Speech & NLP (Fall 2014): Formal Knowledge Representation & Semantics
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Transcript of Speech & NLP (Fall 2014): Formal Knowledge Representation & Semantics
Speech & NLP
www.vkedco.blogspot.com
Knowledge Representation & Semantics
Conceptualization, Syntax & Semantics of First-Order Predicate Calculus,
Interpretation, Variable Assignment, Satisfaction
Vladimir Kulyukin
Department of Computer Science
Utah State University
Outline
● Conceptualization
● First-Order Predicate Calculus
– Syntax
– Semantics: Interpretation, Variable Assignment, Satisfaction
● Natural Language Examples
Introduction
● Intelligent behavior depends on an agent’s knowledge about
its world
● This knowledge is, to a great extent, descriptive (declarative)
● If this knowledge is to be used by a computer, this descriptive
knowledge must be formalized
● Knowledge representation is an area of AI the studies
methods to formalize the existing bodies of knowledge
What Does this Agent Need to Act in the World?
World
Agent
Agent Needs Conceptualization. What Else?
Conceptualization
World
Some ideas about how the world works
Agent
The Agent Needs Some Place to Store Knowledge
Conceptualization
World Knowledge Repository Some accessible place to store knowledge
Agent
Some ideas about how the world works
The Agent Needs a Way to Write Knowledge Down
Conceptualization
World Knowledge Repository Some accessible place to store knowledge
Agent
Some ideas about how the world works
Agents need formalisms to encode & manipulate
knowledge about the world.
Fundamental Tenet of Symbolic AI
Conceptualization
Objects & Relations
● Knowledge formalization begins with a conceptualization of the
world
● Conceptualization, generally speaking, analyzes the world in terms of
objects and relations
● Functions are also relations
● Objects can be concrete (book, pen, block) or abstract (number 2,
honesty, love)
● Objects can be primitive (number 2) or abstract (algebraic
expression)
Universe of Discourse
● It is impossible for any conceptualization to include all objects in the world
● Conceptualizations reside inside observers (human or mechanical) and
include only those objects that present some interest to the observers
● Beekeepers conceptualize the world in terms bees, swarms, beehives, honey
extractors, bee disease treatments, etc
● Number theorists conceptualize the world in terms of numbers, sets, properties
of numbers, etc.
● The set of objects covered by a conceptualization is called the universe of
discourse
Functions & Relations
● Once a conceptualization has objects, the observer must
establish relations among those objects
● There are two types of relations most conceptualizations
contain: functions and relations
● A set of functions in the conceptualization is called functional
basis
● A set of relations in the conceptualization is called relational
basis
Blocks World
A
A
A
A
A
A
c
b
a
d
e
Which objects do you conceptualize in this world?
Blocks World: Objects
A
A
A
A
A
A
c
b
a
d
e
Many human observers conceptualize five blocks {a, b, c, d, e}
}and the table t
Blocks World: Objects
A
A
A
A
A
A
c
b
a
d
e
One can conceptualize five blocks {a, b, c, d, e} and the table t
Blocks World: Functions & Relations
A
A
A
A
A
A
c
b
a
d
e
What functions & relations do you see in this world?
