Alan Turing and the Turing Machine: The Foundation of Modern Computing

ByGeorges Lteif
1 Comment

1. Introduction

Alan Turing is widely recognized as one of the most influential figures in the development of modern computing. His groundbreaking cryptography and computer science work laid the foundation for today’s technology. At the heart of his contributions is the concept of the Turing Machine, a theoretical device that revolutionized the way we think about computation.

In this article, we will explore Alan Turing’s life and achievements, the concept of the Turing Machine, and its importance in modern computing.

2. Series Agenda

This is the second article in this series on computation. Links to the rest can be found below.

1

Finite State Machines

An Introduction to FSMs and their Role in Computer Science

2

Alan Turing and the Turing Machine

Universal Turing Machines and The Foundation of Modern Computing

3. Alan Turing: A Life of Achievements

Alan Turing was born in 1912 in London, England. He attended King’s College, Cambridge, where he studied mathematics and became interested in the foundations of logic and computation.

During World War II, Turing worked as a codebreaker at Bletchley Park, where he played a crucial role in breaking the German Enigma code. After the war, Turing worked on developing the first electronic computers and made essential contributions to the emerging field of artificial intelligence.

Tragically, Turing’s life was cut short when he died in 1954 at the age of 41.

Turing’s work on the Turing Machine laid the foundation for developing computer science as a discipline. His paper “On Computable Numbers” demonstrated that some problems are not computable, meaning that any algorithm cannot solve them. This concept has important implications for developing computer algorithms and the limits of what computers can do.

Turing is also famous for his work on the Turing Test, which measures a machine’s ability to exhibit intelligent behaviour indistinguishable from a human’s. The Turing Test is still used today as a benchmark for measuring the progress of artificial intelligence research.

4. The Turing Machine

4.1 What Is a Turing Machine?

Alan Turing puzzled over the following problem: Can a machine programmed with a set of instructions perform any computation that a mathematician can do? The answer was the Turing machine, which demonstrates that all computation can be performed by a machine that follows a set of rules. This is known as the Church-Turing thesis, which states that a Turing Machine can compute any function an algorithm can.

The Turing Machine is a theoretical model of a simple computing device that can perform calculations and manipulate symbols on an infinite tape. It was first described by Alan Turing in his 1936 paper “On Computable Numbers, with an Application to the Entscheidungsproblem“.

A Turing machine is an improved version of the classical finite state machine. The key to this improvement is its ability to use an infinite tape from which data can be read and written and move on that tape in both directions.

4.2 How Is a Turing Machine Constructed?

The Anatomy of a Turing Machine

The Turing Machine comprises three main components: the tape, the head, and the state table.

  • The tape:
  • The tape is divided into cells, each holding a symbol from a finite alphabet.
  • The head:
  • The head is a device that can read and write symbols on the tape and move left or right along the tape.
  • The state table:
  • The state table is a set of instructions that tells the head what to do based on the symbol it reads and its current state.

4.3 How Do Turing Machines Work?

The image below shows the design of a Turing machine with an internal state table comprising three states, a head that can move left or right, and a tape with some symbols on it.

A Turing machine comprises a head, a tape where symbolic data can be read and written, and a state table defining the transitions between states.
A Turing machine comprises a head, a tape where symbolic data can be read and written, and a state table defining the transitions between states.

The Turing machine works as follows:

  • Step 1: Preparing the tape and positioning the head
  • A finite number of cells contain data that the machine will process. On both sides, the tape is ideally blank and infinite.
  • The machine’s head starts above a cell containing the first element to be processed. For example, the symbol in that cell can be START.
  • The machine’s state table, which provides the transition rules of its internal states, is programmed into the machine during the construction phase.
  • Step 2: Processing the symbols
  • The machine starts by reading the symbol on the tape.
  • The symbol on the tape and the machine’s current state determines the next state, the output, and the direction the head will move.
  • The output symbol is written to the current cell, and the head moves left or right.
  • Step 3: Halting
  • The Turning machine must be programmed to halt when the computation is completed.
  • This can be done by having a special symbol on the tape, for example, END.

4.4 Turing Machines vs Finite State Machines

In a previous article on Finite State Machines (FSMs), we discussed how an FSM could help us solve problems like binary addition, parity checking, or pattern matching. An FSM, however, cannot perform complex pattern matching or multiplication. These are problems that a Turing machine can solve. A finite state machine and a Turing machine are both models of computation, but they differ significantly in terms of computational power and complexity.

