Deduction - Computer Science and...

Wang: CIS 630 : Artificial Intelligence

Logic,

Deduction

An important part of human reasoning

Goal: Representing Knowledge in Logicand Mechanizing Logical Reasoning

• Given a set of

• Assumptions (Facts)

• Universal Laws of Logic (Deduction)

• We can find all new facts which logically follow fromthe assumptions

• Can be used to prove an assertion

• Or disprove an assertion


Logic,

Basic Concepts

• Logical expressions are built out of components

Objects (Constants)

Variables

Functions

The above three are called Terms

Predicates

Connectives

Quantifiers


Logic,

Objects

• Symbols that denote specific things or individuals

JOHN

MARY

BASKETBALL

TRIANGLE

Quantifiers

Variables

• Unspecific references to objects

x

y

z


Logic,

Functions

• An argument to a function is either an object or a variable

- Starting with a lowercase letter

• The value of a function is either an object or a variable

exp(0) = 1

next-day(THURSDAY) = FRIDAY

brother(JOHN) = JIM

Predicates(or functions returning True or False)

• Functions which denote attributes of objects or relationshipsbetween individuals

- Starting with a uppercase letter

Loves(JOHN, MARY)

Man(SOCRATES)

Sunny(THURDAY)


Logic,

Connectives

• Logical operators which computes truth values

∧ AND

∨ OR

¬ NOT

⊃ IMPLIES

Quantifiers

• Logical operators which assert the scope of a predicate

∀ For All (universal quantifier)

∃ There Exists (existential quantifier)


Logic,

Literals

• A term is an object, a variable, or a function

• An atomic formula (atom) is a predicate with a proper numberof arguments (terms)

• A literal is either an atom or the negation of an atom

• No Quantifiers

Well-formed Formulas (wffs)

Defined recursively

• Literals are wffs

• wffs connected by ¬, ∧ , ∨ , and ⊃ are wffs

• wffs surrounded by quantifiers are wffs


Logic,

Notations: Summary Diagram

Feathers(x)

xObjects

John

Variables Functions

exp

Terms Predicates

Feathers

Atoms Negation

LiteralsConnectives Quantifiers

Well-formed formulas

∀ x[Feathers(x) ⊃ Bird(x)]


Logic,

Logic & Semantics

On-relation

External World Logic World

Block A

Block B

On(A, B)


Logic,

Mathematical Logic (Brief Review)

Theorems of logical operations

• Distributivity

x ∧ ( y ∨ z) = (x ∧ y) ∨ (x ∧ z)

x ∨ ( y ∧ z) = (x ∨ y) ∧ (x ∨ z)

• DeMorgan's Laws

¬ (x ∨ y) = (¬x ∧ ¬y)

¬ (x ∧ y) = (¬x ∨ ¬y)

• Other laws of negation

¬ (¬x) = x

¬∀ x[E(x)] = ∃ x[¬E(x)]

¬∃ x[E(x)] = ∀ x[¬E(x)]

• Verify it youself by using truth tables


Logic,

Mathematical Logic (cont.)

Laws of Inference

• Modus Ponens (Forward)

If P ⊃ Q and P is true

Then Q is true

• Modus Tolens (Backforward)

If P ⊃ Q and ¬Q is true (Q is false)

Then ¬P is true (P is false)

• Syllogism

If P ⊃ Q and Q ⊃ R

Then P ⊃ R


Logic,

How to Use Deduction?

• Decide on Terms, Predicates

• Translate facts from Natural Language to well-formedexpressions

• The rest is mechanical ......


Logic,

Examples for Conversion from NaturalLanguage Sentences to Predicate Logic

1. Marcus was a man

Man(Marcus)

2. Marcus was a Pompeian

Pompeian(Marcus)

3. All Pompeians were Romans

∀ x [Pompeian(x) ⊃ Roman(x)]

4. Caesar was a ruler

Ruler(Caesar)

5. All Romans were either loyal to Caesar or hated him

∀ x [Roman(y) ⊃ (LoyalTo(x,Caesar) ∨ Hate(x,Caesar))]

6. Everyone is loyal to someone

∀ x ∃ y LoyalTo(x,y)


Logic,

Predicate Logic Examples (cont.)

