Fuzzy control system

Logical interpretation of fuzzy control
In spite of the appearance there are several difficulties to give a rigorous logical interpretation of the IF-THEN rules. As an example, interpret a rule as IF (temperature is "cold") THEN (heater is "high") by the first order formula Cold(x)→High(y) and assume that r is an input such that Cold(r) is false. Then the formula Cold(r)→High(t) is true for any t and therefore any t gives a correct control given r. A rigorous logical justification of fuzzy control is given in Hájek's book (see Chapter 7) where fuzzy control is represented as a theory of Hájek's basic logic.^[2]
In Gerla 2005 ^[11] another logical approach to fuzzy control is proposed based on fuzzy logic programming: Denote by f the fuzzy function arising of an IF-THEN systems of rules. Then this system can be translated into a fuzzy program P containing a series of rules whose head is "Good(x,y)". The interpretation of this predicate in the least fuzzy Herbrand model of P coincides with f. This gives further useful tools to fuzzy control.
Fuzzy qualitative simulation
Before an Artificial Intelligence system is able to plan the action sequence, some kind of model is needed. For video games, the model is equal to the game rules. From the programming perspective, the game rules are implemented as a Physics engine which accepts an action from a player and calculates, if the action is valid. After the action was executed, the game is in follow up state. If the aim isn't only to play mathematical games but determine the actions for real world applications, the most obvious bottleneck is, that no game rules are available. The first step is to model the domain. System identification can be realized with precise mathematical equations or with Fuzzy rules.^[12]
Using Fuzzy logic and ANFIS systems (Adaptive network based fuzzy inference system) for creating the forward model for a domain has many disadvantages.^[13] A qualitative simulation isn't able to determine the correct follow up state, but the system will only guess what will happen if the action was taken. The Fuzzy qualitative simulation can't predict the exact numerical values, but it's using imprecise natural language to speculate about the future. It takes the current situation plus the actions from the past and generates the expected follow up state of the game.
The output of the ANFIS system isn't providing correct information, but only a Fuzzy set notation, for example [0,0.2,0.4,0]. After converting the set notation back into numerical values the accuracy get worse. This makes Fuzzy qualitative simulation a bad choice for practical applications