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ITCS 3153
Artificial Intelligence
Lecture 13
First-Order Log ic
Chapter 8
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First-order logic
We saw how propo si t ional logic can create
intel l igent behavior
Bu t propos i t ional log ic is a poor representat ionfor complex envi ronm ents
Why?
First -order log ic is a more exp ressive andpowerfu l representation
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What do we like about propositional
logic?
It is :
Declarative
Relationships between variables are described
A method for propagating relationships Expressive
Can represent partial information using disjunction
Compositional
IfA means foo and B means bar,A ^ Bmeans foo and ba
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What dont we like about
propositional logic?
Lacks expressive power to descr ibe the
envi ronm ent conc isely
Separate rules for every square/square relationship in
Wumpus world
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Natural Language
Engl ish appears to be express ive
Squares adjacent to pits are breezy
Bu t natural language is a medium o f communicat ion, not
a know ledge representat ion
Much of the information and logic conveyed by language is dependent
on context
Information exchange is not well defined
Not compositional (combining sentences may mean something different
It is ambiguous
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But we borrow representational
ideas from natural language
Natural language syn tax
Nouns and noun phrases refer to objects
People, houses, cars
Verbs and verb phrases refer to relationshipsbtw objects
Red, round, nearby, eaten
Some relationships are clearly defined functions where there is only one
output for a given input
Best friend, first thing, plus
We bui ld f i rst order logic around objects and relat ions
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Ontology
a specification of a conceptualization
A description of the objects and relationships that can exist
Propositional logic had only true/false relationships
First-order logic has many more relationships
The ontological commitmentof languages is different
How much can you infer from what you know?
Temporal logicdefines additional ontologicalcommitments because of timing constraints
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Higher-order logic
First-order logic is first because you relate objects (thef i rst-order enti t ies that actual ly exist in the world )
There are 10 chickens chickens.number=10
There are 10 ducks ducks.number=10
You canno t bu i ld relationships between relat ions or
funct ions
There are as many chickens as ducks chickens.number =
ducks.number
the number of objects belonging to a group must be a property of the
group, and not the objects themselves
Cannot represent Leibnizs law: Ifxand yshare all properties,xisy
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Another characterization of a logic
Epistemolog ica l comm itments
The possible states of knowledge permitted with respect to
each fact
In first-order logic, each sentence is a statement that is True, false, or unknown
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Formal structure of first-order logic
Models of f i rst-order logic contain: A set of objects (its domain)
Alice, Alices left arm, Bob, Bobs hat
Relationshipsbetween objects
Represented as tuples
Sibling (Alice, Bob), Sibling (Bob, Alice)
On head (Bob, hat)
Person (Bob), Person (Alice)
Some relationships are functionsif a given object is related to exactl
one object in a certain way
Alice -> Alices left arm
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First-order logic syntax
Constant Symbols
A, B, Bob, Alice, Hat
Predicate Symbols
Is, onHead, hasColor, person
Funct ion Symbols
Mother, leftLeg
Each predicate and func t ion s ymbol has an ar i ty
A constant the fixes the number of arguments
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First-order logic syntax
Names of thin gs are abitrary
Knowledge base adds meaning
Number o f poss ib le domain elements is
unbounded
Number of models is unbounded
Checking enumeration by entailment is impossible
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Syntax
Term
A logical expression that refers to an object
Constants
We could assign names to all objects, like providing aname for every shoe in your closet
Function symbols
Used in place of a constant symbol OnLeftFoot(John))
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Atomic Sentences
Formed by a predicatesym bo l fo l lowed by parenthesizedl ist of terms
Sibling (Alice, Bob)
Married (Father(Alice), Mother(Bob))
An atom ic sentence is true in a given m odel , under a
given interpretat ion , i f the relat ion referred to by the
predicate sym bol ho lds among the objects referred to by
the arguments
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Complex sentences
We can use logical connect ives
~Sibling(LeftLeg(Alice), Bob)
Sibling(Alice, Bob) ^ Sibling (Bob, Alice)
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Quantifiers
A way to express propert ies o f ent i re co l lect ions
of objects
Universal quantification (forall, )
The power of first-order logic
ForallxKing(x) => Person(x)
xis a variable
)()(: xPersonxKingx
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Universal Quantification
Forall x, P
P is true for every object x
Forall x, King(x) => Person(x)
Richard the Lionheart King John
Richards left leg
Johns left leg
The crown
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Universal Quantification
Note that al l of th ese are true Implication is true if premise is false
Using AND instead of implication is overly strong
By assert ing a un iversal ly quant i f ied sentence,you assert a who le l is t of indiv idual impl ications
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Existential Quantification
There exists ,
There exists an x such that Crown(x) ^ OnHead(x, John)
It is true for at leastone object
AND, ^, is the appropriate connective
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Existential Quantification
What if we used impl icat ion as the connect ive?
Implication is true if both premise and conclusion are true or
if premise is false
Richard the Lionheart is not a crown, first assertion is true
and existential is satisfied
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Nested Quantifiers
Bui ld ing m ore complex sentences
Everybody loves somebody
There is someone who is loved by everyone
Use un ique variable names and parenth eses whenappropriate
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Combining
Everyone who dislikes parsnips ==there does not exist someone who likes parsnips
Everyone likes ice cream ==
there is no one who does not like ice cream
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Combining
De Morgans rules apply
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Equality
Two terms refer to the same object
Father (John) = Henry
Richard has at least two brothers
Notice this sentence is not the same
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An Example
A Tel l/Ask interface for a first-order know ledge base
Sentences are added with Tell
These are called assertions
Tell (KB, King(John))
Tell (KB, Forall x: King(x) => Person(x))
Queries are made with Ask
Ask (KB, King(John))
Ask (KB, Person(John))
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An Example
Quantified queries
Ask (KB, exist x: Person(x))
KB should return a list of variable/term pairs that satisfy the
query
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The Wumpus World
More precise axiom s than with pro pos i t ional logic Percept has five values
Time is important
A typical sentence
Percept ([Stench, Breeze, Glitter, None, None], 5)
The actions are terms
Turn(right), Turn(left), Forward, Shoot, Grab, Release
Computing best action with a query
Exist a: BestAction(a, 5)
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The Wumpus World
After executing query, KB responds with variable/term list:{a/Grab}
Then tell the KB the action taken
Raw percept data is easily encoded
Reflexes are easily encoded
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Wumpus World
Defining the environment with[x,y] reference instead ofalternative atomic name
Adjacency between two squares
Location of Wumpus is constant: Home (Wumpus)
Location of agent changes: At (Agent, [ ], t)
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Diagnostic Rules
Rules leading from observed ef fects to hiddencauses
Breezy implies pits
Not breezy implies no pits
Combining
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Causal Rules
Some hidd en property causes percepts to begenerated
A pit causes adjacent squares to be breezy
If all squares adjacent to a square a pitless, it will not be
breezy
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Causal Rules
The causal ru les fo rmulate a model
Knowledge of how the environment operates
Model can be very useful and important and replace
straightforward diagnostic approaches
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Conclusion
If the axiom s correct ly and completely descr ibe the waythe wor ld works and the way percepts are produced,
then any com plete logical inference procedure w i l l infer
the strongest possib le descr ipt ion of the wo r ld state
given the avai lable percepts
The agent designer can focus on gett ing the knowledge
r ight wi thout worry ing about the processes of deduct ion
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