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P1-6.1 Problems with Incomplete Information and Many Exceptions

Section ID: P1-6.1 Version: v2026.07.20

Chapter 5 distinguished learning from inference. Now we turn to the next question: why did AI end up using rules, search, probability, and learning together?

The question here is simple: why are some problems hard to handle with explicit rules alone?

Rules remain important. Permissions, policies, forbidden conditions, work procedures, and deployment conditions should still be managed explicitly. But real-world problems are full of situations where it is difficult to write every condition and every exception in advance.

This section uses examples to show where rule-only approaches begin to shake.

In this section, we look at why rule sets do not close neatly when there is incomplete information, partial observability, noise, exceptions, or an explosion of candidates. The distinction among probability, uncertainty, and stochastic continues in 6.2, and the structure of search and heuristics continues in P1-7.

P1-6.1 is the introduction to P1-6. Detailed probability calculation, conditional probability, random variables, and the meaning of stochastic are handled in P1-6.2. Search algorithms and heuristics are handled in P1-7.

The point here is narrower:

rules are still necessary,
but real-world problems often have conditions that make rules alone insufficient

Problem Conditions That Rules Alone Cannot Close

  • Understand the conditions under which rule-based approaches work well.
  • Distinguish the problem conditions that are difficult to handle with rules alone.
  • Connect incomplete information, partial observability, noise, exceptions, and computational limits at an introductory level.
  • Prepare for why uncertainty, probability, heuristics, and learning appear next.

Three Standards

Standard Why it matters Level of understanding needed here
rules are still necessary This helps us read probability and learning as complements rather than replacements. Keep the idea that policies, forbidden conditions, and procedures still need rules.
real-world problems often mix missing information, exceptions, and noise This shows why explicit rules quickly become insufficient. Understand that even the same sentence or scene can be interpreted differently depending on context.
that is why uncertainty, probability, heuristics, and learning appear next This ties P1-6 and P1-7 together. Get an intuition for why probabilistic thinking is needed after rules.

A short role split is useful here:

Term Very short meaning Role in this section
lack of information required information is absent from the start one reason rules alone cannot settle the case
partial observability only part of the full state can be seen what is visible right now may be insufficient
noise observed values may be unstable or wrong the input itself may not be fully trustworthy
exception cases keep appearing outside the existing rules one reason the rule set becomes rapidly complex
explosion of candidates too many possible options exist one reason search and heuristics become necessary

At this stage, one practical intuition is enough: when information is missing, only partly visible, noisy, full of exceptions, or when candidates are too many, rules alone often do not close the problem.

A First Example

Suppose we process the following sentence with a rule:

I ordered yesterday, but tracking still does not work.

One simple rule could be written like this:

if someone says tracking does not work, send the case to the delivery team

This is not a bad rule. In many cases it can classify messages quickly. But the moment the following situations are added, the judgment becomes harder:

Additional situation that becomes visible Is the first rule alone enough?
only 10 minutes have passed since the order it may just be system update delay
payment exists but there is no order number it may be a payment or order-creation issue
the product is preorder delivery it may not be a delivery problem at all
the customer ordered from another account it may be an account-verification issue
a carrier invoice exists but tracking fails it may be an external integration issue

The point is not that the original rule was wrong. The point is that it stops being sufficient.

Now consider face recognition:

if the positions of the eyes, nose, and mouth fit a certain ratio, classify it as a face

That rule is easy to understand. But real photographs include lighting, angle, occlusion, facial expression, resolution, and background.

Image condition Why the rule becomes difficult
the face is turned sideways the eye-nose-mouth position rule breaks
part of the face is hidden by a mask or hand the required clues are not visible
lighting is too dark or too strong the observed pixels no longer reflect shape clearly
drawings, photos, and screen images appear together it becomes unclear what should count as a real face

Autonomous driving gives a similar picture:

if there is an obstacle ahead, stop

That rule is necessary for safety. But on a real road, the system sees objects it must stop for, shadows it can pass through, plastic bags, lane-changing cars, and people who appear suddenly. Even the phrase ahead depends on sensor type, distance, speed, and road context.

These examples show why AI often cannot close a problem with simple rules alone.

rules are necessary
but real inputs are incomplete, unstable, and full of exceptions

Conditions Where Rules Work Well

Rules are strong when conditions and actions are both clear.

Situation Example rule Why it is easier to handle
permission check only admins can deploy the condition and action are explicit
input validation block signup if the email format is invalid the checking criterion is clear
work routing delivery inquiries go to the delivery team the label and the handling team are predefined
safety policy reject the request if it is forbidden the policy condition can be stated explicitly

These kinds of rules remain necessary even in AI systems because they are explainable, reviewable, and keep intended control explicit.

Conditions That Become Hard with Rules Alone

Summarizing the earlier examples, real-world problems often contain the following conditions:

Condition Meaning Example
incomplete information not all information needed for judgment is given tracking does not work alone does not identify the cause
partial observability only part of the full state can be seen a robot does not know every object location from the start
noise observation is wrong or unstable sensor error, blurred image, typo
growing exceptions exceptions to the rule keep appearing delivery, cancellation, and refund are mixed in one message
explosion of candidates too many possibilities to check exhaustively route finding, scheduling, game search

It is always possible to write more rules. The problem is that each added rule may create new exceptions, which then require even more rules.

So the key question is not should we throw away rules? but this:

how far can we state things explicitly with rules,
and from what point do we need search, probability, learning, or human review?

Why Several Approaches End Up Working Together

The table of contents of Artificial Intelligence: A Modern Approach places search, heuristic search, partially observable environments, and uncertain knowledge and reasoning as major axes of introductory AI. Poole and Mackworth likewise explain search as the problem of finding ways for agents to reach goals, and explain that in uncertain situations agents update beliefs from observed evidence.

That flow can be read like this:

Problem condition Approach that becomes necessary
something must be explicitly forbidden or allowed rules
there are too many candidates search, heuristics
information is incomplete probability, uncertainty handling
input expressions vary too much data-driven learning
responsible judgment is required human review, policy decision 
flowchart TD
  I["Incomplete input"]
  J{"Why is it hard with rules alone?"}

  I --> J
  J --> R["Explicit rules"]
  J --> S["Search / heuristic"]
  J --> P["Probability"]
  J --> L["Learned model"]

  R --> H["Human review or policy decision"]
  S --> H
  P --> H
  L --> H
  H --> O["Result"]

This is not a real service architecture diagram. In 6.1, it is only meant to show why AI often combines several approaches rather than relying on one rule set alone.

This picture is not asking us to memorize one final architecture. The point is to connect problem conditions with the kinds of approaches they tend to call for. Different conditions push us toward different combinations.

Checklist

  • Explain the conditions where rule-based approaches work well.
  • Divide rule-resistant problem conditions into incomplete information, partial observability, noise, exceptions, and explosion of candidates.
  • Explain that rules did not disappear, but are combined with other approaches.
  • Explain why uncertainty, probability, search, heuristics, and learning appear next.
  • Explain which of incomplete information, partial observability, noise, exceptions, and explosion of candidates is the key condition in a given problem.
  • Explain what should be stated as a rule and what should be passed to search, probability, learning, or human review.

Sources and Further Reading