knowledge and reasoning – second part knowledge representation logic and representation...

Knowledge and reasoning – second part

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

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

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.

Wumpus world example

Wumpus world characterization

• Deterministic?

• Accessible?

• Static?

• Discrete?

• Episodic?

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.

Exploring a Wumpus world

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

Exploring a Wumpus world

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

Exploring a Wumpus world

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

Exploring a Wumpus world

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

Exploring a Wumpus world

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

Exploring a Wumpus world

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

Exploring a Wumpus world

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

Exploring a Wumpus world

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

Other tight spots

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)

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

Logic in general

Types of logic

The Semantic Wall

Physical Symbol System World

+BLOCKA+

+BLOCKB+

+BLOCKC+

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

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

Entailment

Entailment is different than inference

Logic as a representation of the World

FactsWorld Factfollows

Refers to (Semantics)

Representation: Sentences Sentenceentails

Models

Inference

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

Propositional logic: syntax

Propositional logic: semantics

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

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

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• …

Propositional inference: enumeration method

Enumeration: Solution

Propositional inference: normal forms

“sum of products of simple variables ornegated simple variables”

“product of sums of simple variables ornegated simple variables”

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)

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.

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:

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)

Validity and satisfiability

Theorem

Proof methods

Inference Rules

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

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

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

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

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

-

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

•

•

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• …

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

Summary