First-Order Logic
Chapter 8
Fall 2009
Copyright, 1996 © Dale Carnegie & Associates, Inc.
Why and what
FOL makes a stronger set of ontological
commitments (more than facts) shown in Fig 8.1
The world consists of objects and relations.




Objects - things with individual identities
Properties - sth distinguishing them from others
Relations - sth between objects
Functions - special relations with one value
Facts refer to objects, properties or relations


Suns beat Lakers
Squares neighboring the wumpus are smelly
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FOL
FOL is universal - it can express
anything that can be programmed
- what else do we want?
FOL is the most studied and best
understood scheme yet devised.
Its syntax and semantics
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Domain and domain elements
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Syntax
Symbols



(Fig 8.3, BNF, Page 247)
Constant symbols
Predicate symbols - relations, tuples
Functional symbols - relations
Terms - objects, ground (constant symbols) &
complex (functions) terms
Atomic sentences

Brother(Richard, John),
Married(Father(Richard),Mother(John))
Complex sentences formed by connectives

!Brother(Robin,John)
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Quantifiers
Universal quantification () - to avoid
enumerating the objects by name
combining with variables, we can do that:
x Cat(x)  Mammal(x)
 x P(x)  Q(x) makes a statement about
everything if P(x) is true, but not when
P(x) is false
 x P(x) ^ Q(x) leads to a too strong
statement

 x King(x) ^ Person(x)
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Existential quantification () - make a
statement about some object without
naming it.



x P(x) ^ Q(x) - at least one x such that
P(x) and Q(x) is true
 x P(x)  Q(x) leads to a too weak
statement
No uniqueness is claimed
 is used with , and ^ with 
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Nested quantifiers
Multiple quantifiers can be used.
The order of quantification is important.


x y Loves (y,x)
y x Loves (y,x)
When there is confusion, the variable belongs
to the innermost quantifier that mentions it.

x [Cat(x) v (x Brother(Richard,x))]
Well-formed formula (wff) - sentences that
have all their variables properly introduced.
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Connections, Equality
The two quantifiers are connected via
negation.

De Morgan’s rules
Do we really need both quantifiers?

Some examples
x Likes(x, Icecream) ≡ !x !Likes(x, Icecream)
! x P ≡ x !P
Equality symbol: two terms refer the same
object or not the same object

Some examples (x = y, !(x = y))
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Using FOL
Domain – some part of the world

The kinship domain (P 254)
 Parent, Sibling, Brother, Sister, Child, Daughter, Son, …
Axioms - basic facts


One’s mother is one’s female parent
One’s husband is one’s male spouse
Definitions - concepts defined by axioms

x,y P(x,y)  …
Theorems - that are proved using axioms and definitions,
or entailed by axioms

x,y Sibling(x,y)  Sibling(y,x), as it can be derived from the
definition of Sibling(x,y)
Two important questions in building a KB
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Are axioms in the KB sufficient?
Are all axioms in the KB necessary?
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What should be in a KB


From a purely logical point of view, …
From a practical point of view, …
Adding sentences (assertions) to a KB
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Tell(KB, King(John))
Tell(KB, x King(x)  Person(x))
Asking questions (queries) and getting
answers

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Ask(KB, King(John)) - True
Ask(KB, x Child(x,Spot)) – substitution x/Wonder
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The domain of sets and lists in FOL
EpmtySet – constant {}
Member , Subset - predicates
Intersection, Union, Adjoin - functions
Eight axioms of sets (page 257):
1.
the only sets are EmptySet and those made
by adjoining something to a set {x|s2}
where Set(s2).
The differences between lists and sets

Order and repetition of an element
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Logical agents for Wumpus
Reflex agents classify percepts and act
Model-based agents have an internal
representation
Goal-based agents form goals and achieve
them
The first-order axioms are much more concise
than propositional logic axioms
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Constructing a W-logical agent
Define the interface (percepts) between the
environment and the agent

