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1Decidable and Undecidable Languages

The Halting Problem and

The Return of Diagonalization

CS235 Languages and Automata

Tuesday, November 23, and Wednesday, November 24, 2010

Reading: Sipser 4; Kozen 31; Stoughton 5.2 & 5.3

CS235 Languages and AutomataDepartment of Computer Science

Wellesley College

Recursively Enumerable Languages

L(M) = {w | w is accepted by the

Turing Machine M}

The recursively enumerableThe recursively enumerable (r.e.) languages = the set of all languages that are the language of some Turing Machine.

These are also called

Turing-acceptableand

Turin g-recognizablelanguages.

RE = Recursively Enumerable

(Turing-Recognizable/Acceptable)

Languages

CFL = Context-Free Languages

a n b n c n ww 32-2
gg ggWe will use REto name this set.

There are many languages in

REthat are not in CFL.

ggReg = Regular Languages a*b*(a+b)*bbb(a+b)*a n b n ww R

Decidable and Undecidable Languages

2

Decidability and Semi-Decidability

RE = Recursively Enumerable

(Turing-Recognizable/Acceptable)

Languages

A Turing Machine decidesa language if it

rejects every string it doesn't accept - i.e., it never loops

The recursivelanguages = the set of all

languages that are decided by some Turing

M hi ll l d ib d b

Dec = Recursive (Turing-Decidable)

Languages

CFL = Context-Free Languages

a n b n ww R a n b n c n ww semi-decidable+ decidable

Machine = all languages described by a non-

looping TM.

These are also called theTuring-decidableor

decidable languages.

We will use Decto name this set.

We'll soon see examples of languages that are

in REbut not in Dec. We call these languages semi-decidable+.

Reg = Regular Languages

a*b*(a+b)*bbb(a+b)*Every TM for a semi-decidable+ language halts in the accept state for strings in the language but loops for some strings not in the language.

Any language outside Decis undecidable.

All semi-decidable+ languages are undecidable,

but we'll see there are undecidable languages that aren't semi-decidable+!

32-3Decidable and Undecidable Languages

Dec vs. RE

accept pipe For every language Lin Dec, there is a deciding machineM that for an input string w is guaranteed to deliver a ball to either the accept pipe or reject pipe.

Turing Machine Mfor

a language L in Dec accept pipe reject pipe input string w For every language Lin RE, there is an accepting machineM that for an input string w is guaranteed to deliver a ball to the acce pt pipe if w L. However, if w L, a ball mightnot

Turing Machine Mfor

a language L in RE accept pipe reject pipe input string w ppp,,g be delivered to the reject pipe (Mmight loop).

32-4Decidable and Undecidable Languages

3

Game Plan for the Rest of this Lecture

RE = Recursively Enumerable

(Turing-Recognizable/Acceptable)

Languages

Our main goal is to exhibit a language L

that's semi-decidable+: L in RE - Dec.

But first:

Dec = Recursive (Turing-Decidable)

Languages

CFL = Context-Free Languages

a n b n ww R a n b n c n ww semi-decidable+ decidable

1. we'll need to get some practice

describing decidable languages that involve language encodings. Then:

2. we'll define a language HALT

TM that's in RE - Dec. Fi ll

Reg = Regular Languages

a*b*(a+b)*bbb(a+b)*

Finally:

3. we'll argue that there are languages

that aren't even in RE!

32-5Decidable and Undecidable Languages

Language Encodings

We will consider many languages whose strings contain encodings of DFAs, FAs, NPDAs, CFGs, and TMs. Think of such encodings as Forlan-like specifications for these machines and grammars. E.g. : b a X; W,b -> Z;

X,a -> X; X,b -> Y;

Y,a -> X; Y,b -> Y;

Za > Z; Z b > Z})"

states start states final states alphabet 32-6
DFA 1

Z,a -> Z; Z,b -> Z})

transition function

SĺAB

Aĺ0A1 |%

Bĺ1B0 |%

1variables (start var first) terminals productions

Decidable and Undecidable Languages

4

Warm-Up: Some Decidable Languages

Show that the following languages are decidable by describing (at a high level) an algorithm that decides them (see more in Sipser 4.1)

ACCEPT {( DFA ) | i i L(DFA)}

string describing a pair of (1) a deterministic finite automaton DFA and (2) an input string w •ACCEPT DFA = {(, w) | w is in L(DFA)} •EMPTY DFA = { | L(DFA) = } •ALL DFA = { | L(DFA) = DFA •EQ DFA = {(, ) | L(DFA 1 ) = L(DFA 2 •ACCEPT CFG = {(, w) | w is in L(CFG)} •EMPTY CFG = { | L(CFG) = } (Warning: EQ CFG and ALL CFG are notdecidable! )

32-7Decidable and Undecidable Languages

What about ACCEPT

TM

ACCEPT

TM = {(, w) | w is in L(M) (i.e. M accepts w)}

A Turing Machine M

UTM can acceptACCEPT TM as follows:

1. Check if is a well-formed Turing Machine description.

If not, M

UTM rejects(, w) string describing a Turing Machine M and an input string w

If not, M

UTM rejects(M, w)

2. If is well-formed, M

UTM simulates the running of M on w, e.g., via a 4-tape machine:quotesdbs_dbs17.pdfusesText_23