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Prepositional Logic
Prepositional Logic
(First-(First-order Logic)order Logic)
Lecture #13Lecture
Announcements
Announcements
Writing Assignment #2 is due now Writing Assignment #2 is due now
- – Two printed copies, one anonymousTwo printed copies, one anonymous
Midterm is one week from today
Midterm is one week from today
Read Chapter 9
Read Chapter 9
- – This will be the last chapter on the midtermThis will be the last chapter on the midterm
Programming assignment #1 Programming assignment
- Is being gradedIs being graded
- – Many of you wereMany of you were ““looseloose”” in your output;in your output;
some ignored the specification altogether.some ignored the specification altogether.
Writing Assignment
Writing Assignment
What are the propositions?
What are the propositions?
What are the initial axioms?
What are the initial axioms?
What axioms are added with every guess?What axioms are added with every guess?
How do we detect cheating?How do we detect cheating?
- – Inconsistency implies lyingInconsistency implies lying
- – Inconsistency detected through resolutionInconsistency detected through resolution
These are the four elements that you should look for
in your critique. They should be clearly and correctly
identified, and not merely implicit
Propositional logic
Propositional logic
Propositional logic is declarative
Propositional logic is declarative
☺☺ Propositional logic is compositional:Propositional logic is compositional:
- – meaning of Bmeaning of B
1,11,
∧∧ PP
1,21,
is derived from meaning of Bis derived from meaning of B
1,11,
andand
of Pof P
1,21,
☺☺ Meaning in propositional logic is contextMeaning in propositional logic is context--independentindependent
- – unlike natural language, where meaning depends onunlike natural language, where meaning depends on
context
context
Propositional logic has limited expressive power
Propositional logic has limited expressive power
- – Unlike natural languageUnlike natural language
- – E.g., cannot say "pits cause breezes in adjacent squaresE.g., cannot say "pits cause breezes in adjacent squares““
(except by writing one sentence for each square)(except by writing one sentence for each square)
First Order LogicFirst Order Logic
Examples of things we can say:Examples of things we can say:
- – All men are mortal:All men are mortal:
∀∀x Man(x)x Man(x) ⇒⇒ Mortal(xMortal(x))
- – Everybody loves somebodyEverybody loves somebody
x
x
y Loves(x, y)
y Loves(x, y)
- – The meaning of the wordThe meaning of the word ““aboveabove””
x
x
y above(x,y)
y above(x,y)
(on(x,y)
(on(x,y)
z (on(x,z)
z (on(x,z)
above(z,y))above(z,y))
First Order logicFirst Order logic
Whereas propositional logic assumes the world
Whereas propositional logic assumes the world
contains facts,contains facts,
first- first-order logic (like natural language) assumesorder logic (like natural language) assumes
the world contains
the world contains
- Objects: people, houses, numbers, colors,Objects: people, houses, numbers, colors, ……
- – Relations: red, round, prime, brother of, bigger than,Relations: red, round, prime, brother of, bigger than,
part of,part of, ……
- – Functions: fatherFunctions: father--of, plus,of, plus, ……
Logics in General
Logics in General
Ontological Commitment: What exists in the world
Ontological Commitment: What exists in the world
—— TRUTHTRUTH
- – PL : facts hold or do not hold.PL : facts hold or do not hold.
- – FOL : objects with relations between them that hold or doFOL : objects with relations between them that hold or do
not holdnot hold
EpistemoligicalEpistemoligical Commitment: state of knowledgeCommitment: state of knowledge
allowed with respect to a factallowed with respect to a fact
Syntax of FOL
Syntax of FOL
User defines these primitives: User defines these primitives:
- – Constant symbols (i.e., the "individuals" in the world)Constant symbols (i.e., the "individuals" in the world)
E.g., Mary, 3E.g., Mary, 3
- – Function symbols (mapping individuals to individuals)Function symbols (mapping individuals to individuals)
E.g., fatherE.g., father--of(Mary) = John, colorof(Mary) = John, color--of(Sky) = Blueof(Sky) = Blue
- – Predicate/relation symbols (mapping from individualsPredicate/relation symbols (mapping from individuals
to truth values) E.g., greater(5,3), green(Grass),to truth values) E.g., greater(5,3), green(Grass),
color(Grass, Green)color(Grass, Green)
Syntax (cont.)
