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Church’s Thesis for Turing Machine

In 1936, A method named as lambda-calculus was created by Alonzo Church in which the Church numerals are well defined, i.e. the encoding of natural numbers. Also in 1936, Turing machines (earlier called theoretical model for machines) was created by Alan Turing, that is used for manipulating the symbols of string with the help of tape.

Church Turing Thesis :

Turing machine is defined as an abstract representation of a computing device such as hardware in computers. Alan Turing proposed Logical Computing Machines (LCMs), i.e. Turing’s expressions for Turing Machines. This was done to define algorithms properly. So, Church made a mechanical method named as ‘M’ for manipulation of strings by using logic and mathematics. This method M must pass the following statements:

• Number of instructions in M must be finite.
• Output should be produced after performing finite number of steps.
• It should not be imaginary, i.e. can be made in real life.
• It should not require any complex understanding.

Using these statements Church proposed a hypothesis called

Church’s Turing thesis

that can be stated as: “The assumption that the intuitive notion of computable functions can be identified with partial recursive functions.”

Or in simple words we can say that “Every computation that can be carried out in the real world can be effectively performed by a Turing Machine.”

In 1930, this statement was first formulated by Alonzo Church and is usually referred to as Church’s thesis, or the Church-Turing thesis. However, this hypothesis cannot be proved. The recursive functions can be computable after taking following assumptions:

• Each and every function must be computable.
• Let ‘F’ be the computable function and after performing some elementary operations to ‘F’, it will transform a new function ‘G’ then this function ‘G’ automatically becomes the computable function.
• If any functions that follow above two assumptions must be states as computable function.

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What is The Church-Turing Thesis in TOC?

The Church-Turing thesis says that every solvable decision problem can be transformed into an equivalent Turing machine problem.

It can be explained in two ways, as given below −

The Church-Turing thesis for decision problems.

The extended Church-Turing thesis for decision problems.

Let us understand these two ways.

The Church-Turing thesis for decision problems

There is some effective procedure to solve any decision problem if and only if there is a Turing machine which halts for all input strings and solves the problem.

The extended Church-Turing thesis for decision problems

A decision problem Q is said to be partially solvable if and only if there is a Turing machine which accepts precisely the elements of Q whose answer is yes.

A proof by the Church-Turing thesis is a shortcut often taken in establishing the existence of a decision algorithm.

For any decision problem, rather than constructing a Turing machine solution, let us describe an effective procedure which solves the problem.

The Church-Turing thesis explains that a decision problem Q has a solution if and only if there is a Turing machine that determines the answer for every q ϵ Q. If no such Turing machine exists, the problem is said to be undecidable.

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Church Turing Thesis in Theory of Computation

Theory of computation.

Get this book -> Problems on Array: For Interviews and Competitive Programming

In this article, we have explain the meaning and importance of Church Turing Thesis in Theory of Computation along with its applications and limitations.

Table of contents:

• Introduction to Turing Church Thesis

Applications of Church Turing Thesis

Limitations of church turing thesis.

Prerequisites: Algorithms, Turing Machine

Let us get started with Church Turing Thesis in Theory of Computation.

Definition of Church Turing Thesis

Church Turing Thesis states that:

A computation process that can be represented by an algorithm can be converted to a Turing Machine.

In simple words, any thing that can be done by an Algorithm can be done by a Turing Machine as well. So, all algorithms can be implemented in a Turing Machine.

Specific Computation Models are equivalent which means any one model can be coverted to another model. These Computation Models include:

• One tape Turing Machine
• K tape Turing Machine where K >= 1
• Non Deterministic Turing Machine
• Programs in Programming Languages such as Java, C++, Lisp and others.

So, a program in C++ can be converted to a K tape Turing Machine and vice versa.

The applications of Church Turing Thesis are as follows:

• Church Turing Thesis is used to define an Algorithm (traditionally)
• Used in solving 10th Problem by Hilbert.
• Used in defining modern computing devices including Molecular and Quantum Computers.

