Extensions to FOL. When the form of the statements provides useful information Rule-based systems Frame systems When FOL isn’t enough Default reasoning.

Extensions to FOL

Extensions to FOL•When the form of the statements provides useful information

•Rule-based systems

•Frame systems

•When FOL isn’t enough

•Default reasoning and circumscription

•Reasoning with uncertainty

•Degrees of membership (fuzzy logic)

•Reasoning about belief

What Does Really Mean?

HasChildren Mother A B definition of B

Raining Wet A B A causes B

Fever Infection A B A is a symptom of BB causes A

LikeBig GetHummer A B whenever A occurs,B usually does too

So how should we reason with these very different things?

Rule-Based SystemsThe logic:

a b is equivalent to: a b

So, given: fever infection fever infection fever

Conclude: infection

Given: fever infection fever infection infection

Conclude: fever

But are these two inferences equally useful?

An Example for a Design Task: XCON (1982)

From XCON (1982):If: the most current active context is distributing

massbus devices, and there is a single-port disk drive that has not been

assigned to a massbus, and there are no unassigned dual-port disk drives, and the number of devices that each massbus should support is known, and there is a massbus that has been assigned at least one disk drive that should support additional disk drives, and the type of cable needed to connect the disk drive to the previous

device on the massbus is knownThen: assign the disk drive to the massbus.

An Example for a Diagnosis Task: Mycin (1975)

If: (1) the stain of the ogranism is gram-positive, and (2) the morphology of the organism is coccus, and (3) the growth conformation of the organism is clumps

Then: there is suggestive evidence (0.7) that the identity of the organism is staphylococcus.

Simple Examples Today

eXpertise2Go: http://www.expertise2go.com/

AcquiredIntelligence: http://www.aiinc.ca/demos/

(whales, graduate school)

DecisionScript: http://www.vanguardsw.com/decisionscript/Examples.htm

Implementation of Rule-Based Systems

•Prolog: The KB: reply(sampcor) :- a, b

A query: ?- reply(X)

Use backward chaining to answer the question.

•Expert system shells: Typically combine methods

If (1) the suggested technique category is correlation and regression analysis, and (2) one of the values of the desired correlation/regression result is a measure of the degree to which 2 variables move together Then the suggested analysis approach is to calculate a sample correlation coefficient

If a b reply(sampcor)

Expert System Shells

Some rules are best used in forward chaining mode. For example, data collection and reasoning from symptoms.

Other rules (e.g., how to achieve goals) are best used in backward chaining mode.

All these rules may also want to exploit other kinds of knowledge, like default information associated with classes of objects:

Inheritance, Again

birds canfly T

ISA ISA

robins ostriches canfly F Instance-of Instance-of

Tweety Richy

Scooter is a bird. Can Scooter fly?

Inheritance

Objects inherit from their parents:•Scooter inherits from Bird the facts that:

•its birthmode is eggs, and•it has two wings

Should Scooter inherit from Bird the fact that it can fly?

Default Reasoning

•The importance of default reasoning

•Default reasoning is nonmonotonic.

•Techniques for default reasoning

•Inheritance

•The closed world assumption

•Circumscription

•Maintaining consistency in nonmonotonic reasoning systems

Default Reasoning - Examples

Inheritance from superclasses:x bird(x) canfly(x) UNLESS ostrich(x)

The “normal” case:x bird(x) canfly(x) UNLESS (broken-wing(x)

sick(x) in(oil-slick, x))

The closed world assumption:can cats fly?

Abduction:infection fever Given fever, can we

conclude infection?

Default Reasoning in Nonmonotonic

Inference in FOL systems is monotonic:

The addition of any new assertion that is consistent with the KB will never cause a formula that was previously true to become false.

Default reasoning may be nonmonotonic:Birds can fly.Tweety is a bird. Tweety can fly.

But what if we now learn:

Tweety is an ostrich. orTweety has a broken wing.

Implementing Inheritance

birds canfly T

ISA ISA

robins ostriches canfly F Instance-of Instance-of

Tweety Richy

If we implement inheritance procedurally, we don’t have to write the UNLESS clauses. We assume Tweety isn’t an ostrich.

The Closed World Assumption

The CWA: Any ground atomic sentences that are not asserted to be true in the KB can be assumed to be false.

We make the closed world assumption for two reasons:•We have to. In any complex domain, there may be a huge number of possible facts and there isn’t time to mention each of them explicitly:

•A database of classes mentions the ones that are offered.•An inventory database mentions all the objects on hand.•An airline scheduling system assumes that it will be told if the power is out or the terminal has burned down or is held by terrorists or there is a storm.

•It is consistent with felicitous human communication.

Implementing the CWA: Negation as Failure

A common way to implement the CWA: Interpret failure to prove p as a proof of p.

Example:

hasonhand(x) uses(x) mustorder(x)

How do we prove hasonhand(x)?

