Lecture 20
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Reminders: Project 2 and research paper
outline with references due today.
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Posted: Homework 4, due in two weeks
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Questions?
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Outline
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Chapter 7 – Logical Agents
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Agents Based on Propositional Logic
Chapter 8 – First-Order Logic
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Representation Revisited
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Syntax and Semantics
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Recall: Knowledge-Based Agents
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Simple KB-Agent:
Receives: percept; Returns: action
Static data: KB – knowledge base
t – time counter initially 0
1. Tell (KB, Make-Percept-Sentence (percept, t))
2. action = Ask (KB, Make-Action-Query (t))
3. Tell (KB, Make-Action-Sentence (action, t))
4. Increment t
5. Return action
Thursday, March 15
CS 430 Artificial Intelligence - Lecture 18
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Current State of WW
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A logical agent operates by deducing what to do
from a KB of sentences about the world. In
particular, need to deduce the current state of the
world – where am I, is that square safe, etc.
Previously, started collecting axioms. Starting
square has no pit (P1,1) and no wumpus (W1,1).
Collection of sentences about breezy and smelly
squares of the form:
B1,1  P1,2  P2,1
Thursday, March 22
S1,1  W1,2  W2,1
CS 430 Artificial Intelligence - Lecture 20
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Current State of WW
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Exactly one wumpus:
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At least one wumpus: W1,1  W1,2 … W4,3  W4,4
For every pair of squares, at least one of them must
be wumpus-free: W1,1  W1,2
W1,1  W1,3
…
W4,3  W4,4
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Current State of WW
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How are percepts translated into sentences?
Suppose currently there is a stench. Might
propose proposition Stench be added to KB.
But this may contradict previous step where
there was no stench asserting Stench.
Problem is that a percept asserts something
only about the current time. This is why MakePercept-Sentence receives both percept and a
time. E.g., if t=4 above, adds Stench4, which
does not contradict previous step that added
Stench3
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Fluents
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Aspects of the world that changes are called
fluents and require a time superscript. Can
think of them as state variables. Aspects of the
world that are permanent do not need a time
superscript and are called atemporal
variables.
Initial location of agent is L01,1– agent is in [1,1]
at time 0 – and FacingEast 0, HaveArrow 0, and
0
WumpusAlive .
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Fluents
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Connect the percepts directly to properties of
squares where they are experienced through
the location fluent. For any time step t and any
square [x,y]:
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Ltx,y  (Breeze t  Bx,y)
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Ltx,y  (Stench t  Sx,y)
Now need to keep track of fluents by writing the
transition model as a set of logical sentences.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Effect Axioms
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Define proposition symbols for the occurrence
of actions. E.g., Forward 0 means agent
executes Forward action at time 0.
To describe how world changes, define effect
axioms that specify the outcome of an action in
the next time step. E.g., if agent is at [1,1]
facing east at time 0 and goes Forward, the
result is that agent is in [2,1] and no longer in
[1,1]
L01,1 FacingEast0 Forward0  (L12,1  L11,1)
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Frame Problem
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Works as far as it goes, but what if we want to
know if the agent has an arrow in time t=1?
1
Cannot prove or disprove HasArrow ! Effect
axiom does not state what has not changed
between time steps.
Called the frame problem. One possible
solution is to add frame axioms:
Forward t  (HaveArrow t  HaveArrow t+1)
Forward t  (WumpusAlive t  WumpusAlive t+1)
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Frame Problem
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Proliferation of frame axioms is inefficient. For
m different actions and n fluents, the set of
frame axioms is O(mn).
Fortunately, each action typically changes only
a small number k fluents, i.e., the world exhibits
locality. Want to define transition model with
set of axioms of size O(mk).
