FOL-Logic, Reasoning and Uncertainty , Exams of Artificial Intelligence

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Logic, Reasoning, and UncertaintyLogic, Reasoning, and Uncertainty

CSEP 573

© CSE AI Faculty

What’s on our menu today?What’s on our menu today?

Propositional Logic

Resolution

WalkSAT

Reasoning with First-Order Logic

Unification

Forward/Backward Chaining

Resolution

Wumpus again

Uncertainty

Bayesian networks

Understanding ResolutionUnderstanding Resolution

IDEA: To show

KB

α

, use proof by

contradiction, i.e., show

KB

∧ ¬ α

unsatisfiable

KB is in

Conjunctive Normal Form (CNF):

KB is conjunction of clauses E.g., (A

B)

(B

C

D)

Literals

Clause

Generating new clausesGenerating new clauses

General Resolution inference rule (for CNF):

l^1

l^

k

m

1

m

n

l^1

li-

l

i+

l^

k

m

1

m

j-

m

j+

m

n

where

li

and

m

j^

are complementary

literals (

l^

i^

m

)j

E.g.,

P

1,

P

2,

P

2,

P

1,

Resolution exampleResolution example

Empty clause

Recall that KB is a

conjunction

of all these clauses

Is

P

1,

P

1,

satisfiable? No!

Therefore, KB

∧ ¬ α

is unsatisfiable, i.e.,

KB

α

You got a literal and its negation What does this mean?

KB

¬α

Back to Inference/Proof TechniquesBack to Inference/Proof Techniques

Two kinds (roughly):

Successive application of inference rules

Generate new sentences from old in a sound way

Proof = a sequence of inference rule applications

Use inference rules as

successor function in a

standard search algorithm

E.g., Resolution

Model checking

Done by checking satisfiability: the SAT problem

Recursive depth-first enumeration of models usingheuristics: DPLL algorithm

(sec. 7.6.1 in text)

Local search algorithms (sound but incomplete)

e.g., randomized hill-climbing (WalkSAT)

Why Satisfiability?Why Satisfiability?

Recall:

KB

α

iff

KB

α

is unsatisfiable

Thus, algorithms for satisfiability can be used for

inference by showing

KB

α

is unsatisfiable

BUT… showing a sentence issatisfiable (the SAT problem)

is NP-complete!

Finding a fast algorithm for SAT automatically yields fast algorithms

for hundreds of difficult (NP-

complete) problems

I really can’t get

¬

satisfaction

Satisfiability ExamplesSatisfiability Examples

E.g. 2-CNF sentences (2 literals per clause):(

A

B)

(A

B)

(A

B)

Satisfiable?Yes (e.g., A = true, B = false)(

A

B)

(A

B)

(A

B)

A

B)

Satisfiable?No

The

WalkSAT

algorithm

The

WalkSAT

algorithm

Greed

Randomness

Hard Satisfiability ProblemsHard Satisfiability Problems

Consider random 3-CNF sentences. e.g.,

D

B

C)

(B

A

C)

C

B

E)

(E

D

B)

(B

E

C)

m = number of clausesn = number of symbols •

Hard instances of SAT seem to cluster nearm/n = 4.3 (critical point)

Hard Satisfiability ProblemsHard Satisfiability Problems Median runtime for random 3-CNF sentences,

n = 50

What about me?What about me?

KB contains "physics" sentences for every single

square For every time step

t and every location [

x,y],

we need to add to the KB:

L

x,y

FacingRight

t^

Forward

t^

L

x+1,y

Rapid proliferation of sentences!

t+

t

Limitations of propositional logicLimitations of propositional logic

What we’d like is a way to talk aboutobjects and

groups of objects, and to

define relationships between them

What we’d like is a way to talk aboutobjects and

groups of objects, and to

define relationships between them

Enter…First-Order Logic

(aka “Predicate logic”)