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Grammars and languages

Grammars and languages. Hybrid and uncertain systems. Language. A language L is a set of strings over the alphabet T alphabet = finite set of symbols (letters). Example : T = { a,b,c } L = { abc, abbc, ab }. Gram mars. Grammar is a quaternition:. where :

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Grammars and languages

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  1. Grammars and languages Hybrid and uncertain systems České vysoké učení technické v Praze Fakulta dopravní

  2. Language A language L is a set of strings over the alphabet T • alphabet = finite set of symbols (letters) Example: T = {a,b,c} L = {abc, abbc, ab} České vysoké učení technické v Praze Fakulta dopravní

  3. Grammars Grammar is a quaternition: where: N- a set of non-terminal symbols T- a set of terminal symbols P - a set of rules S - start symbol of the grammar SN České vysoké učení technické v Praze Fakulta dopravní

  4. Rules • set of rules: the left sideof the rule the right side of the rule is a arbitrary string consisting terminal and non-terminal symbols České vysoké učení technické v Praze Fakulta dopravní

  5. Rules • the rule (,)Pis written in the form of    • the sense: „ is transcribed to “ the left side contains always at least one non-terminal symbol (it is possible to rewrite non-terminal symbol) České vysoké učení technické v Praze Fakulta dopravní

  6. An example of simple grammar • the grammar generating symmetric strings of zeros and ones 0000…01…11111 G = (N,T,P,S) N = { S, A } T = { 0, 1 } P = { S 0A1, A 0A1, A  } (symbol  is an empty symbol) České vysoké učení technické v Praze Fakulta dopravní

  7. An example of simple grammar • generated string (sentence): S 0A1  00A11  000A111  000111 Terminology: •  = γ1αγ2 generates  =γ1βγ2 directly, if the rule α  β exists • it is denoted  • example: 00A11  000A111 České vysoké učení technické v Praze Fakulta dopravní

  8. Terminology •  generates , if the sequence α1, α2,…, αn exists such = α1,  = αn a αi  αi+1, i = 1 …n • it is denoted  *  • the sequence of string is a derivation • example: 0A1 * 000111 • derivation description • the sequence of rules – the previous slide • derivation tree České vysoké učení technické v Praze Fakulta dopravní

  9. Derivation tree S 0 A 1 0 A 1 0 A 1  České vysoké učení technické v Praze Fakulta dopravní

  10. Languages and grammars • the languageLG is by the grammarG • the grammar G and the language LG generated by the grammar are equivalent Note: • sentences of the languages are composed only by terminal symbols České vysoké učení technické v Praze Fakulta dopravní

  11. Grammar Classification by Chomski • grammars are classified by the shape of rules • general (unlimited) • context • context-free • regular České vysoké učení technické v Praze Fakulta dopravní

  12. Grammar classification • unlimited - L(0) • rules are general • context – L(1) • γ1Aγ2  γ1βγ2, AN, γ1,γ2 is a context, γ1,γ2 (N  T)*,β (N  T)+ • context-free • A  β, AN, β  (N  T)+ • regular České vysoké učení technické v Praze Fakulta dopravní

  13. Grammar classification • unlimited grammars generate unlimited languages - L(0) • context grammars generate context languages - L(1) • context-free grammars generate context-free languages • regular grammars generate regular languages České vysoké učení technické v Praze Fakulta dopravní

  14. Example of unlimited grammar • G = { N, T, P, S } • N = { S, B } • T = { a, b, c } • P = { S abc, S aSBc, cB Bc, bB bb } • the grammar generates language: České vysoké učení technické v Praze Fakulta dopravní

  15. Example of context grammar • the third rule, cB Bc, of the previous example is not the rule of context grammar, others are valid • we transform the previous grammar to the context one • the rule AB  BAis replaced with the set of rules of context grammar: • the context is denote by the blue letter • AB  XB • XB  XA • XA  BA České vysoké učení technické v Praze Fakulta dopravní

  16. Example of context grammar • but the swapping of symbols can not be applied to the third rule Why? Because the terminal symbol can not be replaced. • we add a new terminal symbol C, the rule cC cc and we modify other rules České vysoké učení technické v Praze Fakulta dopravní

  17. Example of context language G = { N, T, P, S } • N = { S, B, C, X} • T = { a, b, c } • P = { S abC, S aSBC, CB XB, XB  XC, XC  BC, bB bb, bC  bc, cC  c } • the grammar generates the same language České vysoké učení technické v Praze Fakulta dopravní