Including time using a time stamp
 Percept([Stench, Breeze,Glitter, None,None], 5)
Define actions

Actions: Turn(Right), Forward, Shoot, Grab, Release, Climb
Provide an action: a BestAction(a, 5) - a/Grab
Modify the environment
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t,s,b,m,c Percept([s,b,Glitter,m,c],t)Glitter(t)
t Glitter(t)  BestAction(Grab,t)
 This implements a simple reflex behavior
Define adjacency of any two squares (x, y) and (a, b)

x,y,a,b Adjacent([x,y],[a,b])  [a,b] is in …
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Deducing hidden properties
Two major types of rules: causal and diagnostic
Causal rules specify the assumed direction of
causality - model-based reasoning


Squares adjacent to pits are breezy
A pit causes …
Diagnostic rules infer hidden properties from the
percept-derived information


If a location is smelly, the wumpus must either be in
that location or in an adjacent location
If there is breeze, …
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Representing change
Storing a complete percept sequence is
tedious and inefficient to search for actions
An internal model allows an agent to know its
current status

having gold and at home square
Representing change is one of the most
important tasks in knowledge representation

How to represent change?
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Ways of representing change
The latest case only, forget about the past
= having a shallow memory and no history
= repeating errors
Situation calculus (Chapter 10.3)


A particular way of describing change in FOL
representing situations and actions as representing
objects
 Fluents are functions and predicates that vary from
one situation to the next

E.g., location, aliveness
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Situation calculus
Each situation is a
snapshot of the state

Situations are generated
from previous situations by
actions (Fig 10.2, p329)
Give an extra situation
argument for every
relation/property that can
change over time

it’s always the last one
argument
 At(Agent,[1,1],S0)^
At(Agent,[1,2],S1)

using Result(action,
situation)
 Result(Forward, S2) = S3
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Special axioms
Effect axioms - actions are described by
stating their effects


Holding-gold via Grab, !Holing-gold via Release
Are the above enough?
Frame axioms - describing how the world
stays the same


Holding-sth not releasing it, then holding it next
state
!Holding-sth not (grab or present or portable)
The two types of axioms together describe the world in
change.
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Successor-state (SS) axioms - resulting from
the combining of the E- and F- axioms
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true afterwards  [an action made it true v
true already and no action made it false]
One SS axiom is needed for each predicate
changing with time
A Successor-State axiom must list all the ways in
which the predicate can become true or false
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Keep track of location
What direction an agent is facing

Orientation(Agent,S0) = 0
How locations are arranged (via a map)
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x,y LocationToward([x,y],90)=[x,y+1]
Location l ahead of agent p: p,l,s At(p,l,s) 
x,y Adjacent(x,y)  d x=LocationToward(y,d)
What’s known about the map

x,y Wall([x,y])  (x=0 or x=5 or y=0 or y=5)
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What actions change locations

Going forward changes location
What actions change orientations

Turning changes orientation
There are still many research issues:
frame problems - the property remains unchanged
qualification problem - an action guaranteed to work, the
impossibility to list all the preconditions
ramification problem - implicit consequences of an action
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Which action
Different actions can achieve the same goal
depending on constraints
Separating facts about actions from facts
about goals as goals describe the desirability
of outcome states

desirability scale: great, good, medium,risky,
deadly
Defining the desirability of actions, leaving
the inference to choose an action that has
the highest desirability
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A goal-based agent
Certain actions lead to radical policy
change: getting the gold -> returning

s Holding(Gold,s)  GoalLocation([1,1],s)
Explicit goals allow many ways to work out
a sequence of actions
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Inference
Search
Planning
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Summary
FOL is a general-purpose representation
language based on objects and relations
BNF of FOL
A logical agent using FOL
Situation calculus to handle changes
Causal rules are often more flexible and entail
a wider range of consequences
We’re now ready to infer in FOL ...
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