Church TuringThesis

 

⚙️ Why the Thesis Was Needed

After Turing introduced his machine model (and Church introduced the lambda calculus around the same time), mathematicians and logicians faced a problem:

They wanted a precise definition of what it means for a function or process to be computable — but there were many ways to describe an algorithm:

• pseudocode,

• flowcharts,

• block diagrams,

• high-level programs,

• or low-level mechanical steps.

Writing full Turing machine programs for all of them is theoretically possible — but in practice it’s tedious and unhelpful.
So, the question became:

Can we say that anything that can be done on a computer can also be done by a Turing machine?

The Church–Turing Thesis answers yes.

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🧩  The Thesis in Simple Words

If you can describe a process (algorithm) clearly enough that a human or a computer can follow it step by step, then there exists a Turing machine that can do the same thing.

So, the Turing machine becomes the mathematical definition of an algorithm or computable process.

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📘 Why It’s Called a “Thesis” (Not a Theorem)

Kozen emphasizes that:

The Church–Turing Thesis cannot be proved.

Why?

Because to prove it, we’d need a formal definition of what “mechanical computation” means — and that’s exactly what the thesis tries to define.
It’s not a statement within mathematics but a definition that connects mathematics with our intuitive understanding of computation.

So, it’s more like an axiom or a law, similar to Newton’s laws in physics:

• They cannot be proven true.

• But they are accepted because they agree with all known evidence.

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💡  Supporting Arguments for the Thesis

Three strong reasons for believing in the Church–Turing Thesis:

1. 🖥️ Everything that can be done on a digital computer can be done on a Turing machine.
   
   

2. 📜 No counterexamples exist.
   No one has ever found a task that we intuitively call “computable” that a Turing machine cannot perform.

3. 🔁 All equivalent models of computation have the same power.
   Many alternative models have been proposed (e.g., lambda calculus, recursive functions, register machines), and all have been shown to be equivalent to Turing machines in computational power.

Hence, even though it’s not provable, the evidence is overwhelmingly in its favor.

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⚖️ Why It Matters

The Church–Turing Thesis forms the foundation of computer science.
It lets us define what an algorithm is, and what it means for a problem to be computable.

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🧮 Formal Definition of an Algorithm

Definition:
An algorithm for a function

$$f : D \to R$$

is a Turing machine $M$ such that, for every input $d \in D$,the machine halts with output $f(d) \in R$ on its tape.

In simpler words:

• If the Turing machine always halts and gives the correct output, it represents an algorithm for that function.

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🔍 8. Church–Turing Thesis as a “Law of Computer Science”

Kozen beautifully compares it to Newton’s laws in physics:

• Newton’s laws are not logically necessary, but they explain real-world behavior accurately.

• The Church–Turing Thesis plays a similar role:
  it defines what “computation” means and matches every observation about what computers can actually do.

It might someday be refined or replaced, but no contradiction has been found so far.

Hence, the Church–Turing Thesis can be viewed as a basic law of computer science.

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🧭  Summary

Concept	Meaning
Turing machine	A mathematical model that defines computation.
Church–Turing Thesis	Anything computable by an algorithm can be computed by a Turing machine.
Proof status	Not provable — accepted as a definition or axiom.
Importance	Defines the boundary of what is mechanically computable.
Implication	If something cannot be computed by a Turing machine, then it cannot be computed by any digital computer.

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💬 10. Intuitive Summary (in Kozen’s spirit)

• Writing low-level Turing programs is tedious.

• But we can safely assume that any algorithm or program we can describe can be expressed as a Turing machine.

• Hence, we use higher-level languages or descriptions while trusting that, by the Church–Turing Thesis, they are equivalent in computational power.