Syntax Analysis - Part 1. Syntax Analysis

Transcription

1 Syntax Analysis Outline Context-Free Grammars (CFGs) Parsing Top-Down Recursive Descent Table-Driven Bottom-Up LR Parsing Algorithm How to Build LR Tables Parser Generators Grammar Issues for Programming Languages Harry H. Porter,

2 Top-Down Parsing LL Grammars - A subclass of all CFGs Recursive-Descent Parsers - Programmed by hand Non-Recursive Predictive Parsers - Table Driven Simple, Easy to Build, Better Error Handling Bottom-Up Parsing LR Grammars - A larger subclass of CFGs Complex Parsing Algorithms - Table Driven Table Construction Techniques Parser Generators use this technique Faster Parsing, Larger Set of Grammars Complex Error Reporting is Tricky Harry H. Porter,

3 Succeed is string is recognized... and fail if syntax errors Syntax Errors? Good, descriptive, helpful message! Recover and continue parsing! Output of Parser? Build a Parse Tree (also called derivation tree ) Build Abstract Syntax Tree (AST) In memory (with objects, pointers) Output to a file Execute Semantic Actions Build AST Type Checking Generate Code Don t build a tree at all! * x + y 1 Harry H. Porter,

4 Lexical if x<1 thenn y = 5: Typos Syntactic if ((x<1) & (y>5)))... {... { } Semantic if (x+5) then... Type Errors Undefined IDs, etc. Errors in Programs Logical Errors if (i<9) then... Should be <= not < Bugs Compiler cannot detect Logical Errors Harry H. Porter,

5 Compiler Always halts Any checks guaranteed to terminate Decidable Other Program Checking Techniques Debugging Testing Correctness Proofs Partially Decidable Okay? The test terminates. Not Okay? The test may not terminate! You may need to run some programs to see if they are okay. Harry H. Porter,

6 Requirements Detect All Errors (Except Logical!) Messages should be helpful. Difficult to produce clear messages! Example: Syntax Error Example: Line 23: Unmatched Paren if ((x == 1) then ^ Compiler Should Recover Keep going to find more errors Example: x := (a + 5)) * (b + 7)); We re in the middle of a statement This error missed Skip tokens until we see a ; Error detected here Resume Parsing Misses a second error... Oh, well... Checks most of the source Harry H. Porter,

7 Difficult to generate clear and accurate error messages. Example function foo () {... if (...) {... } else { } <eof> Missing } here Not detected until here Example var myvarr: Integer;... x := myvar; Misspelled ID here... Detected here as Undeclared ID Harry H. Porter,

8 For Mature Languages Catalog common errors Statistical studies Tailor compiler to handle common errors well Statement terminators versus separators Terminators: C, Java, PCAT {A;B;C;} Separators: Pascal, Smalltalk, Haskell Pascal Examples begin var t: Integer; t := x; x := y; y := t end if (...) then x := 1 else y := 2; z := 3; Tend to insert a ; here Tend to insert a ; here function foo (x: Integer; y: Integer)... Tend to put a comma here Harry H. Porter,

9 Error-Correcting Compilers Issue an error message Fix the problem Produce an executable Example Error on line 23: myvarr undefined. myvar was used. Is this a good idea??? Compiler guesses the programmer s intent A shifting notion of what constitutes a correct / legal / valid program May encourage programmers to get sloppy Declarations provide redundancy Increased reliability Harry H. Porter,

10 Error Avalanche One error generates a cascade of messages Example x := 5 while ( a == b ) do ^ Expecting ; ^ Expecting ; ^ Expecting ; The real messages may be buried under the avalanche. Missing #include or import will also cause an avalanche. Approaches: Only print 1 message per token [ or per line of source ] Only print a particular message once Error: Variable myvarr is undeclared All future notices for this ID have been suppressed Abort the compiler after 50 errors. Harry H. Porter,

11 Error Recovery Approaches: Panic Mode Discard tokens until we see a synchronizing token. Example Skip to next occurrence of } end ; Resume by parsing the next statement Simple to implement Commonly used The key... Good set of synchronizing tokens Knowing what to do then May skip over large sections of source Harry H. Porter,

12 Error Recovery Approaches: Phrase-Level Recovery Compiler corrects the program by deleting or inserting tokens...so it can proceed to parse from where it was. Example while (x = 4) y := a+b;... Insert do to fix the statement. The key... Don t get into an infinite loop...constantly inserting tokens...and never scanning the actual source Harry H. Porter,

13 Error Recovery Approaches: Error Productions Augment the CFG with Error Productions Now the CFG accepts anything! If error productions are used... Their actions: { print ( Error... ) } Used with... LR (Bottom-up) parsing Parser Generators Error Recovery Approaches: Global Correction Theoretical Approach Find the minimum change to the source to yield a valid program (Insert tokens, delete tokens, swap adjacent tokens) Impractical algorithms - too time consuming Harry H. Porter,

