Artificial Intelligence

Artificial IntelligenceLecture 2: Propositional Logic 2 solve Problems

Problem I - A Story

❑ You roommate comes home; he/she is completely wet

❑ You know the following things:○ Your roommate is wet○ If your roommate is wet, it is because of rain, sprinklers, or both○ If your roommate is wet because of sprinklers, the sprinklers must be on○ If your roommate is wet because of rain, your roommate must not be carrying

the umbrella○ The umbrella is not in the umbrella holder○ If the umbrella is not in the umbrella holder, either you must be carrying the

umbrella, or your roommate must be carrying it○ You are not carrying the umbrella

❑ Can you conclude that the sprinklers are on?

❑ Can AI conclude that the sprinklers are on?

Problem II- Are you ‘Sherlock Holmes’?

❑ There was a robbery in which a lot of goods were stolen. The robber(s) left in a truck. It is known that :

(1) Nobody else could have been involved other than A, B and C.

(2) C never commits a crime without A's participation.

(3) B does not know how to drive.

❑ Is A innocent or guilty?

Problem III- Knights and Knaves

❑ A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie.

❑ You meet two inhabitants: Zoey and Mel.

○ Zoey tells you that Mel is a knave.

○ Mel says ‘Neither Zoey nor I are knaves’.

❑ Can you determine what are they?

❑ who is a knight and who is a knave?

Logics

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

❑ Syntax defines the sentences in the language.

❑ Semantics defines the "meaning" of sentences.

o defines the truth of each sentence w.r.t. each possible world (model)

❑ E.g., the language of arithmetic

o x+2 ≥ y is a sentence; x2+y > {} is not a sentence

o x+2 ≥ y is true iff the number x+2 is no less than the number y

o Sentence x+2 ≥ y is true in a world where x = 7, y = 1

o Sentence x+2 ≥ y is false in a world where x = 0, y = 6

Propositional logic: Syntax

❑ Propositional logic is the simplest logic and illustrates basic ideas of logic

❑ Logical constants

–True, false

❑ Atomic sentence

–A proposition symbol(P, Q, S, ...) representing a true or false statement

❑ The proposition symbols P1, P2,…are sentences (formulae)

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)

Examples of PL sentences❑ “If it is hot and humid, then it is raining”

(P Q) → R

❑ “If it is humid, then it is hot”

Q → P

❑ “It is humid.”

Q

❑ We’re free to choose better symbols, btw:Ho = “It is hot”

Hu = “It is humid”

R = “It is raining”

Propositional logic: Semantics❑ Each model specifies true/false values for each proposition symbol (8

possible models)E.g. P1,2 P2,2 P3,1

false true false

❑ Rules for evaluating truth with respect to an interpretation m:

S is true iff S is false

S1 S2 is true iff S1 is true and S2 is true

S1 S2 is true iff S1 is true or S2 is true

S1 S2 is true iff S1 is false or S2 is true

S1 S2 is false iff S1 is true and S2 is false

S1 S2 is true iff S1S2 is true and S2S1 is true

❑ Simple recursive process evaluates an arbitrary sentence w.r.t. an interpretation, e.g.,

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

Truth tables for connectives

Some Terms

❑ The meaning or semantics of a sentence determines its interpretation.

❑ Given the truth values of all symbols in a sentence, it can be “evaluated” to determine its truth value (True or False).

❑ A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True.

Model for a KBLet the KB be [PQ→R, Q → P]

What are the possible models? Consider all possible assignments of T|F to P, Q and R and check truth tables

FFF: OK

FFT: OK

FTF: NO

FTT: NO

TFF: OK

TFT: OK

TTF: NO

TTT: OK

If KB is [PQ→R, Q → P, Q], then the only model is TTT

P: it’s hotQ: it’s humid R: it’s raining

Validity and satisfiability

❑ A sentence is valid or tautology if it is true under all interpretations, no matter what the world is actually like or what the semantics is. ❑ Example: “It’s raining or it’s not raining.”, True, A A, A A, (A (A B)) B

❑ An inconsistent sentence (contradiction, or unsatisfiable) is a sentence that is False under all interpretations. The world is never like what it describes❑ Example : “It’s raining and it’s not raining.”, A A

❑A sentence is satisfiable if it is true in some model❑ Example: A B, C

Quiz

List the models for the following formulas:

1. A ∧ ¬B

2. (A ∧ B) ∨ (B ∧ C)

3. A ∨ B → C

4. ¬A ↔ B ↔ C

Knowledge base for the story

❑ RoommateWet

❑ RoommateWet➔(RoommateWetBecauseOfRain ∨RoommateWetBecauseOfSprinklers)

