics in which programs can be evaluated and hypotheses can be tested. To carry out such activities we propose a formal framework based on Belnap's four-valued logic. In this framework we work with the class of programs defined by Fitting and we develop a theory for information integration. We also establish an intuitively appealing connection between our hypothesis testing mechanism on the one hand, and the well-founded semantics and Kripke-Kleene semantics of Datalog programs with negation, on the other hand.

Four-valued logic is generalized from the traditional binary logic in order to describe the dynamic logical behavior of objective phenomena. We are going to study its mathematical features, especially the so-called STAR algorithm, the technique for deriving its component expressions, then to show how it can be applied to the static and dynamic test generations for both combinational and sequential circuits as well as to transitional logics.

We introduce a four‐valued logic that includes, in addition to true and false, the values unknown and non‐existent. We introduce the idea of presupposition in fuzzy logic and then use this to relate this four valued logic to the binary logic.

Bilattices and d-frames are two different kinds of structures with a four-valued interpretation. Whereas d-frames were introduced with their topological semantics in mind, the theory of bilattices has a closer connection with logic. We consider a common generalisation of both structures and show that this not only still has a clear bitopological semantics, but that it also preserves most of the original bilattice logic. Moreover, we also obtain a new bitopological interpretation for the connectives of four-valued logic.

It is argued that a sophisticated question-answering machine that has the capability of making inferences from its data base should employ a certain four-valued logic, the motivating consideration being that minor inconsistencies in its data should not be allowed to lead (as in classical logic) to irrelevant conclusions. The actual form of the four-valued logic is ‘deduced’ from an interplay of this motivating consideration with certain ideas of Dana Scott concerning ‘approximation lattices.’