Lesson Plan – Wednesday
October 15
1. Desiderata for Knowledge Representation
2. Propositional Logic - Syntax and Semantics
3. The Curious Incident of the Dog in the Night
4. Logical Entailment
5. How hard is deduction
6. Algorithms for deductive reasoning
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DesideraTA for Knowledge
Representation
Although hierarchical planning allows us considerable flexibility in representing action, other aspects of our representation of the world are quite impoverished.
What about:
Distinguish false facts from unknown facts?
Incomplete knowledge – disjunctions of facts?
Simple rules:
Politicians cannot be trusted.
Objects and relations? “Any friend of yours is a friend of mine”
Numeric uncertainty?
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What is needed: a precise REPRESENTATION LANGUAGE
== LOGIC!
We will see that PROBABILITY also builds on LOGIC, but for
now we just consider TRUTH.
Represent WHAT? The WORLD. LOGIC allows us to make a PRECISE CONNECTION between SYMBOLS on paper or in a COMPUTER and
the WORLD!
PROPOSITIONAL LOGIC:
Syntax: Propositions, connectives AND, OR, NOT, IMPLIES, EQUIVALENT
Proposition: Symbol or List of Symbols
on(A,B) V on(B,A)
touching(A,B) <=> [ on(A,B) V on(B,A) ]
What is important is how we assign MEANING to a sentence!
A POSSIBLE WORLD assigns true or false to every PROPOSITION.
Equivalently: the MEANING of a PROPOSITION is the set of ALL WORLDS in which it is TRUE!
By compositional semantics, each world assigns every SENTENCE true or false.
HOW? By obvious recursive definition!
THUS: the meaning of a SENTENCE is also is the set of ALL WORLDS in which it is true.
EXAMPLES: VENN Diagrams
A, B, A and B, A or B
TRICKY: A => B
ONLY REALLY NEED AND, OR, and NOT
A => B is the same as ~A V B
A <=> B is the same as (~A v B) & (~B v A)
WHAT ABOUT THE SENTENCE
A & not A
It has NO models! == Is UNSATISFIABLE!
INCREASING EXPRESSIVE POWER OF PROPOSITIONAL LOGIC
Propositional logic + schemas:
touching(A,B) <=> [ on(A,B) V on(B,A) ]
touching(A,C) <=> [ on(A,C) V on(C,A) ]
SCHEMAS: ADD SET OF CONSTANTS:
blocks = { A, B, C, D }
UNIVERSAL QUANTIFIER IS AN IMPLICIT AND
covered(A) <=> [ on(B,A) V on(C,A) V on(D,A) ]
EXISTENTIAL QUANTIFIER IS AN IMPLICIT OR
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THE CURIOUS INCIDENT OF THE DOG IN THE NIGHT
A racehorse “Silver Blaze” was stolen from the stable, and a man named Simpson was accused. Sherlock Holmes proved that the horse’s trainer Straker was the true thief by reasoning from the following premises:
1. The horse was stolen by Simpson or by Straker.
2. The thief entered the stable the night of the theft.
3. The dog barks if a stranger enters the stable.
4. Simpson was a stranger.
5. The dog did not bark
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WHY IS LOGIC A GOOD THING?
1. A short sentence can represent an EXPONENTIALLY LARGE set of POSSIBILITIES!
2. Enables INFERENCE: uncover the CONSEQUENCES of our beliefs!
S1 ENTAILS S2 IF AND ONLY IF
Whenever S1 is true, S2 must be true
== The MEANING OF S1 is a SUPERSET of the MEANING of S2
Thus we get DEDUCTIVE REASONING.
ARISTOTLE – 350 BC
GEORGE BOOLE – 1850
LEWIS CARROL - 1880
GRIEDRICH FREGE – 1900
KURT GODEL - 1930
HILARY PUTNAM - 1959
ALAN ROBINSON – 1965
Amazing property:
X entails Y
just in case
X & not Y is unsatisfiable!
WHY???????
Two basic deductive reasoning problems:
Is a sentence unsatisfiable? == theorem proving
Can a sentence be satisfied? == constraint satisfaction
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How can we REASON deductively? Eg: A entails B ?
1. Model enumeration (dumb!) - explicitly check Venn diagram!
for (m in truth assignments){
if (m makes A true && m makes B false) return “NO!”
}
return “yes!”
2. Heuristic search through space of PARTIAL truth assignments – Davis-Putnam-Loveland-Logmann
3. Heuristic search through space of TOTAL truth assignments - Walksat
4. Manipulate sentences - Resolution
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HOW HARD IS DEDUCTIVE REASONING?
First-order logic: add unbounded quantification (allows infinite sets)
Second-order logic: quantify over predicate, not just arguments (allows sets of sets)
Logic |
Unsat (theorem proving) |
Sat (constraint satisfaction) |
propositional logic |
co-NP-complete |
NP-complete |
prop logic + schemas |
|
|
first-order logic |
decidable |
undecidable |
higher-order logic |
undecidable |
undecidable |