CS252 Midterm 2

{ w | w {a, b} and there are twice as many a's as b's in w }

CFG

Chomksy Hierarchy Type 2

Context free language (pda)

Chomsky Hierarchy Type 1

Context sensitive languages (linear bounded automata)

For every non-deterministic PDA that accepts a language L, there exists a deterministic PDA that accepts L.

False

If a CFG G generates the same string using a leftmost, as well as a rightmost, derivation, then G is ambiguously.

False

EQ_DFA how to create new language

L_new=(L2comp intersect L1) union (L1comp intersect L2)

Chomsky Hierachry type 3

Regular languages (finite automata)

(True of False) We can design a TM to accept a language or perform a particular function.

True

8. (True or False) All of the following types of TMs are equivalent in expressive power: a) Single-tape TMs b) Multi-tape TMs c) Deterministic single-tape TMs d) Non-deterministic multi-tape TMs

True

B = { ww | w is a string over {a, b} and |w| > 0 } is not context-free

True

During the process of converting a CFG into its Chomsky normal form, a unit rule is introduced when the new rule S0 S is created, where S0 is the new start variable and S is the original start variable.

True

For each CFG G that generates strings in a language L, there exists a PDA that accepts L.

True

The derivation tree for a string generated by a CFG in Chomsky normal form is a binary tree.

True

The pumping lemma for CFLs states that for sufficiently long strings in a CFL, we can find two, short, nearby substrings that we can "pump" in tandem and the resulting string must also be in the language.

True

Chomsky Hierarchy Type 0

Turing marchines (Recursively enumerable languages and decidable languages)

U = On input <M, w>, where M is a TM and w is a string 1.Simulate M on input w 2.If M ever enters its accept state, accept, if M ever enters its reject state, reject

U is also called a Universal Turing Machine, since it is capable of simulating any other TM from the description of that machine. M halts on w --> accept. M does not halt --> loop

A_CFG

decidable

A_DFA

decidable

A_NFA

decidable

A_REX

decidable

EQ_DFA

decidable

E_CFG

decidable

E_DFA

decidable

A language L is Turing-recognizable if there exists a TM M such that given an input string w over the alphabet of L, M a) Halts and accepts w, or b) Halts and rejects w, or c) Loops forever and returns no answer d) Both (a) and (b) e) All of (a), (b), and (c)

e

What are the possible scenarios that can occur during the steps of a TM M in processing an input string: a) M enters the accept state and accepts the given input string b) M rejects the given input when there are no more transitions to take c) M accepts the input string when it halts d) M rejects the input string when it loops forever e) All of the above

e

Pumping lemma can be used as a tool to show that a language is context-free

false

◦Turing-recognizable languages are not closed under set difference or complementation, i.e., given two Turing-recognizable languages L and P

oIf L is also Turing-recognizable, then L is decidable oL - P may or may not be Turing-recognizable, since L - P = L Ç L Ç P

Given that L and P are two Turing-recognizable languages, then the following languages are Turing recognizable:

oThe union, L È P oThe intersection, L Ç P oThe concatenation LP of L and P oThe Kleene star L* of L

◦Given that L is a decidable language and P is a Turing-recognizable language, then the following languages are Turing-recognizable as well:

oThe union, L È P oThe intersection, L Ç P oThe concatenation LP of L and P oThe difference, P - L (but not L - P)

◦Given that L and P are two decidable languages, then the following languages are decidable as well:

oThe union, L È P oThe intersection, L Ç P oThe difference, L - P oThe complement of L, L oThe concatenation LP of L and P oThe Kleene star L* of L ◦Decidable languages L and P are closed under set difference, since L - P = L Ç L Ç P

A = { a pb q c r| p < q < r } is not context-free

true

Given that BxqjyB is obtained from BuqivB by a single transition of a TM. If the transition causes the tape head to move one cell to the left, then x is a prefix of u

true

L = { a pb p c q | p, q > 0 } { a bb q c q | p, q > 0 } is not context-free

true

EQ_CFG

undecidable

A_TM

undecidable (halting problem)

A_CFG number of steps in chomsky

w can be derived in 2n-1 steps, which allows deciability