3203. Introduction to Artificial Intelligence

Reasoning with Uncertainty

### 1. Introduction

Though there are various types of uncertainty in various aspects of a reasoning system, the "reasoning with uncertainty" (or "reasoning under uncertainty") research in AI has been focused on the uncertainty of truth value, that is, to allow and process truth values other than "true" and "false".

Generally speaking, to develop a system that reasons with uncertainty means to provide the following:

• a semantic explanation about the origin and nature of the uncertainty
• a way to represent uncertainty in a formal language
• a set of inference rules that derive uncertain (though well-justified) conclusions
• an efficient memory-control mechanism for uncertainty management
This is a large and active subfield of AI research. For further information, see Association for Uncertainty in Artificial Intelligence.

### 2. Nonmonotonic logics

A reasoning system is monotonic if the truthfulness of a conclusion does not change when new information is added to the system — the set of theorem can only monotonically grows when new axioms are added. In contrast, in a system doing non-monotonic reasoning the set of conclusions may either grow or shrink when new information is obtained.

Nonmonotonic logics are used to formalize plausible reasoning, such as the following inference step:

```		Birds typically fly.
Tweety is a bird.
--------------------------
Tweety (presumably) flies.
```
Such reasoning is characteristic of commonsense reasoning, where default rules are applied when case-specific information is not available.

The conclusion of nonmonotonic argument may turn out to be wrong. For example, if Tweety is a penguin, it is incorrect to conclude that Tweety flies. Nonmonotonic reasoning often requires jumping to a conclusion and subsequently retracting that conclusion as further information becomes available.

All systems of nonmonotonic reasoning are concerned with the issue of consistency. Inconsistency is resolved by removing the relevant conclusion(s) derived previously by default rules. Simply speaking, the truth value of propositions in a nonmonotonic logic can be classified into the following types:

1. facts that are definitely true, such as "Tweety is a bird"
2. default rules that are normally true, such as "Birds fly"
3. tentative conclusions that are presumably true, such as "Tweety flies"
When an inconsistency is recognized, only the truth value of the last type is changed.

A related issue is belief revision. Revising a knowledge base often follows the principle of minimal change: one conserves as much information as possible.

One approach towards this problem is truth maintenance system, in which a "justification" for each proposition is stored, so that when some propositions are rejected, some others may need to be removed, too.

Major problems in these approaches:

• conflicts in defaults, such as in the "Nixon Diamond"
• computational expense: to maintain the consistency in a huge knowledge base is hard, if not impossible

### 3. Probabilistic reasoning

Basic idea: to use probability theory to represent and process uncertainty. In probabilistic reasoning, the truth value of a proposition is extended from {0, 1} to [0, 1], with binary logic as its special case.

Justification: though no conclusion is absolutely true, the one with the highest probability is preferred. Under certain assumptions, probability theory gives the optimum solutions.

To extend the basic Boolean connectives to probabilty functions:

• negation: P(¬A) = 1 − P(A)
• conjunction: P(A∧B) = P(A) * P(B) if A and B are independent of each other
• disjunction: P(A∨B) = P(A) + P(B) if A and B never happen at the same time
Furthermore, the conditional probability of B given A is P(B|A) = P(B∧A) / P(A), from which Bayes' Theorem is derived, and it is often used to update a system's belief according to new information: P(H|E) = P(E|H) * P(H) / P(E).

Bayesian Networks are directed acyclic graphs in which the nodes represent variables of interest and the links represent informational or causal dependencies among the variables. The strength of dependency is represented by conditional probabilities. Compared to other approaches of probabilistic reasoning, Bayesian network is more efficient, though its actual computational cost is still high for complicated problems.

Challenges to probabilistic approaches:

• unknown probability values
• inconsistent probability assignments
• computational expense
Considering the uncertainty in probability judgments, some people go further to study imprecise probability.

### 4. Fuzzy logic

Fuzzy logic is a generalization of classical logic, and reflects the impression of human language and reasoning.

Examples of fuzzy concepts: "young", "furniture", "most", "cloudy", and so on.

According to fuzzy logic, whether an instance belongs to a concept is usually not a matter of "yes/no", but a matter of degree. Fuzzy logic uses a degree of membership, which is a real number in [0, 1].

A major difference between this number and probability is: the uncertainty in fuzzy concepts usually does not get reduced with the coming of new information. Compare the following two cases:

• I'm afraid that tomorrow will be cloudy, so let's take the picture today.
• I'm not sure whether the current weather should be classified as "cloudy" or not.
Basic fuzzy operators:
• negation: M(¬A) = 1 − M(A).
• conjunction: M(A∧B) = min{M(A), M(B)}.
• disjunction: M(A∨B) = max{M(A), M(B)}.
Typically, in building a fuzzy system, the designer needs to provide all membership functions included in it, by considering how the concepts are used by average people. Most successful applications of fuzzy logic so far are in fuzzy control systems, where expert knowledge is coded into fuzzy rules.

Challenges to fuzzy approaches:

• degree of membership is often context dependent
• general-purpose fuzzy rules are hard to get

### 5. Truth-value as evidential support

This approach is taken in the NARS project, an intelligent reasoning system.

The basic idea is to see the truth-value of a statement as measureing the evidential support the statement gets from the system's experience. Such a truth-value consists of two factors: frequency (the proportion of positive evidence among available evidence) and confidence (the proportion of currently available evidence among all evidence at a near future).

This approach attempts to uniformly represent various types of uncertainty.