What do you mean, What do I mean continued...

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What do you mean, What do I mean? continued...

• Lecture 10-1
• November 30th, 1999
• CS250

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Steps in Building

• Decide what to talk about
• Decide on a vocabulary
• Encode general rules
• Encode an instance
• Pose queries

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General Ontologies

• Categories
• Measures
• Composite Objects
• Time, Space and Change
• Events and Processes
• Physical Objects
• Substances
• Mental Objects and Beliefs

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Categories

• Categories
• Reification
• How many people live on Earth?
• Inheritance
• Creating taxonomies
• Kentucky Fried Chicken
• Dewey decimal
• LoC
• MeSh

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Measures

• Examples Height, mass, cost
• Measure Units function a Number

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Composite Objects

• Not inheritance
• Difference between subclass and member
• Schema
• Script

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Composite Objects

• Not inheritance
• Difference between subclass and member
• General event descriptions
• Schema
• Script

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Using Events to Represent Change

• Whats the problem?
• Continuous time
• Multiple agents
• Actions of different durations
• Event calculus - Reify events

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Event Calculus Vocabulary

• Events are splotches in the space-time continuum
• Events have subevents
• Some events are intervals

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Examples

• Suppose we wish to represent facts about market
  manias

?f f?BulbEating ? SubEvent(f,TulipMania) ?
PartOf(Location(f), Holland)
?s s?StockFrenzy ? SubEvent(s,USBullMarket) ?
PartOf(Location(f), ??)
?s s?StockFrenzy ? SubEvent(s,USBullMarket) ?
TradedOn(Exchange(s), NASDAQ)
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Place

• How are places like intervals?
• Relation In holds among places
• Location function Maps an object to the smallest
  place that contains it

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Processes
Kurt D. Fenstermacher Sonnenfeld directed Men
in Black (1997) Get Shorty (1995) The Addams
Family (1991)

• Why do we need processes when we have events?

• How can we say
• Barry Sonnenfeld was flying some time yesterday
• Barry was flying all day yesterday

E(Flying(Barry), Yesterday)
T(Flying(Barry), Yesterday)
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A Logical Blender

• Suppose Bill is accused of killing a zucchini,
  and when the cold, but efficient, Detective
  Frigerator (known to his pals as simply Re)
  questions the orange juice pitcher in FOPL, the
  orange juice has no idea how to say
• Bill was in the kitchen with the tomato all day
  yesterday

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Composite Events

• Use And to combine two events with the usual
  semantics
• And isnt so bad, but disjunction is a bit more
  complicated -- how do we say
• I saw the whole thing, the beef or the broccoli
  stabbed the zucchini all afternoon.

? p,q,e T(And(p, q), e) ? T(p, e) ? T(q, e)
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Time Intervals

• Time is pretty important
• Divvy up time into Moments and ExtendedIntervals
• Define a couple handy functions
• Start
• End
• Time
• Date

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When Intervals Get Together

• Meet
• Before
• After
• During
• Overlap

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Objects in the Space-Time Continuum

• Remember that events are splotches of space-time
• Some events have coherence through time
• Need to capture the idea of an object existing
  through time

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Roman Empire

• Roman Empire spread across much of Eurasia,
  expanding and contracting, from 753 B.C. until
  the 5th century A.D.

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Roman Empire at 218 B.C.
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Roman Empire at 117 A.D.
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Roman Empire at 395 A.D.
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Fluents

• Roman Empire is an event
• Subevents include
• First, Second and Third Punic Wars
• One of the first known hammer and anvil movements
  in battle (216 BC _at_ Cannae)
• A fluent allows us to capture the notion of the
  Roman Empire throughout time

T(In(Gaul, Roman Empire), AD12)
T(Male(Emperor(RomanEmpire)), 1stCenturyAD)
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Fluent Flavors

• Fluent is a function, fSituations Fvalues
• Domain is the set of all situations (states of
  the world)
• If Fvalues is TRUE, FALSE then its a
  Propositional fluent
• If Fvalues is All situations then its a
  Situational fluent

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Substances

• Less vs. fewer
• Intrinsic vs. extrinsic properties
• Substances are those things that are fungible

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Going, Like, Totally Mental

• What are other agents know, and what are they
  thinking?
• Everybodys looking at me
• Theyre trying to kill me
• You look like someone who knows where I can find
  extra virgin olive oil
• Start with a Believes predicate

Believes(Agent, x)
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Reification You

• A good first pass
• Treat Flies(Superman) as a propositional fluent
• Relationships like Believes, Know and When
  between agents and propositions are propositional
  attitudes
• The problem Can Clark fly?

