1 problem solving cs 331 dr m m awais representational methods formal methods propositional logic...

CS 331 Dr M M Awais 1

Problem Solving

Representational Methods

Formal Methods

• Propositional Logic

• Predicate Logic


Problem Solving

Propositional Logic

• Represent FACTS only• Is declarative in nature• Can infer the truth value of fact only• Context free• Compositional: Can combine facts


Problem Solving

Propositional Logic

• Symbols P,Q,R,S ……..• Truth Symbols True (T), False (F)• Connectives

^ (and), v (or), (Implication), (Equality, equivalence) (not)

Statements (Propositions) could be true/falseFACTS are also called Atomic Proposition


Problem Solving

Truth TableP Q P Q P v Q P Q

T T F F T T

T F F T F F

F T T F T T

F F T T T T

Are same


Problem Solving

Possible Sentences (Well Formed Formulas-WFF)

• P ^ Q Conjuncts

• P v Q Disjuncts

• P Q P=Premise/Antecedent

Q=conclusion/Consequent P

• (P v Q) = ( P Q) Inter conversion


Problem Solving

Propositional logic: Syntax

• Propositional logic is the simplest logic – illustrates basic ideas

• The proposition symbols P1, P2 etc are sentences

– If S is a sentence, S is a sentence (negation)– If S1 and S2 are sentences, S1 S2 is a sentence (conjunction)– If S1 and S2 are sentences, S1 S2 is a sentence (disjunction)– If S1 and S2 are sentences, S1 S2 is a sentence (implication)– If S1 and S2 are sentences, S1 S2 is a sentence (biconditional)


Problem Solving

Propositional logic: SemanticsEach model specifies true/false for each proposition symbol

E.g. P1,2 P2,2 P3,1

false true false

With these symbols, 8 possible models, can be enumerated automatically.Rules for evaluating truth with respect to a model m:

S is true iff S is false S1 S2 is true iff S1 is true and S2 is trueS1 S2 is true iff S1is true or S2 is trueS1 S2 is true iff S1 is false or S2 is true i.e., is false iff S1 is true and S2 is falseS1 S2 is true iff S1S2 is true and S2S1 is true

Simple recursive process evaluates an arbitrary sentence, e.g.,

P1,2 (P2,2 P3,1) = true (true false) = true true = true

•

•


Problem Solving

Laws of Propositional Expressions

• Demorgan’s Law (P v Q)=( P ^ Q)

• Distributive Law P v (Q ^ R) = (P v Q) ^ (P v R)

• Commutative Law P ^ Q= Q ^ P

• Associative Law (P ^ Q) ^ R=P ^ (Q ^ R)

• Contrapositive Law P Q= Q P


Problem Solving

Inferencing

• If simple facts are known to be true one can find the truth value for the expressions

• Thus INTERPRETATIONS can be done.

• Interpretation is the assignment of truth values to the sentences

Symbols T/F

Mapping


Problem Solving

Expressions for KR

• Fact 1: Ali likes cars P

• Fact 2: Ali drives cars Q

• P v Q : Ali likes cars or drives cars

• P ^ Q : Ali likes cars and drives cars Q : Ali does not like cars

• P Q: If Ali likes cars then he drives cars

• P Q: ?????– Above and vice versa


Problem Solving

Implicative Statements

• P Q: If Ali likes cars then he drives cars

• If P is FALSE and Q is TRUE, it means:

• ALI DOESNOT LIKE CARS BUT STILL HE DRIVES THEM

• Truth Table:

P Q P Q

T T T

F F T (of our interest)

T F F

F T T (bizarre)


Problem Solving

Model

• Model:If a sentence “S” is satisfied under a given interpretation “I” for all possible variables bindings (values) then I is a model of S

• Satisfy: If S maps to T (true) under an interpretation I then I satisfy S (interpretation that makes the sentence true)


Problem Solving

Definitions

• Logically Follows: X logically follows from a set of predicate calculus expressions S if every interpretation that satisfy S also satisfy X (X F S)

• Satisfiable: S is satisfiable iff there exists an interpretation and variable assignments that satisfy it


Problem Solving

S F XS: All birds fly

S: Sparrow is bird

X: Sparrow fliesAll humans are mortal

Shahid is a human

Shahid is mortal

All students at LUMS are hardworking

Farooq is a student at LUMS

Farooq is a hard working student


Problem Solving

Definitions• Inconsistent: If a set of expressions are not satisfiable

• Valid: If any expression has a value T for all possible interpertations

• Sound: If an expression logically follows from another expression then the inferential rule is sound

• Complete: When inferential rule produces every expression that logically follows a particular expression


Problem Solving

Logic in general

• Logics are formal languages for representing information such that conclusions can be drawn