14 CFG: Context Free Grammars Example Rule: Stmt if Expr then Stmt else Stmt Terminals Keywords else else Token Classes ID INTEGER REAL Punctuation ; ; ; Non-terminals Any symbol appearing on the lefthand side of any rule Start Symbol Usually the non-terminal on the lefthand side of the first rule Rules (or Productions ) BNF: Backus-Naur Form / Backus-Normal Form Stmt ::= if Expr then Stmt else Stmt Harry H. Porter,

15 E E + E E ( E ) E - E E ID Rule Alternatives E E E + E ( E ) - E ID E + E ( E ) - E ID All Notations are Equivalent E E + E ( E ) - E ID Harry H. Porter,

16 Terminals a b c... Nonterminals A B C... S Expr Notational Conventions Grammar Symbols (Terminals or Nonterminals) X Y Z U V W... Strings of Symbols α β γ... Strings of Terminals x y z u v w... A sequence of zero Or more terminals And nonterminals Including ε Examples A α B A rule whose righthand side ends with a nonterminal A x α A rule whose righthand side begins with a string of terminals (call it x ) Harry H. Porter,

17 Derivations 1. E E + E 2. E * E 3. ( E ) 4. - E 5. ID A Derivation of (id*id) E (E) (E*E) (id*e) (id*id) Sentential Forms A sequence of terminals and nonterminals in a derivation (id*e) Harry H. Porter,

18 Derives in one step If A β is a rule, then we can write αaγ αβγ Any sentential form containing a nonterminal (call it A)... such that A matches the nonterminal in some rule. Derives in zero-or-more steps * E * (id*id) If α * β and β γ, then α * γ Derives in one-or-more steps + Harry H. Porter,

19 Given G S A grammar The Start Symbol Define L(G) The language generated L(G) = { w S + w} Equivalence of CFG s If two CFG s generate the same language, we say they are equivalent. G 1 G 2 whenever L(G 1 ) = L(G 2 ) In making a derivation... Choose which nonterminal to expand Choose which rule to apply Harry H. Porter,

20 Leftmost Derivations In a derivation... always expand the leftmost nonterminal. E E+E (E)+E (E*E)+E (id*e)+e (id*id)+e (id*id)+id 1. E E + E 2. E * E 3. ( E ) 4. - E 5. ID Let LM denote a step in a leftmost derivation ( LM * means zero-or-more steps ) At each step in a leftmost derivation, we have waγ LM wβγ where A β is a rule (Recall that w is a string of terminals.) Each sentential form in a leftmost derivation is called a left-sentential form. If S LM * α then we say α is a left-sentential form. Harry H. Porter,

21 Rightmost Derivations In a derivation... always expand the rightmost nonterminal. E E+E 1. E E + E E+id 2. E * E (E)+id 3. ( E ) (E*E)+id 4. - E (E*id)+id 5. ID (id*id)+id Let RM denote a step in a rightmost derivation ( RM * means zero-or-more steps ) At each step in a rightmost derivation, we have αaw RM αβw (Recall that w is a string of terminals.) where A β is a rule Each sentential form in a rightmost derivation is called a right-sentential form. If S RM * α then we say α is a right-sentential form. Harry H. Porter,

22 Bottom-Up Parsing Bottom-up parsers discover rightmost derivations! Parser moves from input string back to S. Follow S RM * w in reverse. At each step in a rightmost derivation, we have αaw RM αβw where A β is a rule String of terminals (i.e., the rest of the input, which we have not yet seen) Harry H. Porter,

23 Parse Trees Two choices at each step in a derivation... Which non-terminal to expand Which rule to use in replacing it Leftmost Derivation: E E+E (E)+E (E*E)+E (id*e)+e (id*id)+e (id*id)+id E The parse tree remembers only this E + E 1. E E + E 2. E * E 3. ( E ) 4. - E 5. ID ( E id E * ) E id id Harry H. Porter,

24 Parse Trees Two choices at each step in a derivation... Which non-terminal to expand Which rule to use in replacing it E The parse tree remembers only this Rightmost Derivation: E E+E E+id (E)+id (E*E)+id (E*id)+id (id*id)+id E + E 1. E E + E 2. E * E 3. ( E ) 4. - E 5. ID ( E id E * ) E id id Harry H. Porter,

25 Parse Trees Two choices at each step in a derivation... Which non-terminal to expand Which rule to use in replacing it Leftmost Derivation: E E+E (E)+E (E*E)+E (id*e)+e (id*id)+e (id*id)+id E The parse tree remembers only this Rightmost Derivation: E E+E E+id (E)+id (E*E)+id (E*id)+id (id*id)+id E + E 1. E E + E 2. E * E 3. ( E ) 4. - E 5. ID ( E id E * E ) id id Harry H. Porter,

26 Given a leftmost derivation, we can build a parse tree. Given a rightmost derivation, we can build a parse tree. Lefttmost Derivation of (id*id)+id Rightmost Derivation of (id*id)+id Same Parse Tree Every parse tree corresponds to... A single, unique leftmost derivation A single, unique rightmost derivation Ambiguity: However, one input string may have several parse trees!!! Therefore: Several leftmost derivations Several rightmost derivations Harry H. Porter,

