Recursive and Recursively Enumerable Languages in TOC

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Recursively Enumerable (RE) Language is an interesting and important concept in Automata Theory. In this chapter, we will provide an overview on the basics of Recursively Enumerable languages, and their properties and also see why we need this in theoretical computer science.

Recursively Enumerable Languages

In simple words, a "language" is a collection of strings, like words in a dictionary. A recursively enumerable language is a language where we can create a computer program (or a Turing machine) that can systematically list out all the strings that belong to the language.

Consider a machine that can generate all the possible sentences in the English language, one by one. This machine wouldn't necessarily know which sentences are not in the English language, but it could list out all the valid sentences. This is the idea of a recursively enumerable language. It can enumerate all the strings that are part of the language.

Turing Machines and Its Acceptance

The Turing machine can be used in defining Recursively Enumerable languages. It is a theoretical model that can perform basic operations like reading, writing, and moving on a tape. A Turing machine can be programmed to accept or reject strings based on certain rules.

A Turing machine recognizes a language if, for every string that belongs to the language, the machine eventually enters a special "accept" state. For strings that are not in the language, the machine might either enter a "reject" state or continue running indefinitely without stopping.

A Turing machine can loop forever on strings that are not part of the language. This means it might never stop processing, never reaching a "reject" state. This unique characteristic is a defining feature of RE languages.

Recursive Languages: A Subset of RE Languages

Another important subset of RE languages is recursive language. In a recursive language, the Turing machine not only accepts strings belonging to the language but also always halts for strings that are not in the language.

For instance, consider a machine that can recognize all the valid sentences in the English language and, for any string that is not a sentence, will say "This is not a sentence" and then stop. This machine would be capable of recognizing a recursive language.

As an example, Consider the language $\mathrm{L \:=\: \{a^{n}b^{n}c^{n}\:|\: n\:\geq \: 0\}}$. This language consists of strings where the number of a's, b's, and c's are equal.

• RE Language − We can build a Turing machine that starts at the beginning of the string and systematically checks if the number of 'a's, 'b's, and 'c's are equal. If they are, it accepts the string. However, if the string is not of this form, the machine may never halt, potentially looping forever. This makes L a recursively enumerable language.
• Recursive Language − We can also construct a Turing machine that checks if the number of 'a's, 'b's, and 'c's are equal. If they are, it accepts the string. If they are not, it reaches a "reject" state and halts. This makes L a recursive language as well.

Closure Properties of Recursive Languages

Recursive languages possess an interesting property called closure. This means that certain operations performed on recursive languages result in another recursive language.

Here are some key closure properties −

Union

If L1 and L2 are two recursive languages, their union (L1 ∪ L2) is also recursive. Imagine a machine that checks if a string belongs to L1 or L2; if it does, it accepts. Since both machines for L1 and L2 will eventually halt, this combined machine will also halt, making the union recursive.

Concatenation

If L1 and L2 are two recursive languages, their concatenation (L1.L2) is also recursive. Imagine a machine that first checks if the first part of the string belongs to L1, and if it does, it checks the remaining part of the string for L2. Since both L1 and L2 machines halt, this combined machine will also halt.

Kleen Closure

If L1 is a recursive language, its Kleen closure (L1*) is also recursive. This means the language including all possible combinations of strings from L1 concatenated together, including the empty string, is also recursive.

Intersection

If L1 and L2 are two recursive languages, their intersection (L1 ∩ L2) is also recursive. Imagine a machine that checks if a string belongs to both L1 and L2. Since both L1 and L2 machines halt, this combined machine will also halt.

Complement

If L1 is a recursive language, its complement (L1') is also recursive. This means the language containing all strings *not* in L1 is also recursive.

Conclusion

In this chapter, we presented a basic overview of recursively enumerable languages which gives a powerful framework for understanding what can be computed by machines. We covered their properties, examples, and closure properties as well, for a clear understanding of these concepts.