# Higman's Theorem in Theory of Computation

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In this article, we have presented Higman's Theorem in Theory of Computation which was formulated in 1952. We have outlined the overview of proof of Higman's Theorem.

Table of contents:

- Higman's Theorem in Theory of Computation
- Proof of Higman's Theorem

Let us get started with Higman's Theorem in Theory of Computation.

# Higman's Theorem in Theory of Computation

Basic terms to understand Higman's Theorem in Theory of Computation:

Î£ is a finite alphabet.

For two given strings x and y which belongs to Î£*, x is a subsequence of y if x can be obtained from y by deleting zero or more alphabets in y.

L be a language which is a proper subset of Î£*.

We define subsequence as:

SUBSEQ(L) = {x : there exists y âˆˆ L such that x is a subsequence of y}.

In simple terms, **SUBSEQ(L)** is a language that consist of subsequence of all strings in L.

Highman's Theorem states that: For any finite alphabet Î£ and for a given language L which is a proper subset of Î£*, then the **language SUBSEQ(L) is a regular language**.

Higman's Theorem was found in 1952.

# Proof of Higman's Theorem

To prove Higman's Theorem, we need to prove or use an existing result which was first formulated in 1913:

Consider the case Î£ = {0, 1}.

If L = null or SUBSEQ(L) = {0, 1}* , then SUBSEQ(L) is a Regular Language by default. Therefore, assume L is non-empty language and SUBSEQ(L) is a proper subset of {0, 1} *.

Prove the following first:

- SUBSEQ(L) is a proper subset of A.
- not SUBSEQ(L) = (A intersection (not SUBSEQ(L)) union (not A)
- The Language (not A) is a Regular Language.
- If b belongs to {0, 1} and o <= k <= N and x is an element of M
_{bk}, then the Language { y belongs to A_{bk}: x is a subsequence of y} is described by the regular expression 11111* 0000* 11* 0000* - For each b belonging to {0, 1} and o <= k <= N, the set M
_{bk}is a finite set.

You will eventually arrive at: Since not SUBSEQ(L) is a Regular Language, the Language SUBSEQ(L) is a Regular Language based on the Higman's Theorem.

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