- Automata Theory - Applications
- Automata Terminology
- Basics of String in Automata
- Set Theory for Automata
- Finite Sets and Infinite Sets
- Algebraic Operations on Sets
- Relations Sets in Automata Theory
- Graph and Tree in Automata Theory
- Transition Table in Automata
- What is Queue Automata?
- Compound Finite Automata
- Complementation Process in DFA
- Closure Properties in Automata
- Concatenation Process in DFA
- Language and Grammars
- Language and Grammar
- Grammars in Theory of Computation
- Language Generated by a Grammar
- Chomsky Classification of Grammars
- Context-Sensitive Languages
- Finite Automata
- What is Finite Automata?
- Finite Automata Types
- Applications of Finite Automata
- Limitations of Finite Automata
- Two-way Deterministic Finite Automata
- Deterministic Finite Automaton (DFA)
- Non-deterministic Finite Automaton (NFA)
- NDFA to DFA Conversion
- Equivalence of NFA and DFA
- Dead State in Finite Automata
- Minimization of DFA
- Automata Moore Machine
- Automata Mealy Machine
- Moore vs Mealy Machines
- Moore to Mealy Machine
- Mealy to Moore Machine
- Myhill–Nerode Theorem
- Mealy Machine for 1’s Complement
- Finite Automata Exercises
- Complement of DFA
- Regular Expressions
- Regular Expression in Automata
- Regular Expression Identities
- Applications of Regular Expression
- Regular Expressions vs Regular Grammar
- Kleene Closure in Automata
- Arden’s Theorem in Automata
- Convert Regular Expression to Finite Automata
- Conversion of Regular Expression to DFA
- Equivalence of Two Finite Automata
- Equivalence of Two Regular Expressions
- Convert Regular Expression to Regular Grammar
- Convert Regular Grammar to Finite Automata
- Pumping Lemma in Theory of Computation
- Pumping Lemma for Regular Grammar
- Pumping Lemma for Regular Expression
- Pumping Lemma for Regular Languages
- Applications of Pumping Lemma
- Closure Properties of Regular Set
- Closure Properties of Regular Language
- Decision Problems for Regular Languages
- Decision Problems for Automata and Grammars
- Conversion of Epsilon-NFA to DFA
- Regular Sets in Theory of Computation
- Context-Free Grammars
- Context-Free Grammars (CFG)
- Derivation Tree
- Parse Tree
- Ambiguity in Context-Free Grammar
- CFG vs Regular Grammar
- Applications of Context-Free Grammar
- Left Recursion and Left Factoring
- Closure Properties of Context Free Languages
- Simplifying Context Free Grammars
- Removal of Useless Symbols in CFG
- Removal Unit Production in CFG
- Removal of Null Productions in CFG
- Linear Grammar
- Chomsky Normal Form (CNF)
- Greibach Normal Form (GNF)
- Pumping Lemma for Context-Free Grammars
- Decision Problems of CFG
- Pushdown Automata
- Pushdown Automata (PDA)
- Pushdown Automata Acceptance
- Deterministic Pushdown Automata
- Non-deterministic Pushdown Automata
- Construction of PDA from CFG
- CFG Equivalent to PDA Conversion
- Pushdown Automata Graphical Notation
- Pushdown Automata and Parsing
- Two-stack Pushdown Automata
- Turing Machines
- Basics of Turing Machine (TM)
- Representation of Turing Machine
- Examples of Turing Machine
- Turing Machine Accepted Languages
- Variations of Turing Machine
- Multi-tape Turing Machine
- Multi-head Turing Machine
- Multitrack Turing Machine
- Non-Deterministic Turing Machine
- Semi-Infinite Tape Turing Machine
- K-dimensional Turing Machine
- Enumerator Turing Machine
- Universal Turing Machine
- Restricted Turing Machine
- Convert Regular Expression to Turing Machine
- Two-stack PDA and Turing Machine
- Turing Machine as Integer Function
- Post–Turing Machine
- Turing Machine for Addition
- Turing Machine for Copying Data
- Turing Machine as Comparator
- Turing Machine for Multiplication
- Turing Machine for Subtraction
- Modifications to Standard Turing Machine
- Linear-Bounded Automata (LBA)
- Church's Thesis for Turing Machine
- Recursively Enumerable Language
- Computability & Undecidability
- Turing Language Decidability
- Undecidable Languages
- Turing Machine and Grammar
- Kuroda Normal Form
- Converting Grammar to Kuroda Normal Form
- Decidability
- Undecidability
- Reducibility
- Halting Problem
- Turing Machine Halting Problem
- Rice's Theorem in Theory of Computation
- Post’s Correspondence Problem (PCP)
- Types of Functions
- Recursive Functions
- Injective Functions
- Surjective Function
- Bijective Function
- Partial Recursive Function
- Total Recursive Function
- Primitive Recursive Function
- μ Recursive Function
- Ackermann’s Function
- Russell’s Paradox
- Gödel Numbering
- Recursive Enumerations
- Kleene's Theorem
- Kleene's Recursion Theorem
- Advanced Concepts
- Matrix Grammars
- Probabilistic Finite Automata
- Cellular Automata
- Reduction of CFG
- Reduction Theorem
- Regular expression to ∈-NFA
- Quotient Operation
- Parikh’s Theorem
- Ladner’s Theorem
Restricted Turing Machine in Automata Theory
There are several types of Turing Machines and they are quite powerful and useful in several cases. They can simulate any algorithm and are more powerful than any other automaton, such as finite automata (FA), pushdown automata (PDA), or linear bounded automata (LBA).
