Languages And Natural Languages Communications Essay


Language or known as formal language was invented by Stephen Cole Kleene, an American mathematician, in the 1950s. Formal languages are studied in computer science and linguistics. In computer science, formal languages are used for the precise definition of data formats and syntax of programming language. Formal languages are also used in logic and foundation of mathematics to represent syntax of formal theories.


Language means a set of words, for example, finite strings of letters (a, b, c, d, A, B, C, D, …) or symbols(+, -, *, /, =, …). Natural language is known as "body of words and methods of combining words used and understood by a considerable community", which is actually the language that we use daily to communicate with each other. The set which these letters or symbols are taken is known as alphabet over which the language is defined. The alphabet is represent by the symbol Σ. Whereas Σ* is the set of all finite strings including the empty string, λ(lambda), and Σ+ is all the elements in Σ* except the empty string, λ. From the definitions, Σ and Σ+ are both subset of Σ*. Given an example, Σ = {a, b, c, d}. The following shows some example of languages over Σ.

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{aa, abc, abd, aabbddc}


, The empty set


{anbndn | n > 0}

A language L is a proper language if and only if the empty string, λis not an element of L. Otherwise, it is known as improper language. An example of proper language over Σ, L = {"a", "b", "cd", "abc", "abcd'}. And an example of improper language over Σ, L = {"a", "abc", "aabbc", λ}.

Results Of The Research

1. Operations on Languages

1.1 Concatenation

The concatenation of languages is the concatenations of all strings from two languages. Let L1 and L2 be two languages over the alphabet Σ. And x is a string in L1, y is a string in L2. After the concatenation L1L2, the new set of elements in L1L2 consists of all strings in the form of xy. For example,

Σ= {a,b,c,d}

L1 = {"aa", "bb", "cc", "dd"}

L2 = {"bb", "aa", "ddc", "ccd"}

L1L2 = {"aabb", "bbaa", "ccddc", "ddccd"}

L2L1 = {"bbaa", "aabb", "ddccc", "ccddd"}

1.2 Intersection

The intersection of languages is all the strings which are contained in both languages. Let L1 and L2 be two languages over the alphabet Σ. And x is an element of the intersection of L1 and L2 (L1 ∩ L2) if and only if x ∈ L1 and x ∈ L2. For example,

Σ= {a,b,c,d}

L1 = {"aa", "bb", "cc", "dd"}

L2 = {"bb", "aa", "ddc", "ccd"}

L1 ∩ L2 = {"aa", "bb"}

1.3 Complement

The complement of a language with respect to a given alphabet, Σ consists of all strings over the alphabet that are not in the language. It consists of infinite strings unless a specific condition is mentioned. For example, given

Σ= {a,b,c,d}

L = {"aa", "bb", "abc", "abcd"}

And the question: "What is the complement of L?"

¬ L = { λ, "aaa", "aaaa", "abb", "bbbbb", "abbcddd", …}

This is an infinite of strings over the alphabet, Σ. If the question: "What is the complement of L of length 2"

¬ L = {"ab", "ac", "ad", "ba", "bc", "bd", "ca", "cb", "cc", "cd", "da", "db", "dc", "dd"}

This is finite answer with respect to the alphabet, Σ.

1.4 Kleene star

The Kleene star (or known as Kleene operator or Kleene closure) is the language consisting of all words that are concatenations of 0 or more words in the original language. The application of the Kleene star to a language, say L, is written as L*, which is the set of all strings over symbols in L, including the empty string, λ. For example,

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Σ= {a,b,c,d}

L = {"c", "d", "ab", "dc"}

L* = { λ, "cd", "cdd", "ab", "cab", "abc", "abd", "ddc", "abdc", "abababab", "ababdcdcdc", …}

1.5 Reversal

The reversal of a language is the reverse order of characters of all strings in the language. Let L be a language, and s be a string in L. From the definition, the reversal of L,

LR = {sR | s ∈ L}.

For example,

Σ= {a,b,c,d}

L1 = {"abc", "ddbca", "abcdabc", "dcdabdcbba"}

L1R = {"cba", "acbdd", "cbadcba", "abbcdbadcd"}

2. Regular Language

Regular language is the set of formal language which are finite. The following shows the collection of regular language,

The empty language Ø is a regular language.

The empty string language {ε} is a regular language.