Blocks World: Functions & Relations
A
A
A
A
A
A
c
b
a
d
e
We can define the partial function hat that maps a block into the block on top of
it. Formally, hat consists of the following tuples: hat: {<b, a>, <c, b>, <e, d>}
Blocks World: Functions & Relations
A
A
A
A
A
A
c
b
a
d
e
We can define the relations on or above with the obvious interpretations. Formally, these
relations consist of the following tuples:
on: {<a, b>, <b, c>, <d, e>}
above: {<a, b>, <b, c>, <a, c>, <d, e>}
Blocks World: Functions & Relations
A
A
A
A
A
A
c
b
a
d
e
We can define the relation clear that holds for a block if and only if there is no block on top of
it: clear: {a, d}
Blocks World: One Conceptualization
A
A
A
A
A
A
c
b
a
d
e
.,,,,,,,, clearaboveonhatedcba
Upper Bound on Number of N-ary Relations
subsets. possible 2 are There tuples.- theseofsubset a
isrelation ary -Every tuples.-distinct are There
objects. contains Discourse of Universe theSuppose
nO
n
n
nnO
O
Notes on Conceptualizations
● Conceptualizations, although they are written down, consists
of the objects and relations the observer actually sees in
the world
● The same world may have multiple conceptualizations (e.g.,
blocks world can be conceptualized in terms of line segments,
curves, and their relations)
● Different conceptualizations allow/inhibit certain kinds of
knowledge (light as a wave vs. light as a particle; geocentric
vs. heliocentric universe)
Realism vs. Nominalism
● Realism takes a stand that objects & relations in one’s
conceptualization really exist in the world
● Nominalism takes a stand that objects & relations in
one’s conceptualization do not necessarily exist in the
world
● AI takes a standpoint that conceptualizations are
justified by their utility to the system (this is, strictly
speaking, neither realism nor nominalism)
Brief Introduction
to
First-Order Predicate Calculus
Alphabets & Symbols
● Since FOPC is a formal language, it must start with an
alphabet
● Chapter 2 in Logical Foundations of AI contains one such
alphabet (typically it consists of the standard ASCII
augmented with specific mathematical symbols)
● FOPC has two types of symbols: variables and constants
● Constants consists of object constants, function
constants, and relation constants
Variables & Constants
● A variable is a sequence of lowercase alphanumeric characters and numeric characters
such that the first character is lowercase alphabetic
● An object constant names a specific element in the universe of discourse and is a
sequence of alphabetic characters or digits such that the first character is either uppercase
alphabetic or digit
● A function constant names a function on the members of the universe of discourse and is
a mathematical operator or a sequence of alphabetic characters or digits in which the first
character is uppercase alphabetic
● A relation constant names a relation on the members of the universe of discourse and is a
mathematical operator or a sequence of alphabetic characters or digits in which the first
character is uppercase alphabetic
Variables & Constants: Examples
● Variables: x, y, z, x10, y15, z500
● Object constants: Logan, Aristotle, Hallway100
● Function constants: Age, Cosine, Tangent, +, -, *
● Relation constants: Above, Clear, Below
Terms
● A term is an object’s name
● A term can be a variable, an object constant, or a
functional expression
● A functional expression is an expression of the form
f(t1, t2, …, tn) , where f is an n-ary function constant
and are t1, t2, …, tn terms (this is a recursive definition)
Terms: Examples
● A, B, C, D, E are object constants and, therefore, terms
● Hat is a function constant
● Hat(C) is a term (functional expression)
● Hat(Hat(C)) is a term (functional expression)
● Hat(x) is a term (functional expression)
● Hat(Hat(x)) is a term (functional expression)
Well-Formed Formulas (WFFs)
● In FOPC, facts are stated in sentences (aka well-
formed formulas or wffs)
● Three types of sentences:
– Atomic sentences (aka atoms);
– Logical sentences;
– Quantified sentences
Atoms
at(C))Above(A, H
Hat(C))On(Hat(B),
On(A, B)
tttt nn
:Examples
termsare ,..., andconstant relation a is where,,..., 11
Atoms
21
21
21
:Examples atoms. are sexpression
subset es,inequaliti ,equalities almathematic All
tt
tt
tt
Logical Sentences
atomsor sentences logical are , where,:eequivalenc 6.
atomsor sentences logical are , where, :nimplicatio reverse 5.
atomsor sentences logical are , where, :nimplicatio 4.
atomor sentence logical a is ,... :ndisjunctio 3.
atomor sentence logical a is ,... :nconjunctio 2.
sentence logical a is where :negation 1.
sentences. logicalother or
atoms tooperators logical applyingby formed are sentences Logical
11
11
nin
nin
Logical Sentences: Examples
AEOnADOnACOnABOnAClear
yxOnyxAbove
yxAboveyxOn
xxHatAbovexxHatAbove
EDAboveBAOn
BAOn
,,,, 6.
,, 5.
,, 4.
,, 3.
,, 2.