  • Finite State Machine (FSM):
  • Definition: An FSM is a mathematical model consisting of a finite number of states, transitions between these states, and input and output symbols. It operates by transitioning between states based on input symbols.
  • Limitation: A state machine’s main limitation is its finite number of states. This makes it hard to solve problems where the inputs are of arbitrary lengths, requiring an arbitrarily large number of states. A state machine’s memory is finite.
  • Turing Machine (TM):
  • Definition: A Turing machine consists of an infinite tape divided into cells, a read/write head that can move left or right along the tape, a finite set of states, and a transition function that determines the machine’s behaviour based on the current state and the symbol read from the tape.
  • Capabilities: TMs are more powerful than FSMs because they can traverse a tape of arbitrary length (in other words, it has unbounded memory) and go back and forth on it. This bidirectional movement allows Turing machines to traverse back and forth on the tape, facilitating operations such as backtracking, which is essential for various computational tasks like parsing, searching, and problem-solving.

5. Examples of Turing Machines

5.1 Implementing an AND Operator

We want to build a Turing machine that performs an AND operation on two bits. This machine will look as follows.

  • The tape:
  • The tape is prepared such that the symbol where the head is positioned is START.
  • The two symbols after START are our operands. These are two binary bits, each taking a value of either 0 or 1.
  • The fourth cell on the tape contains a blank reserved for the result.
  • The fifth cell has the HALT symbol, where the machine would halt.
  • The state table:
  • The state table requires two states, a START and an END. The starting state is START, and the END state is reached once the HALT symbol is read.
  • If the machine reads START, it moves to the right without changing the contents of the current cell.
  • The head then reads the contents of the current cell, which could be either 0 or 1.
  • If the symbol in the current cell is 0, the head moves two cells to the right, writes a 0 (the result of our operation) and moves right again.
  • If the symbol in the current cell is 1, the head moves one cell to the right. Two scenarios are possible: the new cell might contain 0 or 1. If it is the former, the head moves to the right, writes 0 in the result, and moves right again. If it is the latter, it will do the same, except write 1 in the result cell.
  • Finally, the head will be above the HALT symbol, which, after reading and processing, will signal the end of the computation.

The AND operation is one of the most straightforward problems a Finite State Machine could effectively resolve, even for inputs of arbitrarily large sizes. The solution is to search the input from left to right for the first 0 and then keep moving to the right until it hits the first blank cell (the result cell), where it would write 0. If it arrives at the blank cell without encountering a 0, it writes a 1. A few internal states and transition rules are enough for such a machine.

A more interesting example would be a regular expression search with wildcards, such as 1.*7. This regular expression matches any substring that starts with one and ends with seven, with anything in between.

5.2 Implementing a REGEX Search

Like the AND operator, we will specify what’s on the tape and the machine’s internal states.

An implementation of a Turing machine that searches a string trying to match a regular expression with wildcards, such as 1.*7
An implementation of a Turing machine that searches a string trying to match a regular expression with wildcards, such as 1.*7
  • The tape:
  • The tape is prepared such that the symbol where the head is positioned is START.
  • To keep things simple, we assume all the subsequent cells after the START have random numbers between 0 and 9.
  • We place an END symbol at the end of the random number sequence, delimiting the input.
  • After the END symbol, we have an empty cell for writing the result.
  • The last cell will have the HALT symbol.
  • The state table:
  • Like before, the machine’s head is placed at the START symbol, and its internal state is also the initial state. Next, the head moves to the right.
  • While moving through the tape, if the symbol on the current cell is not a 1 or 7, the machine overwrites it with an X.
  • While moving to the right, if the machine encounters a 7, it replaces it with a Y, reverses direction, looks for a 1, and ignores any X symbols in between.
  • While moving to the left, the machine would reverse course if it hits START or Y.
  • After locating a 7, the machine scans for a 1. If a 1 is found, the machine moves again to the right, ignoring anything other than the END symbol. It will then write a 1 in the result cell, indicating that a match has been found.
  • If the END symbol is reached, the machine will conclude the computation, write a 0 (match not found) and halt.

This problem, because it requires backtracking and updating the input dynamically during the search process, cannot be easily implemented with a finite state machine without exploding the number of internal states.

6. Computation and the Halting Problem

6.1 The Halting Problem

The halting problem, formulated by Alan Turing in 1936, is fundamental in computer science and seeks to determine whether a given program will halt (terminate) or run indefinitely for a particular input. Formally, the halting problem can be stated as follows:

Given a description of a program P and its input x, determine whether P halts (terminates) when executed on input x.

In other words, the halting problem asks whether an algorithm exists that can correctly decide whether any arbitrary program, when given any input, will eventually halt or continue running indefinitely.

6.2 Solving the Halting Problem

Alan Turing proved that solving the halting problem for all possible programs and inputs is impossible. He did so by employing a proof by contradiction, assuming the existence of a Turing machine H can solve the halting problem. Using H, one could construct another Turing machine D (the “diagonalization machine”) that takes its own description as input and behaves as follows:

  • If H predicts that ( D ) will halt when given its own description as input, then D enters an infinite loop.
  • If H predicts that D will run indefinitely when given its own description as input, then D halts.

By construction, D behaves contrary to what H predicts, leading to a contradiction. Hence, H cannot exist, and the halting problem is undecidable.

