Outline

1
Outline

• Recap
• Knowledge Representation I
• Textbook Chapters 6, 7, 9 and 10

2
Some KR Languages

• Propositional Logic
• Predicate Calculus
• Frame Systems
• Rules with Certainty Factors
• Bayesian Belief Networks
• Influence Diagrams
• Semantic Networks
• Concept Description Languages
• Nonmonotonic Logic

3
In Fact

• All popular knowledge representation systems are
  equivalent to (or a subset of)
• Logic (Propositional Logic or Predicate Calculus)
• Probability Theory

4
Propositional Logic

• Syntax
• Atomic sentences P, Q,
• Connectives ? , ?, ?, ?
• Semantics
• Truth Tables
• Inference
• Modus Ponens
• Resolution
• DPLL
• GSAT
• Resolution
• Complexity

5
Notation
? ? ? ?? ?
Inference
Entailment

• Sound ?? implies ?
• Complete ? implies ??

6
Propositional Logic SEMANTICS

• Multiple interpretations
• Assignment to each variable either T or F
• Assignment of T or F to each connective via defns

Note (P ? Q) equivalent to ? P ? Q
7
FOL Definitions

• Constants a,b, dog33.
• Name a specific object.
• Variables X, Y.
• Refer to an object without naming it.
• Functions father-of
• Mapping from objects to objects.
• Terms father-of(father-of(dog33))
• Refer to objects
• Atomic Sentences in(father-of(dog33), food6)
• Can be true or false
• Correspond to propositional symbols P, Q

8
Terminology

• Literal u or ?u, where u is a variable
• Clause disjunction of literals
• Formula, ?, conjunction of clauses
• ?(u) take ? and set all instances of u true
  simplify
• e.g. ?((P, ?Q)(R, Q)) then ?(Q)P
• Pure literal var appearing in a formula either as
  a negative literal or a positive literal (but not
  both)
• Unit clause clause with only one literal

9
Definitions

• valid tautology always true
• satisfiable sometimes true
• unsatisfiable never true

1) smoke ? smoke 2) smoke ? fire 3) (smoke ?
fire) ? (?smoke ? ?fire) 4) smoke ? fire ? ?fire
?smoke ? smoke valid
? smoke ? fire satisfiable
? (? smoke ? fire) ? (?smoke ? ?fire)
(smoke ? ?fire) ? ? smoke ? ?fire valid
valid
10
Inference

• Backward Chaining (Goal Reduction)
• Based on rule of modus ponens
• If know P1 ?... ? Pn and know (P1 ?... ? Pn )gt
  Q
• Then can conclude Q
• Resolution (Proof by Contradiction)
• GSAT

11
Student-Prof Example

• Some students like all professors. No student
  likes any tough professors. Thus, no professor
  is tough.

12
Unification and Substitution

• Substitution
• a set of pairs sxa, yb
• Instance of a substitution
• Fp(x,y,f(a)), Fsapplying s on Fp(a,b,f(a)
• Replacement is simultaneous txa,yx
• Composition of Substitutions st?
• Unifier a substitution that makes two
  expressions the same
• Most General Unifier MGU is a smallest unifier
• Example unify p(f(x), h(y), a) and p(f(x), z, a)

13
Normal Forms (Chapter 9, page 281)

• CNF Conjunctive Normal Form
• Conjunction of disjuncts (each disjunct
  clause)

(P ? Q) ? R (P ? Q) ? R
?(P ? Q) ? R
?P ? ?Q ? R
(?P ? ?Q) ? R
(?P ? R) ? (?Q ? R)
14
Removing Existential

• Skolem Constants (page 281)
• Skolem Functions (page 282)

15
Conversion to Normal Form

• Remove implications
• Move negation inwards
• Standardize variables
• Move quantifiers left
• Skolemization (every body has a heart)
• Distribute and, ors
• Clausal Form

16
Resolution

• Refutation Complete
• Given an unsatisfiable KB in CNF,
• Resolution will eventually deduce the empty
  clause
• Proof by Contradiction
• To show ? ? Q
• Show ? ? ?Q is unsatisfiable!

17
Resolution Refutation Procedure

• Page 281 of text
• Negating theorem
• Normal Form Conversion
• Derive an empty clause
• Answer Extraction

18
Student-Prof Example

• FOL sentences
• Conclusion clause negate
• Use refutation to prove.

19
Finding Answers

• Fathers father is a grandfarther
• John is Kens father
• Larry is Johs father
• Question who is Kens grandfather?

20
Application Logic Programming

• Prolog (page 304)
• Sequence of sentences
• Horn clauses
• Queries
• Negation as failure
• Distinct names distinct objects
• Built-in predicates for math, etc.
• Example membership function

21
Logic Programming (page 304)

• Defining membership
• member(X, XL).
• member(X, YL) - member(X,L).
• How does Logic Programming Systems find answers?

22
Semantic Networks (page 317)

• Graphically represent the following
• Birds are animals
• Mammals are animals
• Penguins are birds
• Cats are mammals
• Birds fly
• Penguins dont fly

• Animals are alive
• Animals dont fly
• Birds have two legs
• Mammals have 4 legs
• Semantic Networks have
• Properties
• Subset links
• Member links

23
GSAT
1992
Procedure GSAT (CNF formula ?, max-restarts,
max-climbs) For i 1 to max-restarts do A
randomly generated truth assignment for j
1 to max-climbs do if A satisfies ? then
return yes A random choice of one of best
successors to A successor means only 1
(var,val) changes from A best means making
the most clauses true