Church's Thesis for Turing Machine

In this chapter, we will cover an important concept in Automata Theory, the Church's Thesis, which is sometimes also called as the Church-Turing Thesis. This is the fundamental concept based on which a class of problems have been introduced including decidability problems, etc.

What is Computability?

Through this Church's Thesis, we will learn what we mean by the term "computable". In this chapter, we will start discussing its origins and the concept in detail for a better understanding.

The term "computable" was a fuzzy concept, with no proper definition or standard before. Mathematicians and logicians Alonzo Church and Alan Turing gave an idea about this terminology, to find a way to explain what could or could not be computed.

Alonzo Church and Lambda Calculus

Alonzo Church was an American mathematician and logician who made significant contributions to mathematical logic and the foundations of theoretical computer science. He introduced a formal system called the Lambda Calculus, which is a mathematical model of computation.

Church's idea was that if something could be computed using Lambda Calculus, then it could be considered "computable".

Alan Turing and Turing Machines

Turing developed the concept of Turing machine, which is a theoretical device that manipulates symbols on a strip of tape according to a set of rules. He proposed that if something could be computed by a Turing machine, then it is "computable."

Alan Turing was a student of Alonzo Church. Through Lambda Calculus and Turing machines both Church and Turing arrived at the same conclusion: anything that could be computed by either of these models is what we call "computable."

The Church-Turing Thesis

The Church-Turing Thesis states that any function that can be computed algorithmically can be computed by a Turing machine. This thesis ties together the work of Church and Turing, providing a foundation for the modern understanding of computation.

The thesis implies that the Turing machine is a universal model of computation, meaning that anything that can be computed, in theory, can be computed by a Turing machine.

Importance of Church-Turing Thesis

The Church-Turing Thesis is a hypothesis based on b evidence, providing a standard for defining "computable" in computer science. It defines "algorithmically computable" as something that can be computed by a Turing machine.

Variations of Turing Machines

The Church-Turing Thesis focuses on the potential of adding specific features to Turing machines to enhance their computational capabilities.

Equivalence of Turing Machines and Lambda Calculus

Lambda Calculus are little variation of Turing Machine, although they operate in different ways, they are both equally powerful in terms of what they can compute. This equivalence forms the basis of the Church-Turing Thesis, which tells us that anything that can be computed using an algorithm can be computed by a Turing machine.

Decidable and Recognizable Languages

One of the most important concepts is Decidability problem. Through the Church-Turing Thesis that it can conclude that the problem Turing machines can recognize or decide. The languages can be categorized into decidable languages and Turing-recognizable languages.

Decidable Languages

A language is considered decidable if there is a Turing machine that will always halt and give a yes or no answer for every string in the language. If a string belongs to the language, the machine accepts it and halts. If the string does not belong to the language, the machine rejects it and halts.

Turing-Recognizable Languages

A language is considered Turing-recognizable (also known as recursively enumerable) if there is a Turing machine that will halt and accept a string if the string belongs to the language. However, if the string does not belong to the language, the Turing machine may either reject it and halt or go into an infinite loop and never halt.

Difference between Decidable and Recognizable Languages

The key difference between decidable and Turing-recognizable languages is in the halting behaviour of the Turing machine. For decidable languages, the Turing machine always halts, whether the string is in the language or not. For Turing-recognizable languages, the machine may not halt if the string is not in the language.

Undecidable Languages

Beyond decidable and Turing-recognizable languages, there are undecidable languages. For these languages, no Turing machine can decide whether a string belongs to the language because the machine never halts for some inputs. These languages lie outside the scope of what can be computed by any Turing machine.

Conclusion

In this chapter, we presented the basic concept of Church-Turing Thesis. The Lambda Calculus and the Turing Machines are equivalent. It defines what we mean by "computable" and establishes the Turing machine as the standard model of computation.