The equivalence of Cognitive Map, Fuzzy Cognitive Map and Multi Value Fuzzy Cognitive Map

Cognitive Map (CM), Fuzzy Cognitive Map (FCM) are two related tools for modeling human being's cognition and facilitating machine deduction over the cognitive model accordingly. FCM has extended CM by modeling the strength of the causal relationship. To acquire the ability of modeling the strength of cause and effect, Multi-Value FCM (MVFCM) is formally presented in this paper, which uses mutli-value concepts. Not surprisingly, FCM is more complex than CM and MVFCM is more complex than FCM. This paper proves that there exists a theoretical equivalence among MVFCM, FCM and CM. It shows that for every MVFCM, there exists a FCM, or a CM that represents the MVFCM. This result allows domain experts to model applications with more descriptive MVFCM form and perform theoretical analysis with the simpler CM form. The equivalence among the models also provides strong support to the theoretical analysis of the models and flexibility to their applications.