# Revisit the example. Transformed DFA 10/1/16 A B C D E. Start

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1 Revisit the example ε 0 ε 1 Start ε a ε 2 3 ε b ε 4 5 ε a b b ε-closure(0)={0, 1, 2, 4, 7} = A Trans(A, a) = {1, 2, 3, 4, 6, 7, 8} = B Trans(A, b) = {1, 2, 4, 5, 6, 7} = C Trans(B, a) = {1, 2, 3, 4, 6, 7, 8} = B Trans(B, b) = {1, 2, 4, 5, 6, 7, 9} = D Trans(C, a) = {1, 2, 3, 4, 6, 7, 8} = B Trans(C, b) = {1, 2, 4, 5, 6, 7} = C Trans(D, a) = {1, 2, 3, 4, 6, 7, 8} = B Trans(D, b) = {1, 2, 4, 5, 6, 7, 10} = E (mark as end state because 10 is included) Trans(E, a) = {1, 2, 3, 4, 6, 7, 8} = B Trans(E, b) = {1, 2, 4, 5, 6, 7} = C 1 ε Transformed DFA b a b a a b A B C D E a b Start b a 2 1

2 Minimizing the DFA Insight Identify equivalent state sets if all states have the same transitions Merge equivalent states as a new state in the refined DFA 3 Minimizing the DFA Initially partition the state set into two groups: (1) Group I: the states with only final states, and (2) Group II: the states with only non-final states for each group G do partition G into subgroups such that for any states s and t in G, for all input symbols, they have transitions to states in the same group replace G with identified subgroups 4 2

3 Initially, two partitions: G1 = {E}, G2 = {A, B, C, D} Revisit the example G1 cannot be further partitioned State Input symbol Trans(G2, a) = {B} G2 a b Trans(G2, b) = {C, D, E}, the resulting set A B C does not belong to the same group B B D partition G2 into {A, B, C}=G3, {D}=G4 C B C Trans(G3, a) = {B} G3 D B E Trans(G3, b) = {C, D}, the resulting set E B C does not belong to the same group Partition G3 into {A, C} = G5, {B} = G6 Trans(G5, a) = {B} = G6 Trans(G5, b) = {C} G5 Therefore, the resulting partition is: {A, C}, {B}, {D}, {E} 5 Refined DFA a b a b b S0 S1 S2 S3 a a Start b 6 3

4 State Transition Diagram (DFA) Le-er/Digit Start Le-er id Digit int_lit Digit 7 Constructing the Lexical Analysis Convenient utility subprograms: getchar - gets the next character of input, puts it in nextchar, determines its class and puts the class in charclass addchar - puts the character from nextchar into the place the token is being accumulated: nexttoken 8 4

5 Implementation Pseudo-code static TOKEN nexttoken; static CHAR_CLASS charclass; void lex() { getchar(); switch (charclass) { case LETTER: addchar(); getchar(); while (charclass == LETTER charclass == DIGIT) { addchar(); getchar(); } return; //nexttoken = ID 9 Implementation (Cont d) case DIGIT: addchar(); getchar(); while (charclass == DIGIT) { addchar(); getchar(); } return;//nexttoken = INT_LIT default: report error(); } /* End of switch */ } /* End of function lex */ 10 5

6 Key Points about Scanner Nearly universal rule Always take the longest possible token from the input foobar is never parsed to foo or fooba Regular expressions generate a regular language, while DFAs recognize it 11 Parser By analogy to RE and DFAs, a contextfree grammar (CFG) is a generator for a context-free language, while a parser is a language recognizer Responsibilities Generate a parse tree, report syntax errors if any 12 6

7 Two Classes of Grammars Left-to-right, Leftmost derivation (LL) Left-to-right, Rightmost derivation (LR) We can build parsers for these grammars that run in linear time 13 Grammar Comparison LL E -> T E E -> + T E ε T -> F T T -> * F T ε F -> id LR E -> E + T T T -> T * F F F -> id 14 7

