(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Reasoning/Inference

Given a set of facts/beliefs/rules/evidence Evaluate a given statement

Determine the truth of a statement Determine the probability of a statement

Find a statement that satisfies a set of constraints SAT

Find a statement that optimizes a set of constraints MAX-SAT (Assignment that maximizes the number of

satisfied constraints.) Most probable explanation (MPE) (Setting of hidden

variables that best explains observations.)

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Examples of Reasoning Problems

Evaluate a given statement Chess: status(position,LOST)? Backgammon: Pr(game-is-lost)?

Find a satisfying assignment Chess: Find a sequence of moves that will win the

game Optimize

Backgammon: Find the move that is most likely to win

Medical Diagnosis: Find the most likely disease of the patient

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Facts, Beliefs, Evidence must be

represented somehow

Propositional Logic Statements about a fixed, finite number of objects

First-Order Logic Statements about a variable, possibly-infinite, set of

objects and relations among them Probabilistic Propositional Logic

Statements of probability over the rows of the truth table

Probabilistic First-Order Logic Statements of probability over the possible models

of the axioms

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Propositional Logic

Sentence ::= AtomicSentence | ComplexSentence

AtomicSentence ::= True | False | symbolSymbol ::= P | Q | R | …ComplexSentence ::= : Sentence

| (Sentence Æ Sentence)| (Sentence Ç Sentence)| (Sentence ) Sentence)| (Sentence , Sentence)

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Application: WUMPUS

Maze of caves A WUMPUS is in one of the caves Some of the caves have pits One of the caves has gold Agent has an arrow

Performance measure: +1000 for picking up the goal; –1000 for being eaten by the WUMPUS or falling into a pit; –1 for each action; –10 for shooting the arrow

Actions: forward, turn left, turn right, shoot arrow, grab gold

Sensors: Stench (cave containing WUMPUS and its

four neighbors) Breeze (cave containing pit and its four

neighbors) Glitter (cave containing gold) Scream (if arrow kills WUMPUS) Bump (if agent hits wall)

Exactly one WUMPUS in cave choosen uniformly at random (except for Start state)

Each cave has probability 0.2 of pit

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Inference from Sensors

Reasoning problem: Given sensors, what can we infer about the state of the world?

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Some Sentences

There is no wumpus in 1,1: : W1,1

If there is a wumpus in 2,2, then there is a stench in 1,2 and 2,3 and 3,2 and 2,1 W2,2 ) S1,2 Æ S2,3 Æ S3,2 Æ S2,1

There is gold in 3,3 iff there is glitter in 3,3: Go3,3 , Gl3,3

There is only one wumpus: W1,1 Ç W1,2 Ç … Ç W4,4 W1,1 ) : W1,2 Æ : W1,3 Æ … Æ : W4,4 W1,2 ) : W1,1 Æ : W1,3 Æ … Æ : W4,4

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Sensor Readings = Sentences

Starting state: no glitter, no stench, no breeze : Gl1,1

: S1,1

: B1,1

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Is there a WUMPUS in 2,1?

Logical Reasoning allows us to draw inferences: : B1,1

W1,2 ) B1,1 Æ B2,2 Æ B1,3

These imply (by the rule of “deny consequent”) : W1,2

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Rules of Logical Inference

Modus Ponens Given: ) and Conclude:

Deny Consequent Given: ) and : Conclude: :

AND Elimination Given: Æ Conclude:

Deny Disjunct Given: Ç and : Conclude

Resolution Given: Ç and : Ç Conclude: Ç

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Resolution: A useful inference rule

for computation

Convert all statements to conjunctive normal form (CNF) ) becomes {: Ç } Æ becomes {}, {} , becomes {: Ç }, { Ç : }

Negate query Apply resolution to search for the empty clause

(contradiction). Useful primarily for First-Order Logic (see

below)

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Satisfiability

Given a set of sentences, is there a satisfying assignment of True and False to each proposition symbol that makes the sentences true?

To decide if is true given , we check if Æ is satisfiable

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Complete Inference Procedure

The Davis-Putnam algorithm is a complete inference procedure for propositional logic If there exists a satisfying assignment, it will

find it. Can be very efficient. But can be very slow,

too. SAT is NP-Complete

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Incomplete Inference Procedure

WALKSAT is a stochastic procedure (a form of stochastic hill climbing) that finds a satisfying assignment rapidly but only with high probability

Can be extended to handle MAX-SAT problems where the goal is to find an assignment that maximizes the number of satisfied clauses

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI First-Order Logic

Propositional Logic requires that the number of objects in the world be fixed so that we can give each one a name: W1,1, W1,2, … B1,1, B1,2, … G1,1, G1,2, …

This does not scale to worlds of variable or unknown size

It is also very tedious to write down all of the clauses describing the Wumpus world

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI First-Order Logic permits variables

that range over objects

Sentence ::= AtomicSentence | Sentence Connective Sentence| Quantifier Variable, … Sentence| : Sentence| (Sentence)

AtomicSentence ::= Predicate(Term),…) | Term = term

Term ::= Function(Term,…)| Constant| Variable

Connective ::= ) | Æ | Ç | ,Quantifier ::= 8 | 9Constant ::= A | X1 | JohnVariable ::= a | x | sPredicate ::= Before | HasColor | Raining | …Function ::= Mother | LeftLegOf | …

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Compact Description of

Wumpus Odor

If a wumpus is in a cave, then all adjacent caves are smelly 8 ℓ1, ℓ2 At(Wumpus, ℓ1) Æ Adjacent(ℓ1, ℓ2) )

Smelly(ℓ2)

Compare propositional logic: W2,2 ) S2,1 Æ S2,3 Æ S1,2 Æ S3,2

(and 15 similar sentences in the 4x4 world)

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Definition of Adjacent

8 ℓ1, ℓ2 Adjacent(ℓ1, ℓ2) ,

(row(ℓ1) = row(ℓ2) Æ

(col(ℓ1) = col(ℓ2) + 1 Ç

col(ℓ1) = col(ℓ2) – 1)) Ç

(col(ℓ1) = col(ℓ2) Æ

(row(ℓ1) = row(ℓ2) + 1 Ç

row(ℓ1) = row(ℓ2) – 1))

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Inference Rules for Quantifiers

Universal Elimination: Given 8 x Conclude SUBST({x/g},)

[g must be term that does not contain variables]

Example: Given 8 x Likes(x,IceCream) Conclude Likes(Ben,IceCream)SUBST(x/Ben, Likes(x,IceCream)) ´

Likes(Ben,IceCream) Many other rules…

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Unification

Unification is a pattern matching operation that finds a substitution that makes two sentences match: UNIFY(p,q) = iff SUBST(,p) = SUBST(,q)

Example: UNIFY(Knows(John,x), Knows(John,Jane)) =

{x/Jane} UNIFY(Knows(John,x), Knows(y,Mother(y))) =

{y/John,x/Mother(John)}

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI First-Order Resolution

Given: Ç , : Ç and UNIFY(,)= Conclude: SUBST(, Ç )

Resolution is a refutation complete inference procedure for First-Order Logic If a set of sentences contains a contradiction, then a

finite sequence of resolutions will prove this. If not, resolution may loop forever (“semi-

decidable”)

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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AI Summary

Propositional logic finite worlds logical entailment is decidable Davis-Putnam is complete inference

procedure First-Order logic

infinite worlds logical entailment is semi-decidable Resolution procedure is refutation complete