Syntax (cont.)
FOL supplies these primitives:
FOL supplies these primitives:
- – Variable symbols. E.g., x,yVariable symbols. E.g., x,y
- Connectives. Same as in PL:Connectives. Same as in PL:
- – Equality =Equality =
- – Quantifiers: Universal (Quantifiers: Universal (∀∀) and Existential () and Existential (∃∃))
Atomic sentences
Atomic sentences
TermTerm == functionfunction (( termterm
11
,...,,..., termterm
nn
oror constantconstant oror variablevariable
Atomic sentence =Atomic sentence = predicatepredicate (( termterm
1
1
,...,,..., termterm
n
n
oror termterm
1
1
= term= term
2
2
Example terms:Example terms:
Brother(KingJohn,RichardTheLionheartBrother(KingJohn,RichardTheLionheart))
Greater(Length(LeftLegOf(RichardGreater(Length(LeftLegOf(Richard)),)),
Length(LeftLegOf(KingJohn
Length(LeftLegOf(KingJohn))))))
Complex sentencesComplex sentences
Complex sentences are made from atomicComplex sentences are made from atomic
sentences using connectivessentences using connectives
S
S
S
S
1
1
S
S
2
2
S
S
1
1
S
S
2
2
S
S
1
1
S
S
2
2
S
S
1
1
S
S
2
2
and by applying quantifiers.and by applying quantifiers.
Examples:Examples:
Sibling(KingJohn,RichardSibling(KingJohn,Richard)) ⇒⇒ Sibling(Richard,KingJohnSibling(Richard,KingJohn))
greater(1,2)greater(1,2) ∨∨ less-less-oror--equalequal(1,2)(1,2)
∀∀x,y Sibling(x,y)x,y Sibling(x,y) ⇒⇒ Sibling(y,x)Sibling(y,x)
Truth in firstTruth in first--order logicorder logic
Recall that a model is the set of all possible worlds under cons
Recall that a model is the set of all possible worlds under consideration.ideration.
Sentences are true with respect to a
Sentences are true with respect to a modelmodel and anand an interpretationinterpretation
Model contains objects (domain elements) and relations among them Model contains objects (domain elements) and relations among them
Interpretation specifies referents for Interpretation specifies referents for
constant symbolsconstant symbols →→ objectsobjects
predicate symbolspredicate symbols →→ relationsrelations
function symbolsfunction symbols →→ functional relationsfunctional relations
An atomic sentence An atomic sentence predicate(termpredicate(term
11
,...,,...,termterm
nn
)) is trueis true
iff theiffthe objectsobjects referred to byreferred to by termterm
11
,...,,...,termterm
nn
are in the relation are in therelation referred to byreferred to by predicatepredicate ..
Existential quantification (cont.)
Existential quantification (cont.)
Typically,Typically, ∧∧ is the main connective withis the main connective with ∃∃
Common mistake: using
Common mistake: using ⇒
with
with ∃
∃ ∃ xx At(x,At(x, CSUCSU )) ⇒⇒ Smart(x)Smart(x)
When is this true?When is this true?