7. People only try to assassinate rulers they aren't loyal to

∀ x∀ y[(Person(x) ∧ Ruler(y) ∧ TryAssassinate(x,y)) ⊃ ¬LoyalTo(x,y)]

8. Marcus tried to assassinate Caesar

TryAssassinate(Marcus, Caesar)

Can you prove: ¬LoyalTo(x,Caesar)


Logic,

Resolution

A way of mechanizing deduction

• Convert to Clause Form

• Each clause is a conjunct

Clause-1 ∧ Clause-2 ∧ ... ∧ Clause-n

• Each conjunct is a disjunction of literals

Clause-k = Literal-1 ∨ Literal-2 ∨ ... ∨ Literal-m

• Unification

A matching procedure which finds a set of most generalsubstitutions to make two literals identical

• Theorem proving

Backforward reasoning to discover contradictions


Logic,

Conversion to Clause Form

Example:∀ x[Brick(x) ⊃ (∃ y[On(x,y) ∧ ¬Pyramid(y)]

∧ ¬∃ y[On(x,y) ∧ On(y,x)] ∧ ∀ y[¬Brick(y) ⊃ ¬Equal(x,y)])]

1. Eliminate implications (⊃) , using (P ⊃ Q) = (¬P ∨ Q)

∀ x[¬Brick(x) ∨ (∃ y[On(x,y) ∧ ¬Pyramid(y)] ∧ ¬∃ y[On(x,y) ∧ On(y,x)] ∧ ∀ y[¬(¬Brick(y)) ∨ ¬Equal(x,y)])]

2. Move negation (¬) down to atoms

∀ x[¬Brick(x) ∨ (∃ y[On(x,y) ∧ ¬Pyramid(y)]∧ ∀ y[¬On(x,y) ∨ ¬On(y,x)]∧ ∀ y[Brick(y) ∨ ¬Equal(x,y)])]

3. Eliminate existential quantifiers by inserting Skolemfunctions

∀ x[¬Brick(x) ∨ ((On(x,Support(x)) ∧ ¬Pyramid(Support(x))∧ ∀ y[¬On(x,y) ∨ ¬On(y,x)]∧ ∀ y[Brick(y) ∨ ¬Equal(x,y)])]


Logic,

Conversion to Clause Formcontinued

4. Rename variables so that no two variables are the same

∀ x[¬Brick(x) ∨ ((On(x,(Support(x)) ∧ ¬Pyramid(Support(x))∧ ∀ y[¬On(x,y) ∨ ¬On(y,x)]∧ ∀ z[Brick(z) ∨ ¬Equal(x,z)])]

5. Eliminate the universal quantifiers

[¬Brick(x)∨ ((On(x,(Support(x)) ∧ ¬Pyramid(Support(x))∧ [¬On(x,y) ∨ ¬On(y,x)]∧ [Brick(z) ∨ ¬Equal(x,z)])]

6. Move disjunctions down to the literals

[(¬Brick(x) ∨ (On(x,(Support(x))∧ ¬Pyramid(Support(x)))∧ (¬Brick(x) ∨ ¬On(x,y) ∨ ¬On(y,x))∧ (¬Brick(x) ∨ Brick(z) ∨ ¬Equal(x,z))]

in another step

[(¬Brick(x) ∨ On(x,Support(x))∧ (¬Brick(x) ∨ ¬Pyramid(Support(x))∧ (¬Brick(x) ∨ ¬On(x,y) ∨ ¬On(y,x))∧ (¬Brick(x) ∨ Brick(z) ∨ ¬Equal(x,z))]


Logic,

Conversion to Clause Formcontinued

7. Eliminate conjunctions by convention

(¬Brick(x) ∨ On(x,Support(x))(¬Brick(x) ∨ ¬Pyramid(Support(x))(¬Brick(x) ∨ ¬On(x,y) ∨ ¬On(y,x))(¬Brick(x) ∨ Brick(z) ∨ ¬Equal(x,z))

8. Rename all variables again so that no two are the same

By the fact: ∀ x[P(x) ∧ Q(x)] = ∀ x [P(x)] ∧ ∀ x[Q(x)]

So we finally arrive at

(¬Brick(x) ∨ On(x,(Support(x))(¬Brick(w) ∨ ¬Pyramid((Support(w)))(¬Brick(u) ∨ ¬On(u,y) ∨ ¬On(y,u))(¬Brick(v) ∨ Brick(z) ∨ ¬Equal(v,z))


Logic,

Basics of Resolution

Given

¬P ∨ QP ∨ R

We can conclude

Q ∨ R

OR given

¬Feathers(Squigs) ∨ Bird(Squigs)Feathers(Squigs)

We can conclude

Bird(Squigs)

• In general, resolution cancels a pair of literals with oppositesigns from two clauses, and combines the rest to a newclause.


