Outline Recap Knowledge Representation I Textbook: Chapters 6, 7, 9 and 10.
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Transcript of Outline Recap Knowledge Representation I Textbook: Chapters 6, 7, 9 and 10.
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
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
• Sound implies =
• Complete = implies
=
Inference Entailment
Implication (syntactic symbol)}
Propositional Logic: SEMANTICS
• Multiple interpretations– Assignment to each variable either T or F– Assignment of T or F to each connective via defns
PT
T
F
F
Q
PT
T
F
F
Q
PT
T
F
F
Q
PT
F
P Q P Q P Q P
Note: (P Q) equivalent to P Q
T
F F
F
F
T T
T
T
T TF
T
F
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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
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
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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)
valid
(smoke fire) smoke fire valid
Inference
• Backward Chaining (Goal Reduction)– Based on rule of modus ponens
– If know P1 ... Pn and know (P1 ... Pn )=> Q
– Then can conclude Q
• Resolution (Proof by Contradiction)
• GSAT
Student-Prof Example
• Some students like all professors. No student likes any tough professors. Thus, no professor is tough.
Unification and Substitution
• Substitution – a set of pairs s={x=a, y=b}
– Instance of a substitution • F=p(x,y,f(a)), Fs=applying s on F={p(a,b,f(a)}
• Replacement is simultaneous t={x=a,y=x}
– 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)
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)
Conversion to Normal Form
• Remove implications
• Move negation inwards
• Standardize variables
• Move quantifiers left
• Skolemization (every body has a heart)
• Distribute and, or’s
• Clausal Form
Resolution
A B C, C D E A B D E
• 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!
Resolution Refutation Procedure
• Page 281 of text– Negating theorem– Normal Form Conversion– Derive an empty clause– Answer Extraction
Finding Answers
• Father’s father is a grandfarther
• John is Ken’s father
• Larry is Joh’s father
• Question: who is Ken’s grandfather?
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
Logic Programming (page 304)
• Defining membership– member(X, [X|L]).– member(X, [Y|L]) :- member(X,L).
• How does Logic Programming Systems find answers?
Semantic Networks (page 317)
• Graphically represent the following– Birds are animals
– Mammals are animals
– Penguins are birds
– Cats are mammals
– Birds fly
– Penguins don’t fly
– Animals are alive
– Animals don’t fly
– Birds have two legs
– Mammals have 4 legs
• Semantic Networks have– Properties
– Subset links
– Member links
GSAT
Procedure GSAT (CNF formula: , max-restarts, max-climbs) For i := 1 to max-restarts do
A := randomly generated truth assignmentfor 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
[1992]