Circumscription

x bird(x) canfly(x) UNLESS (broken-wing(x) sick(x) in(oil-slick, x))

Is different from:x bird(x) broken-wing(x) sick(x) in(oil-slick, x)

canfly(x)Or:x bird(x) adult(x) withmother(x)

One way to implement this is to create the predicate Abnormal:

x bird(x) canfly(x) UNLESS Abnormal(x)(broken-wing(x) sick(x) in(oil-slick, x)) Abnormal(x)

Circumscription

Then we circumscribe Abnormal, i.e., we prefer models in which Abnormal is true of the smallest possible number of individuals consistent with the rest of the KB.

But what happens if we are told just:bird(Tweety) and then we conclude canfly(Tweety)

Then we are told:broken-wing(Tweety)

How do we undo the conclusion canfly(Tweety)?

Abbott, Babbitt, and Cabot

Truth Maintenance Systems

The basic idea: Associate with each assertion one or more justifications. Believe any assertion with at least one valid justification.

Each justification is composed of two parts:•An IN-list•An OUT-list

We will define the operation of a TMS that operates as a service to a separate reasoning system. The TMS doesn’t make choices. It is just a bookkeeper.

The Structure of a Justification

Before Alibis

Abbott’s Situation, with Alibi

Babbitt’s Situation

Cabot’s Situation

The Big Picture

New Facts Come In

Deciding How to Resolve the Conflict

Abduction

Examples:

infection fever measles spotsraining wetsidewalks

If given:

fever, can we conclude infection?spots, can we conclude measles?wetsidewalks, can we conclude raining?

Uncertainty and Fuzziness

•Degrees of truthJohn is tall.John is very tall.

•Probability of truthJohn is in Austin (p = .6)Coin is heads (p = .5)

•Certainty of beliefJohn is in Austin (c = .2) a wild guessCoin is heads (c = 1) sure it’s 50/50

When Must We Deal with Uncertainty?•Diagnosis:

Observe: spots, fever, headache. What’s wrong with the patient?Observe: clothes are wrinkled and hot. What’s wrong with the dryer?

•Interpretation:

•Speech understanding•Language understanding•Image understanding•Data interpretation

•Planning:

If I turn the steering wheel, where will the car go?

Probabilistic Reasoning

P(strep) = x (the probability that a random person has strep right now)

P(staph) = y (similar)

Suppose that we can use the same drug in either case, so we want to know

P(strep staph) =

Probabilistic Reasoning

P(strep) = x (the probability that a random person has strep right now)

P(staph) = y (similar)

Suppose that we can use the same drug in either case, so we want to know

P(strep staph) = P(strep) + P(staph) - P(strep staph)

Probabilistic ReasoningSuppose There are Three Factors

P(a b c) = P(a) + P(b) + P(c) - P(a b) - P(a c) - P(b c)+P(a b c)

P(a b c) = P(a b c)- P(a) - P(b) - P(c) + P(a b) + P(a c) + P(b c)

Conditional Probability

P(measles spots) = P(measles | spots) P(spots) definitionP(measles spots) = P(spots | measles) P(measles)

P(measles | spots) = P(measles spots) definition P(spots)

P(measles | spots) = P(spots | measles) P(measles) Bayes Rule

P(spots)

Examples from Diagnosis and Interpretation

P(measles | spots) = P(spots | measles) P(measles) P(spots)

P(word x | sound y) = P(sound y | word x) P(word x) P(sound y)

Word = Argmax(P(sound y | word x) P(word x)) x

Naïve Bayes ClassifierWhat if Multiple Observations Are Available? P(measles | spots fever)

= P(spots fever | measles) P(measles) P(spots fever)

Assume spots and fever are independent:

= P(spots fever | measles) P(measles) P(spots fever)

= P(spots|measles) P(fever | measles) P(measles) P(spots) P(fever)

Disease = Argmax(P(spots|x) P(fever |x) P(x) ) x

Not Quite So Naïve Bayes ClassifierComparing chicken pox (pox) to measles:

P(measles | spots fever)

= P(spots fever | measles) P(measles) P(spots fever)

Assume spots and fever are independent given measles or pox and measles and pox are independent:

= P(spots|measles) P(fever | measles) P(measles) P(spots fever)

= P(spots|measles) P(fever | measles) P(measles) P(spots|measles)*P(fever|measles)*P(measles) + P(spots|pox)* P(fever|pox)*P(pox)

Disease = Argmax(P(spots|x) P(fever |x) P(x) ) x

Learning Naïve Bayes Classification

An important aspect of naïve Bayes classification is that a classifier can be learned.

Where do numbers like p(spots | measles) come from?

Answer:

patient diagnosis fever cough spots sore throat sneezes achy

1 Measles Yes Yes Yes No No Yes

2 Measles Yes No Yes No No Yes

3 Measles Yes No No No No No

4 Chickenpox Yes No Yes Yes No No

5 Chickenpox Yes No Yes No No No

Various ad hoc Approaches

Unfortunately, it often happens that we don’t have all the joint probabilities required to compute true probabilities for our conclusions. So a variety of approximate methods are used.

http://www.expertise2go.com/webesie/tutorials/Inference/Confidence1.htm