❑ RoommateWetBecauseOfSprinklers➔SprinklersOn

❑ RoommateWetBecauseOfRain➔¬(RoommateCarryingUmbrella)

❑ UmbrellaGone

❑ UmbrellaGone➔(YouCarryingUmbrella ∨ RoommateCarryingUmbrella)

❑ ¬ (YouCarryingUmbrella)

Knowledge base for the story

❑ W

❑ W →(R ∨ S)

❑ S → O

❑ R → ¬(C)

❑ G

❑ G → (Y ∨ C)

❑ ¬(Y)

Entailment

❑ Entailment means that one thing follows from another

KB╞ α

❑ Knowledge base KB entails sentence α if and only if α is true in all worldswhere KB is true

❑ E.g., the KB containing “Baramkeh is in Damascus” and “Damascus is in Syria” entails “Baramkeh is in Syria”

❑ E.g., the KB containing x+y = 4 entails 4 = x+y

❑ Entailment is a relationship between sentences (i.e., syntax) that is based on semantics

Exercise: Prove using truth tables, the following deductions

1. Double negative elimination

¬¬P ⊨P

2.Conjunction introduction/ elimination

(a) {P, Q} ⊨P∧Q; (b) P ∧ Q ⊨P; (c) P ∧ Q ⊨Q.

3.Disjunction introduction/ elimination

(a) P ⊨ P ∨ Q; (b) Q ⊨P ∨ Q;

(c) {P ∨ Q, P → R, Q → R} ⊨R

4.Bi-conditional introduction/ elimination

(P → Q) ∧ (Q → P) ⊨(P ↔ Q)

5.De Morgan

(a) ¬(P ∧ Q) ⊨¬P ∨ ¬Q;

(b) ¬(P ∨ Q) ⊨¬P ∧ ¬Q

Proofs of the Deduction Rules

Logical equivalence

Two sentences are logically equivalent iff they are true in same models: α ≡ ß iff α╞ β and β╞ α

Quiz

Given that:P = (A ∨ B ) ∧ (¬C ∨ ¬D ∨ E)Q1 = A ∨ BQ2= (A ∨ B ∨ C) ∧ ((B ∧ C ∧ D) → E)Q3= (A ∨ B) ∧ (¬D ∨ E)

Does :1- P ⊨ Q1?2- P ⊨ Q2?3- P ⊨ Q3?

Inference

❑ KB ├i α : sentence α can be derived from KB by procedure i, i proves αLogical inference is used to create new sentences that logically follow from a given set of sentences (KB).

❑ Soundness: i is sound if whenever KB ├i α, it is also true that KB╞ α❑ An inference rule is sound if every sentence X produced by an inference rule

operating on a KB logically follows from the KB. (That is, the inference rule does not create any contradictions)

❑ Completeness: i is complete if whenever KB╞ α, it is also true that KB ├i α ❑ An inference rule is complete if it is able to produce every expression that logically

follows from (is entailed by) the KB

Sound rules of inference

❑Here are some examples of sound rules of inferenceA rule is sound if its conclusion is true whenever the premise is true

❑Each can be shown to be sound using a truth table

RULE PREMISE CONCLUSION

Modus Ponens A, A → B B

And Introduction A, B A B

And Elimination A B A

Double Negation A A

Unit Resolution A B, B A

Resolution A B, B C A C

Other Inference Rules

❑Modus Tollens

P →Q , Q

P

❑ Chaining (Syllogismal Hypothetic)

P →Q, Q →R

P →R

Example: Modus Ponens

❑ Latin for "mode that affirms"

❑Whenever sentences of form a=> b and a are given, the sentence b can be inferred

❑ R1: Green => Martian

❑ R2: Green

❑ Inferred: Martian

Example: And-Elimination

❑Any of conjuncts can be inferred

❑ R1: Martian ^ Green

❑ Inferred: Martian

❑ Inferred: Green

❑Use truth tables if you want to confirm inference rules

Soundness of modus ponens

A B A → B OK?

True True True

True False False

False True True

False False True

Soundness of modus Tollens(the way that denies by denying)

A B A → B B OK?

True True True

True False False

False True True

False False True

Soundness of the resolution inference rule

Resolution is not complete

❑ Not completeness of resolution inference rule:

P R ⊨ P R

But we cannot use resolution to prove P R from P R????

Proving things

❑ A proof is a sequence of sentences, where each sentence is either a premise or a sentence derived from earlier sentences in the proof by one of the rules of inference.

❑ The last sentence is the theorem (also called goal or query) that we want to prove.