For each x ∈Σ, the singleton language {x} is a regular language.

If L1 and L2 are regular language, then L1 ∪ L2 (union), L1 • L2 (cancatenation), L1*, L2* (Kleene star) are regular languages.

Besides the above, no other languages over Σ are regular.

As a summary, all finite languages are regular. For example,

The set of strings: {anbn | n < 3}.

3. Regular Expression

A regular expression, often called a pattern, is an expression that describes a set of strings. They are usually used to give a simple description of a set, without having to list all elements. The following operations are used to construct regular expression.

3.1 Boolean "or"

A vertical bar used to separate alternatives. For example, love | live can match "love" or "live".

3.2 Grouping

Parentheses (or bracket - "( )") are used to define the scope and precedence of the operators. For example, love | live and l(o|i)ve are equivalent patterns which both describe the set of "love" and "live".

3.3 Quantification

A quantifier after a token (such as a character) or group specifies how often that preceding element is allowed to occur. The most common quantifiers are the question mark ?, the asterisk * (derived from the Kleene star), and the plus sign + (Kleene cross).

? The question mark indicates there is zero or one of the preceding element. For example, mone?y matches both "mony" and "money".

* The asterisk indicates there is zero or more of the preceding element. For example, ab*c matches "ac", "abc", "abbc", "abbbc", and so on.

+ The plus sign indicates that there is one or more of the preceding element. For example, ab+c matches "abc", "abbc", "abbbc", and so on.

These operations can be used simultaneously in a regular expression, for example,

a | b* denotes {ε, a, b, bb, bbb, ...}

(a | b)* denotes the set of all strings with no symbols other than a and b, including the empty string: {ε, a, b, aa, bb, aaa, bbb, aaaa, bbbb, ...}

ab*(c | ε) denotes the set of strings starting with 'a', then zero or more 'b' and finally optionally a 'c': {a, ab, ac, abc, abb, abbb, abbc, abbbbbc, ...}

4. Natural Language

A natural language (or ordinary language) is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written. Natural language is distinguished from constructed languages and formal languages such as computer-programming languages or the "languages" used in the study of formal logic, especially mathematical logic.

It is possible to view the relations between mathematics and natural language from different aspects. Language is towards determining and perceiving the phenomenal relationships, and mathematics attempts to display abstract relations. We use natural language in every phase of mathematics to express mathematical symbols.

4.1 Relation Between Mathematics And Natural Language

Mathematics is not a natural language like Malay, English and Chinese. There is not such a society of which mother tongue is mathematics, besides it is not a dialect of any language like English and others. Yet mathematics is the common language of science that whole world can understand each other.

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Mathematics statement, can be read in several ways of natural language, for example:

3 + 3 = 9, this symbolic statement can be read in several ways, for example in English:

Three plus three is nine.

Three plus three makes nine.

If we add three to three, we have nine, etc.

During right time, an individual who uses natural language should use certain mathematical symbols accurately. For example, many numbers in mathematics can be used in function of adjectives describing words:

" Twelve ringgits, or the fifth anniversary, etc".

The difference between the language of mathematics and natural language mostly occurs within written language rather than in spoken language. The writing form of natural language is alphabetical, while the form of expression of mathematics is carried out through symbols.

Conclusions And Recommendations

From our research, language in mathematics do has its operations. The operations including concatenation, intersection, complement, Kleene star and reversal. By understanding all these operations on language, students would be able to do well in this topic. In the topic of language, it contains subtopic such as Regular Language and Regular Expression, students can further understand the topic of language if they have learn these subtopics.

It would be a significant mistake to consider mathematics is not a language regarding that the language of mathematics is entirely based on symbols. But, these symbols bring great meaning that people should understand and after that should be able to translate in natural languages.

Natural language, which we use to communication with each other, no matter English, Malay, Chinese, Tamil and others, is actually related to language in mathematics. In natural language, we write in alphabetical form, whereas in mathematics language, we write in mathematics expression. No matter in alphabetical form or expression form, the main purpose of language is to be able to express the ideational and sentimental concepts. However, to explain the mathematical expressions in natural language, it requires individual abilities, knowledge, and understanding of mathematics.

Our recommendation is that, student should learn to relate mathematics language and natural language, not only to understand the topic Language in mathematics, but also to be able to express mathematical expressions in natural language. With this, student will be able further understand this topic and gain better knowledge about Language.