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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.)

- Evaluate a given statement

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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

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

Propositional Logic

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

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

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

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) S1,2Æ 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

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

Is there a WUMPUS in 2,1?

- 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

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

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

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

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

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

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| 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

Definition of Adjacent

- 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

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

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

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

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

(c) 2003 Thomas G. Dietterich and Devika Subramanian

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