ies 503 1 knowledge and reasoning – second part knowledge representation logic and representation...

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Knowledge and reasoning – second part

• Knowledge representation• Logic and representation• Propositional (Boolean) logic• Normal forms• Inference in propositional logic• Wumpus world example

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Knowledge-Based Agent

• Agent that uses prior or acquired knowledge to achieve its goals• Can make more efficient decisions• Can make informed decisions

• Knowledge Base (KB): contains a set of representations of facts about the Agent’s environment

• Each representation is called a sentence

• Use some knowledge representation language, to TELL it what to know e.g., (temperature 72F)

• ASK agent to query what to do• Agent can use inference to deduce

new facts from TELLed facts

Knowledge Base

Inference engine

Domain independent algorithms

Domain specific content

TELL

ASK

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Generic knowledge-based agent

1. TELL KB what was perceivedUses a KRL to insert new sentences, representations of facts, into KB

2. ASK KB what to do.Uses logical reasoning to examine actions and select best.

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Wumpus world example

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Wumpus world characterization

• Deterministic?

• Accessible?

• Static?

• Discrete?

• Episodic?

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Wumpus world characterization

• Deterministic? Yes – outcome exactly specified.

• Accessible? No – only local perception.

• Static? Yes – Wumpus and pits do not move.

• Discrete? Yes

• Episodic? (Yes) – because static.

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Exploring a Wumpus world

A= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG = Glitter

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Other tight spots

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Another example solution

No perception 1,2 and 2,1 OK

Move to 2,1

B in 2,1 2,2 or 3,1 P?

1,1 V no P in 1,1

Move to 1,2 (only option)

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Example solution

S and No S when in 2,1 1,3 or 1,2 has W

1,2 OK 1,3 W

No B in 1,2 2,2 OK & 3,1 P

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Logic in general

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Types of logic

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The Semantic Wall

Physical Symbol System World

+BLOCKA+

+BLOCKB+

+BLOCKC+

P1:(IS_ON +BLOCKA+ +BLOCKB+)P2:((IS_RED +BLOCKA+)

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Truth depends on Interpretation

Representation 1 World

A

BON(A,B) TON(A,B) F

ON(A,B) F A

ON(A,B) T B

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Entailment

Entailment is different than inference

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Logic as a representation of the World

FactsWorld Factfollows

Refers to (Semantics)

Representation: Sentences Sentenceentails

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Models

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Inference

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Basic symbols

• Expressions only evaluate to either “true” or “false.”

• P “P is true”• ¬P “P is false” negation• P V Q “either P is true or Q is true or both” disjunction• P ^ Q “both P and Q are true” conjunction• P => Q “if P is true, the Q is true” implication• P Q “P and Q are either both true or both false”

equivalence

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Propositional logic: syntax

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Propositional logic: semantics

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Truth tables

• Truth value: whether a statement is true or false.• Truth table: complete list of truth values for a statement

given all possible values of the individual atomic expressions.

Example:

P Q P V QT T TT F TF T TF F F

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Truth tables for basic connectives

P Q ¬P ¬Q P V Q P ^ Q P=>Q PQ

T T F F T T T TT F F T T F F FF T T F T F T FF F T T F F T T

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Propositional logic: basic manipulation rules

• ¬(¬A) = A Double negation

• ¬(A ^ B) = (¬A) V (¬B) Negated “and”• ¬(A V B) = (¬A) ^ (¬B) Negated “or”

• A ^ (B V C) = (A ^ B) V (A ^ C) Distributivity of ^ on V• A => B = (¬A) V B by definition• ¬(A => B) = A ^ (¬B) using negated or• A B = (A => B) ^ (B => A) by definition• ¬(A B) = (A ^ (¬B))V(B ^ (¬A)) using negated and & or• …

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Propositional inference: enumeration method

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Enumeration: Solution

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Propositional inference: normal forms

“sum of products of simple variables ornegated simple variables”

“product of sums of simple variables ornegated simple variables”

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Deriving expressions from functions

• Given a boolean function in truth table form, find a propositional logic expression for it that uses only V, ^ and ¬.

• Idea: We can easily do it by disjoining the “T” rows of the truth table.

Example: XOR function

P Q RESULTT T FT F T P ^ (¬Q)F T T (¬P) ^ QF F F

RESULT = (P ^ (¬Q)) V ((¬P) ^ Q)

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A more formal approach

• To construct a logical expression in disjunctive normal form from a truth table:

- Build a “minterm” for each row of the table, where:

- For each variable whose value is T in that row, include

the variable in the minterm

- For each variable whose value is F in that row, include

the negation of the variable in the minterm

- Link variables in minterm by conjunctions

- The expression consists of the disjunction of all minterms.

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Example: adder with carry

Takes 3 variables in: x, y and ci (carry-in); yields 2 results: sum (s) and carry-out (co). To get you used to other notations, here we assume T = 1, F = 0, V = OR, ^ = AND, ¬ = NOT.

co is:

s is:

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Tautologies

• Logical expressions that are always true. Can be simplified out.