27 Leftmost Derivation #1 E E+E id+e id+e*e id+id*e id+id*id Ambuiguous Grammars E id E + E id E * E id 1. E E + E 2. E * E 3. ( E ) 4. - E 5. ID Input: id+id*id Leftmost Derivation #2 E E*E E+E*E id+e*e id+id*e id+id*id E E + E * E E id id id Harry H. Porter,

28 Ambiguous Grammar More than one Parse Tree for some sentence. The grammar for a programming language may be ambiguous Need to modify it for parsing. Also: Grammar may be left recursive. Need to modify it for parsing. Harry H. Porter,

29 Translating a Regular Expression into a CFG First build the NFA. For every state in the NFA... Make a nonterminal in the grammar For every edge labeled c from A to B... Add the rule A cb For every edge labeled ε from A to B... Add the rule A B For every final state B... Add the rule B ε Harry H. Porter,

30 Recursive Transition Networks Regular Expressions NFA DFA Context-Free Grammar Recursive Transition Networks Exactly as expressive a CFGs... But clearer for humans! IfStmt if Expr then Stmt Expr... elseif Expr then Stmt Stmt else Stmt... end ; Terminal Symbols: Nonterminal Symbols: Harry H. Porter,

31 The Dangling Else Problem This grammar is ambiguous! Stmt if Expr then Stmt if Expr then Stmt else Stmt...Other Stmt Forms... Example String: if E 1 then if E 2 then S 1 else S 2 Interpretation #1: if E 1 then (if E 2 then S 1 else S 2 ) S if E 1 then S if E 2 then S 1 else S 2 Interpretation #2: if E 1 then (if E 2 then S 1 ) else S 2 S if E 1 then S else S 2 if E 2 then S 1 Harry H. Porter,

32 Goal: Match else-clause to the closest if without an else-clause already. Solution: Stmt WithElse The Dangling Else Problem if Expr then Stmt if Expr then WithElse else Stmt...Other Stmt Forms... if Expr then WithElse else WithElse...Other Stmt Forms... Any Stmt occurring between then and else must have an else. i.e., the Stmt must not end with then Stmt. Interpretation #1: if E 1 then (if E 2 then S 1 else S 2 ) Stmt if E 1 then Stmt if then WithElse else WithElse E 2 S 1 S 2 Harry H. Porter,

33 Goal: Match else-clause to the closest if without an else-clause already. Solution: Stmt WithElse The Dangling Else Problem if Expr then Stmt if Expr then WithElse else Stmt...Other Stmt Forms... if Expr then WithElse else WithElse...Other Stmt Forms... Any Stmt occurring between then and else must have an else. i.e., the Stmt must not end with then Stmt. Interpretation #2: if E 1 then (if E 2 then S 1 ) else S 2 Stmt if E 1 then WithElse else Stmt Harry H. Porter,

34 Goal: Match else-clause to the closest if without an else-clause already. Solution: Stmt WithElse The Dangling Else Problem if Expr then Stmt if Expr then WithElse else Stmt...Other Stmt Forms... if Expr then WithElse else WithElse...Other Stmt Forms... Any Stmt occurring between then and else must have an else. i.e., the Stmt must not end with then Stmt. Interpretation #2: if E 1 then (if E 2 then S 1 ) else S 2 Stmt if E 1 then WithElse else Stmt if then WithElse else WithElse E 2 Harry H. Porter,

35 Goal: Match else-clause to the closest if without an else-clause already. Solution: Stmt WithElse The Dangling Else Problem if Expr then Stmt if Expr then WithElse else Stmt...Other Stmt Forms... if Expr then WithElse else WithElse...Other Stmt Forms... Any Stmt occurring between then and else must have an else. i.e., the Stmt must not end with then Stmt. Interpretation #2: if E 1 then (if E 2 then S 1 ) else S 2 Stmt if E 1 then WithElse else Stmt if then WithElse else WithElse E 2 Oops, No Match! Harry H. Porter,

36 Top-Down Parsing Find a left-most derivation Find (build) a parse tree Start building from the root and work down... As we search for a derivation... Must make choices: Which rule to use Where to use it May run into problems! Option 1: Backtracking Made a bad decision Back up and try another choice Option 2: Always make the right choice. Never have to backtrack: Predictive Parser Possible for some grammars (LL Grammars) May be able to fix some grammars (but not others) Eliminate Left Recursion Left-Factor the Rules Harry H. Porter,

37 Input: aabbde S Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd Harry H. Porter,

38 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd Harry H. Porter,

39 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a B Harry H. Porter,

40 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a B Harry H. Porter,

41 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a B Harry H. Porter,

42 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a B b b b Harry H. Porter,

43 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a B b b b Harry H. Porter,

44 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a B b b b Harry H. Porter,

45 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a B b b b Failure Occurs Here!!! Harry H. Porter,

46 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a B We need an ability to back up in the input!!! Harry H. Porter,

47 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd Harry H. Porter,

48 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a b a Harry H. Porter,

49 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a b a Harry H. Porter,

50 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a b a Harry H. Porter,

51 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a b a Harry H. Porter,

52 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a b a Failure Occurs Here!!! Harry H. Porter,