  18. Using grammars in programming • lexical elements of programming languages (keyword, constants) are defined by the regular grammars • programming languages are defined by context-free grammars České vysoké učení technické v Praze Fakulta dopravní

  19. Regular grammars • the shape for rules: A aB orA a, where A, B  N, a T Note: • rules of shape A aBare members of the right regular grammar • rules of shape A Baare members of the left regular grammar České vysoké učení technické v Praze Fakulta dopravní

  20. Example • grammar that generates positive integer constants in C programming language • decimal constants start with 1-9 • octal constants start with 0 • hexadecimal constants start with 0x G = (N, T, P, S) N = { S, X, D, H, O } T = { 0,...,9,x,A,..., F } České vysoké učení technické v Praze Fakulta dopravní

  21. Example České vysoké učení technické v Praze Fakulta dopravní

  22. Finite State Machines • regular language generated by the regular grammar can be accepted by the finite state machine • FSM is a model of lexical analyzer that recognizes if the input string belongs to the language • FSM is equivalent to the regular grammar České vysoké učení technické v Praze Fakulta dopravní

  23. FSMs • FSM is a five-tuple where T is a finite set of input symbols Q is a finite set of internal states  is a transition: • function: Q  T  Q for deterministic FSM • relation  Q  T  Q for nondeterministic FSM České vysoké učení technické v Praze Fakulta dopravní

  24. FSMs • Kis a set of final states • q0is the initial state Note: • FSM has no output function • if FSM accepts a string from the language the present state is s K • FSM can be nondeterministic • it is transformable to the deterministic one České vysoké učení technické v Praze Fakulta dopravní

  25. FSMs České vysoké učení technické v Praze Fakulta dopravní

  26. Algorithm of constructing a FSM from the regular grammar • the set of input symbol is given X = T • the set of internal states is given Q = N {U},U N • each rule A aB implicates the transition (A,a)=B, each rule A a implicates the transition (A,a)=U • the set of final statesK= {U}, orK={U,S}, if the ruleS   exists České vysoké učení technické v Praze Fakulta dopravní

  27. Equivalent FSM to the regular grammar • the FSM is nondeterministic S U final state initial state České vysoké učení technické v Praze Fakulta dopravní

  28. Equivalent FSM to the regular grammar • corresponding deterministic FSM S initial state final states České vysoké učení technické v Praze Fakulta dopravní

  29. Regular expressions • a finite alphabet T is given • regular expressions generate regular language, they defined recursively, using operations „*“ (iteration), „·“ (concatenation) a „+“ (union) Definition: 1) Each letter x Tis a regular expression 2) If E1, E2 are regular expressions, then E1 · E2, E1+ E2, E1 *, (E1) are regular expressions too. České vysoké učení technické v Praze Fakulta dopravní

  30. Regular expression generating constants in C language (1+...+9)(0+...+9)*+0(0+...+7)*+ +0x(0+...+9+A+...+F)(0+...+9+A+...+F)* České vysoké učení technické v Praze Fakulta dopravní

  31. Equivalence • regular grammars, regular expressions and FSMs are equivalent and convertible Regular grammars FSMs Regular expressions České vysoké učení technické v Praze Fakulta dopravní

  32. Example of context-free grammar • the grammar generating a simple programming language G = {N,T,P,S} N = {S, Seq, Block, Comm, Cond} T = {main,{,},;,read_x,write_x,++,--, if,(,),else,==,!=,0,x,>,<} České vysoké učení technické v Praze Fakulta dopravní

  33. Rules • S main{Seq}, • Seq Comm, Seq CommSeq • Block Comm,Block {Seq} • Comm  read_x;,Comm  write_x; • Comm  x++;,Comm  x—-; • Comm  if(Cond) Block • Comm  if(Cond) Block else Block • Cond  x==0, Cond  x>0, Cond  x<0, • Cond  x!=0 České vysoké učení technické v Praze Fakulta dopravní

  34. Generated sequence S main{Seq}  main{Comm} main{if(Cond) Block }  main{if(x!=0) Block }  main{if(x!=0) Comm }  main{if(x!=0)if(Cond)Block else Block }  main{if(x!=0)if(x<0)Comm else Comm}  main{if(x!=0)if(x<0)x++; else Comm}  main{if(x!=0)if(x<0)x++; else x--;} České vysoké učení technické v Praze Fakulta dopravní

