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