Proximity-based unification theory

Similarity relations are reflexive, symmetric, and transitive fuzzy relations. They help to make approximate inferences, replacing the notion of equality. Similarity-based unification has been quite intensively investigated, as a core computational method for approximate reasoning and declarative programming. In this paper we consider solving constraints over several similarity relations, instead of a single one. Multiple similarities pose challenges to constraint solving, since we can not rely on the transitivity property anymore. Existing methods for unification with fuzzy proximity relations (reflexive, symmetric, non-transitive relations) do not provide a solution that would adequately reflect particularities of dealing with multiple similarities. To address this problem, we develop a constraint solving algorithm for multiple similarity relations, prove its termination, soundness, and completeness properties, and discuss applications.

Proximity relations in the fuzzy relational database model

In order to provide approximate reasoning capabilities, in Gerla G, Sessa MI (1999) Chen G, Ying M, Cai K-Y (Eds) Fuzzy Logic and Soft computing, 19–31, Kluwer Academic Publishers, Boston an extension of Logic Programming has been proposed. Logic programs on function-free languages are considered, and approximate and imprecise information are represented by introducing a similarity relation ? in the set of predicate names and object names of the language. The inference system exploits the classical resolution rule of the Logic Programming paradigm. Moreover, the notion of fuzzy least Herbrand model is also provided. In this paper, by introducing the general notion of structural translation of languages, we generalize these results to the case of logic programs with function symbols. Some properties of the similarity relations are also proven.

6 - From Unification in Combination of Equational Theories to a New AC-Unification Algorithm

Some 2000 years ago, the average annual distance a person would normally travel, was approximately 500 km. The action radius of most people remained rather stable, but it rose gradually after the Industrial Revolution to some 1820 km (by car, bus, railway or aircraft) in the year 1960. Then, a period of rapid increase started, with almost 4390 km per year in 1990. Clearly, air transport, but also technological advances and changing lifestyles formed the background of this megatrend (see Chaturin, 1988; Schafer 1988). Accessibility and proximity have become keywords in understanding the geographical pattern of the 'homo mobilis'. The emerging question is if and how this pattern of physical movement will be affected by the digital revolution. Virtual connectivity through cyberspace has been a source of further unprecedented rise in the action radius of modern man. There is an increasing awareness that cyberspace and physical space are not substitutes. On the contrary these two domains are related with a complex relation, with cyber-place (CP) being the intermediate layer. The latter is essentially an integral part of Batty's (1997) concept of virtual geography, which is identified as the projection of the infrastructural layer of cyberspace on traditional space. CP is defined in accordance with cyberspace, the function of which is supported by CP. Just like other elements of the Internet jargon such as the Internet superhighway, virtual communities, web-surfing, telecommuting, etc., CP and cyberspace are geographic metaphors.

Heuristic methods of fuzzy clustering hold a special place in data mining. They are important in preliminary data analysis when the number of clusters, their structure and mutual arrangement are unknown. The object clustering methods, based on fuzzy binary relations are generalized by developing clear and fuzzy single-level clustering methods, clear and fuzzy sequential multi-level clustering methods. Possible examples of fuzzy binary relations, which characterize similarity of objects by length, angle and distance of their vector features are presented. For this purpose, the Harrington type desirability function and scale, enabling effective analysis of clustering results are suggested. Based on the proposed methods, the software systems that were effectively used for solving applied clustering problems are developed. Also, the study illustrated the clear single-level clustering method on a specific example. It is shown that application of the apparatus of fuzzy binary relations in clustering provides an additional opportunity to study the dynamics of the number of clusters, their structure and determine the degree of similarity of objects in a cluster. The results can be used for preliminary data analysis and for holding the object clustering procedure.