  35. Other sequence S main{Seq}  main{Comm} main{if(Cond) Block else Block }  main{if(Cond) Comm else Block }  main{if(x!=0) Comm else Comm }  main{if(x!=0)if(Cond)Block else Comm }  main{if(x!=0)if(x<0)Comm else Comm}  main{if(x!=0)if(x<0)x++; else Comm}  main{if(x!=0)if(x<0)x++; else x--;} • the left nonterminal symbol is always replaced (left derivation) České vysoké učení technické v Praze Fakulta dopravní

  36. Ambiguity Remark: • two syntactically identical sentences are generated by the two different derivations (the same syntax, but different semantics) • such languages are ambiguous • solutions: • to define additional rules in programming languages, for example else is assigned to the nearest if České vysoké učení technické v Praze Fakulta dopravní

  37. The analysis of context free languages • context-freelanguage is analyzed by FSM with stack (LIFO) – push down automaton • Note: • analysis of context languages and unlimited languages is NP problem České vysoké učení technické v Praze Fakulta dopravní

  38. Translation regular grammars • translation regular grammar G = (N,T,D,P,S) where: N- a set of non-terminal symbols T- a set of terminal (input) symbols D- a set of output symbols P – a set of rules S - start symbol SN České vysoké učení technické v Praze Fakulta dopravní

  39. Rules • rules are of the form: A aB orA a, where A, B  N, a T,   D* (D* is a set of all strings over alphabetD) České vysoké učení technické v Praze Fakulta dopravní

  40. Example of the transl. grammar G= {N,T,D,P,S} N = {S,A,K,X} T = {a,+,*} D = {,,} P = { S aA, A +K, A *X, K a, X a } • example: S aA a+K  a+a • the grammar translates expression a+a in infix form to output expression  in postfix form České vysoké učení technické v Praze Fakulta dopravní

  41. Translation FSM • translation FSM is a six-tuple where Ta set of input symbols Da set of output symbols Qa set of internal states Ka set of terminal states q0is a initial state České vysoké učení technické v Praze Fakulta dopravní

  42. Translation FSM is amapping: • : Q  T  { Mi: Mi  Q  D* } • if a grammar contains rules A ayB, resp.A ay, where y D (the rule contains only one output symbol) and there are no two rules such that A ayB and A ayC then the translation FSM is deterministic and it hold properties of the sequential mapping České vysoké učení technické v Praze Fakulta dopravní

  43. Translation FSM • mapping can be dividedinto: • translation function : Q  T  Q • output function: Q  T  D • then FSM is Mealy one Poznámka: • FSMs in hardwaredomain has usually no set of terminal states K České vysoké učení technické v Praze Fakulta dopravní

  44. Equivalency • there is an equivalency Regular translation grammars TranslationFSMs České vysoké učení technické v Praze Fakulta dopravní

  45. Examples • Construct a regular grammar which generates decimal numbers with sign +/- • Construct a context-free grammar which generates boolean expressionsin disjunctive form using and (*), or (+), negation (-) ansinput variablesa,b,c, output variable is y. The expression is terminated by semicolon ";" České vysoké učení technické v Praze Fakulta dopravní

  46. Notes • grammars are used not only with languages • other generative systems can be defined by grammars • grammars of the "nature" • L – systems (Lindenmayer systems) • a group of fractals defined by grammars České vysoké učení technické v Praze Fakulta dopravní

  47. Sierpinski triangle G = (V,P,S) V={S,G,F,+,-} • a finite set of symbols P = {S FGF + +FF + +FF, F FF, G  + + FGF − −FGF − −FGF + +} • interpretation using "turtle graphics" • "F" – moving turtle forward (drawing a line) • "G" – ignore • "+" – rotate to the left around given angle • "–" – rotate to the right around given angle České vysoké učení technické v Praze Fakulta dopravní

  48. angle = 60 degree- triangles České vysoké učení technické v Praze Fakulta dopravní

  49. Helge von Koch curve G = (V,P,S) V={S,F,+,-} • a finite set of symbols P = {S F +F − − F + F, F F +F − − F + F} • "turtle graphics" • "F" – moving turtle forward (drawing a line) • "+" – rotate to the left around given angle • "–" – rotate to the right around given angle České vysoké učení technické v Praze Fakulta dopravní

  50. angle = 60 degree České vysoké učení technické v Praze Fakulta dopravní

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