Church’s Thesis for Turing Machine

The Church-Turing Thesis is an important idea in the study of computability i.e. the ability to solve problems using a set of rules or procedures. It is an abstract model of a computing device, proposed by Alan Turing and Alonzo Church. It helps define algorithms and computing processes.

The basic idea of the thesis is:

• Any effective calculation or computation that can be done by a human (following specific steps) can also be done by a Turing machine.
• A Turing machine is a simple mathematical model that represents a basic form of a computer. This machine helps explain the logic behind modern computers.

To define these algorithms clearly, Alonzo Church developed a method called "M" for manipulating strings using logic and mathematics. This method must meet the following criteria:

1. Finite instructions: The method must have a limited number of steps.
2. Finite output: The method should produce a result after a certain number of steps.
3. Real-life feasibility: The method should be physically possible.
4. Simple to understand: It should not require complex understanding.

Based on these conditions, Church proposed the Church-Turing Thesis, which states:

"Every computation that can be done in the real world can be effectively performed by a Turing machine."

This idea was first formulated by Church in 1930 and is known as the Church-Turing Thesis. Although it cannot be proven, the hypothesis assumes that all computable functions can be represented by partial recursive functions.

In simpler terms:

• Assumption 1: Every function must be computable.
• Assumption 2: If a function F is computable and you perform basic operations on it to get a new function G, then G is also computable.

Importance of the Church-Turing Thesis

• Defines computability: The Church-Turing Thesis provides a clear definition of what is considered "computable" in computer science.
• Standard for algorithms: It defines "algorithmically computable" as anything that can be computed by a Turing machine.
• Basis for computability: It helps in understanding which problems can be solved using computers.
• Foundation of computer science: The thesis is fundamental to the study of computability and sets the foundation for modern computer science.

Relationship Between Turing Machines and Lambda Calculus

Equivalent Power

Though they look very different, Turing Machines and Lambda Calculus are equivalent in terms of computational power. This means:

• Anything you can compute with a Turing Machine can be computed using Lambda Calculus.
• Anything you can compute with Lambda Calculus can be simulated by a Turing Machine.

This equivalence was crucial to supporting Church’s Thesis because it showed two very different formal systems capture the same notion of “computable function.”

Different Perspectives

• Turing Machines focus on a mechanical step-by-step process involving states and symbols on a tape.
• Lambda Calculus focuses on symbolic manipulation of functions and substitution.

Both models provide a formal way to define algorithms but from different angles: one machine-based, the other function-based.