CSCI 2670Introduction to Theory of

Computing

October 13, 2005

October 13, 2005 2

Agenda• Yesterday

– Decidability and regular languages• Today

– Putting things in perspective– More on decidability and regular

languages– Decidability and context free

grammars

October 13, 2005 3

Course goals• Theoretically explore the capabilities

and limitations of computers– Complexity theory

• What makes some problems computationally hard and others easy?

– Computability theory• What problems can be solved by a

computer?– Automata theory

• How can we mathematically model computation?

October 13, 2005 4

Some perspective• Automata theory

– Introduced DFA’s, NFA’s, RE’s• Showed that they all accept the same

class of languages– Introduced CFG’s, PDA’s

• PDA is essentially an NFA with a stack• PDA’s and CFG’s accept the same class of

languages

October 13, 2005 5

Perspective (cont.)• Computability Theory

– Introduced TM’s• Like PDA’s with more general memory model

– Importance of TM’s• Church-Turing Thesis• Any algorithm can be implemented on a TM

– Use the TM model and Church-Turing Thesis to understand and classify languages

• Decidable languages• Undecidable languages• Recognizable languages• Unrecognizable languages• Complements of languages in these classes

October 13, 2005 6

Perspective (cont.)• Complexity theory (later this

semester)– Use TM model to determine how long

an algorithm takes to run• Function of input length

– Classify algorithms according to their complexity

October 13, 2005 7

Decidable languages• A language is decidable if some

Turing machine decides it– Every string in * is either accepted

or rejected

October 13, 2005 8

Some decidable languages• ADFA = {<B,w> | B is a DFA that

accepts input string w}• ANFA = {<B,w> | B is an NFA that

accepts input string w}• AREX = {<R,w> | R is a regular

expression that generates string w}• EDFA = {<A> | A is a DFA and L(A) =

}• EQDFA = {<A,B> | A and B are DFA’s

and L(A) = L(B)}

October 13, 2005 9

Question• How would we show that the

following language is decidable?ALLDFA = {<A> | A is a DFA that

recognizes *}

October 13, 2005 10

Another question• Let L be any regular language• How would we show L is decidable?

– Assume L is described using a DFA

October 13, 2005 11

Deciders and CFG’s• Consider the following language

ACFG = {<G,w> | G is a CFG that generates string w}

• Is ACFG decidable?– Problem: How can we get a TM to

simulate a CFG?• Must be certain CFG tries a finite number

of steps!– Solution: Use Chomsky normal form

October 13, 2005 12

Chomsky normal form review• All rules are of the form

A →BCA →a

• Where A, B, and C are any variables (B and C cannot be the start variable)

S → ε• is the only εrule, where S is the

start variable

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How many steps to generate w?

• If |w| = 0– 1 step

• If |w| = n > 0?– 2n – 1 steps

October 13, 2005 14

TM simulating ACFG

M = “On input <G>, where G is a CFG1. Convert G into Chomsky normal

form2. If |w| = 0

If there is an S → εrule, accept Otherwise, reject

3. List all derivations with 2|w|-1 steps If any generate w, accept Otherwise, reject”

October 13, 2005 15

Empty CFG’s• Consider the following languageECFG = {<G> | G is a CFG and L(G) = }• Theorem: ECFG is decidable• Can we use the TM in ACFG to prove

this?– No. There are infinitely many possible

strings in Σ*

– Instead, we need to check if there is any way to get from the start variable to some string of terminals

October 13, 2005 16

Work backwardsB = “On input <G>, where G is a CFG1. Mark all terminals2. Repeat until no new variables are

marked• Mark any variable A if G has a rule

A→U1U2…Uk where U1, U2, …, Uk are all marked

3. If S is marked, reject4. Otherwise, accept”

October 13, 2005 17

What about EQCFG?

• Recall for EQDFA, we considered(L(A) L(B)) (L(A) L(B))

• Will this work for CFG’s?– No. CFG’s are not closed under

complementation or intersection• EQCFG is not a decidable language!

– We will see this later

October 13, 2005 18

Decidability of CFL’sTheorem: Every context-free

language L is decidableProof: For each w, we need to

decide whether or not w is in L. Let G be a CFG for L. This problem boils down to ACFG, which we showed is decidable.

Question: Why don’t we just make a TM that simulates a PDA?

October 13, 2005 19

Relationship of classes of languages

Regular

Context-free Decidabl

eTuring-recognizable