Introduction to parsers


 Lydia Lynch
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1 Syntax Analysis Introduction to parsers Contextfree grammars Pushdown automata Topdown parsing LL grammars and parsers Bottomup parsing LR grammars and parsers Bison/Yacc  parser generators Error Handling: Detection & Recovery 1
2 Introduction to parsers source code Lexical Analyzer token next token Parser syntax tree Semantic Analyzer Symbol Table 2
3 Context Free Grammar CFG & Terminology Rewrite vs. Reduce Derivation Language and CFL Equivalence & CNF Parsing vs. Derivation lm/rm derivation & parse tree Ambiguity & resolution Expressive power Derivation is the reverse of Parsing. If we know how sentences are derived, we may find a parsing method in the reversed direction. 3
4 CFG: An Example Terminals: id, +, , *, /, (, ) Nonterminals: expr, op Productions: expr expr op expr expr ( expr ) expr  expr expr id op +  * / The start symbol: expr 4
5 Notational Conventions in CFG a,b,c, [+09], id: symbols in A,B,C,,S, expr,stmt: symbols in N U,V,W,,X,Y,Z: grammar symbols in( +N) denotes strings in( +N) * u,v, w, denotes strings in * A is an abbreviation of Alternatives: at RHS A A A 5
6 ContextFree Grammars A set of terminals: basic symbols from which sentences are formed A set of nonterminals: syntactic variables denoting sets of strings A set of productions: rules specifying how the terminals and nonterminals can be combined to form sentences The start symbol: a distinguished nonterminal denoting the language 7
7 CFG: Components Specification for Structures & Constituency CFG: formal specification of structure (parse trees) G = {, N, P, S} : terminal symbols N: nonterminal symbols P: production rules S: start symbol 8
8 CFG: Components : terminal symbols the input symbols of the language programming language: tokens (reserved words, variables, operators, ) natural languages: words or parts of speech preterminal: parts of speech (when words are regarded as terminals) N: nonterminal symbols groups of terminals and/or other nonterminals S: start symbol: the largest constituent of a parse tree 9
9 CFG: Components P: production (rewriting) rules form: A β (A: nonterminal, β: string of terminals and nonterminals) meaning: A rewrites to ( consists of, derived into )β, or β reduced to A start with Sproductions (S β) 10
10 Derivations A derivation step is an application of a production as a rewriting rule E E A sequence of derivation steps E E (E) (id ) is called a derivation of  ( id ) from E The symbol * denotes derives in zero or more steps ; the symbol + denotes derives in one or more steps 11
11 CFG: Accepted Languages ContextFree Language Language accepted by a CFG L(G) = { S + (strings of terminals that can be derived from start symbol)} Proof of acceptance: by induction On the number of derivation steps On the length of input string 12
12 ContextFree Languages A contextfree language L(G) is the language defined by a contextfree grammar G A string of terminals is in L(G) if and only if S +, is called a sentence of G If S *, where may contain nonterminals, then we call a sentential form of G E E (E) (id ) G 1 is equivalent to G 2 if L(G 1 ) = L(G 2 ) 13
13 CFG: Equivalence Chomsky Normal Form (CNF) (Chomsky, 1963): εfree, and Every production rule is in either of the following form: A A1 A2 A a (A1, A2: nonterminal, a: terminal) i.e., two nonterminals or one terminal at the RHS Properties: Generate binary parse tree Good simplification for some algorithms e.g., grammar training with the insideoutside algorithm (Baker 1979) Good tool for theoretical proving e.g., time complexity 14
14 CFG: Equivalence Every CFG can be converted into a weakly equivalent CNF equivalence: L(G1) = L(G2) strong equivalent: assign the same phrase structure to each sentence (except for renaming nonterminals) weak equivalent: do not assign the same phrase structure to each sentence e.g., A B C D == {A B X, X CD} 15
15 CFG: An Example Terminals: id, +, , *, /, (, ) Nonterminals: expr, op Productions: expr expr op expr [R1] expr ( expr ) [R2] expr  expr [R3] expr id [R4] op +  * / The start symbol: expr 16
16 Left & Rightmost Derivations Each derivation step needs to choose a nonterminal to rewrite an alternative to apply A leftmost derivation always chooses the leftmost nonterminal to rewrite E lm  E lm  ( E ) lm  ( E + E ) lm  ( id + E ) lm  ( id + id ) A rightmost (canonical) derivation always chooses the rightmost nonterminal to rewrite E rm  E rm  ( E ) rm  ( E + E ) rm  (E + id ) rm  ( id + id ) 17