Logic,

Unification

Q(x)P(x) ==> FAIL

P(x)P(y) ==> {y → x}

(Book version: x/y)

P(Marcus)P(y) ==> {y → Marcus} (or Marcus/y)

P(Marcus)P(Julius) ==> FAIL

P(x,x)P(y,z) ==> {x → y, y → z}


Logic,

Find Most General Substitutions

Given:

Hate(x,y)Hate(Marcus,z)

We could produce:

{x → Marcus, y → z}{x → Marcus, z → y}{x → Marcus, y → Caesar, z → Caesar}{x → Marcus, y → Polonius, z → Polonius}


Logic,

Unification

Two literals: L1 and L2

1. If one of them is a term

• If they are identical, return NIL

• If one of them is variable and another one contains thesame variable, return FAIL

E.g.: {x, f(x)} ==> endless unification

else substitute the variable by the other

• Return FAIL

2. If the predicate names are not the same or they have differentnumber of arguments, return FAIL.

3. Recursively call the algorithm with each pair ofcorresponding arguments

• If FAIL, return FAIL

• Else append substitutions together and return it


Logic,

Resolution Algorithm

1. Convert all facts to clause form (product of sums)

2. Negate the goal (theorem to be proven)

3. Convert the negated goal to clause form, and add to the setof facts

4. Until NIL is found, repeat

• Select two clauses

• Compute their resolvent

• Add it to the set of facts


Logic,

Heuristics

Still needed

• Set of support

Choose clauses which contain facts which appear in thegoal, or facts directly derived from the negated goal byprior resolution

• Unit preference

Choose shortest clauses

• Breadth-first

Start from the initial clauses, level by level


Logic,

Example

Whoever can read is literate

Dolphins are not literate

Some Dolphins are intelligent

PROVE

Some who are intelligent cannot read

Translate into Predicate Logic

• Whoever can read is literate

∀ x [R(x) ⊃ L(x)]

• Dolphins are not literate

∀ x [D(x) ⊃ ¬L(x)]

• Some Dolphins are intelligent

∃ x [D(x) ∧ I(x)]

• Goal: Some who are intelligent cannot read

∃ x [I(x) ∧ ¬R(x)]


Logic,

Convert into Clause Form

Forget about reality

• ∀ x [R(x) ⊃ L(x)] convert to ¬R(x) ∨ L(x) ... (1)

• ∀ x [D(x) ⊃ ¬L(x)] convert to ¬D(y) ∨ ¬L(y) ... (2)

• ∃ x [D(x) ∧ I(x)] convert to D(S1) ... (3)I(S1) ... (4)

• Negate conclusion

¬[ ∃ x(I(x) ∧ ¬R(x))] convert to ¬I(z) ∨ R(z) ... (5)


Logic,

Resolve

¬R(x) ∨ L(x) ... (1)¬D(y) ∨ ¬L(y) ... (2)D(S1) ... (3)I(S1) ... (4)

and

¬I(z) ∨ R(z) ... (5)

• (4) and (5) ==> R(S1) ... (6)

• (6) and (1) ==> L(S1) ... (7)

• (7) and (2) ==> ¬D(S1) ... (8)

• (8) and (3) ==> NIL DONE


Logic,

Conclusions about Resolution

• It is powerful, and but subject to exponential explosion

• Theoretically, first-order predicate logic is

• Undecidable. No guarantee to halt (terminate)

• Acceptable. If the goal is true, guaranteed to halt

• Together, it is called semidecidable

• It is a uniform framework, and a general problem solvingmethod.

• But logic is weak for representing certain kinds of knowledge


Logic,

Other Forms of Inference

• Induction

• Abduction

• Analogy

• Induction

• Reason from specific cases to general rules

• Very difficult to mechanize

• A major characteristic of learning(problem for machine learning)

Robins flySparrows fly ==> Birds flyHawks fly (what about ostrich?)


Logic,

Abduction

• Reverse causal reasoning

If P ⊃ Q, and QAbduce P

Example: Anyone who has lung problems is shortof breath.

Bill feels short of breath.Therefore, Bill has lung problems

• Approximate reasoning

• A useful form of reasoning, particularly for findingexplanations (diagnosis)


Logic,

Analogy

• A powerful inference tool for people

• Frequently used in Natural Language

• Seeing clouds from a plane is like seeing ocean

• All modern cities are alike

• Problems in electromagnetism are just like problems influid flow

• How is it used?

• Find an old problem similar to the new one

• Recall the solution to the old problem

• Apply it to the new problem

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