Believes(Agent, Flies(Superman))
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It is clear.

• Referential transparency
• Any term can be substituted for an equal term
• FOL is referentially transparent

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Knowing for Action

• Knowing preconditions What do you need to know
  to do action a?
• Knowledge effects What effect does performing
  action a have on an agents knowledge?

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Replacing that Zucchini

• Grocery shopping
• Percepts
• Actions
• Goals
• Environment

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You say you wanna resolution?
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Chain of Fools
American(x) ?????????

• Forward chaining
• Start with sentences, apply SdMP (GMP) to derive
  new conclusions
• Good when adding new facts
• Backward chaining
• Start from sentences and derive premises
• Got goal?

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Forward Chaining
for each rule that p unifies with a premise if
the other premises are known then add
conclusion to KB keep on chainin

• Renaming
• Two sentences are renamings of one another if
  they are the same except for variable names

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Composition

• Define COMPOSE(T1, T2) to apply two substitutions
  in a row
• SUBST(COMPOSE(T1, T2), p)
• SUBST(T2, SUBST(T1, p))

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Forward Chaining in Action
1) American(x) ? Weapon(y) ? Nation(z) ?
Hostile(z) ? Sells(x, y, z) ? Criminal(x) 2)
Owns(Nono, x) ? Missile(x) ? Sells(West, Nono,
x) 3) Missile(x) ? Weapon(x) 4) Enemy(x,
America)? Hostile(x)
ForwardChain(KB, American(West)) ForwardChain(KB,
Nation(Nono)) ForwardChain(KB, Enemy(Nono,
America)) ForwardChain(KB, Hostile(Nono)) Forward
Chain(KB, Owns(Nono, M1)) ForwardChain(KB,
Missile(M1)) ForwardChain(KB, Sells(West, Nono,
M1)) ForwardChain(KB, Weapon(M1)) ForwardChain(
KB, Criminal(West))
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Whats the Problem?

• Will-nilly inferencing

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Backward Chaining

• Start from what youre trying to prove, and look
  for support
• When a query q is asked

If a matching fact q is known return the
unifier for each rule whose consequent q matches
q attempt to prove each premise of rule by
backward chaining
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Revisiting Unification

• Can we unify
• Knows(John, x) Knows(x, Elizabeth)

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Now whats wrong?

• Is this complete?
• Inference procedure i is complete iff
• KB i ? whenever KB ?

PhD(x) ? HighlyQualified(x) ?PhD(x) ?
EarlyEarnings(x) HighlyQualified(x) ?
Rich(x) EarlEarnings(x) ? Rich(x)
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Does a Complete Algorithm Exist?

• Kurt says yes
• Any sentence that is entailed by another set of
  sentences can be proved from that set
• In other words We can find a complete inference
  procedure
• What is it?

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Resolution

• Remember Chapter 6?

• Is this an improvement?

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Resolution Procedure

• Resolution is a refutation procedure To prove KB
  ?, show KB ? ?? is unsatisfiable

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Resolution Procedure
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Canonical Forms

• CNF
• Start with a bunch of disjunctions
• Pretend all of them are joined with one big
  conjunct
• INF
• Each sentence is an implication with a
  conjunction of atoms on the left, and a
  disjunction of atoms on the right

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Out of the Frying Pan?

• Created GMP, needed Horn clauses
• But cant always transform sentences into Horn
  clauses!
• Find another procedure
• Stumble upon resolution, which needs CNF or INF
• Can we always transform into CNF or INF?

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CNF vs. Horn

• The diff
• In Horn, RHS must be an atom
• In CNF, RHS is a disjunction
• MP can derive atomic conclusions, what about
  resolution?
• Recast terms as implications of TRUE

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Conversion to CNF

• Can convert any FOL KB into CNF

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Skolemization

• Remove existential quantifiers by elimination
• Like EE, but more general
• Replace existentially quantified variables with
  unique constants
• What happens if theres a universal
  quantification hiding inside?
• Example Everyone has a heart

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Resolution Proof

• To prove ?
• Negate it, ??
• Convert it to CNF
• Add to a CNF KB
• Infer a contradiction

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Da Proof