10th Problem by Hilbert

It has been used to solve the 10th Problem by Hilbert in 8th August 1900 at the Second International Congress of Mathematicians in Paris. These problems were listed as critical problems that should be solved for progress in Mathematics.

The 10th Problem by Hilbert was:

Does there exists a process with finite number of steps that can determine if a given polynomial with integer coefficients has integral roots?

Another way to look at the problem is to find if there is an Algorithm to find if there exists an integral root for a given polynomial or not.

For example: This is a Polynomial:

35x 10 y 2 z 9 + 11x 6 z 7 + 103xyz + 17y 31 z 3 = 0.

Is there an algorithm that can find if there exist a solution in integers?

Note the solution is not needed. Only we need to find if such a solution exists or not.

In 1970, it was proved that no such algorithm exists. This was done by Matiyasevich.

Algorithm = Church Turing Thesis

To solve the 10th Hilbert Problem, one needs to understand what is meant by an algorithm. In fact, there have been different definitions and all have proved to be equivalent. Some definitions were:

• 1936: Algorithm = Turing Machine
• 1936: Algorithm = Lambda Calculus
• 1970+: Algorithm = Implementation in Programming Languages like C and Lisp
• Final: Algorithm = process converted to Turing Machine.

Finally, it was agreed that an Algorithm is based on Church Turing Thesis which said:

"Any computational process can be considered as an Algorithm if it can be converted to a Turing Machine." Note: This does not hold true as of now.

Modern Computing Devices

Traditional Computers which are in use today, are limited by Church Turing Thesis. This is because Church Turing Thesis defines an Algorithm which can be implemented in a real system.

Therefore, the Computing Device you are using is basically a Turing Machine.

The only difference is that Computing Devices are efficient while Turing Machine is inefficient. Theoretically, from a point of view of algorithms, there is no difference.

There are 3 different approaches future computers may take:

• Quantum Computer : Solve Computing Problems using atoms by quantum rules. This is an active area of research.
• Molecular Computer : Solve Computing Problems using Molecules by taking advantage of Physical laws of Moleculars. This includes replicating the idea of DNA.
• Super Recursive Algorithm : This domain has not been realized yet and exists in theory but this is the part where Church Turing Thesis fail. We have covered this in the next section on "Limitations of Church Turing Thesis".

Two different futuristic models of Computer which follows Church Turing Thesis:

• Quantum Computers can be represented as Non Deterministic Turing Machine
• Molecular Computers can be represented by Turing Machine with many tapes and heads

Therefore, Quantum and Molecular Computers are same fundamentally and they are only more efficient than Mechnical Computers.

Super Recursive Algorithms proved Church Turing Thesis wrong. The first Super Recursive algorithm was introduced in 1965 by Mark Gold and Hillary Putnam by using ideas of limit recursive and limit partial recursive functions. It was based on ideas from non standard analysis by Abraham Robinson in 1966 and Inductive Definition of sets by Spector in 1959. This resulted in Inductive Inference by Gasarch and Smith in 1997 and is used in Machine Learning.

Super Recursive Algorithms can solve problems that are unsolvable by Turing Machines. To account for this, a new idea was introduced: Inductive Turing Macine. These were not accepted as Algorithm for a long time as it was refuting Church Turing Thesis and Godel Incompleteness Theorem (as proved in 1987 by Burgin).

The idea of Inductive Turing Machine is as follows:

• Turing Machine has a property that it stops after giving a result.
• Most programs stop after giving a result and this seems to be reasonable as what a program should do once it has found the answer.
• Operating Systems are also programs but it does not give a standard output. It gives some strings to the users during its use but it cannot be considered as a output. The functionality of an Operating System is considered to be the output. It does not stop like standard program. If it stops, it cannot give any output.
• There can be programs which give a result at the moment which is good enough but
• This idea of not stopping after giving a result is the basis.

