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1 CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 6 Mälardalen University 2006
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2 RECAPITULATION: REGULAR LANGUAGES - Languages, Alphabets and Strings - Operations on Strings - Operations on Languages - Regular Expressions - Finite Automata - Regular Grammars - Pumping lemma INTRODUCTION: CONTEXT FREE LANGUAGES
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3 NEXT WEEK! Midterm Exam 1 Regular Languages Place: U2114 Time: Tuesday 2007-04-24, 10:15-12:00 It is OPEN BOOK. (This means you are allowed to bring in one book of your choice.) It will cover lectures 1 through 5 (Regular Languages).
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4 Med det som vi har lärt oss hittills kan vi klara följande tentauppgifter…
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5 Tenta 29 okt 1999; uppgift 2 (L Salling) Konstruera (med motiveringar) vars strängar innehåller samtliga tre tecken! a) Ett reguljärt uttryck över
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6 Lösning eller
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7 Konstruera (med motiveringar) vars strängar innehåller samtliga tre tecken! b) En minimal DFA för ett språk L över
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9 Särskiljningsalgoritm
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10 c) En reguljär grammatik för L
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11 Tenta 24 okt 1994; uppgift 2 (L Salling) Reguljära? vars strängar innehåller ett jämnt antal a:n! a) Språket över Ja, språket är reguljärt och beskrivs med ett reguljärt uttryck:
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12 Tenta 24 okt 1994; uppgift 2 (L Salling) b) De välformade aritmetiska uttrycken formade i alfabetet Nej, språket är inte reguljärt: Ta följande sträng: N stycken a adderas Om språket vore reguljärt skulle det kunna pumpas. Men de N avslutande tecknen består enbart av höger- parenteser och kan inte ändras utan att balansen med vänsterparenteserna förstörs.
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13 Tenta 15 mars 1995; uppgift 3 (L Salling) Reguljära? c) Ja, språket är reguljärt och beskrivs med ett reguljärt uttryck:
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14 Tenta 15 mars 1995; uppgift 3 (L Salling) Reguljära? d) Nej. Strängen vars enda palindromprefix längre än 2 är strängen själv, kan inte pumpas någonstans inuti b-block utan att falla ur språket.
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15 Tenta 15 mars 1995; uppgift 3 (L Salling) Reguljära? e) Nej. Om det vore reguljärt skulle även föregående språk vara det (eftersom det är komplementspråk, och regulariteten bevaras under komplementbildning).
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16 Pumping Lemma is necessary but not sufficient for RL OBS! The pumping lemma does not give a sufficient condition for a language to be regular! You can not use it to show that language is regular. For example, the language (strings over the alphabet {0,1} consisting of a nonempty even palindrome followed by another nonempty string) is not regular but can still be "pumped" with m = 4: Suppose w=uuRv has length at least 4. If u has length 1, then |v| ≥ 2 and we can take y to be the first character in v. Otherwise, take y to be the first character of u and note that yk for k ≥ 2 starts with the nonempty palindrome yy. For a practical test that exactly characterizes regular languages, see the Myhill-Nerode theorem. The typical method for proving that a language is regular is to construct either a Finite State Machine or a Regular Expression for the language.
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17 Minimizing DFA’s By Partitioning (Delmängdskonstruktion)
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18 Minimizing DFA Different methods All involve finding equivalent states: States that go to equivalent states under all inputs We will use the Partitioning Method
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19 Minimizing DFA’s by Partitioning Consider the following DFA (from Forbes Louis): Accepting states are yellow Non-accepting states are blue Are any states really the same?
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20 S 2 and S 7 are really the same: Both Final states Both go to S6 under input b Both go to S3 under an a S 0 and S 5 really the same. Why? We say each pair is equivalent Are there any other equivalent states? We can merge equivalent states into 1 state
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21 Partitioning Algorithm First Divide the set of states into Final and Non-final states Partition I Partition II a b S0S0 S1S1 S4S4 S1S1 S5S5 S2S2 S3S3 S3S3 S3S3 S4S4 S1S1 S4S4 S5S5 S1S1 S4S4 S6S6 S3S3 S7S7 *S 2 S3S3 S6S6 *S 7 S3S3 S6S6
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22 Partitioning Algorithm Now See if states in each partition each go to the same partition S 1 & S 6 are different from the rest of the states in Partition I (but like each other) We will move them to their own partition a b S0S0 S 1 I S 4 I S1S1 S 5 I S 2 II S3S3 S 3 I S4S4 S 1 I S 4 I S5S5 S 1 I S 4 I S6S6 S 3 I S 7 II *S 2 S 3 I S 6 I *S 7 S 3 I S 6 I
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23 Partitioning Algorithm a b S0S0 S1S1 S4S4 S5S5 S1S1 S4S4 S3S3 S3S3 S3S3 S4S4 S1S1 S4S4 S1S1 S5S5 S2S2 S6S6 S3S3 S7S7 *S 2 S3S3 S6S6 *S 7 S3S3 S6S6
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24 Partitioning Algorithm Now again See if states in each partition each go to the same partition In Partition I, S 3 goes to a different partition from S 0, S 5 and S 4 We’ll move S3 to its own partition a b S0S0 S 1 III S 4 I S5S5 S 1 III S 4 I S3S3 S 3 I S4S4 S 1 III S 4 I S1S1 S 5 I S 2 II S6S6 S 3 I S 7 II *S 2 S 3 I S 6 III *S 7 S 3 I S 6 III
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25 Partitioning Algorithm Note changes in S 6, S 2 and S 7 a b S0S0 S 1 III S 4 I S5S5 S 1 III S 4 I S4S4 S 1 III S 4 I S3S3 S 3 IV S1S1 S 5 I S 2 II S6S6 S 3 IV S 7 II *S 2 S 3 IV S 6 III *S 7 S 3 IV S 6 III
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26 Partitioning Algorithm Now S 6 goes to a different partition on an a from S 1 S 6 gets its own partition. We now have 5 partitions Note changes in S 2 and S 7 a b S0S0 S 1 III S 4 I S5S5 S 1 III S 4 I S4S4 S 1 III S 4 I S3S3 S 3 IV S1S1 S5 IS5 I S 2 II S6S6 S 3 IV S 7 II *S 2 S 3 IV S 6 V *S 7 S 3 IV S 6 V
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27 Partitioning Algorithm All states within each of the 5 partitions are identical. We might as well call the states I, II III, IV and V. a b S0S0 S 1 III S 4 I S5S5 S 1 III S 4 I S4S4 S 1 III S 4 I S3S3 S 3 IV S1S1 S 5 I S 2 II S6S6 S 3 IV S 7 II *S 2 S 3 IV S 6 V *S 7 S 3 IV S 6 V
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28 Partitioning Algorithm a b I III I *II IVV III I II IV V II Here they are: V a a a a a b b b b b b
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29 Chomsky Hierarchy
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30 Automata theory: formal languages and formal grammars
