A Four-Valued Logic for Rough Set-Like Approximate Reasoning

Rough Sets and Current Trends in Computing

2008

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Abstract. This paper presents a language for defining four-valued rough sets and to reason about them. Our framework brings together two major fields: rough sets and paraconsistent logic programming. On the one hand it provides a paraconsistent approach, based on four-valued rough sets, for integrating knowledge from different sources and reasoning in the presence of inconsistencies. On the other hand, it also caters for a specific type of uncertainty that originates from the fact that an agent may perceive different objects of the universe as being indiscernible. This paper extends the ideas presented in [9]. Our language allows the user to define similarity relations and use the approximations induced by them in the definition of other four-valued sets. A positive aspect is that it allows users to tune the level of uncertainty or the source of uncertainty that best suits applications.

Section: Logics Reflecting Truth Ordering and Knowledge Orderingmentioning

confidence: 99%

Paraconsistent Logic Programs with Four-Valued Rough Sets

Rough Sets and Current Trends in Computing

2008

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“…Namely, an object o belongs to the lower approximation of a given set A whenever all objects indiscernible from o belong to A and o belongs to its upper approximation, when there are objects indiscernible from o belonging to A. Indiscernibility is modeled by similarity relations reflecting limited perceptual capabilities as well as incomplete and imprecise knowledge. Such approximations naturally lead to three-and four-valued logics (see, e.g., [6,10,14,21,32,22]). …”

Section: Introductionmentioning

confidence: 99%

Modeling and Reasoning with Paraconsistent Rough Sets

Fundamenta Informaticae

2009

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We present a language for defining paraconsistent rough sets and reasoning about them. Our framework relates and brings together two major fields: rough sets [23] and paraconsistent logic programming [9]. To model inconsistent and incomplete information we use a four-valued logic.The language discussed in this paper is based on ideas of our previous work [21, 32, 22] developing a four-valued framework for rough sets. In this approach membership function, set containment and set operations are four-valued, where logical values are t (true), f (false), i (inconsistent) and u (unknown).We investigate properties of paraconsistent rough sets as well as develop a paraconsistent rule language, providing basic computational machinery for our approach.

“…Unlike [11], in our framework, similarity between objects is captured by four-valued relations and all properties correspond to four-valued sets. When considering similarities as well as approximations to be four-valued, as we do in the current paper, it appears that the truth ordering proposed in [14] makes definitions of set containment and approximations problematic. Therefore, we propose to slightly modify the truth ordering (by changing the relationship between f and u only).…”

Section: Introductionmentioning

confidence: 77%

Section: Four-valued Calculusmentioning

confidence: 99%

“…Namely, an object o belongs to the lower approximation of a given set A whenever all objects indiscernible from o belong to A and o belongs to the upper approximation, when there are objects indiscernible from o belonging to A. Indiscernibility is modeled by similarity relations reflecting limited perceptual capabilities as well as incomplete and imprecise knowledge. Such approximations naturally lead to three-and four-valued logics (see, e.g., [3,8,10,14]). The goal of the current paper is to ground Pawlak's ideas in a four-valued framework.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Four-Valued Extension of Rough Sets

Rough Sets and Knowledge Technology

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Abstract. Rough set approximations of Pawlak [15] are sometimes generalized by using similarities between objects rather than elementary sets. In practical applications, both knowledge about properties of objects and knowledge of similarity between objects can be incomplete and inconsistent. The aim of this paper is to define set approximations when all sets, and their approximations, as well as similarity relations are four-valued. A set is four-valued in the sense that its membership function can have one of the four logical values: unknown (u), false (f), inconsistent (i), or true (t). To this end, a new implication operator and set-theoretical operations on four-valued sets, such as set containment, are introduced. Several properties of lower and upper approximations of four-valued sets are also presented.