Propositional Logic

1
Propositional Logic

• Agenda
• Other forms of inference in propositional logic
• Basics of First Order Logic (FOL)
• Vision
• Final Homework now posted on web site

2
Announcements

• Final Exam Date
• Dec. 19th
• 110-4pm
• 833 Mudd
• Getting homeworks back
• Game playing will be returned 12/10 in class
• Machine learning will be returned in final exam
• Class participation grade will be posted by 12/10
• Midterm curve will be given in class 12/10
• Final class will wrap up vision and do whats
  next and review

3
Types of Inference

• Resolution Theorem proving
• Model Checking
• Forward chaining with modus ponens
• Backward chaining with modus ponens

4
One Problem done all ways
5
Model Checking

• Enumerate all possible worlds
• Restrict to possible worlds in which the KB is
  true
• Check whether the goal is true in those worlds or
  not

6
Inference as Search

• State current set of sentences
• Operator sound inference rules to derive new
  entailed sentences from a set of sentences
• Can be goal directed if there is a particular
  goal sentence we have in mind
• Can also try to enumerate every entailed sentence

7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
Example
12
Characteristics of FOL

• Declarative
• Expressive
• Partial information
• Negation
• Compositionality

13
Ontological Commitment

• Propositional logic
• There are facts that either hold or do not hold
  in the world
• Logic constrains facts
• First-order logic
• The world consists of objects and relations
  between objects
• Logic constrains allowable objects, properties of
  objects, relations between objects

14
Ontological commitments of higher order logics

• Temporal logic
• Facts hold at particular times and those times
  are ordered
• Epistemological
• Agents hold beliefs about facts
• Three possible states of knowledge
• The agent believes a fact
• The agent does not believe it
• The agent has no opinion
• Probabilistic
• Facts are true to different degrees (Truth value
  from 0 to 1)

15
Problems with propositional logic
16
(No Transcript)
17
Propositional Logic is lacking in expressiveness

• Cannot represent knowledge of complex
  environments in a concise way
• E.g., Squares adjacent to pits are breezy
• Need objects
• Squares, pits, Kathy
• Need relations
• Adjacent, breezy, smelly, know
• Need functions
• Father-of, mother-of

18
Syntax of FOL basic elements

• Constants Charles, Ken, Victor
• Predicates knows, adjacent, gt
• Functions Sqrt, father-of
• Variables x,y,a,b
• Connectives ?,V,,?,?
• Equality
• Quantifiers ?,?

19
Atomic Sentences

• Atomic sentence predicate (term1termm)
  or term1term2
• Term function (term1, , termm) or
  constant or variable
• E.g. know(Charles,Ken), Adjacent (x,y),
  father-of(Kathy) Michael, Victor, x

20
Complex Sentences

• Complex sentences are made from atomic sentences
  using connectivesS, S1?S2, S1VS2, S1?S2,
  S1?S2
• E.g., adjacent(x,y) ? adjacent (y,x),
  knows(Charles, Michael),

21
Truth in First-order Logic

• Sentences are true with respect to a model and an
  interpretation
• Model contains ? 1 objects (domain elements) and
  relations among them
• Interpretation specifies referents for
• Constant symbols -gt objects
• Predicate symbols -gt relations
• Function symbols -gt functional relations
• An atomic sentence predicate (term1,,termn) is
  true iff the objects referred to by term1,,
  termn are in the relation referred to by
  predicate.

22
Universal quantification

• ?ltvariablesgt ltsentencegt
• Everyone at Columbia is smart?x At(x,Columbia)
  ? Smart(x)
• ?x P is true in a model m iff P with x being each
  possible object in the model
• At (Leia, Columbia) ? Smart(Leia)
• At (Ryan, Columbia) ? Smart (Ryan)
• At (Archana, Columbia) ? Smart (Archana)
• At (Stanley, Columbia) ? Smart (Stanley)
• ..

23
A common mistake

• Typically, ? is the main connective used with ?
• Common mistake using as the main connective ?
  ?x At(x,Columbia) ? Smart(x)

24
Existential Quantification

• ?ltvariablesgt ltsentencegt
• Someone at Columbia is smart?x At(x,Columbia)
  Smart(x)
• ? x P is true in a model m iff P with x being
  each possible object in the model
• Equivalent to the disjunction of instantiations
  of P
• At (Leia, Columbia) ? Smart(Leia)
• V At (Ryan, Columbia) ? ? Smart (Ryan)
• V At (Archana, Columbia) ? ? Smart (Archana)
• V At (Stanley, Columbia) ? ? Smart (Stanley)

25
Another Common Mistake

• Typically, ? is the main connective with ?
• Common mistake using ? as the main connective ?
  x At(x,Columbia) ? Smart(x)

26
Properties of Quantifiers

• ?x ?y is the same ?y ?x
• ?x ? y is the same as ? y ? x
• ? x ? y is not the same as ? y ? x
• ? x? y Loves(x,y)
• ? y ? x Loves(x,y)
• Everyone is loved by someone.
• Quantifier duality each can be expressed using
  the other ? x Likes (x,Icecream) ? x
  Likes(x,IceCream) ? x Likes(x, Broccoli)
  ?x Likes(x,Broccoli)

27
Properties of Quantifiers

• ?x ?y is the same ?y ?x
• ?x ? y is the same as ? y ? x
• ? x ? y is not the same as ? y ? x
• ? x? y Loves(x,y)
• There is a person who loves everyone in the world
• ? y ? x Loves(x,y)
• Quantifier duality each can be expressed using
  the other ? x Likes (x,Icecream) ? x
  Likes(x,IceCream) ? x Likes(x, Broccoli)
  ?x Likes(x,Broccoli)

28
Properties of Quantifiers

• ?x ?y is the same ?y ?x
• ?x ? y is the same as ? y ? x
• ? x ? y is not the same as ? y ? x
• ? x? y Loves(x,y)
• There is a person who loves everyone in the world
• ? y ? x Loves(x,y)
• Everyone is loved by someone.
• Quantifier duality each can be expressed using
  the other ? x Likes (x,Icecream) ? x
  Likes(x,IceCream) ? x Likes(x, Broccoli)
  ?x Likes(x,Broccoli)

29
Translation from English to FOL

• A mother is a female parent
• Andrew likes one of the homework problems
• ?