53 Input: aabbde A S a Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd Harry H. Porter,

54 Input: aabbde S Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd Harry H. Porter,

55 Input: aabbde C S e Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd Harry H. Porter,

56 Input: aabbde S C e Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a D Harry H. Porter,

57 Input: aabbde C S e Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a D b b d Harry H. Porter,

58 Input: aabbde C S e Backtracking 1. S Aa 2. Ce 3. A aab 4. aaba 5. B bbb 6. C aad 7. D bbd a a D b b d Harry H. Porter,

59 Will never backtrack! Predictive Parsing Requirement: For every rule: A α 1 α 2 α 3... α N We must be able to choose the correct alternative by looking only at the next symbol May peek ahead to the next symbol (token). Example A ab cd E Assuming a,c FIRST (E) Example Stmt if Expr... for LValue... while Expr... return Expr... ID... Harry H. Porter,

60 Predictive Parsing LL(1) Grammars Can do predictive parsing Can select the right rule Looking at only the next 1 input symbol Harry H. Porter,

61 Predictive Parsing LL(1) Grammars Can do predictive parsing Can select the right rule Looking at only the next 1 input symbol LL(k) Grammars Can do predictive parsing Can select the right rule Looking at only the next k input symbols Harry H. Porter,

62 Predictive Parsing LL(1) Grammars Can do predictive parsing Can select the right rule Looking at only the next 1 input symbol LL(k) Grammars Can do predictive parsing Can select the right rule Looking at only the next k input symbols Techniques to modify the grammar: Left Factoring Removal of Left Recursion Harry H. Porter,

63 Predictive Parsing LL(1) Grammars Can do predictive parsing Can select the right rule Looking at only the next 1 input symbol LL(k) Grammars Can do predictive parsing Can select the right rule Looking at only the next k input symbols Techniques to modify the grammar: Left Factoring Removal of Left Recursion But these may not be enough! Harry H. Porter,

64 Predictive Parsing LL(1) Grammars Can do predictive parsing Can select the right rule Looking at only the next 1 input symbol LL(k) Grammars Can do predictive parsing Can select the right rule Looking at only the next k input symbols Techniques to modify the grammar: Left Factoring Removal of Left Recursion But these may not be enough! LL(k) Language Can be described with an LL(k) grammar. Harry H. Porter,

65 Problem: Stmt Left-Factoring if Expr then Stmt else Stmt if Expr then Stmt OtherStmt With predictive parsing, we need to know which rule to use! (While looking at just the next token) Harry H. Porter,

66 Problem: Stmt Left-Factoring if Expr then Stmt else Stmt if Expr then Stmt OtherStmt With predictive parsing, we need to know which rule to use! (While looking at just the next token) Solution: Stmt if Expr then Stmt ElsePart OtherStmt ElsePart else Stmt ε Harry H. Porter,

67 Problem: Stmt Left-Factoring if Expr then Stmt else Stmt if Expr then Stmt OtherStmt With predictive parsing, we need to know which rule to use! (While looking at just the next token) Solution: Stmt if Expr then Stmt ElsePart OtherStmt ElsePart else Stmt ε General Approach: Before: A αβ 1 αβ 2 αβ 3... δ 1 δ 2 δ 3... After: A αc δ 1 δ 2 δ 3... C β 1 β 2 β 3... Harry H. Porter,

68 Problem: Stmt A Left-Factoring if Expr then Stmt else Stmt α if Expr then Stmt ε α OtherStmt β 2 β 1 δ 1 With predictive parsing, we need to know which rule to use! (While looking at just the next token) Solution: Stmt if Expr then Stmt ElsePart OtherStmt ElsePart else Stmt ε General Approach: Before: A αβ 1 αβ 2 αβ 3... δ 1 δ 2 δ 3... After: A αc δ 1 δ 2 δ 3... C β 1 β 2 β 3... Harry H. Porter,

69 Problem: Stmt A Left-Factoring if Expr then Stmt else Stmt α if Expr then Stmt ε α OtherStmt β 2 β 1 With predictive parsing, we need to know which rule to use! (While looking at just the next token) Solution: Stmt if Expr then Stmt ElsePart A δ 1 α OtherStmt δ 1 ElsePart else Stmt ε C β General Approach: 1 β 2 Before: A αβ 1 αβ 2 αβ 3... δ 1 δ 2 δ 3... After: A αc δ 1 δ 2 δ 3... C β 1 β 2 β 3... C Harry H. Porter,

70 Whenever Left Recursion A + Aα Simplest Case: Immediate Left Recursion Given: A Aα β Transform into: A βa' A' αa' ε where A' is a new nonterminal More General (but still immediate): A Aα 1 Aα 2 Aα 3... β 1 β 2 β 3... Transform into: A β 1 A' β 2 A' β 3A'... A' α 1 A' α 2 A' α 3 A'... ε Harry H. Porter,

71 Example: S Af b A Ac Sd e Is A left recursive? Yes. Left Recursion in More Than One Step Harry H. Porter,

72 Example: S Af b A Ac Sd e Is A left recursive? Yes. Is S left recursive? Left Recursion in More Than One Step Harry H. Porter,

