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Resolution and Refutation Proofs

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### Resolution and Refutation Proofs

Resolution Proof

Introduction to Artificial Intelligence

CS440/ECE448

Lecture 13

Homework due March 2

Last lecture

- Substitutions and unification
- Generalized Modus Ponens
- Resolution definition
This lecture

- Refutation proofs
- True-or-false questions
- Fill-in-the-blanks questions
- Resolution properties
Reading

- Chapter 9

Generalized Modus Ponens (GMP)

p1’, p2’,…, pn’, (p1p2… pn q) wherepi’ = pi

q for all i

For example, let

p1’ = Faster (Bob, Pat)

p2’ = Faster (Pat, Steve)

Faster (x, y) Faster (y, z) Faster (x, z)

Unify p1’ and p2’ with the premise

= { x/Bob, y/Pat, z/Steve }

Apply substitution to the conclusion

q = Faster(Bob, Steve)

pi and q atomic sentences

Universally quantified variables

Forward Chaining Example

- White: Facts added in turn
- Yellow: The result of implication of rules.
- Buffalo(x) Pig(y) Faster(x, y)
- Pig(y) Slug(z) Faster(y, z)
- Faster(x, y) Faster(y, z) Faster(x, z)
- Buffalo(Bob) [ Unifies with 1-a ]
- Pig(Pat) [ Unifies with 1-b, GMP Fires ]
[ Unifies with 2-a ]

- Faster(Bob,Pat) [ Unifies with 3-a, 3-b ]
- Slug(Steve) [ Unifies with 2-b, GMP Fires ]
- Faster(Pat, Steve) [ Unifies with 3-b and with 6,GMP Fires ]
- Faster (Bob, Steve)
- …

Pig(y) Slug(z) Faster (y, z)

Slimy(a) Creeps(a) Slug(a)

Pig(Pat)

Slimy(Steve)

Creeps(Steve)

Backward Chaining ExampleRefutation Proofs

- Given
- a knowledge base KB (collection of true sentences),
- a proposition P,
we wish to prove that P is true.

- Proof by contradiction (refutation):
- Assume that P is FALSE (i.e., that P is TRUE).
- Show that a contradiction arises.

- A complete approach to refutation can be obtained using a single inference rule: resolution.

Resolution Inference Rule

- Idea: If is true or is true
and is false or is true

then or must be true

- Basic resolution rule from propositional logic:

- Can be expressed in terms of implications
,

- Note that Resolution rule is a generalization of Modus Ponens
, is equivalent to TRUE ,

TRUE

Generalized Resolution

Generalized resolution rule for first order logic (with variables)

If pj can be unified with qk, then we can apply the resolution rule:

p1 … pj … pm

q1 … qk … qn

Subst(, (p1 … pj-1 pj+1 … pm q1 … qk-1 qk+1 … qn))

where = Unify (pj, qk)

- Example:
KB: Rich(x) Unhappy(x)

Rich(Me)

Substitution: = { x/Me }

Conclusion: Unhappy(Me)

Canonical Form

- For generalized Modus Ponens, entire knowledge base is represented as Horn Sentences.
- For resolution, entire database will be represented using Conjunctive Normal Form (CNF)
- Any first order logic sentence can be converted to a Canonical CNF form.
- Note: Can also do resolution with implicative form, but let’s stick to CNF.

Converting any FOL to CNF

- Literal = (possibly negated) atomic sentence, e.g., Rich(Me)
- Clause = disjunction of literals, e.g., Rich(Me) Unhappy(Me)
- The KB is a conjunction of clauses
- Any FOL sentence can be converted to CNF as follows:
- Replace P QbyP Q
- Move inwards to literals, e.g., x P becomes x P
- Standardize variables, e.g., (x P) (x Q) becomes (x P) (y Q)
- Move quantifiers left in order, e.g., x P y Q becomes x y P Q
- Eliminate by Skolemization (next slide)
- Drop universal quantifiers
- Distribute over , e.g., (P Q) R becomes (P R) (Q R)
- Flatten nested conjunctions & disjunctions, e.g. (P Q) R P Q R

Skolemization

(Thoralf Skolem 1920)

- The process of removing existential quantifiers by elimination.
- Simple case: No universal quantifiers.
Existential Elimination Rule

- For example:
x Rich(x)

becomes

Rich(G1)

where G1 is a new ``Skolem constant'‘.

- More tricky when is inside .

Skolemization – continued

- More tricky when is inside
E.g., ``Everyone has a heart''

x Person(x) y Heart(y) Has(x,y)

- Incorrect:
x Person(x) Heart(H1) Has(x,H1)

This means everyone has the same heart calledH1.

- Problem is that for each person, we need another “heart” – i.e., consider the “heart” to be a function of the person.
- Correct:
Person(x) Heart(H(x)) Has(x,H(x))

where H is a new symbol (``Skolem function'')

- Skolem function arguments: all enclosing universally quantified variables.

Resolution proof

p1 … pj … pm

q1 … qk … qn

Subst(, (p1 … pj-1 pj+1 … pm q1 … qk-1 qk+1 … qn))

- To prove :
- Negate .
- Convert to CNF.
- Add to CNF KB.
- Infer contradiction using the resolution rule (a contradiction is detected when resolution derives the empty clause).

- E.g., to prove Rich(Me),add Rich(Me) to the CNF KB, then:
PhD(x) HighlyQualified(x)

PhD(x) EarlySalary(x)

HighlyQualified(x) Rich(x)

EarlySalary(x) Rich(x)

Resolution Proof

Rich(Me)

PhD(x) HighlyQualified(x)

PhD(x) EarlySalary(x)

HighlyQualified(x) Rich(x)

EarlySalary(x) Rich(x)

Resolution Proof

Rich(Me)

PhD(x) HighlyQualified(x)

PhD(x) EarlySalary(x)

HighlyQualified(x) Rich(x)

EarlySalary(x) Rich(x)

Resolution Proof

Rich(Me)

PhD(x) HighlyQualified(x)

PhD(x) EarlySalary(x)

HighlyQualified(x) Rich(x)

EarlySalary(x) Rich(x)

Rich(Me)

PhD(x) HighlyQualified(x)

PhD(x) EarlySalary(x)

HighlyQualified(x) Rich(x)

EarlySalary(x) Rich(x)

No bird swims

Pete is a bird

Does Pete Fly?

The Knowledge base

1. Bird(x) Flies(x)

2. Bird(y)Swims(y)

Bird(Pete)

The query

Flies ( Pete)

Applying Resolution

5. Swims(Pete) 2,3

Bird(Pete) 1,4

{ } 3,6

True-or-False QuestionFill-in-the-blanks Question

- Given a database KB and a sentence with free variables v1, …, vn what are the bindings that make true?
- Prove that given KB, v1, …, vn
- i.e., add to the database, derive a contradiction and find what is the subsitution leading to it.
- Green’s trick: add {, Ans (v1 ,…, vn)} instead.

Decidability

How hard is it to determine if KB entails ?

- Propositional logic (zeroth order) is decidable:
- Can determine whether or not KB entails in finite time.

- Second order logic is undecidable:
- Cannot determine whether KB entails in finite time.

- First order logic is semi-decidable:
- If KB entails or , then a proof will be found in finite time.
- But, if KB neither entails or , then proof process may never terminate.

Resolution properties

- Resolution is sound.
- Resolution refutation is complete. (See Section 9.5 for proof.)
- Resolution (and any proof procedure) is at best semi-decidable.
- Note: checking consistency of a database is also semidecidable.

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