Reasoning inference
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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

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Reasoning/Inference

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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.)

(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


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

  • 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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