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31 Automata theory: formal languages and formal grammars
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32 Regular Languages Context-Free Languages Non-regular languages
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33 Formal Definition Grammar Set of variables Set of terminal symbols Start variable Set of production rules
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34 Regular Grammars A regular grammar is any right-linear or left-linear grammar Examples
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35 A Nonregular Language DFA must have infinite number of states. Statesare distinct for each
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36 Context-Free Languages
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37 Context-Free Languages Pushdown Automata Context-Free Grammars stack automaton
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38 Context-Free Grammars
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39 A context-free grammar A derivation Example
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40 A context-free grammar Another derivation
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42 A context-free grammar A derivation Example
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43 A context-free grammar Another derivation
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45 A context-free grammar A derivation Example
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46 A context-free grammar A derivation
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48 Definition: Context-Free Grammars Grammar Productions of the form: is string of variables and terminals VariablesTerminal symbols Start variables
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49 Definition: Context-Free Languages A language is context-free if and only if there is a grammar with
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50 Derivation Order Leftmost derivation
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51 Derivation Order Rightmost derivation
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52 Leftmost derivation
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53 Rightmost derivation
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54 Derivation Trees
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59 Derivation Tree
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60 yield Derivation Tree
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61 Partial Derivation Trees Partial derivation tree
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62 Partial derivation tree
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63 Partial derivation tree sentential form yield
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64 Same derivation tree Sometimes, derivation order doesn’t matter Leftmost: Rightmost:
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65 Ambiguity
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66 leftmost derivation derivation (* denotes multiplication)
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67 derivation leftmost derivation
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68 Two derivation trees
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69 The grammar is ambiguous! Stringhas two derivation trees
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70 stringhas two leftmost derivations The grammar is ambiguous:
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71 Definition A context-free grammar is ambiguous if some string has two or more derivation trees (two or more leftmost/rightmost derivations)
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72 Why do we care about ambiguity? Let’s see the case
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73 Why do we care about ambiguity?
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74 Why do we care about ambiguity?
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75 Correct result:
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76 Ambiguity is bad for programming languages We want to remove ambiguity!
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77 We fix the ambiguous grammar… …by introducing parentheses () to indicate grouping, (precedence) Non-ambiguous grammar
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79 Unique derivation tree
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80 The grammar is non-ambiguous Every string has a unique derivation tree
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81 Inherent Ambiguity Some context free languages have only ambiguous grammars! Example:
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82 The string has two derivation trees
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83 Compilers
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84 Compiler Program v = 5; if (v>5) x = 12 + v; while (x !=3) { x = x - 3; v = 10; }...... Add v,v,0 cmp v,5 jmplt ELSE THEN: add x, 12,v ELSE: WHILE: cmp x,3... Machine Code
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85 Lexical analyzer parser Compiler program machine code input output
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86 A parser “knows” the grammar of the programming language
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87 Parser PROGRAM STMT_LIST STMT_LIST STMT; STMT_LIST | STMT; STMT EXPR | IF_STMT | WHILE_STMT | { STMT_LIST } EXPR EXPR + EXPR | EXPR - EXPR | ID IF_STMT if (EXPR) then STMT | if (EXPR) then STMT else STMT WHILE_STMT while (EXPR) do STMT
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88 The parser finds the derivation of a particular input 10 + 2 * 5 Parser E E + E | E * E | INT E E + E E + E * E 10 + E*E 10 + 2 * E 10 + 2 * 5 input derivation
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89 derivation derivation tree E E + E E + E * E 10 + E*E 10 + 2 * E 10 + 2 * 5 10 E 2 E 5 E E + E *
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90 derivation tree mult a, 2, 5 add b, 10, a machine code 10 E 2 E 5 E E + E *
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91 Parsing
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92 grammar Parser input string derivation
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93 Example: Parser derivation input ?
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94 Exhaustive Search Phase 1: All possible derivations of length 1 Find derivation of
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96 Phase 2 Phase 1
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97 Phase 2 Phase 3
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98 Final result of exhaustive search Parser derivation input (top-down parsing)
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99 Context Free Art http://www.contextfreeart.org/wiki/index.php?page=AboutPage http://www.contextfreeart.org/wiki/index.php?page=AboutPage
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100 Context Free Art http://www.contextfreeart.org/wiki/index.php?page=AboutPage http://www.contextfreeart.org/wiki/index.php?page=AboutPage
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