• Syntax defines the sentences in the language

• Semantics define the "meaning" of sentences;– i.e., define truth of a sentence in a world


Problem Solving

Validity and satisfiability

VALIDA sentence is valid if it is true in all models,

e.g., True, A A, A A, (A (A B)) B

SATISFIABLEA sentence is satisfiable if it is true in some model

e.g., A B

UNSATISFIABLEA sentence is unsatisfiable if it is true in no models

e.g., A A

–


Problem Solving

Example: Wumpus World• Performance measure

– gold +1000, death -1000– -1 per step, -10 for using the arrow

• Environment– Squares adjacent to wumpus are smelly– Squares adjacent to pit are breezy– Glitter iff gold is in the same square– Shooting kills wumpus if you are facing it– Shooting uses up the only arrow– Grabbing picks up gold if in same square– Releasing drops the gold in same square

• Sensors: Stench, Breeze, Glitter, Bump, Scream• Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot

•–––––

–•


Problem Solving

Wumpus world characterization

• Fully Observable • Deterministic • Episodic • Static • Discrete • Single-agent?

•••••

•


Problem Solving

Wumpus world characterization

• Fully Observable No – only local perception• Deterministic Yes – outcomes exactly specified• Episodic No – sequential at the level of actions• Static Yes – Wumpus and Pits do not move• Discrete Yes• Single-agent? Yes – Wumpus is essentially a

natural feature

•

•

•

•

•

•


Problem Solving

Exploring a wumpus world


Problem Solving



Problem Solving

Finding wumpus world

W

4

3

2

1

1 2 3 4

If Wumpus is in [1,3],

How can we prove it using Propositional logic


Problem Solving


W

4

3

2

1

1 2 3 4

Knowledge base Contains:~S1,1 ~B1,1~S2,1 B2,1 S1,2 B1,2


Problem Solving


W

4

3

2

1

1 2 3 4

Rules:R1: ~ S1,1 ~W1,1 ^ ~W1,2 ^ ~W2,1

R2: ~ S2,1 ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1

R3: S1,2 W1,3 V W1,2

V W2,2 V

~W1,1

Assumption: The square with wumpus is also smelly

s


Problem Solving

Inference rules in PL

• Modus Ponens

• And-elimination: from a conjuction any conjunction can be inferred:

• Resolution:

Unit resolution when

l3 is empty

,


Problem Solving

Proof of W presenceW

4

3

2

11 2 3 4

Knowledge base Contains:~ S1,1 ~B1,1~S2,1 B2,1 S1,2 B1,2Rules:R1: ~ S1,1 ~W1,1 ^ ~W1,2 ^ ~W2,1

R2: ~ S2,1 ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1 R3: S1,2 W1,3

V W1,2 V W2,2

V ~W1,1

Modus Ponen on R1: ~ S1,1 ~W1,1 ^ ~W1,2 ^ ~W2,1 ~W1,1 ^ ~W1,2 ^ ~W2,1

And Elimination:

~W1,1 ~W1,2

~W2,1

Results:

~W1,1 ~W1,2

~W2,1


Problem Solving


4

3

2

11 2 3 4


R2: ~ S2,1 ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1 R3: S1,2 W1,3

V W1,2 V W2,2

V ~W1,1

Modus Ponen on R2: ~ S2,1 ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1

~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1

And Elimination:~W1,1

~W2,1

~W2,2

~W3,1

Results:

~W1,1 ~W1,2

~W2,1

~W1,1

~W2,1

~W2,2

~W3,1


Problem Solving


4

3

2

11 2 3 4


R2: ~ S2,1 ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1 R3: S1,2 W1,3

V W1,2 V W2,2

V W1,1 Modus Ponen on R3: S1,2 W1,3 V W1,2 V W2,2

V W1,1

W1,3 V W1,2 V W2,2 V W1,1

Unit Resolution:Alpha:W1,3 V W1,2 V W2,2 and Beta: W1,1 W1,3 V W1,2 V W2,2

Unit Resolution again with : W1,2 and W2,2

W1,3

Results:

~W1,1 ~W1,2

~W2,1

~W1,1~W2,1~W2,2 ~W3,1

W1,3

Simply drop all facts that can create conflict


Problem Solving

Pros and cons of propositional logic

Propositional logic is declarative Propositional logic allows partial/disjunctive/negated

information– (unlike most data structures and databases)

Propositional logic is compositional:– meaning of (B P ) is derived from meaning of B and of P

Meaning in propositional logic is context-independent– (unlike natural language, where meaning depends on context)

Propositional logic has very limited expressive power– (unlike natural language)– E.g., cannot say “Music cause disturbance to all neighbors“

• except by writing one sentence for each neighbor

•

1 problem solving cs 331 dr m m awais representational methods formal methods propositional logic...

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