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We present a study of confluence and related properties of fuzzy relations defined over similarity spaces. The ordinary confluence is an essential property of relations connected to the idea of rewriting and substituting which appears in abstract rewriting systems. This paper is a continuation of our previous paper, where we have introduced analogous notions related to substitutability in graded setting using residuated lattices as structures of truth degrees, leaving the ordinary notions a particular case when the underlying structure is the two-valued Boolean algebra. In this paper, we further extend our previous results by developing the notions of confluence and related properties respecting a given similarity relation. We also present definitions of confluence and related properties of relations on a generalized pseudometric spaces. Using the well-known link between generalized pseudometrics and similarities, we describe the connection of the notions defined on generalized pseudometric spaces to the corresponding notions on similarity spaces. The introduced notions are further illustrated by example of rewriting based on graded if-then rules.

We study similarity issues of confluence and related properties of fuzzy relations. The ordinary notions of divergence, convergence, convertibility, and confluence are essential properties of relations which appear in abstract rewriting systems. In a graded setting, we can introduce analogous notions related to the idea of approximate rewriting. This paper is a continuation of our previous paper (Belohlavek et al. 2010), where we have introduced such notions using residuated structures of truth degrees, leaving the ordinary notions a particular case when the underlying structure of truth degrees is the two-valued Boolean algebra. In this paper, we focus on similarity issues of confluence and present formulas which provide approximations of degrees of confluence and related properties based on similarity of the original fuzzy relations.

Confluence and termination are essential properties connected to the idea of rewriting and substituting which appear in abstract rewriting systems. The aim of the present paper is to investigate confluence, termination, and related properties from the point of view of fuzzy logic leaving the ordinary notions a particular case when the underlying structure of truth degrees is two-valued Boolean algebra. The main motivation of this study is the fact that in several natural situations, the notion of substitutability is inherently fuzzy rather than crisp.

Confluence and termination are essential properties of relations connected to the idea of rewriting and substituting which appear in abstract rewriting systems. This paper presents a preliminary study of confluence, termination, and related properties from the point of view of fuzzy logic leaving the ordinary notions a particular case when the underlying structure of truth degrees is the two-valued Boolean algebra. The main motivation of this study is the fact that there are natural examples where the notion of substitutability is inherently fuzzy rather than crisp.