8 Two Categories of Parsers LL(1) Parsers L: scanning the input from left to right L: producing a leftmost derivation 1: using one input symbol of lookahead at each step to make parsing action decisions LR(1) Parsers L: scanning the input from left to right R: producing a rightmost derivation in reverse 1: the same as above 15 Two Categories of Parsers LL(1) parsers (predicative parsers) Top down Build the parse tree from the root Find a left most derivation for an input string LR(1) parsers (shift-reduce parsers) Bottom up Build the parse tree from leaves Reducing a string to the start symbol of a grammar 16 8

9 Top-down parsing algorithms Recursive predictive parsing Recursive-descent parsing Non-recursive predictive parsing LL(1): Table-driven parsing 17 Motivating Example Consider the grammar S -> cad A -> ab a Input string: w = cad How to build a parse tree top-down? 18 9

10 Recursive-Descent Parsing Initially create a tree containing a single node S (the start symbol) Apply the S-rule to see whether the first token matches If matches, expand the tree Apply the A-rule to the leftmost nonterminal A Since the first token matches both alternatives (A1 and A2), randomly pick one (e.g., A1) to apply S c A d S c A d a b 19 Recursive-Descent Parsing S c Since the third token d does not match b, report A d failure and go back to A to try another a b alternative Rollback to the state before applying A1 rule, S and then apply the alternative rule The third token matches, so parsing is successfully done c A a d 20 10

11 Recursive-Descent Parsing Algorithm Suppose we have a scanner which generates the next token as needed. Given a string, the parsing process starts with the start symbol rule: if there is only one RHS then for each terminal in the RHS compare it with the next input token if they match, then continue else report an error for each nonterminal in the RHS call its corresponding subprogram and try match if no match is found, then report an error else // there is more than one RHS choose the RHS based on the next input token (the lookahead) for each chosen RHS call the corresponding subprogram and try match if no match is found, then report an error 21 Constructing Parser Utility program match( ) sees what it expects to see, and do corresponding processing 22 11

12 An Example (one RHS) /* Function expr Parses strings in the language generated by the rule: <expr> <term> {(+ -) <term>} */ void expr() { /* Parse the first term */ term(); /* As long as the next token is + or -, call lex to get the next token, and parse the next term */ } while (nexttoken == PLUS_CODE nexttoken == MINUS_CODE){ lex(); term(); } 23 Another Example (multiple RHS) /* Function factor Parses strings in the language generated by the rule: <factor> -> id (<expr>) */ void factor() { if (nexttoken == ID_CODE) { lex(); } else if (nexttoken == LEFT_PAREN_CODE) { lex(); expr(); if (nexttoken == RIGHT_PAREN_CODE) { lex(); } else error(); } else error(); /* Neither RHS matches */ } 24 12

13 Key points about recursive-descent parsing Recursive-descent parsing may require backtracking LL(1) does not allow backtracking By only looking at the next input token, we can always precisely decide which rule to apply By carefully designing a grammar, i.e., LL(1) grammar, we can avoid backtracking 25 Two Obstacles to LL(1)-ness Left recursion E.g., id_list -> id_list_prefix ; id_list_prefix -> id_list_prefix, id id When the next token is id, which rule should we apply? Common prefixes E.g., A -> ab a When the next token is a, which rule should we apply? 26 13

14 LL(1) Grammar Grammar which can be processed with LL(1) parser Non-LL grammar can be converted to LL(1) grammar via: Left-recursion elimination Left factoring by extracting common prefixes 27 Left-Recursion Elimination Replace left-recursion with rightrecursion id_list -> id_list_prefix ; id_list_prefix -> id_list_prefix, id id => id_list -> id id_list_tail id_list_tail -> ;, id id_list_tail 28 14

15 Left Factoring Extract the common prefixes, and introduce new nonterminals as needed A -> ab a => A -> ab B -> b ε 29 Non-LL Languages Simply eliminating left recursion and common prefixes is not guaranteed to make LL(1) An example in Pascal: stmt -> if condition then_clause else_clause other_stmt then_clause -> then stmt else_clause -> else stmt ε How to parse if C1 then if C2 then S1 else S2? 30 15

16 Non-LL Languages Define disambiguating rule, use it together with ambiguous grammar to parse top-down E.g., in the case of a conflict between two possible productions, the one to use is the one that occurs first, textually in the grammar to pair the else with the nearest then Disambiguating rule can be also defined for bottom-up parsing 31 16

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