Inductive Turing Machine is more powerful than Conventional Turing Machine. Inductive Turing Machine can solve the Halting Problem which is known to be unsolvable by Conventional Turing Maching.

There are different types of Inductive Turing Machine:

• Inductive Turing Machine + Structured Memory
• Inductive Turing Machine + Structured Rules (control device)
• Inductive Turing Machine + Structured Head (Operating Device)

Today, Church Turing Thesis is not considered as an Universal Principle. Inductive Turing Machine is the most powerful super recursive algorithm.

This lead to the formulation of "Extended Church Turing Thesis".

There are three open questions:

• How to realize Super Recursive algorithms in technological devices?
• How modern computing devices are related to Super Recursive Algorithm?
• What are the new possibilities with Super Recursive Algorithm?

Think about these research open ended problems in Theory of Computation.

With this article at OpenGenus, you must have the complete idea of Church Turing Thesis in Theory of Computation.

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Many candidates are rejected or down-leveled in technical interviews due to poor performance in behavioral or cultural fit interviews. Ace your interviews with this free course, where you will practice confidently tackling behavioral interview questions.

In 1936, Alan Turing presented a theoretical model for machines, or presently known as Turing machines . We use this machine to manipulate the symbols of input strings on the tape.

We describe a Turing machine as an abstract representation of a computing device.

In the same year, Alonzo Church developed a lambda calculus method A system of mathematical logic used for expressing computation based on abstraction. It is used for the simulation of a Turing machine. . Encoding of natural numbers was introduced and called Church numerals.

Church-Turing thesis

Alan Turing gave the idea for logical computing machines that worked on Turing expressions.

A mechanical method was made for this. Assume this method is M M M , and we use it to manipulate strings by logic and mathematics.

Requirements for the Church-Turing thesis

This defined method M M M should pass the following:

• It shouldn't require any complex processing requirements.
• We obtain output after performing a finite number of steps.
• We can implement the method M M M in the real world.
• The number of instructions for M M M should be finite.

After considering these requirements, Church suggested a hypothesis called the Church-Turing thesis, defined below:

The assertion that partial recursive functions can be used to identify the intuitive concept of computable functions.

Despite this hypothesis, we can't prove this claim.

We describe this thesis under two main categories:

• Church-Turing thesis for decidability problems: A decidability problem A problem that is either true or false. can be solved effectively if there exists a Turing machine that halts for all of its input strings and calculates the solution.
• Extended Church-Turing thesis for decidability problems: A decidability problem is partially solvable if there exists a Turing machine that accepts the elements of the problem whose answer is "yes."

Proof for recursive functions

We can compute recursive functions by considering the following assumptions:

• All functions must be computable.
• Assume a computable function f f f . After performing elementary operations, the function will transform into a new function g g g . g g g is automatically a computable function.

Applications of the thesis

This thesis finds its applications in many calculable and computational fields. Some of these are:

• Lambda calculus
• Single and multiple tape Turing machines
• Counter machine model
• Register machine, a machine similar to the computer
• Markov algorithms
• Combinatory logic
• Pointer machines

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Church-Turing Thesis

The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent computation involving a Turing machine . In Church's original formulation (Church 1935, 1936), the thesis says that real-world calculation can be done using the lambda calculus , which is equivalent to using general recursive functions .

The Church-Turing thesis encompasses more kinds of computations than those originally envisioned, such as those involving cellular automata , combinators , register machines , and substitution systems . It also applies to other kinds of computations found in theoretical computer science such as quantum computing and probabilistic computing.

There are conflicting points of view about the Church-Turing thesis. One says that it can be proven, and the other says that it serves as a definition for computation. There has never been a proof, but the evidence for its validity comes from the fact that every realistic model of computation, yet discovered, has been shown to be equivalent. If there were a device which could answer questions beyond those that a Turing machine can answer, then it would be called an oracle .