Examples:

TT V AA V (¬A)¬(A ^ (¬A))A A((P V Q) P) V (¬P ^ Q)(P Q) => (P => Q)

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Validity and satisfiability

Theorem

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Proof methods

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Inference Rules

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Inference Rules

Resolution

Conjunctive Normal Form (CNF) conjunction of disjunctions of literals clauses

E.g., (A B) (B C D)• Resolution inference rule (for CNF):

li … lk, m1 … mn

li … li-1 li+1 … lk m1 … mj-1 mj+1 ... mn

where li and mj are complementary literals.

E.g., P1,3 P2,2, P2,2

P1,3

• Resolution is sound and complete for propositional logic

Conversion to CNF

B1,1 (P1,2 P2,1)

1) Eliminate , replacing α β with (α β)(β α).(B1,1 (P1,2 P2,1)) ((P1,2 P2,1) B1,1)

2) Eliminate , replacing α β with α β.(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)

3) Move inwards using de Morgan's rules and double-negation:

(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)

4) Apply distributivity law ( over ) and flatten:(B1,1 P1,2 P2,1) (P1,2 B1,1) (P2,1 B1,1)

Resolution algorithm

• Proof by contradiction, i.e., show KBα unsatisfiable

Resolution example

• KB = (B1,1 (P1,2 P2,1)) B1,1 α = P1,2

•

Forward and backward chaining

• Horn Form (restricted)KB = conjunction of Horn clauses

• Horn clause = • proposition symbol; or• (conjunction of symbols) symbol

• E.g., C (B A) (C D B)• Modus Ponens (for Horn Form): complete for Horn KBs

α1, … ,αn α1 … αn ββ

• Can be used with forward chaining or backward chaining.• These algorithms are very natural and run in linear time•

•

•

Forward chaining

• Idea: fire any rule whose premises are satisfied in the KB,• add its conclusion to the KB, until query is found

Forward chaining algorithm

• Forward chaining is sound and complete for Horn KB

Forward chaining example

Proof of completeness

• FC derives every atomic sentence that is entailed by KB

1. FC reaches a fixed point where no new atomic sentences are derived

2. Consider the final state as a model m, assigning true/false to symbols

3. Every clause in the original KB is true in m a1 … ak b

4. Hence m is a model of KB5. If KB╞ q, q is true in every model of KB,

including m•

Backward chaining

Idea: work backwards from the query q:check if q is known already, orprove by BC all premises of some rule concluding q

Avoid loops: check if new subgoal is already on the goal stack

Avoid repeated work: check if new subgoal1. has already been proved true, or2. has already failed

•

• to prove q by BC,

Backward chaining example

Forward vs. backward chaining

• FC is data-driven, automatic, unconscious processing,• e.g., object recognition, routine decisions

• May do lots of work that is irrelevant to the goal

• BC is goal-driven, appropriate for problem-solving,• e.g., Where are my keys? How do I get into a PhD

program?

• Complexity of BC can be much less than linear in size of KB

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Efficient propositional inference

Two families of efficient algorithms for propositional inference:

Complete backtracking search algorithms• DPLL algorithm (Davis, Putnam, Logemann,

Loveland)

• Incomplete local search algorithms• WalkSAT algorithm

•

The DPLL algorithm

Determine if an propositional logic sentence (in CNF) is satisfiable.

Improvements over truth table enumeration:1. Early termination

A clause is true if any literal is true.A sentence is false if any clause is false.

2. Pure symbol heuristicPure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A B), (B C), (C A), A and B are pure,

C is impure. Make a pure symbol literal true.

3. Unit clause heuristicUnit clause: only one literal in the clauseThe only literal in a unit clause must be true.

•

The DPLL algorithm

The WalkSAT algorithm

• Incomplete, local search algorithm

• Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses

• Balance between greediness and randomness

The WalkSAT algorithm

Inference-based agents in the wumpus world

A wumpus-world agent using propositional logic:

P1,1

W1,1

Bx,y (Px,y+1 Px,y-1 Px+1,y Px-1,y)

Sx,y (Wx,y+1 Wx,y-1 Wx+1,y Wx-1,y)

W1,1 W1,2 … W4,4

W1,1 W1,2

W1,1 W1,3 …

64 distinct proposition symbols, 155 sentences•

•

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Wumpus world: example

• Facts: Percepts inject (TELL) facts into the KB• [stench at 1,1 and 2,1] S1,1 ; S2,1

• Rules: if square has no stench then neither the square or adjacent square contain the wumpus• R1: !S1,1 !W1,1 !W1,2 !W2,1

• R2: !S2,1 !W1,1 !W2,1 !W2,2 !W3,1

• …

• Inference: • KB contains !S1,1 then using Modus Ponens we infer

!W1,1 !W1,2 !W2,1

• Using And-Elimination we get: !W1,1 !W1,2 !W2,1• …

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Limitations of Propositional Logic

1. It is too weak, i.e., has very limited expressiveness:

• Each rule has to be represented for each situation:e.g., “don’t go forward if the wumpus is in front of you” takes 64 rules

2. It cannot keep track of changes:• If one needs to track changes, e.g., where the agent has

been before then we need a timed-version of each rule. To track 100 steps we’ll then need 6400 rules for the previous example.

Its hard to write and maintain such a huge rule-base

Inference becomes intractable

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Summary

ies 503 1 knowledge and reasoning – second part knowledge representation logic and representation...

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