Reasoning inference
This presentation is the property of its rightful owner.
Sponsored Links
1 / 22

Reasoning/Inference PowerPoint PPT Presentation


  • 71 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Reasoning/Inference

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Reasoning inference

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


Examples of reasoning problems

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

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

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

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

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

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

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

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


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 

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

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

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

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

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

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

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

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

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

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

  • 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

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

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


  • Login