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Reasoning/Inference - PowerPoint PPT Presentation

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

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• 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

• 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

• 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

Sentence ::= AtomicSentence | ComplexSentence

AtomicSentence ::= True | False | symbol

Symbol ::= P | Q | R | …

ComplexSentence ::= :Sentence

| (SentenceÆSentence)

| (SentenceÇSentence)

| (Sentence)Sentence)

| (Sentence,Sentence)

(c) 2003 Thomas G. Dietterich and Devika Subramanian

• 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

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

(c) 2003 Thomas G. Dietterich and Devika Subramanian

• 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) S1,2Æ S2,3Æ S3,2Æ S2,1

• There is gold in 3,3 iff there is glitter in 3,3:

• Go3,3, Gl3,3

• There is only one wumpus:

• W1,1Ç W1,2Ç … Ç W4,4

• W1,1): W1,2Æ: W1,3Æ … Æ: W4,4

• W1,2): W1,1Æ: W1,3Æ … Æ: W4,4

(c) 2003 Thomas G. Dietterich and Devika Subramanian

• Starting state: no glitter, no stench, no breeze

• : Gl1,1

• : S1,1

• : B1,1

(c) 2003 Thomas G. Dietterich and Devika Subramanian

• Logical Reasoning allows us to draw inferences:

• : B1,1

• W1,2) B1,1Æ B2,2Æ B1,3

• These imply (by the rule of “deny consequent”)

• : W1,2

(c) 2003 Thomas G. Dietterich and Devika Subramanian

Given: ) and 

Conclude: 

Deny Consequent

Given: ) and :

Conclude: :

AND Elimination

Given: Æ

Conclude: 

Deny Disjunct

Given: Ç and :

Conclude 

Rules of Logical Inference

• Resolution

• Given: Ç and :Ç

• Conclude: Ç

(c) 2003 Thomas G. Dietterich and Devika Subramanian

• 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

• 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

• 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

• 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

• 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

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| 9

Constant ::= A | X1 | John

Variable ::= a | x | s

Predicate ::= Before | HasColor | Raining | …

Function ::= Mother | LeftLegOf | …

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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) S2,1Æ S2,3Æ S1,2Æ S3,2

• (and 15 similar sentences in the 4x4 world)

(c) 2003 Thomas G. Dietterich and Devika Subramanian

• 8ℓ1, 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

• 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

• 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

• 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

• 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

(c) 2003 Thomas G. Dietterich and Devika Subramanian