73 Left Recursion in More Than One Step Example: S Af b A Ac Sd e Is A left recursive? Yes. Is S left recursive? Yes, but not immediate left recursion. S Af Sdf Harry H. Porter,

74 Left Recursion in More Than One Step Example: S Af b A Ac Sd e Is A left recursive? Yes. Is S left recursive? Yes, but not immediate left recursion. S Af Sdf Approach: Look at the rules for S only (ignoring other rules)... No left recursion. Harry H. Porter,

75 Left Recursion in More Than One Step Example: S Af b A Ac Sd e Is A left recursive? Yes. Is S left recursive? Yes, but not immediate left recursion. S Af Sdf Approach: Look at the rules for S only (ignoring other rules)... No left recursion. Look at the rules for A... Harry H. Porter,

76 Left Recursion in More Than One Step Example: S Af b A Ac Sd e Is A left recursive? Yes. Is S left recursive? Yes, but not immediate left recursion. S Af Sdf Approach: Look at the rules for S only (ignoring other rules)... No left recursion. Look at the rules for A... Do any of A s rules start with S? Yes. A Sd Harry H. Porter,

77 Left Recursion in More Than One Step Example: S Af b A Ac Sd e Is A left recursive? Yes. Is S left recursive? Yes, but not immediate left recursion. S Af Sdf Approach: Look at the rules for S only (ignoring other rules)... No left recursion. Look at the rules for A... Do any of A s rules start with S? Yes. A Sd Get rid of the S. Substitute in the righthand sides of S. A Afd bd Harry H. Porter,

78 Left Recursion in More Than One Step Example: S Af b A Ac Sd e Is A left recursive? Yes. Is S left recursive? Yes, but not immediate left recursion. S Af Sdf Approach: Look at the rules for S only (ignoring other rules)... No left recursion. Look at the rules for A... Do any of A s rules start with S? Yes. A Sd Get rid of the S. Substitute in the righthand sides of S. A Afd bd The modified grammar: S Af b A Ac Afd bd e Harry H. Porter,

79 Left Recursion in More Than One Step Example: S Af b A Ac Sd e Is A left recursive? Yes. Is S left recursive? Yes, but not immediate left recursion. S Af Sdf Approach: Look at the rules for S only (ignoring other rules)... No left recursion. Look at the rules for A... Do any of A s rules start with S? Yes. A Sd Get rid of the S. Substitute in the righthand sides of S. A Afd bd The modified grammar: S Af b A Ac Afd bd e Now eliminate immediate left recursion involving A. S Af b A bda' ea' A' ca' fda' ε Harry H. Porter,

80 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd e Harry H. Porter,

81 The Original Grammar: S Af b A Ac Sd Be B Ag Sh k Left Recursion in More Than One Step Assume there are still more nonterminals; Look at the next one... Harry H. Porter,

82 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε Harry H. Porter,

83 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε B Ag Sh k Look at the B rules next; Does any righthand side start with S? Harry H. Porter,

84 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε B Ag Afh bh k Substitute, using the rules for S Af... b... Harry H. Porter,

85 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε B Ag Afh bh k Does any righthand side start with A? Harry H. Porter,

86 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε B Ag Afh bh k Do this one first. Harry H. Porter,

87 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε B bda'g BeA'g Afh bh k Substitute, using the rules for A bda'... BeA'... Harry H. Porter,

88 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε B bda'g BeA'g Afh bh k Do this one next. Harry H. Porter,

89 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε B bda'g BeA'g bda'fh BeA'fh bh k Substitute, using the rules for A bda'... BeA'... Harry H. Porter,

90 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε B bda'g BeA'g bda'fh BeA'fh bh k Finally, eliminate any immediate Left recursion involving B Harry H. Porter,

91 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be B Ag Sh k So Far: S Af b A bda' BeA' A' ca' fda' ε B bda'gb' bda'fhb' bhb' kb' B' ea'gb' ea'fhb' ε Finally, eliminate any immediate Left recursion involving B Harry H. Porter,

92 Left Recursion in More Than One Step The Original Grammar: S Af b A Ac Sd Be C If there is another nonterminal, B Ag Sh k then do it next. C BkmA AS j So Far: S Af b A bda' BeA' CA' A' ca' fda' ε B bda'gb' bda'fhb' bhb' kb' CA'gB' CA'fhB' B' ea'gb' ea'fhb' ε Harry H. Porter,

93 Algorithm to Eliminate Left Recursion Assume the nonterminals are ordered A 1, A 2, A 3,... (In the example: S, A, B) for each nonterminal A i (for i = 1 to N) do for each nonterminal A j (for j = 1 to i-1) do Let A j β 1 β 2 β 3... β N be all the rules for A j if there is a rule of the form A i A j α then replace it by A i β 1 α β 2 α β 3 α... β N α endif endfor Eliminate immediate left recursion among the A i rules endfor A 1 A 2 A 3... A j... A 7 A i Inner Loop Outer Loop Harry H. Porter,

