There are various ways of representing uncertainty. Here we consider three different approaches, representing three different areas of uncertainty:

Probability theory:

Probabilistic assertions and queries are not usually about particular possible worlds, but about sets of them.

In probability theory, the set of all possible worlds is called the sample space. The Greek letter Ω (uppercase omega) is used to refer to the sample space, and ω (lowercase omega) refers to elements of the space, that is, particular possible worlds.

A fully specified probability model associates a numerical probability P (ω) with each possible world.1 the basic axioms of probability theory say that every possible world has a probability between 0 and 1 and that the total probability of the set of possible worlds is 1:

0 ≤P(ω) ≤1 for every ω and _ω∈Ω

P(ω) = 1

i. If a coin is flipped there is an equal chance of it landing on head side or tail side, consider H1 is for heads and H2 for tails. This scenario is expressed as P(H1)=0.5 and P(H2)=0.5.

ii. The probability of 1st and 2nd toss both landing on heads is 0.5*0.5=0.25.

iii. We can write this as P(H1^H2)-0.25 and in general two independent events P and Q, P(P^Q)=P(P)*P(Q).

Fuzzy logic:

In the existing expert systems, uncertainty is dealt with through a combination of predicate logic and probability-based methods. A serious shortcoming of these methods is that they are not capable of coming to grips with the pervasive fuzziness of information in the knowledge base, and, as a result, are mostly ad hoc in nature.

An alternative approach to the management of uncertainty which is suggested in this paper is based on the use of fuzzy logic, which is the logic underlying approximate or, equivalently, fuzzy reasoning. A feature of fuzzy logic which is of particular importance to the management of uncertainty in expert systems is that it provides a systematic framework for dealing with fuzzy quantifiers, e.g., most, many, few, not very many, almost all, infrequently, about 0.8, etc.

In this way, fuzzy logic subsumes both predicate logic and probability theory, and makes it possible to deal with different types of uncertainty within a single conceptual framework.

In fuzzy logic, the deduction of a conclusion from a set of premises is reduced, in general, to the solution of a nonlinear program through the application of projection and extension principles. This approach to deduction leads to various basic syllogisms which may be used as rules of combination of evidence in expert systems.

Truth maintenance System:

To choose their actions, reasoning programs must be able to make assumptions and subsequently revise their beliefs when discoveries contradict these assumptions.

The Truth Maintenance System (TMS) is a problem solver subsystem for performing these functions by recording and maintaining the reasons for program beliefs. Such recorded reasons are useful in constructing explanations of program actions and in guiding the course of action of a problem solver

TMS are another form of knowledge representation which is best visualized in terms of graphs.

It stores the latest truth value of any predicate. The system is developed with the idea that truthfulness of a predicate can change with time, as new knowledge is added or exiting knowledge is updated.

It keeps a record showing which items of knowledge is currently believed or disbelieved.