Some computational models are more efficient, in terms of computation time and memory, for different tasks. For example, it is suspected that quantum computers can perform many common tasks with lower time complexity , compared to modern computers, in the sense that for large enough versions of these problems, a quantum computer would solve the problem faster than an ordinary computer. In contrast, there exist questions, such as the halting problem , which an ordinary computer cannot answer, and according to the Church-Turing thesis, no other computational device can answer such a question.

The Church-Turing thesis has been extended to a proposition about the processes in the natural world by Stephen Wolfram in his principle of computational equivalence (Wolfram 2002), which also claims that there are only a small number of intermediate levels of computing power before a system is universal and that most natural systems are universal.

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

More things to try:

• 50 digits of sqrt(2)+sqrt(3)
• expand sin(x+y+z)
• integer partitions of 10

Referenced on Wolfram|Alpha

Cite this as:.

Rowland, Todd . "Church-Turing Thesis." From MathWorld --A Wolfram Web Resource, created by Eric W. Weisstein . https://mathworld.wolfram.com/Church-TuringThesis.html

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Artificial Intelligence Models

The basic idea, theory, meet practice.

TDL is an applied research consultancy. In our work, we leverage the insights of diverse fields—from psychology and economics to machine learning and behavioral data science—to sculpt targeted solutions to nuanced problems.

These days, tech is everywhere. Society has steadily moved towards increased automation and digitization, which has only been exacerbated by the COVID-19 pandemic. Work-from-home orders and storefront closures have solidified modern society as a digital era.

Increased automation and digitization have been made possible thanks to artificial intelligence. Artificial intelligence is all about making computers and machines make decisions like humans. By programming computers to mimic human thinking patterns, they are able to perform aspects of our jobs — although a scary thought (cue the sci-fi movies of robots taking over the world), AI can make processes much more effective and often more accurate.

There are different models of artificial intelligence.

• Artificial intelligence models are the tools and algorithms used to train computers to process and analyze data – just as humans do.
• Machine learning  is a broad category that falls under the artificial intelligence model label, in which computers are taught to think by themselves and develop their own algorithms after processing vast amounts of data.
• Other artificial intelligence models need an algorithm to be programmed into the computer and will learn to adjust the algorithm based on experience.
• Lastly, there are also models that do not have the ability to learn on their own at all – they only function according to the preprogrammed algorithm and need human input. 1

For example, Google Maps and other navigation applications use artificial intelligence models to guide us to our destinations. The machine remembers the edges of buildings that it learned by using data from other travellers and through inputted data via an algorithm. As people use the application on a day-to-day basis, the model incorporates the data gathered from these travels and can give more accurate route information by recognizing changes in traffic flow. 2

However, a big question remains: do artificial intelligence models enhance humanity and society, or do they run the risk of making humans redundant? Here are two different opinions:

“ The development of full artificial intelligence could spell the end of the human race… It would take off on its own, and re-design itself at an ever increasing rate. Humans, who are limited by slow biological evolution, couldn’t compete, and would be superseded. ”

– Stephen Hawking, an English theoretical physicist, who discovered that black holes emit radiation and was the first to discover a theory of relativity and of quantum mechanics. 3

“ Some people call this artificial intelligence, but the reality is that this technology will enhance us. So instead of artificial intelligence, I think we’ll augment our intelligence. ”

– Ginni Rometty, American business executive who was the first woman to serve as the president and CEO of IBM. 3

Artificial Intelligence : a branch of computer science, where the engineering of machines mimics human problem solving and decision-making. It is the opposite of “natural intelligence”, exhibited by humans and animals. 4

The Artificial Intelligence Effect : a phenomenon in which people no longer see artificial intelligence for what it is after becoming a widespread part of daily life. It is seen as a tool because we are so used to technology completing a task and hiding the work behind it. For example, you likely don’t think of using Google Maps as using an artificial intelligence model! 5

Machine Learning : the process of a computer attempting to learn from the past. Data is inputted into a machine, gets passed through an algorithm (an artificial intelligence model) and churns out an output. If the computer returns the correct result, then it affirms the algorithm. If it is wrong, it adjusts its algorithm accordingly. 6