94 Table-Driven Predictive Parsing Algorithm Assume that the grammar is LL(1) i.e., Backtracking will never be needed Always know which righthand side to choose (with one look-ahead) No Left Recursion Grammar is Left-Factored. Example E' + T E' ε T' * F T' ε F ( E ) id Term... +Term +Term Term Factor... * Factor * Factor *... * Factor Step 1: Step 2: From grammar, construct table. Use table to parse strings. Harry H. Porter,

95 Table-Driven Predictive Parsing Algorithm X Y Z $ Stack of grammar symbols ( a * 4 ) + 3 $ Input String Algorithm Output E' + T E' ε T' * F T' ε F ( E ) id Pre-Computed Table: Harry H. Porter,

96 Table-Driven Predictive Parsing Algorithm X Y Z $ Stack of grammar symbols ( a * 4 ) + 3 $ Input String Algorithm Output E' + T E' ε T' * F T' ε F ( E ) id Pre-Computed Table: Input Symbols, plus $ id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε Nonterminals T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

97 Table-Driven Predictive Parsing Algorithm X Y Z $ Stack of grammar symbols Pre-Computed Table: Nonterminals ( a * 4 ) + 3 $ Input String Algorithm Input Symbols, plus $ id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Output E' + T E' ε T' * F T' ε F ( E ) id Blank entries indicate ERROR Harry H. Porter,

98 Predictive Parsing Algorithm Set input ptr to first symbol; Place $ after last input symbol Push $ Push S repeat X = stack top a = current input symbol if X is a terminal or X = $ then if X == a then Pop stack Advance input ptr else Error endif elseif Table[X,a] contains a rule then Pop stack Push Y K... Push Y 2 Push Y 1 Print ( X Y 1 Y 2... Y K ) else Syntax Error endif until X == $ // Table[X,a] is blank X A $ // call it X Y 1 Y 2... Y K Y 1 Y 2... Y K A $ Harry H. Porter,

99 Input: (id*id)+id Output: Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

100 Input: (id*id)+id Output: Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id Add $ to end of input Push $ Push E id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

101 Input: (id*id)+id Output: Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ E $ Add $ to end of input Push $ Push E id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

102 Input: (id*id)+id Output: Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ E $ Look at Table [ E, ( ] Use rule E TE' Pop E Push E' Push T Print E TE' id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

103 Input: (id*id)+id Output: Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ T E $ Look at Table [ E, ( ] Use rule E TE' Pop E Push E' Push T Print E TE' id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

104 Input: (id*id)+id Output: Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ T E $ id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Table [ T, ( ] = T FT' Pop T Push T Push F Print T FT' Harry H. Porter,

105 Input: (id*id)+id Output: Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ F T E $ id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Table [ T, ( ] = T FT' Pop T Push T Push F Print T FT' Harry H. Porter,

106 Input: (id*id)+id Output: Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ F T E $ Table [ F, ( ] = F (E) Pop F Push ) Push E Push ( Print F (E) id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

107 Input: (id*id)+id Output: F ( E ) Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ ( E ) T E $ Table [ F, ( ] = F (E) Pop F Push ) Push E Push ( Print F (E) id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

108 Input: (id*id)+id Output: F ( E ) Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ ( E ) T E $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

109 Input: (id*id)+id Output: F ( E ) Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ E ) T E $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

110 Input: (id*id)+id Output: F ( E ) Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ E ) T E $ Table [ E, id ] = E TE' Pop E Push E' Push T Print E TE' id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

111 Input: (id*id)+id Output: F ( E ) T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Table [ E, id ] = E TE' Pop E Push E' Push T Print E TE' id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

112 Input: (id*id)+id Output: F ( E ) T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ T, id ] = T FT' Pop T Push T' Push F Print T FT' Harry H. Porter,

113 Input: (id*id)+id Output: F ( E ) F T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ T, id ] = T FT' Pop T Push T' Push F Print T FT' Harry H. Porter,

114 Input: (id*id)+id Output: F ( E ) F T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Table [ F, id ] = F id Pop F Push id Print F id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

115 Input: (id*id)+id Output: F ( E ) F id id T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Table [ F, id ] = F id Pop F Push id Print F id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

116 Input: (id*id)+id Output: F ( E ) F id id T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

117 Input: (id*id)+id Output: F ( E ) F id T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

118 Input: (id*id)+id Output: F ( E ) F id T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ T', * ] = T' *FT' Pop T' Push T' Push F Push * Print T' *FT' Harry H. Porter,

119 Input: (id*id)+id Output: F ( E ) F id T' * F T' * F T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ T', * ] = T' *FT' Pop T' Push T' Push F Push * Print T' *FT' Harry H. Porter,

120 Input: (id*id)+id Output: F ( E ) F id T' * F T' * F T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

121 Input: (id*id)+id Output: F ( E ) F id T' * F T' F T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

122 Input: (id*id)+id Output: F ( E ) F id T' * F T' F T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Table [ F, id ] = F id Pop F Push id Print F id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

123 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id id T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Table [ F, id ] = F id Pop F Push id Print F id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

124 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id id T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

125 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

126 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T E ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Table [ T', ) ] = T' ε Pop T' Push <nothing> Print T' ε id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

127 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ E Table [ T', ) ] = T' ε ) Pop T' T E Push <nothing> $ Print T' ε id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

