1. Interval Logic Axioms Meet(i, j)  End(i)= Begin(j)
Before(i, j)  End(i) < Begin(j)
After(j , i)  Before(i, j)
During(i, j)  Begin(j) < Begin(i) < End(i) < End(j)
Overlap(i, j)  Begin(i) < Begin(j) < End(i) < End(j)
Begins (i, j)  Begin(i) = Begin(j)
Finishes (i, j)  End(i) = End(j)
Equals (i, j)  Begin(i) = Begin(j)  End(i) = End(j)

3. RDF examples <?xml version="1.0" encoding="UTF-8"?> <rdf:RDF
<rdf:RDF
xmlns:rdf="
xmlns:dc="
xmlns:region="
<rdf:Description rdf:about="
<dc:title>Oxford</dc:title>
<dc:coverage>Oxfordshire</dc:coverage>
<dc:publisher>Wikipedia</dc:publisher>
<dc:language>English</dc:language>
<region:population>10000</region:population>
<region:principaltown rdf:resource="
</rdf:Description>
</rdf:RDF>

4. RDF Examples
<?xml version="1.0"?> <rdf:RDF xmlns:rdf=" xmlns:cd=" <rdf:Description rdf:about=" Burlesque"> <cd:artist>Bob Dylan</cd:artist> <cd:country>UK</cd:country> <cd:company>Columbia</cd:company> <cd:price>10.90</cd:price> <cd:year>1985</cd:year> </rdf:Description>

5. <rdf:Description rdf:about="http://www.site.com/zoo#bengie">
<rdf:type rdf:resource="
<feature:color>brown</feature:Color>
<friendsWith rdf:about="
</rdf:Description>
does bengie bark? does bonnie know/like bengie?

6. OWL Example
<owl:Class rdf:about="
<rdfs:label>The plant type</rdfs:label>
<rdfs:comment>The class of all plant types.</rdfs:comment>
</owl:Class>
<!-- OWL Subclass Definition - Flower -->
<owl:Class rdf:about="
<!-- Flowers is a subclassification of planttype -->
<rdfs:subClassOf rdf:resource="
<rdfs:label>Flowering plants</rdfs:label>
<rdfs:comment>Flowering plants, also known as angiosperms.</rdfs:comment>
<rdf:Description rdf:about="
<!-- Orchid is an individual (instance) of the flowers class -->
<rdf:type rdf:resource="
<plants:family>Orchidaceae</plants:family>
</rdf:Description>

7. 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(x))years(4) (all???)
"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 (exists???)
"all birds can fly, except penguins, stuffed brids, plastic birds, birds with broken wings..."
The problems with FOL involve:
default rules & exceptions
degrees of truth
strength of rules

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

9. Closed-World Assumption (CWA) in PROLOG
every fact that is not explicitly asserted (or provable) is assumed to be false
can include negated antecedents in rules ("\+" = not)
stench(2,1).
stench(3,2).
stench(4,1).
wumpus_free(X,Y) :- room(X,Y),adjacent(X,Y,P,Q),\+ stench(P,Q).
?- wumpus_free(3,1). No
?- wumpus_free(2,4).
Yes

10. using CWA for default reasoning
bird(X) :- canary(X). ?- bird(tweety). Yes
bird(X) :- penquin(X). ?- canFly(tweety). Yes
canary(tweety). ?- bird(opus). Yes
penquin(opus). ?- canFly(opus). No
canFly(X) :- bird(X),\+ penguin(X).
how is negation-as-failure implemented?
modify back-chaining to handle negative antecedents
when trying to prove ¬P(X) on goal stack, try proving P(X) and if fail then ¬P(X) succeeds
goal stack: canFly(tweety)
goal stack: canFly(opus) X
bird(tweety) ¬penguin(tweety)
canary(tweety)
bird(opus) ¬penguin(opus)
penguin(opus)
succeeds X
fails
penguin(tweety)
*** fails***
penguin(opus)
***succeeds*** X

11. 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

12. 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

13. 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

14. Description Logics decidable subset of FOL Tbox (terminological)
efficient algorithms based on set relationships
Tbox (terminological)
Mother ⊑ Parent
Parent ≡ Father ⊔ Mother
parentOf ⊑ ancestorOf
brotherOf ◦ parentOf ⊑ uncleOf
⊤ ⊑ Male ⊔ Female
Male ⊓ Female ⊑ ⊥
PeopleWithDaughters ≡ ∀parentOf.Female
Person ⊑ (≥2 childOf.Parent ⊓ ≤2 childOf.Parent)
Abox (assertional)
Mother(julia)
parentOf(julia, john)

15. Protege

16. 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...)

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

18. 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

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