Modeling Uncertain Reasoning with Possibilisitic Petri Nets

Approximate reasoning with words is one of the remarkable human capability that manipulates perceptions in a wide variety of physical and mental tasks whether in fuzzy or uncertain surroundings. To model this remarkable human capability, L.A. Zadeh (1999) proposed a new concept of "computing with words", which is a methodology in which the objects of computation are words and propositions drawn from a natural language. It provides a basis for a computational theory to imitate how humans make perception-based rational decisions in a fuzzy environment. Besides fuzziness, humans also perform perception-based reasoning well under uncertain circumstances. To deal with uncertain information in reasoning methods, several formalisms have been proposed, such as certainty factor (Shortliffe 1976), probabilistic logic (Nilsson 1986), Dempster-Shafer theory of evidence (Shafer 1976), possibilistic logic (Dubois et al. 1994), and possibilistic reasoning (Lee et al. 2000), etc. An adequate management of uncertainty for reasoning methods has become a significant issue (Bonissone 1985). To increase the efficiency of rule-based reasoning with uncertain information, two issues are particularly relevant: the possibility of exploiting concurrency, and the use of smart control strategies (Giordana 1985). To achieve these goals, a number of researchers have reported progress towards the modeling of rule-based reasoning with Petri nets (Bugarin et al. 1994, Chen et al. 1990, Konar 1996, Loony 1988, Scarprlli et al. 1996). Petri nets are a graphical and mathematical modeling tool to describe and study information processing systems. Petri nets with a powerful modeling and analysis ability are capable of providing a basis for variant purposes, such as knowledge representation, reasoning mechanism, knowledge acquisition, and knowledge verification. There are several rationales behind to base a computational paradigm for rule-based reasoning on Petri net theory: (1) Petri nets achieve the structuring of knowledge within rule bases, which can express the relationships among rules and help experts construct and modify rule bases. (2) The Petri net's graphic nature provides the visualization of the dynamic behavior of rule-based reasoning. (3) Petri nets make it easier to design an efficient reasoning algorithm. (4) The Petri net's analytic capability provides a basis for developing a knowledge verification technique. (5) The underlying relationship of concurrency among rule activations can be modeled by Petri nets, which is an important aspect where real-time performance is crucial. In this paper, we combine a highlevel Petri nets model with possibilistic reasoning to model uncertainty with possibilistic information. Possibilistic reasoning (Lee et al. 2000), inspired by Nilsson's probabilistic entailment, is an uncertain reasoning for classical propositions weighted by the lower bounds of necessity measures and the upper bounds of possibility measures. We model our uncertainty about the actual world by defining a possibility distribution over all possible worlds to specify the degree of possibility that the actual world is in each possible world. A degree of possibility is interpreted as a degree of ease and suitable for representing human confidence level. With the aid of semantic trees to determine all possible worlds, the necessity and possibility degrees of conclusions can be measured. In our approach, a possibilistic token carries information to describe a proposition and the corresponding possibility and necessity measures. Four types of possibilistic transitions, inference, aggregation, duplication and aggregation-duplication transitions, are introduced to fulfill the mechanism of possibilistic reasoning. The inference transitions perform possibilistic reasoning; duplication transitions duplicate a possibilistic token to several tokens representing the same proposition and possibility and necessity measures; aggregation transitions combine several possibilistic tokens with the same classical proposition; and aggregation-duplication transitions combine aggregation transitions and duplication transitions. A reasoning algorithm based on possibilistic Petri nets is also outlined to improve the efficiency of possibilistic reasoning.