Properties of quantifiers
Properties of quantifiers
∀∀xx ∀∀yy is the same asis the same as ∀∀yy ∀∀xx
∃∃xx ∃∃yy is the same asis the same as ∃∃yy ∃∃xx
∃∃xx ∀∀yy isis notnot the same asthe same as ∀∀yy ∃∃x:x:
∃∃xx ∀∀yy Loves(x,y)Loves(x,y)
- – ““There is a person who loves everyone in the worldThere is a person who loves everyone in the world””
∀∀yy ∃∃x Loves(x,y)x Loves(x,y)
- – ““Everyone in the world is loved by at least one personEveryone in the world is loved by at least one person””
Quantifier duality
Quantifier duality: each can be expressed using the other: each can be expressed using the other
∀ ∀xx Likes(x,IceCreamLikes(x,IceCream)) ¬∃¬∃xx ¬¬Likes(x,IceCream)Likes(x,IceCream)
∃ ∃x Likes(x,Broccoli)x Likes(x,Broccoli) ¬∀¬∀xx ¬¬Likes(x,Broccoli)Likes(x,Broccoli)
Equality
Equality
term
term
1
1
= term
= term
2
2
is true under a given interpretation
is true under a given interpretation
if and only ifif and only if termterm
11
andand termterm
22
refer to the samerefer to the same
objectobject
E.g., definition ofE.g., definition of SiblingSibling in terms ofin terms of ParentParent ::
x,y
x,y Sibling(x,y)Sibling(x,y)
[
[
(x = y)
(x = y)
m,f
m,f
(m = f)
(m = f)
Parent(m,x)Parent(m,x) ∧∧ Parent(f,x)Parent(f,x) ∧∧ Parent(m,y)Parent(m,y) ∧∧
Parent(f,y)]Parent(f,y)]
WumpusWumpus World (World (reduxredux))
An agent (e.g. in
An agent (e.g. in wumpuswumpus world) may keepworld) may keep
a knowledge base of objects, predicates &a knowledge base of objects, predicates &
inference rulesinference rules
- Objects in this world are grid squares [Objects in this world are grid squares [w,hw,h]]
- – Some predicates are sensationsSome predicates are sensations
E.g. Stench([w,h E.g.Stench([w,h]),]), Glitter([w,hGlitter([w,h]),]), Breeze([w.hBreeze([w.h])])
These are asserted over time, as the agent
These are asserted over time, as the agent
explores
explores
- – Other predicates are facts about locationsOther predicates are facts about locations
E.g. Wumpus([w,h E.g.Wumpus([w,h]),]), Pit([w,hPit([w,h]),]), Gold([w,hGold([w,h])])
Warning: my presentation varies from
the book’s (which uses time, not location)
Interacting with FOL
Interacting with FOL KBsKBs
Suppose aSuppose a wumpuswumpus--world agent perceives a smell and aworld agent perceives a smell and a
breeze (but no glitter) in square [1,2]breeze (but no glitter) in square [1,2]
Tell(KB, Stench([1,2])Tell(KB, Stench([1,2])
Tell(KB
Tell(KB, Breeze([1,2])), Breeze([1,2]))
Tell(KB
Tell(KB,, ¬
Glitter([1,2]))
Glitter([1,2]))
This asserts a set of facts (axioms)
This asserts a set of facts (axioms)
Ask(KB, Safe([1,3]))Ask(KB, Safe([1,3]))
This asks the KB whether the statement Safe([1,3])This asks the KB whether the statement Safe([1,3])
is entailed by the known facts (axioms)
is entailed by the known facts (axioms)
DeductionsDeductions
These use a set of inference rules stored in These use a set of inference rules stored in
the KB
the KB
- – ∀∀w,h Breeze([w,hw,hBreeze([w,h])]) ⇒⇒ Pit([w+1,h]) vPit([w+1,h]) v
Pit([w
Pit([w--1,h]) v Pit([w,h+1]) v Pit([w,h1,h]) v Pit([w,h+1]) v Pit([w,h--1])1])
- – ∀∀w,h Safe([w,hw,hSafe([w,h])]) ⇔⇔ ((¬¬Pit([w,hPit([w,h]) ^]) ^
¬¬Wumpus([w,hWumpus([w,h]))]))
- – ∃∃ w,h,w,h, Wumpus([w,hWumpus([w,h])])
- – ∀∀w,h,x,y ((w=x) ^ ( h=y))w,h,x,y((w=x) ^ ( h=y))
v
v
(Wumpus([w,hWumpus([w,h]) ^]) ^ Wumpus([x,yWumpus([x,y]))]))
Creating a KB using FOL
Creating a KB using FOL
Identify the task (what will the KB be used for)
Identify the task (what will the KB be used for)
2.2. Assemble the relevant knowledge
Assemble the relevant knowledge
Knowledge acquisition.Knowledge acquisition.
3.3. Decide on a vocabulary of predicates, functions, and constantsDecide on a vocabulary of predicates, functions, and constants
Translate domainTranslate domain--level knowledge into logiclevel knowledge into logic--level names.level names.
4.4. Encode general knowledge about the domainEncode general knowledge about the domain
define axiomsdefine axioms
Encode a description of the specific problem instance
Encode a description of the specific problem instance
6.6. Pose queries to the inference procedure and get answers
Pose queries to the inference procedure and get answers
7.7. Debug the knowledge baseDebug the knowledge base
Examples
Examples
The
The kinship
kinship domain
domain
Basic predicates: Female, Parent.
