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  1. (10 pts.) Construct a Turing machine that copies a string of zeros over a one, that is, where is the initial state and is the final state. (You may use a tape alphabet of 4 symbols, say .)

  2. (5 pts.) Given the deterministic finite automaton defined by the transition table below, find an equivalent minimal state finite automaton. (State is the final state.)

    The initial partition is

    The second partition is

    The transition table of the minimal state machine is (state 2 is the final state):

  3. (5 pts.) Convert the following context-free grammar into Chomsky normal form.

First make each production of the form or

Next make each production of the form into the form

Short Answer Questions: Answer each of the following questions in a sentence or two. Try to use your own words rather than direct copying from the text or notes. Each correct answer earns 5 points.

  1. What is the pumping lemma for context-free languages and for what purpose is it usually used? If is a context-free language, then there exists a natural number such that if and , then can be written in the form where and for all .
  2. When is a grammar in Greibach normal form? If all productions have the from where
  3. When is a grammar ambiguous? If some word in the language has two different derivation (parse) trees
  4. What is a derivation (parse) tree? (1) A parse tree is a tree where every vertex has a label from , (2) the root is labeled , (3) interior vertices are labeled with variables, (4) If vertex labeled has children labeled , then is a production in the grammar, and (5) if is the label of a node then the node is a leaf and is the only son of its father.
  5. What is the universal language and why is it called ``universal''?
  6. What is the Chomsky hierarchy?
  7. What is the Church-Turing hypothesis and what implications does it have? Every computable function is a partial recursive function, or every computable language is recursively enumerable.
  8. What does Rice's theorem state and what does it mean? Any nontrivial property of the recursively enumerable language is undecidable. A property is trivial if no languages satisfy it or if all languages satisfy it.
  9. What is the -closure of a state ?

    It is the set of all states reachable from by following lambda transitions.

  10. Algorithmically construct a finite automaton for the regular expression

True and False Questions: Place a T next to the statement if it is true and an F otherwise. Each correct answer earns 2 points.

T 1.
If string is accepted by a nondeterministic finite state automaton, then it is accepted by some deterministic finite state automaton.
F 2.
If string is accepted by a nondeterministic pushdown automaton, then it is accepted by some deterministic pushdown automaton.
F 3.
Nondeterministic Turing machines can accept more languages than deterministic Turing machines.
F 4.
The complement of every context-free language is context-free.
F 5.
The productions can be used in a context-free grammar.
F 6.
Every context-free language can be derived from some grammar that is not ambiguous.
T 7.
Regular languages are closed under intersection.
F 8.
DFA's use a stack to store information.
F 9.
Every language can be recognized by a Turing machine.
F 10.
If and are regular expressions then
T 11.
Moore and Mealy machines are finite automata with output.
T 12.
If is a move for a Turing machine , then could occur in 's computation.
F 14.
The universal language is recursive.
T 15.
A recursive language is recognized by some Turing machine that halts on all input.
If string is accepted by a nondeterministic finite state automaton, then it is accepted by some deterministic finite state automaton.
If string is accepted by a nondeterministic pushdown automaton, then it is accepted by some deterministic pushdown automaton.
The complement of every context-free language is context-free.
The productions can be used in a context-free grammar.
Every context-free language can be derived from some grammar that is not ambiguous.
Regular languages are closed under intersection.
Nondeterministic Turing machines can accept more languages than deterministic Turing machines.
DFA's use a stack to store information.
Every language can be recognized by a Turing machine.
If and are regular expressions then
Moore and Mealy machines are finite automata with output.
If is a move for a Turing machine , then could occur in 's computation.
The set of all strings with an equal number of 0's and 1's is regular.
The universal language is recursive.
A recursive language is recognized by some Turing machine that may not halt on all input.
What is the pumping lemma for regular languages and how is it often used? If is a regular language, then there exists a natural number such that if and , then can be written in the form where and for all . Fall 1992 CS 5083 Quiz 1 Name:

(8 pts.) Let denote the real numbers. Define a relation for all by: if and only if is an integer. Show that is an equivalence relation.

    is reflexive: for all real numbers , since in an integer.
    is symmetric: if and are real numbers and , then in an integer. It follows that is an integer, so .
    is transitive: if and for real numbers then and are integers. Thus is an integer and .
Since the relation is reflexive, symmetric and transitive, is an equivalence relation.
(8 pts.) Find the transitive closure of the relation

(5 pts.) Show that the complement of a recursively enumerable language need not be recursively enumerable. (Hint: Use the fact that there are languages that are recursively enumerable, but not recursive and a theorem proved in class.)

Suppose that the complement of a recursively enumerable language is recursively enumerable. Then by a theorem proved in class, and are recursive. Since there are languages that are recursively enumerable but not recursive, there must be recursively enumerable languages whose complements are not recursively enumerable.

(7 pts.) Given the transitions below for a Turing machine with final state and the string show the sequence of instantaneous descriptions as the machine processes the string. Is in the language of the machine?

Since we halt in the final state , the string .

(7 pts.) Use primitive recursion to define the exponential function . (You may use the fact that the product function prod is recursive.)

First, notice that , that is, the sucessor composed with the zero function yields for all natural number .

Next, notice that Thus,

(15 pts.) Define a function from the natural numbers N to (the strings over ), by

In general, for , let

a)
Show by induction on the length of a string that if , then .
b)
Show by induction on the length of a string that if , then there exists a natural number such that .
c)
What can be concluded about the set ?

a)
Let , so that . If then . Assume that if then for all words of length or less. Let . Then or where has length . If then there exists a such that . So and by the inductive hypothesis. Thus is a one-to-one function.
b)
Let , so that and . Assume that for all words of length or less there is a natural number such that . Let . Then or where has length . By the inductive hypothesis, there exists such that and for some power . If , let so that . If , let so that . Thus is a onto function.
c)
The set of all strings over 0 and 1 is countably infinite; it has the cardinality of the integers.

Fall 1992 CS 5083 Quiz 2 Name:

  1. (10 pts.) Show how to construct a nondeterministic finite automaton with transitions that accepts all strings represented by the regular expression

  2. (10 pts.) Given the following nondeterministic finite automaton, show how to construct an equivalent deterministic finite automaton.

    denotes accept (final) states.

  3. (10 pts.) Given the following nondeterministic finite automaton with transtions, find an equivalent nondeterministic finite automaton without transitions.

    Note that -closure and -closure, thus

    Similarly,

    Continuing for the other states, we find:

  4. (10 pts.) Show that the language is not regular.

    Suppose that is regular, let be the integer of the pumping lemma, and let . By the pumping lemma can be decomposed as where , and for all natural numbers . Now for some and for some . Thus, has length . We want to show that this length can not be a factorial for all natural numbers . Consider . We have which has length , and where the last strict inequality is true for all . Since a one state machine could not recognize the language, the langugage must not be regular.

  5. (10 pts.) Given the following deterministic finite automaton, with final states and , find the first three partitions generated by the state minimazation algorithm.

    The initial partition is

    The second partition is

    The third partition is




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wds@cs.fit.edu
Fri Apr 28 13:00:33 EDT 1995