, 1.
sentences. logicalother or atoms to
operators logical applyingby formed are sentences Logical
Quantified Sentences
vTablevBlockv
vBlockvBluev
vBlockvBluev
xBlockxBluex
vv
vv
:Examples
sentence a is and
variablea is where :tionquantifica lexistentia 2.
sentence a is and
variablea is where :tionquantifica universal 1.
s.quantifier lexistentiaor universal the
withsentencesother prefixingby formed are sentences Quantified
More Examples of Quantified Sentences
yxyx
yxyx
yxyx
yxLovesyx
yxLovesyx
yxLovesyx
,
,
,
Semantics
How Does the Agent Know What is True?
Conceptualization
World Knowledge Repository
Agent
Worlds, Conceptualizations, Knowledge
Repositories, & Agents
● Sentences are written in a knowledge repository (book, smartphone,
database, etc.)
● Conceptualizations of the world exist in the agent’s head (some true,
some false, some partially true)
● Truth of each sentence is evaluated with respect to a specific
conceptualization
● As the agent acts in the world, the agent may modify or abandon
conceptualizations or adopt new ones
Interpretation as a Function
● Interpretation is a mapping b/w the elements of a formal
language (FOPC in our case) and the elements of a
conceptualization
● Formally, an interpretation is the function I(σ) where σ is an
element of the language
● The value of I(σ) is an element of a given conceptualization
● The universe of discourse is denoted |I|
Formal Properties of Interpretation
constantrelation a is if ,
constantfunction a is if , :
constantobject an is if
n
n
II
III
II
Blocks World Interpretation I
daClearI
edcacbbaAboveI
edcbbaOnI
debcabHatI
eEIdDIcCIbBIaAI
,
,,,,,,,
,,,,,
,,,,,
;;;;;
Blocks World Interpretation J
daClearI
deacbcabAboveI
debcabOnJ
debcabHatJ
eEJdDJcCJbBJaAJ
Above
OnClear
HatIJ
,
,,,,,,,
,,,,,
,,,,,
;;;;;
:below'' as
and under'' as interpretsbut ,relation unary
and ,function constants,object on with agrees
Variable Assignment
CzUByUAxU
U
;;:Example
constants.object
to variablesmapping function a is assignment Variable
symbols.other from separately dinterprete are Variables
Term Interpretation
nIUii
n
IU
IU
IU
xxftTxtI
fIttt
tUtTt
tItTt
T
UI
,...,then ,
, and ,..., form theof terma is If .3
then variable,a is If .2
then constant,object an is If .1
:follows as defined
objects to termsmapping assignment terma is the
,assignment variablea is and tion,interpretaan is If
1
1
Term Interpretation: Example
then ,let
and ,previously definedtion interpreta theis If
bCHatwUHatIwHatT
CwU
I
IU
Satisfaction
General Notation
U
IU
U
IU
I
I
assignment variablea
and tion interpretaan under satisfiednot is Sentence:|
assignment variablea
and tion interpretaan under satisfied is Sentence:|
Case 1
2121 iff | tTtTUtt IUIUI
Case 2
edcacbbaAboveIcaCTAT
UCAAbove
edcacbbaAboveI
cCIaAI
ItTtTUtt
IUIU
I
nIUnIUI
,,,,,,,,,
because satisfied is ,|
,,,,,,,
;;
:Example
,..., iff ,...,| 21
Case 3
UttUtt II 2121 ,...,| iff ,...,|
Case 4
niUU iInI ,...,1,| iff ...| 1
Case 5
niUU iInI ,...,1 somefor ,| iff ...| 1
Case 6
UUU III 2121 | | iff |
Case 7
UUU III 122121 | | iff |
Case 8
.|,in with
replaced is after |,| allfor iff |
Ud
vIdUv
I
I
Case 9
Udv
IdUv
I
I
|,in with replaced is after
|,| somefor iff |
Examples
xPoisonousxMushroomxPurplex
xPoisonousxMushroomxPurplex
poisonous. are mushrooms purple All
Examples
xPurplexPoisonousxMushroomx
purple. isit ifonly poisonous is mushroomA
Examples
xPoisonousxMushroomxPurplex
xPoisonousxMushroomxPurplex
poisonous. is mushroom purple No
References
● Ch 02, M. Genesereth & N. Nilsson. Logical Foundations
of AI, Morgan Kaufmann