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Fluent Axioms
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Solution is to write axioms about fluents rather
than about actions. For each fluent F, have an
t+1
axiom that defines value of F in terms of
fluents at time t and actions that may have
occurred at time t.
t+1
Truth value of F is set either because action
at t caused it to become true, or it was already
true and the action at t does not cause it to
become false:
F t+1  ActionCausesF t  (F t  ActionCausesNotF t)
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Fluent Axioms
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Axiom of this form is called a successor-state
axiom. Simple example is for HaveArrow,
since once it is shot, arrow disappears:
HaveArrow t+1  (HaveArrow t  Shoot t)
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Location fluent is more complex:
Lt+11,1  (Lt1,1  (Forward t  Bump t+1 ))
 (Lt1,2  (South t  Forward t ))
 (Lt2,1  (West t  Forward t ))
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Relevant fluent axioms are added at each time
step.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Hybrid Agent
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Logical deduction can be combined with
problem-solving algorithms in a fairly
straightforward manner to create a hybrid agent
(Fig. 7.20).
Propositional knowledge is used to infer the
state of the world, domain specific code is used
to determine what kind of plan to make,
problem-solving search is used to create plan
of actions.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Representation Revisited
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Propositional logic has many advantages as a
representation for knowledge.
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It is declarative. Its syntax corresponds to facts and
inference is separate and domain-independent
It allows partial, disjunctive, and negated
information unlike programming languages.
It is compositional. The meaning of B1,1  P1,2 is
derived from the meanings of B1,1 and P1,2.
It is context independent and unambiguous unlike
natural languages.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Representation Revisited
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Main problem with propositional logic is that it is
not expressive enough. E.g., "physics" of WW
must be input as a sentence per square for
Breeze and Smell.
Add representational ideas from natural
language:
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Objects: nouns and noun phrases
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Relations: verb and verb phrases
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Functions: relations that specify a particular
"value" for a given "input"
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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First-Order Logic: Models
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First-order logic (FOL) is built around objects
and relations. It can also express facts about
some or all of the objects in the universe.
An FOL model contains objects. The domain
of a model is the set of object (domain
elements) it contains. The domain is required
to be nonempty, i.e., every possible world
must contain at least one object.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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First-Order Logic: Models
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Example model
with five objects:
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Richard the
Lionhearted
King John
Crown
Richard the
Lionheart's left
leg
King John's left
leg
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Relations
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Object in a model may be related in some way.
Formally, a relation is a set of tuples of objects
that are related.
Previous picture indicates that Richard is the
brother of John. The "brother" relation is
{ < Richard the Lionhearted, King John >,
< King John, Richard the Lionhearted > }
The crown is on John's head, so the "on head"
relation is { < The Crown, King John > }
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Relations
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"brother" and "on head" are binary relations.
Model also has unary relations (or properties).
E.g., "person" property is true of both Richard
the Lionhearted and King John. "king" property
is true only of King John. "crown" property is
true only of the The Crown.
Some relations are best considered as
functions – a given object must be related to
exactly one object in this way. E.g., each
person has one left leg. Model has a "left leg"
function.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Symbols and Interpretations
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FOL has three kinds of symbols
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constant symbols representing objects, e.g.,
Richard, John, TheCrown
predicate symbols representing relations, e.g.,
Brother, OnHead, Person, King, Crown
function symbols representing functions, e.g.,
LeftLeg
By convention, these symbols start with
uppercase letter
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Symbols and Interpretations
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Each model includes an interpretation that
specifies exactly which objects, relations and
functions are referred to by the constant,
predicate, and function symbols.
Usually use the intended interpretation, e.g.,
John refers to King John, etc., but model may
have a different interpretation.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Terms
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A term is a logical expression that refers to an
object. A constant symbol is a term, but it is not
convenient to a constant symbol to name every
objects.
A function symbol followed by an "argument"
list is a term. E.g. LeftLeg(John) refers to King
John's left leg under the intended interpretation,
but this object has no other name.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Atomic Sentences
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An atomic sentence (or atom for short) is
formed from a predicate symbol optionally
followed by a list of terms as "arguments". E.g.,
Brother (Richard, John).
Atoms can have complex terms as arguments.
E.g. Married (Father(Richard), Mother(John))
An atomic sentence is true in a given model if
the relation referred to by the predicate symbol
holds among the objects referred to by the
arguments.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Complex Sentences
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Can construct more complex sentences by
using logical connectives. These have the
same syntax and semantics as in propositional
logic.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Universal Quantification ()
"For all"
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Express properties of entire collections objects
using quantifiers. Universal quantification
makes statements about every object.
