1. Limitations of First-Order Logic
FOL is very expressive, but...
consider how to translate these:
"most students graduate in 4 years"
x student(x) → duration(undergrad)years(4) ???
"only a few students switch majors"
s,m1,m2,t1,t2 student(s)^major(s,m1,t1)major(s,m2,t2) m1m2  t1t2 ???
"all students must take Communications, except Joe"
The problems involve:
default rules & exceptions
degrees of truth
strength of rules

2. Solutions Closed-World Assumption in PROLOG Non-monotonic logics
Semantic Networks
Fuzzy Logic
Bayesian Probability

3. Closed-World Assumption (CWA) in PROLOG
every fact that is not asserted is assumed to be false - very handy
example facts:
query with negation:
how is this implemented? how does this affect reasoning/inference?
modify back-chaining to handle negative antecedents: "Negation-as-failure"
when trying to prove ¬P(X) on goal stack, try proving P(X) and if fail then ¬P(X) succeeds

4. Non-monotonic Logics
allow retractions later (popular for truth- maintenance systems)
"birds fly", "penguins are birds that don't fly"
x bird(x)→fly(x)
x penguin(x)→bird(x), x penguin(x)→¬fly(x)
bird(tweety), bird(opus) |= fly(opus)
later, add that opus is a penguin, change inference
penguin(opus) |= ¬fly(opus)
Definition: A logic is monotonic if everything that is entailed by a set of sentences a is entailed by any superset of sentences ab
opus example is non-monotonic

5. example syntax of default rule
bird(x): fly(x) / fly(x) or bird(x) ≻ fly(x)
semantics: "if PRECOND is satisfied and it is not inconsistent to believe CONSEQ, then CONSEQ"
Circumscription
add abnormal predicates to rules
x bird(x)¬abnormal1(x)→fly(x)
x penguin(x) ¬abnormal2(x) →bird(x)
x penguin(x) ¬abnormal3(x) →¬fly(x)
algorithm: minimize number of abnormals needed to make KB consistent
{bird(tweety),fly(tweety),bird(opus),penguin(opus), ¬fly(opus)} is INCONSISTENT
{bird(tweety),fly(tweety),bird(opus),penguin(opus), ¬fly(opus), abnormal1(opus)} is CONSISTENT

6. Semantic Networks graphical representation of knowledge
nodes, slots, edges, "isa" links
procedural mechanism for answering queries
follow links
different than formal definition of "entailment"
inheritance
can override defaults
related to Description Logics
popular for large medical
knowledge bases of anatomy,
diseases, injuries, procedures...

7. Fuzzy Logic
some expressions involve "degrees" of truth, like "John is tall"
membership function
"most students with high SATs have high GPAs"
inference by computing with membership funcs.
"only days that are warm and not windy are good for playing frisbee"
suppose today is 85 and the wind is 15 kts NE
T(A^B) = min(T(A),T(B))
T(AvB) = max(T(A),T(B))
popular for control applications (like thermostats...)

8. warm
not windy
windy
1.0
wind speed
0.4 15 85
temp:
0kts kts
wind: speed
temp

9. Probability conditional probabilities play role of rules
people with a toothache are likely to have a cavity
p(cavity|toothache) = 0.6
joint probabilities, priors
Bayes Rule

10. Bayesian Networks
graphical models where edges represent conditional probabilities
popular for modern AI systems (expert systems)
important for handling uncertainty =