❑ Example for the “weather problem” given above.

1. Humid Premise “It is humid”

2. Humid→Hot Premise “If it is humid, it is hot”

3. Hot Modus Ponens(1,2) “It is hot”

4. (HotHumid)→Rain Premise “If it’s hot & humid, it’s raining”

5. HotHumid And Introduction(1,2) “It is hot and humid”

6. Rain Modus Ponens(4,5) “It is raining”

Inference Rules: Resolution

❑ Conjunctive Normal Form (CNF)conjunction of disjunctions of literals, often called clausesE.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

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:(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)

Quiz: Conversion to CNF

1. (PQ)(RP)

2. (P Q)(RP)

3. (P R P) (Q R P)

4. (P R) (Q R P)

(P Q) (R P)

Using Inference Rules

1. It is not sunny this afternoon and it is colder than yesterday. 2. If we go swimming it is sunny. 3. If we do not go swimming then we will take a canoe trip. 4. If we take a canoe trip then we will be home by sunset.

We will be home by sunset ?

PropositionsP It is sunny this afternoonq it is colder than yesterdayR we go swimming S we will take a canoe tripT We will be home by sunset

Sentences Representations1. p q2. r → p3. r → s4. s → t

Using Inference Rules

Propositionsp it is sunny this afternoonq it is colder than yesterdayr we go swimming s we will take a canoe tript we will be home by sunset

We will be home by sunset ?

Sentences Representations1. p q2. r → p3. r → s4. s → t

Using the resolution rule (an example)

1. Anna is skiing or it is not snowing.

2. It is snowing or Bart is playing hockey.

Anna is skiing or Bart is playing hockey ??

Propositionsp Anna is skiing

q Bart is playing hockey

r it is snowing

Sentences Representations

1. p r

2. r qResolution Rule

p qAnna is skiing or Bart is playing hockey

Resolution Refutation Principle

❑ Resolution refutation proves a theorem by negating the statement to be proved and adding this negated goal to the set of axioms that are known to be true.

❑ Use the resolution rule of inference to show that this leads to a contradiction.

❑ Once the theorem prover shows that the negated goal is inconsistent with the given set of axioms, it follows that the original goal must be consistent.

Example (resolution refutation)

❑Using resolution refutation principle, show that C D is a logical consequence of:

S = {A B, A D, C B}

❑First, we add negation of logical consequence (i.e., (CVD) ≡ C D) to the set S.

❑Get S’={A B, A D, C B, C, D}.

❑Now show that S’ is unsatisfiable by deriving contradiction using resolution principle.

Example (resolution refutation)

Resolution Refutation Example

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

The moving robot example

Bat-ok(of robot) and Liftable(block) then Moves(robot-arm)

Bat_ok, Liftable→Moves

Moves, Bat_ok

Example

1. Battery-OK Bulbs-OK Headlights-Work

2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts

3. Engine-Starts Flat-Tire Car-OK

4. Headlights-Work

5. Battery-OK

6. Starter-OK

7. Empty-Gas-Tank

8. Car-OK

9. Flat-Tire

10. Starter-OK Empty-Gas-Tank Engine-Starts

11. Battery-OK Empty-Gas-Tank Engine-Starts

12. Battery-OK Starter-OK Engine-Starts

13. Engine-Starts Flat-Tire

14. Engine-Starts Car-OK

15. .....

Example…

Battery-OK

Starter-OK

Empty-Gas-Tank

Car-OK

Flat-Tire

…

Battery-OK Starter-OK Engine-Starts

Engine-Starts Flat-Tire

…

Battery-OK Starter-OK Flat-Tire

…

Starter-OK Flat-Tire

…

Flat-Tire

…

False (empty clause)

Resolution Heuristics

❑ Set-of-support heuristics:

❑ At least one ancestor of every inferred clause comes from a

❑ Shortest-clause heuristics:

❑Generate a clause with the fewest literals first

❑Unit Resolution

❑ Simplifications heuristics:

❑ Remove any clause containing two complementary literals (tautology)

❑ If a symbol always appears with the same “sign”, remove all the clauses that contain it (pure symbol)

Example (Set-of-Support)




4. Headlight-Work

5. Battery-OK

6. Starter-OK

7. Empty-Gas-Tank

8. Car-OK

9. Flat-Tire

Example (Set-of-Support)




4. Headlight-Work

5. Battery-OK

6. Starter-OK

7. Empty-Gas-Tank

8. Car-OK

9. Flat-Tire


11. Engine-Starts

12. Battery-OK Starter-OK Empty-Gas-Tank

13. Starter-OK Empty-Gas-Tank

14. Empty-Gas-Tank

15. False

Note the goal-directedflavor

Example (Shortest-Clause)