Neural Networks : Artificial models are designed with neural networks. Neural networks mimic how neurons in our brain interact with one another — an input triggers a response and creates an output. 6

Deep Learning : Deep learning is the way that machine learning functions. While some artificial intelligence models are built by first inputting an algorithm, deep learning is a technique where the machine develops an algorithm after encountering vast amounts of data. 4

Turing Machine : a Turing machine is a hypothetical machine developed by mathematician Alan Turing in 1936. It was a machine that, by changing data in 0’s and 1’s (simplifying data to its essentials) could simulate any computer algorithm. 7

Supervised Machine Learning Models : artificial intelligence models that require human training. People will tag sets of data, and the model will learn from the way that humans are analyzing the data. 8

Unsupervised Machine Learning Models:  artificial intelligence models which require no human input. These models are trained by software instead, which identifies patterns so that the computer can mimic it. 8

Semi-supervised Machine Learning Models : artificial intelligence models which combines both supervised and unsupervised machine learning approaches, using both human training and software training. 8

Mathematicians Alonzo Church and Alan Turing were the first to use computation as a device to conduct formal reasoning. They developed the Church-Turing thesis in 1936, which suggests that any real-world computation can be translated into an equivalent computation involving a Turing machine. The thesis was developed shortly after Turing developed the Turing machine, and opened the realm of possibilities for computer learning. People began to believe that it might be possible to build an electronic brain. 9

Since access to computers wasn’t widespread in 1936, it took a few years for the “electronic brain” to become a nuanced theory. The Turing model was only hypothetical, but in 1943, neuroscientist Warren Sturgis McCulloch and logician Walter Harry Pitts formalized it and created the first computation theory of mind and brain. In their paper titled “A Logical Calculus of the Ideas Immanent in Nervous Activity,” they explained how neural mechanisms in computers could realize mental functions. 10

However, artificial intelligence wasn’t a reality until 1949, because computers could not store commands. Although they could execute them, they could not retain an artificial intelligence model. No one had yet made that reality a possibility because computing was very expensive. The term artificial intelligence wasn’t even coined until 1955, and it was in that same year that computer scientist and cognitive psychologists Allen Newell, Cliff Shaw, and  Herbert Simon  created a proof of concept for artificial intelligence. They developed the Logic Theorist, a program that used artificial intelligence to mimic the problem-solving skills of a human. 11

From that moment on, many became interested in developing artificial intelligence models. In 1997, American computer scientist Tom Mitchell gave a more refined definition of artificial intelligence than had previously been expressed. He defined it as “ a computer program is said to learn from experience E with respect to some task T and some performance measure P, if its performance on T, as measured by P, improves with experience E .”  12

Let’s represent this using the Google Maps Example. If you want a computer to predict traffic patterns (task T), you would run a program through an artificial intelligence model with data about past traffic patterns (experience E) and once it has successfully learned, it will do better at predicting future traffic patterns (performance P). 12

Consequences

There are hundreds of practical, important uses of artificial intelligence models. AI models help make the analysis and processing of data more efficient and increase automation. Both deep learning and artificial intelligence models are revolutionizing society.

Initially, artificial intelligence models were reactive machines that couldn’t store any memory, which means they couldn’t learn from experience. These days, all artificial intelligence computers can store memory, which means machines are constantly getting better and better at analyzing data. While deep learning machines learn completely from experience, computers that abide by artificial intelligence models continue to refine their algorithms through experience. 13  These machines are making processes more efficient, reducing the need for human intervention (and therefore reducing human error), and can help organizations understand how to improve their functions. 14  

There are advantages of both machine learning models and artificial intelligence models that do not learn solely from experience but instead use pre-programmed algorithms. Those who use pre-programmed algorithms can quickly process data and deliver desired results. It doesn’t require additional time to “learn” what to do, only to refine its processes, thereby requiring simpler and cheaper machinery. Machine learning, although more expensive, can process more complex data and is self-sufficient thus not requiring as much human input.