E.g.,  x King(x)  Person(x) "If x is a king, x is
a person"
x is a variable, by convention starting with
lowercase, and is a term all by itself. Can use as
an argument to a function. E.g. LeftLeg(x). A
term with no variables is a ground term.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Universal Quantification ()
"For all"
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Intuitively,  x P(x) means that P is true for
every object x. More precisely,  x P(x) is true
in a given model if P is true in all possible
extended interpretations constructed from the
interpretation given in the model, where each
extended interpretation specifies a domain
element to which x refers.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Universal Quantification ()
"For all"
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E.g., x can be 5 possible objects, so equivalent
is asserting following sentences are true:
King(Richard)  Person(Richard)
King(John)  Person(John)
King(LeftLeg(Richard))  Person(LeftLeg(Richard))
King(LeftLeg(John))  Person(LeftLeg(John))
King(TheCrown)  Person(TheCrown)
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Second one is true since both sides are true.
Rest are vacuously true with false premises.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Universal Quantification ()
"For all"
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Common mistake is to use conjunction instead
of implication:  x King(x)  Person(x)
King(Richard)  Person(Richard)
King(John)  Person(John)
King(LeftLeg(Richard))  Person(LeftLeg(Richard))
King(LeftLeg(John))  Person(LeftLeg(John))
King(TheCrown)  Person(TheCrown)
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Says that every object is a King and a Person.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Existential Quantification ()
"There exists"
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Existential quantification makes a statement
about some object in the universe without
naming it.
E.g., x Crown(x)  OnHead(x, John) "King
John has a crown on his head"
Intuitively, x P says P is true for at least one
object x. More precisely, P is true in at least
one extended interpretation that assigns x to a
domain element.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Existential Quantification ()
"There exists"
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E.g. one of the following sentences must be true
Crown(Richard)  OnHead(Richard, John)
Crown(John)  OnHead(John, John)
Crown(LeftLeg(Richard))  OnHead(LeftLeg(Richard), John)
Crown(LeftLeg(John))  OnHead(LeftLeg(John), John)
Crown(TheCrown)  OnHead(TheCrown, John)
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The last sentence is true in this model, so the
original existentially quantified sentence is true in
this model.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Existential Quantification ()
"There exists"
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The natural connective to use with  is . Using
 with  leads to an overly strong statement;
using  with  leads to a very weak statement.
Consider: x Crown(x)  OnHead(x, John)
Crown(Richard)  OnHead(Richard, John)
Crown(John)  OnHead(John, John)
…
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Existentially quantified implications are true
whenever any object fails to satisfy the
premise, so not very interesting.
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Nested Quantifiers
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Consecutive quantifiers of the same type can
be written as one quantifier with several
variables.
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Order matters for quantifiers of different type
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x, y Brother(x,y)  Sibling(x,y)
x y Loves(x,y) "Everybody loves somebody"
y x Loves(x,y) "Someone is loved by everyone"
Variables bind to the innermost quantifier that
uses it
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x (Crown(x)  (x Brother(Richard, x)))
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Connections between  and 
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 is a conjunction of all extended
interpretations, while  is a disjunction of all
extended interpretations, so DeMorgan's rule
applies
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x P  x P
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x P  x P
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x P  x P
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x P  x P
Thursday, March 22
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Example:
x Likes(x, IceCream) 
x Likes(x, IceCream)
CS 430 Artificial Intelligence - Lecture 20
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Equality
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Last type of atomic sentence uses the equality
symbol to signify that two terms refer to the
same object.
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Father(John) = Henry
Also used with negation to insist that two terms
are not the same object. E.g. "Richard has at
least two brothers"
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x,y Brother(x,Richard)  Brother(y,Richard)
  (x=y)
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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Using First-Order Logic
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Sentences, called assertions, are added to a
KB using Tell as in propositional logic.
Questions, called queries or goals, are made
using Ask.
When query has a variable, would like to know
what values make query sentence true. E.g.
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AskVars(KB, Person(x))
Response is a substitution or binding list
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{ x/John }, { x/Richard }
Thursday, March 22
CS 430 Artificial Intelligence - Lecture 20
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