Basic predicates: Female, Parent.
Other predicates in this domain:Other predicates in this domain:
One's mother is one's female parentOne's mother is one's female parent
∀∀m,cm,c Mother(c)Mother(c) = m= m ⇔⇔ (Female(m)(Female(m) ∧∧ Parent(m,c))Parent(m,c))
∀∀ x,yx,y Sibling(x,y)Sibling(x,y) ⇔⇔ [[¬¬(x = y)(x = y) ∧∧ ∃∃m,fm,f ¬¬(m = f)(m = f) ∧∧ Parent(m,x)Parent(m,x) ∧∧
Parent(f,x)
Parent(f,x) ∧
Parent(m,y)
Parent(m,y) ∧
Parent(f,y)]
Parent(f,y)]
A first cousin is a child of a parent
A first cousin is a child of a parent’’s siblings sibling
x,y
x,y FirstCousin(x,y
FirstCousin(x,y)) ⇔
p,ps Parent(p,x)
p,ps Parent(p,x) ∧
Sibling(ps,p)
Sibling(ps,p) ∧
Parent(ps,y)Parent(ps,y)
These are the axioms of the domain (they are also definitions
These are the axioms of the domain (they are also definitions
since they usesince they use biconditionalsbiconditionals).).
Some sentences areSome sentences are ““theoremstheorems”” ---- they can be derived fromthey can be derived from
the axioms:the axioms:
- – ““SiblingSibling”” is symmetricis symmetric
∀∀x,yx,y Sibling(x,y)Sibling(x,y) ⇔⇔ Sibling(y,x)Sibling(y,x)
Examples (cont)
Examples (cont)
TheThe setset domaindomain
Notation:Notation: {x|s} is the set resulting from adding x to the set s.{x|s} is the set resulting from adding x to the set s.
∀∀s Set(s)s Set(s) ⇔⇔ (s = {} )(s = {} ) ∨∨ ((∃∃xx ∃∃ss
2
2
Set(sSet(s
2
2
)) ∧∧ s = {x|ss = {x|s
2
2
x
x ∃
s {x|s} = {}
s {x|s} = {}
x
x ∀
s x
s x ∈
s
s ⇔
[
[
y,s
y,s
22
} (s = {y|s
} (s = {y|s
22
(x = y
(x = y ∨
x
x ∈
s
s
22
))]
))]
x
x ∀
s x
s x ∈
s
s ⇔
s = {x|s}
s = {x|s}
s
s
11
s
s
22
s
s
11
s
s
22
x x
x x ∈
s
s
11
x
x ∈
s
s
22
∀∀ss
11
∀∀ss
22
(s(s
11
= s= s
22
)) ⇔⇔ (s(s
11
⊆⊆ ss
22
∧∧ ss
22
⊆⊆ ss
11
∀∀xx ∀∀ss
11
,s,s
22
xx ∈∈ (s(s
11
∩∩ ss
22
)) ⇔⇔ (x(x ∈∈ ss
11
∧∧ xx ∈∈ ss
22
∀∀xx ∀∀ss
11
,s,s
22
xx ∈∈ (s(s
11
∪∪ ss
22
)) ⇔⇔ (x(x ∈∈ ss
11
∨∨ xx ∈∈ ss
22
Examples (cont)
Examples (cont)
TheThe natural numbersnatural numbers domaindomain
0 is a natural number: 0 is a natural number:
NatNum(0)NatNum(0)
The successor of a natural number is a natural number: The successor of a natural number is a natural number:
∀ ∀nn NatNum(nNatNum(n)) ⇒⇒ NatNum(S(nNatNum(S(n))))
Constraints on the successor function: Constraints on the successor function:
∀ ∀nn ¬¬(0 = S(n))(0 = S(n))
∀ ∀mm ∀∀nn ¬¬(m = n)(m = n) ⇒⇒ ¬¬(S(m) = S(n))(S(m) = S(n))
Defining addition:
Defining addition:
n
n NatNum(nNatNum(n)) ⇒
+(0, n) = n
+(0, n) = n
m
m ∀
n
n NatNum(mNatNum(m)) ∧
NatNum(n
NatNum(n)) ⇒
+(S(m), n) = S(+(m,n))
+(S(m), n) = S(+(m,n))