A universal Turing machine plays a crucial role in understanding the halting problem. A universal Turing machine is a theoretical construct that can simulate the behaviour of any other Turing machine when provided with the appropriate description of the machine and its input. Since Turing machines can represent any algorithm or computer program, a universal Turing machine can effectively simulate the execution of any program.

The connection to the halting problem arises because if we had an algorithm to determine whether any program halts on a specific input, we could use it to simulate that program on the universal Turing machine and decide whether or not it halts. However, Turing’s proof shows that no such algorithm exists, thereby establishing the undecidability of the halting problem. Thus, the existence of the halting problem highlights the limitations of computation and the boundaries of what can be computed algorithmically.

6.3 The Halting Problem and Gödel’s Incompleteness Theorem

Gödel’s incompleteness theorems, formulated by mathematician Kurt Gödel in the 1930s, establish limitations on formal mathematical systems. The first incompleteness theorem states that in any consistent formal system sufficiently strong to express arithmetic, true but unprovable statements exist within the system itself. The second incompleteness theorem shows that such a system cannot prove its own consistency if it is consistent.

The halting problem, on the other hand, is a problem in computability theory that deals with computation’s limitations. It states that no algorithm can determine whether a program will halt or run indefinitely for all possible inputs.

The connection between Gödel’s incompleteness theorems and the halting problem is that both demonstrate the existence of limitations or undecidable problems within formal systems. Gödel’s incompleteness theorems illustrate the existence of true but unprovable statements within formal mathematical systems. At the same time, the halting problem demonstrates the existence of problems that cannot be solved algorithmically by any computer program.

These results collectively underscore the inherent limitations of formal systems and computation, contributing to our understanding of the boundaries and constraints of mathematics, logic, and computer science.

7. The Church-Turing Test

The Church-Turing thesis, also known as the Church-Turing hypothesis, is a fundamental concept in computer science and mathematics. It asserts that a Turing machine can compute any function that a mechanical procedure can and vice versa. This hypothesis was proposed by Alonzo Church in 1936 and later expanded upon by Alan Turing in 1937.

The Church-Turing thesis provides a framework for analyzing the limits of computation and has profoundly impacted the development of theoretical computer science. One necessary consequence of the thesis is that it implies the existence of undecidable problems, meaning that no algorithm can solve them for all possible inputs.

The Church-Turing test is a practical way to apply the Church-Turing thesis. It involves comparing the computational capabilities of two systems to determine whether they are equivalent. For example, one could compare the capabilities of classical and quantum computers to decide whether or not they can solve the same set of problems.

To perform a Church-Turing test, one first identifies a problem that is known to be solvable by a Turing machine. Then, one implements a program for the problem on the tested system and a reference Turing machine. If the program can be run on both systems and produces the same output for the same inputs, then the two systems are considered equivalent regarding the tested problem. If the program produces different outputs on the two systems, they are not equivalent.

The Church-Turing test has been used to study various computational systems, including classical computers, quantum computers, neural networks, and cellular automata. It is an essential tool for understanding computation limits and developing new computational models.

8. Quantum Computers and the Church-Turing Test

One example of how the Church-Turing test has been applied is in the study of quantum computers. Quantum computers are computing systems that use quantum mechanical phenomena, such as superposition and entanglement, to perform certain computations more efficiently than classical computers.

However, it is unclear whether quantum computers are more powerful than classical computers or if they can be simulated by a classical computer using a classical algorithm. The Church-Turing test provides a way to compare the computational capabilities of quantum and classical computers.

Human or machine? The Church-Turing test helps determine whether machines can exhibit human-like intelligence.
Human or machine? The Church-Turing test helps determine whether machines can exhibit human-like intelligence.

One way to perform a Church-Turing test for quantum computers is to consider a problem known to be solvable by a quantum computer, such as factoring large numbers. Then, one can implement a program for the problem on both a quantum and classical computer.

If the program can be run on both systems and produces the same output for the same inputs, then the two systems are considered equivalent regarding the tested problem. If the program produces different outputs on the two systems, they are not equivalent.

For example, in 2017, a team of researchers at IBM used the Church-Turing test to compare the computational capabilities of a five-qubit quantum computer and a classical computer. They implemented a quantum algorithm for a specific problem on both systems and compared the outputs. They found that the quantum computer could perform the computation more efficiently than the classical computer, indicating that it was more powerful for this problem.

This is just one example of how the Church-Turing test can be applied to compare the computational capabilities of different systems. It is an important tool for understanding computation limits and developing new computational models.

9. Conclusion

Alan Turing’s legacy as a pioneer in computer science is inextricably linked to the concept of the Turing Machine. His groundbreaking work on the theoretical foundations of computation paved the way for the development of modern computing technology.

The Turing Machine is a powerful tool for reasoning about computation and has been instrumental in the development of modern computer technology. Turing’s contributions to cryptography, artificial intelligence, and the limits of computation have profoundly impacted the digital age.

10. References

One Comment

Leave a Reply

Your email address will not be published. Required fields are marked *