4. Headlight-Work

5. Battery-OK

6. Starter-OK

7. Empty-Gas-Tank

8. Car-OK

9. Flat-Tire

Example (Shortest-Clause)




4. Headlight-Work

5. Battery-OK

6. Starter-OK

7. Empty-Gas-Tank

8. Car-OK

9. Flat-Tire


11. Engine-Starts

12. Bulbs-OK Headlights-Work

13. Battery-OK Starter-OK Empty-Gas-Tank

14. Starter-OK Empty-Gas-Tank

15. Empty-Gas-Tank

16. False

Example (Pure Literal)




4. Headlights-Work

5. Battery-OK

6. Starter-OK

7. Empty-Gas-Tank

8. Car-OK

9. Flat-Tire

Horn sentences

❑A Horn sentence or Horn clause has the form:P1 P2 P3 ... Pn → Q

or alternativelyP1 P2 P3 ... Pn Q

where Ps and Q are non-negated atoms

❑To get a proof for Horn sentences, apply Modus Ponens repeatedly until nothing can be done

(P → Q) = (P Q)

Forward and backward chainingHorn 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 β

β

Can be used with forward chaining or backward chaining.

These algorithms are very natural and run in linear time

Forward and backward chaining

❑ Horn clauses: disjunction of literals of which at most one is positive

❑ Important because Horn clauses can be written as an implication whose premise is a conjuction of positive literals and whose conclusion is a single positive literal

❑ Definite clauses: exactly one positive literal

▪ Positive literal forms the head

▪ Negative literals form the body

Inference with Horn clauses can be done by forward or backward chaining in a time that is linear in the size of the KB.

Forward chaining

Fire any rule whose premises are satisfied in the KB

Add its conclusion to the KB, until query is found

Forward chaining example

Backward chaining

❑ Work backwards from the query q

❑To prove q by BC, ❑ check if q is known already, or

❑ prove by BC all premises of some rule concluding q

❑Avoid loops:

❑check if new sub-goal is already on the goal stack

❑Avoid repeated work: ❑ check if new sub-goal has already been proved true, or

❑ has already failed

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 (only relevant facts are considered)

Forward and backward chaining ❑ Forward chaining is often preferable in cases where there are many rules with the same

conclusions.

❑ Rule systems of taxonomic hierarchies: the taxonomy of the animal kingdom :

❑ animal(X) :- sponge(X).

❑ animal(X) :- arthopod(X).

❑ animal(X) :- vertebrate(X). ...

❑ vertebrate(X) :- fish(X).

❑ Now, suppose we have the fact "dog(fido)" and we query whether "animal(fido)".

❑ In forward chaining, we will successively add "carnivore(fido)", "mammal(fido)", "vertebrate(fido)", and "animal(fido)". The query will then succeed immediately. The total work is proportional to the height of the hierarchy.

❑ In backward chaining, the query " animal(fido)" will unify with the first rule, and generate the subquery " sponge(fido)", which will initiate a search for Fido through all the subdivisions of sponges, and so on. Ultimately, it searches the entire taxonomy of animals looking for Fido.

❑ vertebrate(X) :- mammal(X) ...

❑ mammal(X) :- carnivore(X) ...

❑ carnivore(X) :- dog(X).

❑ carnivore(X) :- cat(X). ...

Propositional logic is a weak language

❑ Hard to identify “individuals” (e.g., Mary, 3)

❑ Can’t directly talk about properties of individuals or relations between individuals (e.g., “Bill is tall”)

❑ Generalizations, patterns, regularities can’t easily be represented (e.g., “all triangles have 3 sides”)

❑ First-Order Logic (abbreviated FOL or FOPC) is expressive enough to concisely represent this kind of information

❑ FOL adds relations, variables, and quantifiers, e.g.,

❑ “Every elephant is gray”: x (elephant(x) → gray(x))

❑ “There is a white alligator”: x (alligator(X) ^ white(X))

Summary

Logical agents apply inference to a knowledge base to derive new information and make decisions

Basic concepts of logic:syntax: formal structure of sentencessemantics: truth of sentences wrt modelsentailment: necessary truth of one sentence given anotherinference: deriving sentences from other sentencessoundness: derivations produce only entailed sentencescompleteness: derivations can produce all entailed sentences

Resolution for propositional logic

Forward, backward chaining are linear-time, complete for Horn clauses

Propositional logic lacks expressive power

Artificial Intelligence

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