Controversies

There are quite a few ethical controversies when it comes to artificial intelligence models.

One is that a lot of artificial intelligence models are used to “survey” our behavior, whether it be our digital footprint or facial recognition, and we don’t know how exactly the data is being used or stored.

In machine learning, since artificial intelligence models learn by themselves, ethical concern is that there is a lack of transparency with artificial intelligence tools. 15  For models that don’t abide by machine learning, there can exist biases in the algorithms that are inputted into the programs. For example, there has been a lot of controversy surrounding facial recognition after it became apparent that this technology was significantly less accurate in recognizing the faces of Black people. This occurred from a majority-white team creating the models, who themselves were not as accurate when distinguishing between people of color. Their bias became embedded within the artificial intelligence model. 16

There is also the question of whether artificial intelligence models can have morality included as part of their programming. If an autonomous (self-driving) car finds itself in a situation where a jaywalker will be hit if it doesn’t slam down on the brakes, it must decide between the safety of the people in the car and the safety of the pedestrian — how can a computer make that decision? 15

Some people also think that artificial intelligence is reducing our humanity and what is natural. Phenomena like “designer babies”, where people can choose what genes a child will have, are being debated as to whether they take away from what is natural. Such innovations require us to consider the moral and ethical aspects of artificial intelligence. As stated by the Executive Chairman of the World Economic Forum, Klaus Schwab, “ We must address, individually and collectively, moral and ethical issues raised by cutting-edge research in artificial intelligence and biotechnology, which will enable significant life extension, designer babies, and memory extraction .”  3

Where are artificial intelligence models used?

In Medicine

Artificial intelligence models can detect cancer in patients. By analyzing X-Rays and CRT images, they are able to detect abnormalities in the human body related to cancer. Since these days, those models abide by machine learning, they are becoming more accurate and able to recognize even abnormal cancers because it has learned through experience. 17

In Communication

Ever wonder how our phones predict what we’re about to say next? Our phones give us suggestions for the next word in a text message or predict the end of the sentence in an email. They also give us suggestions when they think we’ve misspelled a word. All of this is possible through artificial intelligence models, where our phones analyze our previous communication (and the communication patterns of the general population) to predict what we want to say next. 2

Chatbots & Digital Assistants

Chatbots have taken over customer service agents, whether it be an Alexa or Siri. Chatbots can efficiently answer frequently asked questions by analyzing the customer’s question and matching it to past experiences. Digital assistants listen to your voice, process, and analyze the data, and perform the desired function. 2

Targeted Ads

We’ve all heard of the conspiracy theories that our phones are listening to us, but our phones store so much data about us that they don’t need to listen to us to populate our social media with targeted ads. Based on your previous searches, the searches of people in your network, and demographic markers, artificial intelligence models predict what products you are most likely to buy and shows you them on your feeds. 2

Related Content

Combining AI and Behavioral Science Responsibly

Artificial intelligence is used to revolutionize all fields, including behavioral science. Since “artificial intelligence” is a bit of a buzzword and encompasses many variations of computer learning, this article helps break down what exactly artificial intelligence is and how it is used both positively and negatively. Our contributor Julian Hazell explores whether artificial intelligence really gives us greater insight into human behavior, or whether we program it to reinforce our pre-existing beliefs.

The AI Governance of AI

These days, data is one of the most valuable resources (sometimes more than money) and it governs much of our lives. Data determines what ads we are shown, what products we buy, and shapes our likes and dislikes. All of our choices are somewhat guided by data. However, in this article, our contributors Mark Esposito, Danny Goh, Josh Entsminger and Terence Tse question whether we should be comfortable living in a society where our behavior is shaped by AI, or more importantly, by the people controlling the AI. They ask the question — who is being held accountable for ensuring that the way AI is used, and data is shared is ethical?

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