128 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ E Table [ E', ) ] = E' ε ) Pop E' T E Push <nothing> $ Print E' ε id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

129 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Table [ E', ) ] = E' ε Pop E' Push <nothing> Print E' ε id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

130 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε ) T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

131 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Top of Stack matches next input Pop and Scan id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

132 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T E $ Example E' + T E' ε T' * F T' ε F ( E ) id ( id * id ) + id $ Table [ T', + ] = T' ε Pop T' Push <nothing> Print T' ε id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) Harry H. Porter,

133 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ T', + ] = T' ε Pop T' Push <nothing> Print T' ε Harry H. Porter,

134 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ E', + ] = E' +TE' Pop E' Push E' Push T Push + Print E' +TE' Harry H. Porter,

135 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' + T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ E', + ] = E' +TE' Pop E' Push E' Push T Push + Print E' +TE' Harry H. Porter,

136 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' + T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Top of Stack matches next input Pop and Scan Harry H. Porter,

137 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Top of Stack matches next input Pop and Scan Harry H. Porter,

138 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ T, id ] = T FT' Pop T Push T' Push F Print T FT' Harry H. Porter,

139 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ T, id ] = T FT' Pop T Push T' Push F Print T FT' Harry H. Porter,

140 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ F, id ] = F id Pop F Push id Print F id Harry H. Porter,

141 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id id T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ F, id ] = F id Pop F Push id Print F id Harry H. Porter,

142 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id id T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Top of Stack matches next input Pop and Scan Harry H. Porter,

143 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Top of Stack matches next input Pop and Scan Harry H. Porter,

144 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id T E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ T', $ ] = T' ε Pop T' Push <nothing> Print T' ε Harry H. Porter,

145 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id T' ε E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ T', $ ] = T' ε Pop T' Push <nothing> Print T' ε Harry H. Porter,

146 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id T' ε E $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ E', $ ] = E' ε Pop E' Push <nothing> Print E' ε Harry H. Porter,

147 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id T' ε E' ε $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Table [ E', $ ] = E' ε Pop E' Push <nothing> Print E' ε Harry H. Porter,

148 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id T' ε E' ε $ Example E' + T E' ε T' * F T' ε F ( E ) id id + * ( ) $ E E TE' E TE' E' E' +TE' E' ε E' ε T T FT' T FT' T' T' ε T' *FT' T' ε T' ε F F id F (E) ( id * id ) + id $ Input symbol == $ Top of stack == $ Loop terminates with success Harry H. Porter,

149 Input: (id*id)+id Output: Reconstructing the Parse Tree E T E' E' + T E' ε T' * F T' ε F ( E ) id Harry H. Porter,

150 Input: (id*id)+id Output: Reconstructing the Parse Tree E T E' E' + T E' ε T' * F T' ε F ( E ) id F T' Harry H. Porter,

151 Input: (id*id)+id Output: F ( E ) Reconstructing the Parse Tree E T E' E' + T E' ε T' * F T' ε F ( E ) id F T' ( E ) Harry H. Porter,

152 Input: (id*id)+id Output: F ( E ) Reconstructing the Parse Tree E T E' F T' E' + T E' ε T' * F T' ε F ( E ) id ( E ) T E' Harry H. Porter,

153 Input: (id*id)+id Output: F ( E ) Reconstructing the Parse Tree E T E' F T' E' + T E' ε T' * F T' ε F ( E ) id ( E ) T E' F T' Harry H. Porter,

154 Input: (id*id)+id Output: F ( E ) F id Reconstructing the Parse Tree E T E' F T' ( E ) E' + T E' ε T' * F T' ε F ( E ) id T E' F T' id Harry H. Porter,

155 Input: (id*id)+id Output: F ( E ) F id T' * F T' Reconstructing the Parse Tree E T E' F T' ( E ) E' + T E' ε T' * F T' ε F ( E ) id T E' F T' id * F T' Harry H. Porter,

156 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id Reconstructing the Parse Tree E T E' F T' ( E ) E' + T E' ε T' * F T' ε F ( E ) id T E' F T' id * F T' id Harry H. Porter,

157 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε Reconstructing the Parse Tree T E' F T' ( E ) T E' E E' + T E' ε T' * F T' ε F ( E ) id F T' id * F T' id ε Harry H. Porter,

158 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε Reconstructing the Parse Tree T E' F T' ( E ) T E' E E' + T E' ε T' * F T' ε F ( E ) id F T' ε id * F T' id ε Harry H. Porter,

159 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε Reconstructing the Parse Tree T E' F T' ( E ) ε T E' F T' ε E E' + T E' ε T' * F T' ε F ( E ) id id * F T' id ε Harry H. Porter,

160 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' Reconstructing the Parse Tree T E' F T' + T ( E ) ε T E' ε F T' E E' + T E' ε T' * F T' ε F ( E ) id E' id * F T' id ε Harry H. Porter,

161 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' Reconstructing the Parse Tree F T T ( E ) ε T' F E' ε T' E + F E' T T' E' + T E' ε T' * F T' ε F ( E ) id E' id * F T' id ε Harry H. Porter,

