Languages and Machines

Unit one: Formal Languages

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What is a language?

• means of communication• linear• can translate between two languages• has containment structure of

symbols• has atomic (terminal) symbols from

which all others can be made

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Hierarchy of structures

Paragraph

Sentence

Word

Letter

1..

1..

1..

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What is a machine?

• a language* processor• interprets a language• translates one language to another• modifies an expression in a language

* a language can have graphical symbols

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Examples of languages

• English• German• Java• Mathematics• UML

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The cat sat on the mat.

Example of language: English

symbol

string

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Example of language: German

Die Katze saß auf dem Teppich.

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Example of language: Java

while (sunny) mat.sitOn(myCat);

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Example of language: Maths

c : Cat ● isOnMat(c)

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Example of language: UML

• Note: this two-dimensional diagram could be translated to a one-dimensional string (e.g. XML)

Cat

feedstroke

Mat

sitOn(c : Cat)

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UML in XML<uml> <class> <height>2.1</height> <width>3.5</width> <position x=0.6 y=3.1> <name>'Cat'</name> <attributes></attributes> <operations> <operation>'feed'</operation>

<operation>'stroke'</operation> </operations> </class> <class> ... </class></uml>

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UML in XML without layout

<uml> <class> <height> 2.1 </height> <width> 3.5 </width> <position x=0.6 y=3.1> <name> 'Cat' </name> <attributes> </attributes> <operations> <operation> 'feed' </operation> <operation> 'stroke' </operation> </operations> </class> <class> ... </class> </uml>

• no information lost• a linear string of symbols

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Activity

• write down an example of a language, and state what its symbols are

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Formal language

• has a collection of symbols• is this a set or sequence?• a set• is it a finite or infinite set?• finite• we call this set the alphabet

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Activity

Write down the alphabet for 1. the language of binary numbers2. the language of ordinary decimal

numbers3. the language of traffic lights4. the language of simple diagrams

made up of circles, rectangles and straight lines

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Suggested answer

1. {0,1}2. {0,1,2,3,4,5,6,7,8,9}3. {red, amber, green}4. {circle, rectangle, line}

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Formal language

• a formal language is just a set of strings• is the set finite?• maybe, it depends on the language• e.g.1. the language consisting of all two-

bit strings is finite: {00,01,10,11}• e.g.2. the language consisting of all binary

strings is infinite: given any finite set of strings, we can always add another string by taking the longest string and adding one more bit to it

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Classes of formal language

regular

phrase structure

context-freecontext-sensitive

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Classes of language: examples

• The man bites the dog. (Context-free)

• ((5 + 3) / (5 – 1)) * (7 - 5) (Context-free)

• NUMBER_OF_PAWS (Regular)

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Regular languages

1. the empty set is a regular language2. the set consisting of the empty string () is

a regular language3. the set consisting of a one-symbol string

is a regular language4. a new regular language can be made by

taking a string from a regular language and concatenating it with a string from a regular language

5. a new regular language can be made by taking the union of two regular languages

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Examples of regular languages

1. the set of all two-bit strings2. the set of all English words3. the set of all decimal integers4. the set of Java identifiers

You don't believe me?...

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Activity

How many strings do the following regular languages contain?

1. all the possible three-bit strings2. all the single-digit decimal numbers3. all the possible repetitions of the

traffic-light sequence (red, amber, green, amber)

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Suggested Answers

1. 82. 103. infinite

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Recognizing regular languages

• regular languages can be recognized and interpreted by a finite-state machine

• for example, here is a machine to recognize a two-bit string:

0

1

0

1

acceptor states

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Finite-state machines

• Here is a machine that recognizes a bit-string of any length:

0

1

can you simplify this machine?

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Summary

• A language, formally speaking, is a set of strings. The set may be finite or infinite.

• A string is a finite sequence of symbols. The sequence has a minimal length of zero.

• A symbol is just a mark or shape that conveys meaning. It is a member of a finite alphabet.

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Summary

• A regular language has very simple formation rules involving sequences, repetitions and alternatives.

• A context-free language is a language in which each kind of phrase has the same structure, irrespective of where it is in the string of phrases

• We speak in natural language, which are not strictly formal languages. The grammar rules of natural languages are more-or-less context free.

• We use computer languages. All the usual computer languages are context-free.