However, not all Turing machines are created equal. In certain scenarios, the full power of a standard Turing machine might be too much, or unnecessary. Here we talk about the Restricted Turing Machines. These machines are variations of the standard Turing machine but with specific limitations. In this chapter, we will highlight the concept of restricted machines in detail with examples for a clear understanding.
Basics of Turing Machines
Let us see the basics of Turing machines for a recap to understand the restricted machines in a better way. As we know, a Turing machine can accept recursively enumerable languages and compute partial recursive functions. It consists of a tape that acts as both input and memory, a tape head that reads and writes symbols on the tape, and a finite state control that guides the machine's operations. There are a few important terms associated with them.
- Recursively Enumerable Language − A language that a Turing machine can recognize or enumerate.
- Deterministic vs. Non-Deterministic Turing Machines (DTM vs. NTM) − By default, Turing machines are deterministic (DTM), but they can also be non-deterministic (NTM). It can be proved that, DTM and NTM have the same computational power.
A standard Turing machine can act as both a Recognizer (accepting or rejecting strings) and an Enumerator (listing out all strings in a language).
Restricted Turing Machines
Now let us see the idea of restricted Turing machines. These are variations of the standard Turing machine but with specific limitations. These restrictions generally reduce the computational power of the machine, making them less powerful than a full Turing machine.
Here are some common types of restricted Turing machines −
- Halting Turing Machine
- Linear Bounded Automata
- Unidirectional Turing Machine
- Read-Only Turing Machine
- Read-Only Unidirectional Turing Machine
Read the following sections to get an overview of each of these restricted Turing machines.
Halting Turing Machine
A Halting Turing Machine is always halts for every input string. This means that no matter what input it receives, the machine will eventually stop its computation. This type of machine can accept recursive languages, which are a subset of recursively enumerable languages.
If we remember the definitions, recursive languages are those for which there exists a Turing machine that halts on all inputs and correctly decides whether each string belongs to the language.
Example
Consider a Turing machine designed to determine whether a given number is even or odd. A halting Turing machine for this task would, read the input number. Then, determine if it's even or odd. Halt after making the determination, ensuring that the machine always stops for any input.
Linear Bounded Automata
A Linear Bounded Automata (LBA) is a special type of Turing machine where the tape's storage space is limited to the length of the input string.
Unlike a standard Turing machine, which has an infinite tape, an LBA has a tape that is only as long as the input, making it less powerful but still more powerful than a pushdown automaton (PDA). LBAs are particularly important because they can accept context-sensitive languages.
Example
Consider the language $\mathrm{L \:=\: \{ a^n b^n c^n\: \colon\: n\: \geq\: 1 \}}$. This language consists of strings where the number of 'a's, 'b's, and 'c's are all equal.
An LBA can be constructed to: verify that the number of 'a's, 'b's, and 'c's are the same. Ensure that the tape head does not exceed the input length, effectively using the limited tape space.
Unidirectional Turing Machine
A Unidirectional Turing Machine has a tape head that can only move in one direction, either left or right but not both. This restriction limits the machine's power, making it equivalent to a finite automaton (FA), which can only recognize regular languages.
Example
Consider a unidirectional Turing machine designed to recognize the language $\mathrm{L \:=\: \{ w\: \mid \: w\:\in\: (0,\: 1)^* }$ and w ends with 1}.
Here if can start at the leftmost '0' or 1 and move right. If there is 1 it can move to another state but for 0 come back to previous state, since it will end with 1, if input ends and it is from the next state it will accept that
Read-Only Turing Machine
Another variation is a Read-Only Turing Machine. In this type of Turing machine where the tape head can only read symbols and cannot write on the tape. This restriction makes the machine equivalent to a finite automaton, meaning it can only accept regular languages.
Example
A read-only Turing machine might be used to check if a string belongs to the language $\mathrm{L \:=\: \{ 0^n\: \colon\: n\: \geq\: 1 \}}$. The machine would −
Read each symbol on the tape. Since it cannot write, it simply checks if all symbols are '0' and halts, accepting the string if true.
Read-Only Unidirectional Turing Machine
Another variation of read only machine is Read-Only Unidirectional Turing Machine, which combines the restrictions of being read-only and unidirectional. The tape head can only read and move in one direction, making this machine very similar to a finite automaton. It can only recognize regular languages.
Example
Consider a machine tasked with checking if a string belongs to the language $\mathrm{L \:=\: \{01^n \:\colon\: n\: \geq\: 1 \}}$. This machine would:
Start at the first '0' and move right. Ensure that all following symbols are '1'. Since it can only move in one direction and cannot write, it checks the sequence and accepts or rejects based on the pattern.
Conclusion
Restricted Turing machines are constrained over normal Turing Machines. There are several variations in such machines including linear bounded automata, restricted machines, etc.
In this chapter, we explained the concept of restricted machines in detail where we highlighted their types and presented some examples to show what type of language they can accept and an overview on how they can accept these languages.