Prove that string concatenation cannot be checked by finite
automata by showing that the following language...
Prove that string concatenation cannot be checked by finite
automata by showing that the following language over Σ = {0, 1} is
not regular: L3 = {w1#w2#w1w2 : w1, w2 ∈ Σ ∗ }
Description:
In this assignment, you will implement a deterministic finite
automata (DFA) using C++ programming language to extract all
matching patterns (substrings) from a given input DNA sequence
string.
The alphabet for generating DNA sequences is {A, T, G, C}.
Write a regular expression that represents all DNA strings
that begin with ‘A’ and end with ‘T’. Note: assume empty
string is not a valid string.
Design a deterministic finite automaton to recognize the
regular expression.
Write a program which...
Prove that if a language is not recursively enumerable, then its
complement cannot be recursive. In this problem, you can use
diagrams (black boxes with inputs and outputs to represent
procedures and algorithms) as we used in class, in your proof. What
can the complement be?
Automata Theory and Formal Languages
Instructions:
Draw the DFA (Deterministic Finite Automaton) of the
following:
DFA in which set of all strings can be accepted which end with
‘a’
DFA in which set of all strings can be accepted which start
with ab
DFA in which set of all strings can be accepted which contain
ab
DFA in which set of all strings can be accepted which ends with
ab
Automata Theory and Formal Languages
Instructions:
Draw the DFA (Deterministic Finite Automaton) of the
following:
DFA for binary number divisible by 2
DFA for binary number divisible by 3
DFA in which set of all strings can be accepted which start
with a
DFA in which set of all strings can be accepted which contains
‘a’
5. Prove the Following:
a. Let {v1, . . . , vn} be a finite collection of vectors in a
vector space V and suppose that it is not a linearly independent
set.
i. Show that one can find a vector w ∈ {v1, . . . , vn} such
that w ∈ Span(S) for S := {v1, . . . , vn} \ {w}. Conclude that
Span(S) = Span(v1, . . . , vn).
ii. Suppose T ⊂ {v1,...
Question 4 Prove that the following language is not regular. ? =
{ 0 ?1 ? | ?, ? ≥ 0, ? ≠ 2? + 1 }
Question 5 Prove that the following language is not regular. ? =
{ ? ∈ { 0, 1, 2} ∗ | #0 (?) + #1 (?) = #2 (?) } where #? (?)
denotes the number of occurrences of symbol a in string w.
Unless otherwise noted, all sets in this module are finite.
Prove the following statements...
1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B|
and (b) |A × B| = |A||B|.
2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b)
surjective then |A| ≥ |B|.
3. For each part below, there is a function f : R→R that is (a)
injective and surjective. (b)...