162 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id Reconstructing the Parse Tree F T T ( E ) ε T' F E' ε T' E + F id E' T T' E' + T E' ε T' * F T' ε F ( E ) id E' id * F T' id ε Harry H. Porter,

163 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id T' ε Reconstructing the Parse Tree F id T * T ( E ) ε T' F F T' E' ε T' E + F id E' T T' ε E' + T E' ε T' * F T' ε F ( E ) id E' id ε Harry H. Porter,

164 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id T' ε E' ε Reconstructing the Parse Tree F id T * T ( E ) ε F T' ε T' F F T' E' ε T' E + id E' T ε E' + T E' ε T' * F T' ε F ( E ) id E' id ε Harry H. Porter,

165 Input: (id*id)+id Output: F ( E ) F id T' * F T' F id T' ε E' ε T' ε E' + T E' F id T' ε E' ε Reconstructing the Parse Tree Leftmost Derivation: E T E' F T' E' ( E ) T' E' ( T E' ) T' E' ( F T' E' ) T' E' ( id T' E' ) T' E' ( id * F T' E' ) T' E' ( id * id T' E' ) T' E' ( id * id E' ) T' E' ( id * id ) T' E' ( id * id ) E' ( id * id ) + T E' ( id * id ) + F T' E' ( id * id ) + id T' E' ( id * id ) + id E' ( id * id ) + id E' + T E' ε T' * F T' ε F ( E ) id Harry H. Porter,

166 FIRST Function Let α be a string of symbols (terminals and nonterminals) Define: FIRST (α) = The set of terminals that could occur first in any string derivable from α = { a α * aw, plus ε if α * ε } Harry H. Porter,

167 FIRST Function Let α be a string of symbols (terminals and nonterminals) Define: FIRST (α) = The set of terminals that could occur first in any string derivable from α = { a α * aw, plus ε if α * ε } Example: E' + T E' ε T' * F T' ε F ( E ) id FIRST (F) =? Harry H. Porter,

168 FIRST Function Let α be a string of symbols (terminals and nonterminals) Define: FIRST (α) = The set of terminals that could occur first in any string derivable from α = { a α * aw, plus ε if α * ε } Example: E' + T E' ε T' * F T' ε F ( E ) id FIRST (F) = { (, id } FIRST (T') =? Harry H. Porter,

169 FIRST Function Let α be a string of symbols (terminals and nonterminals) Define: FIRST (α) = The set of terminals that could occur first in any string derivable from α = { a α * aw, plus ε if α * ε } Example: E' + T E' ε T' * F T' ε F ( E ) id FIRST (F) = { (, id } FIRST (T') = { *, ε} FIRST (T) =? Harry H. Porter,

170 FIRST Function Let α be a string of symbols (terminals and nonterminals) Define: FIRST (α) = The set of terminals that could occur first in any string derivable from α = { a α * aw, plus ε if α * ε } Example: E' + T E' ε T' * F T' ε F ( E ) id FIRST (F) = { (, id } FIRST (T') = { *, ε} FIRST (T) = { (, id } FIRST (E') =? Harry H. Porter,

171 FIRST Function Let α be a string of symbols (terminals and nonterminals) Define: FIRST (α) = The set of terminals that could occur first in any string derivable from α = { a α * aw, plus ε if α * ε } Example: E' + T E' ε T' * F T' ε F ( E ) id FIRST (F) = { (, id } FIRST (T') = { *, ε} FIRST (T) = { (, id } FIRST (E') = { +, ε} FIRST (E) =? Harry H. Porter,

172 FIRST Function Let α be a string of symbols (terminals and nonterminals) Define: FIRST (α) = The set of terminals that could occur first in any string derivable from α = { a α * aw, plus ε if α * ε } Example: E' + T E' ε T' * F T' ε F ( E ) id FIRST (F) = { (, id } FIRST (T') = { *, ε} FIRST (T) = { (, id } FIRST (E') = { +, ε} FIRST (E) = { (, id } Harry H. Porter,

173 To Compute the FIRST Function For all symbols X in the grammar... if X is a terminal then FIRST(X) = { X } if X ε is a rule then add ε to FIRST(X) if X Y 1 Y 2 Y 3... Y K is a rule then if a FIRST(Y 1 ) then add a to FIRST(X) if ε FIRST(Y 1 ) and a FIRST(Y 2 ) then add a to FIRST(X) if ε FIRST(Y 1 ) and ε FIRST(Y 2 ) and a FIRST(Y 3 ) then add a to FIRST(X)... if ε FIRST(Y i ) for all Y i then add ε to FIRST(X) Repeat until nothing more can be added to any sets. Harry H. Porter,

174 To Compute the FIRST(X 1 X 2 X 3...X N ) Result = {} Add everything in FIRST(X 1 ), except ε, to result Harry H. Porter,

175 To Compute the FIRST(X 1 X 2 X 3...X N ) Result = {} Add everything in FIRST(X 1 ), except ε, to result if ε FIRST(X 1 ) then Add everything in FIRST(X 2 ), except ε, to result endif Harry H. Porter,