Languages and Machines Unit one: Formal Languages

2 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

3 Hierarchy of structures Paragraph Sentence Word Letter 1..

4 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

5 Examples of languages English German Java Mathematics UML

6 The cat sat on the mat. Example of language: English symbol string

7 Example of language: German Die Katze saß auf dem Teppich.

8 Example of language: Java while (sunny) mat.sitOn(myCat);

9 Example of language: Maths  c : Cat ● isOnMat(c)

10 Example of language: UML Note: this two-dimensional diagram could be translated to a one-dimensional string (e.g. XML) Cat feed stroke Mat sitOn(c : Cat)

11 UML in XML 'Cat' 'feed' 'stroke'...

12 UML in XML without layout 'Cat' 'feed' 'stroke'... no information lost a linear string of symbols

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

14 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

15 Activity Write down the alphabet for 1.the language of binary numbers 2.the language of ordinary decimal numbers 3.the language of traffic lights 4.the language of simple diagrams made up of circles, rectangles and straight lines

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

17 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

18 Classes of formal language regular phrase structure context-free context-sensitive

19 Classes of language: examples The man bites the dog. (Context-free) ((5 + 3) / (5 – 1)) * (7 - 5) (Context-free) NUMBER_OF_PAWS(Regular)

20 Regular languages 1.the empty set is a regular language 2.the set consisting of the empty string (  ) is a regular language 3.the set consisting of a one-symbol string is a regular language 4.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

21 Examples of regular languages 1.the set of all two-bit strings 2.the set of all English words 3.the set of all decimal integers 4.the set of Java identifiers You don't believe me?...

22 Activity How many strings do the following regular languages contain? 1.all the possible three-bit strings 2.all the single-digit decimal numbers 3.all the possible repetitions of the traffic-light sequence (red, amber, green, amber)

23 Suggested Answers infinite

24 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: acceptor states

25 Finite-state machines Here is a machine that recognizes a bit-string of any length: 0 1 can you simplify this machine?

26 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.

27 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.