Unification is a basic operation in theorem proving, in type inference algorithms, and in logic programming languages such as Prolog. Prolog will play a major role in software development for the Fifth Generation project, and thus developing fast algorithms for unification is an important goal. This paper shows that the running time for a linear unification algorithm can often be improved substantially by use of parallel processing. The same is true for algorithms for some other complete problems in P, namely, the monotone circuit value problem and the path system accessibility problem. Previous theoretical work in computational complexity has suggested that these problems are not parallelizable; in practice this is not the case. To resolve this paradox, we introduce new complexity classes PC and PC* that capture the practical notion of parallelizability we discuss in this paper. We pose several open questions concerning PC and PC*.

Fuzzy graph representation of a fuzzy concept lattice

Properties of the fuzzy composition based on aggregation operators

Nominal logic is a variant of first-order logic with special facilities for reasoning about names and binding based on the underlying concepts of swapping and freshness. It serves as the basis of logic programming and term rewriting techniques that provide similar advantages to, but remain simpler than, higher-order logic programming or term rewriting systems. Previous work on nominal rewriting and logic programming has relied on nominal unification, that is, unification up to equality in nominal logic. However, because of nominal logic's equivariance property, these applications require a stronger form of unification, which we call equivariant unification. Unfortunately, equivariant unification and matching are NP-hard decision problems. This paper presents an algorithm for equivariant unification that produces a complete set of finitely many solutions, as well as NP decision procedure and a version that enumerates solutions one at a time. In addition, we present a polynomial time algorithm for swapping-free equivariant matching, that is, for matching problems in which the swapping operation does not appear.

Aggregation of fuzzy quasi-metrics

Linguistic modelling based on semantic similarity relation among linguistic labels

As the number of fatalities on the roads rises, analysts forced to look for models that might foretell a driver's tendency for traffic accidents (RTAs). This study intends to investigate the relationship between drivers' abilities to judge speed and available space in relation to the occurrence of RTAs. The Circulant Composition of Fuzzy Relations Inference System employed as the foundation for the approach used for this goal (CCFRIS). The first CCFRIS's inputs related to drivers' capacity for speed assessment. 200 novice drivers participated in the trial, which tested these skills at test speeds of 35, 55, and 75 km/h. In the driving simulator, the participants evaluated the aforementioned speed numbers from four distinct vantage points. On the other side, space evaluation capabilities of drivers make up the second CCFRIS's inputs. The identical set of drivers underwent both 2D and 3D space evaluation exams. The quantity of RTAs a motorist has experienced is the third CCFRIS structure taken into account. The research found that the space assessment qualities explained participation in RTAs better than the drivers' speed assessment abilities after testing three proposed CCFRIS on empirical data. This study also looks into binary fuzzy relations and fuzzy relation composition characteristics.

We consider the problem of characterizing and computing a minimal context for a given fuzzy context. That is, for a given binary fuzzy relation between a set of objects and a set of attributes, we want to obtain a new fuzzy relation in such a way that its concept lattice is isomorphic to the concept lattice of the original fuzzy relation and such that the new fuzzy relation is minimal. It turns out that the essence of this problem may be rephrased as the problem of finding bases of fuzzy closure systems. In the Boolean case, the problem is relatively simple and its solution is well known. In a fuzzy setting, the problem is more complex, basically because there are two generating operations involved: intersection, which is the only operation involved in the Boolean case, and shift, which is degenerate in the Boolean case. In this paper, we present some first results, examples, and issues for future research in this problem. Proofs and some technical material is left to a full version of this paper.

On a set H equipped with a fuzzy binary relation R which is reflexive, we first introduce a family of hyperoperations *Rp where p ∈ [0, 1]. As a result we obtain a family of commutative quasi-hypergroup . Then we show that for every a, b ∈ H the family can be considered as the p-cuts of a fuzzy set a *R b and in this manner we synthesize a fuzzy hyperoperation *R and we show is a p-fuzzy quasi-hypergroup. Lastly, we prove that any fuzzy hyperoperation satisfying some given properties also has the same properties as this fuzzy hyperoperation.