AUTOMATA THEORY
5 th Module
Q1 ) Write a note on the Turing machine and Multi Tape Turing Machine ?
Ans )
A Turing machine is a theoretical model in automata theory introduced
by Alan Turing in 1936.
It serves as a fundamental concept for understanding the limits and
capabilities of computation.
The machine operates based on transitions between these states,
reading symbols from the tape, writing new symbols, and moving the
head according to the transition rules.
It can compute any function that can be algorithmically computed,
making it a powerful theoretical tool in computer science.
A multi-tape Turing machine is an extension of the basic Turing machine
model. Instead of a single tape, it has multiple tapes that run in parallel.
Each tape has its own read/write head, and the heads can move
independently.
They offer a way to express complex computations more concisely, but
they are still equivalent in computational power to the original Turing
machine model, meaning anything computable by one can be computed
by the other.
In summary, a Turing machine is a foundational concept in automata
theory that abstractly models computation.
A multi-tape Turing machine extends this model by allowing multiple
tapes to run in parallel, enhancing its computational efficiency while
maintaining equivalent computational power to the single-tape variant.
Q2) Briefly explain about the post correspondence problem and class N
and NP problem ?
Ans )
The Post Correspondence Problem (PCP) is a well-known undecidable
problem in automata theory. Given a set of string pairs, the problem is to
determine whether there exists a sequence of indices that, when used to
select strings from the pairs, results in two equal strings.
The complexity classes P and NP are fundamental concepts in
computational complexity theory:
1.Class P: This class consists of decision problems that can be solved
in polynomial time by a deterministic Turing machine. In other words,
these are problems for which an algorithm exists that can solve them
efficiently. Algorithms in P are considered "tractable" in terms of
computational complexity.
2.Class NP: The term "NP" stands for "nondeterministic polynomial
time," though it's worth noting that the class doesn't necessarily require
non deterministic computation.
This class consists of decision problems for which a given solution can
be verified for correctness in polynomial time by a deterministic Turing
machine. This means that if someone claims to have a solution, you can
check its validity relatively quickly.
In summary, the Post Correspondence Problem involves finding equal
strings by selecting and concatenating pairs of strings, and it's
undecidable. Class P represents problems solvable in polynomial time,
while class NP represents problems whose solutions can be efficiently
verified.
Q3 ) Design a turing machine for A raise to n and B raised to n?
Ans)
Designing a complete Turing machine for a specific problem typically
requires more detailed steps and transitions than can be provided in a
concise response. However, I can give you a high-level overview of how
you might approach designing a Turing machine to recognize the
language of "A^n B^n" for a limited scope:
Turing Machine Description:
Initialization: Start in the initial state and move the tape head to the
rightmost end of the input string. Place a special blank symbol '#' to
mark the end of the input.
Scan for 'A's: Move the tape head to the left, scanning for 'A's. When you
find an 'A', replace it with a special symbol 'X' (or any other symbol of
your choice) and move to the right end of the string.
Scan for 'B's: Move the tape head to the left again, this time scanning for
'B's. When you find a 'B', replace it with a special symbol 'Y' (or another
symbol) and move to the right end of the string.
Matching 'X's and 'Y's: Now, move the tape head to the leftmost end of
the string. Compare each 'X' with the corresponding 'Y'. If they match,
move to the right and continue the comparison. If at any point you find a
mismatch or reach the end of the tape without encountering a mismatch,
the input does not belong to the language. In this case, reject the input.
Acceptance: If you successfully match all 'X's and 'Y's without any
mismatches, and you reach the right end of the tape, the input belongs
to the language "A^n B^n." In this case, accept the input.
Q4)Briefly explain about linear bounded automata and turing machines
with infinite tape?
Ans)
Linear Bounded Automata (LBA):
In an LBA, the tape head movement is limited to the portion of the tape
containing the input plus a constant amount of additional space.
This restriction ensures that the tape head doesn't move beyond a
linearly increasing amount of space relative to the input length.
LBAs are equivalent in computational power to Turing machines,
meaning they can recognize the same set of languages.
It is similar to a Turing machine but with a restricted tape length.
Turing Machine with Infinite Tape:
A Turing machine is a fundamental abstract model of computation. It
consists of an infinite tape, a read/write head, and a finite set of states.
The tape can be thought of as an unbounded memory that extends
infinitely in both directions.
This infinite tape allows the Turing machine to theoretically perform
computations of arbitrary length.
The machine operates by reading symbols from the tape, writing new
symbols, changing states, and moving the head left or right based on a
set of predefined transition rules.
Q5 ) Write a note on Decide ability and Undecide ability?
Ans)
Decidability and undecidability are fundamental concepts in automata theory
that refer to the ability to determine the membership of a given string in a
particular language using a computational process.
Decidability: A language is said to be decidable if there exists an algorithm or
a Turing machine that, for any given input string, can determine whether the
string belongs to the language or not.
Undecidability: A language is considered undecidable if there is no algorithm
or Turing machine that can determine whether a given input string belongs to
the language or not. This means that for some languages, there is no general,
systematic procedure that can always give a definite answer. Undecidability
highlights the existence of languages that are inherently complex and cannot
be completely analysed or categorised by computational processes.
In summary, decidability refers to the property of a language that allows us to
determine membership using an algorithm or Turing machine. Undecidability,
on the other hand, indicates that there are languages for which no such
algorithmic procedure exists, revealing the inherent limitations of computation.