17 Left & Rightmost Derivations Representation of leftmost/rightmost derivations: Use the sequence of productions (or production numbers) to represent a derivation sequence. Example: E rm  E rm  ( E ) rm  ( E + E ) rm  (E + id ) rm  ( id + id ) => [3], [2], [1], [4], [4] (~ R3, R2, R1, R4, R4) Advantage: A compact representation for parse tree (data compression) Each parse tree has a unique leftmost/rightmost derivation 18
18 Parse Trees A parse tree is a graphical representation for a derivation that filters out the order of choosing nonterminals for rewriting NP NP PP NP girl in the park 19
19 Context Free Grammar (CFG): Specification for Structures & Constituency Parse Tree: graphical representation of structure Root node (S): a sentencial level structure Internal nodes: constituents of the sentence Arcs: relationship between parent nodes and their children (constituents) Terminal nodes: surface forms of the input symbols (e.g., words) Bracketed notation: Alternative representation e.g., [I saw [the [girl [in [the park]]]]] 20
20 Parse Tree: I saw the girl in the park S NP VP 1st parse NP PP NP pron v det n p det n I saw the girl in the park 21
21 Parse Tree: I saw the girl in the park S NP VP 2nd parse NP NP PP NP pron v det n p det n I saw the girl in the park 22
22 LM & RM: An Example E  E E lm  E lm  ( E ) lm  ( E + E ) lm  ( id + E ) lm  ( id + id ) ( E ) E + E id id E rm  E rm  ( E ) rm  ( E + E ) rm  ( E + id ) rm  ( id + id ) 23
23 Parse Trees & Derivations Many derivations may correspond to the same parse tree, but every parse tree has associated with it a unique leftmost and a unique rightmost derivation 24
24 Ambiguous Grammar A grammar is ambiguous if it produces more than one parse tree for some sentence more than one leftmost/rightmost derivation E E + E id + E id + E * E id + id * E id + id * id E E * E E + E * E id + E * E id + id * E id + id * id 25
25 Ambiguous Grammar E E E + E E * E id E * E E + E id id id id id 26
26 Resolving Ambiguity Use disambiguating rules to throw away undesirable parse trees Rewrite grammars by incorporating disambiguating rules into grammars 27
27 An Example The danglingelse grammar stmt if expr then stmt if expr then stmt else stmt other Two parse trees for if E 1 then if E 2 then S 1 else S 2 28
28 An Example S if E then S else S if E then S S if E then S Preferred parse: closest then if E then S else S 29
29 Disambiguating Rules Rule: match each else with the closest previous unmatched then Remove undesired state transitions in the pushdown automaton shift/reduce conflict on else 1 st parse: reduce 2 nd parse: shift 30
30 Grammar Rewriting stmt m_stmt unm_stmt ; with only paired thenelse m_stmt if expr then m_stmt else m_stmt other unm_stmt if expr then stmt if expr then m_stmt else unm_stmt 31
31 RE vs. CFG Every language described by a RE can also be described by a CFG Example: (a b)*abb A0 a A0 b A0 a A1 A1 ba2 A2 ba3 A3 (1) Right branching (2) Starts with a terminal symbol 32
32 a( b) A0 A0 RE vs. CFG Regular Grammar: Right branching Starts with a a( b) A0 terminal symbol a A1 b A2 (a b)* abb b A3 33
33 RE vs. CFG A0 a A0 b A0 a A1 A1 ba2 A2 ba3 A3 A0 a RE: (a b) * abb A2 start a b b b A1 A3 34
34 RE vs. CFG a DFA for (a b) * abb A2 A0 start A0 b A0 a A1 A1 aa1 b A2 A2 aa1 b A3 A3 aa1 b A0 b a b b 0 a a b a A1 A3 35
35 CFG: Expressive Power (cont.) Writing a CFG for a FSA (RE) define a nonterminal Ni for a state with state number i start symbol S = N0 (assuming that state 0 is the initial state) for each transition δ(i,a)=j (from state i to stet j on input alphabet a), add a new production Ni a Nj to P (a== ε Ni Nj) for each final state i, add a new production Ni εto P 36
36 CFG: Expressive Power CFG vs. Regular Expression (R.E.) Every R.E. can be recognized by a FSA Every FSA can be represented by a CFG with production rules of the form: A a B ε Therefore, L(RE) L(CFG) 38
37 CFG: Expressive Power (cont.) Chomsky Hierarchy: R.E.: Regular set (recognized by FSAs) CFG: Contextfree (Pushdown automata) CSG: Contextsensitive (Linear bounded automata) Unrestricted: Recursively enumerable (Tuning Machine) 39
38 PushDown Automata Input Stack Finite Automata Output 40
39 RE vs. CFG Why use REs for lexical syntax? do not need a notation as powerful as CFGs are more concise and easier to understand than CFGs More efficient lexical analyzers can be constructed from REs than from CFGs Provide a way for modularizing the front end into two manageablesized components 41
40 CFG vs. FiniteState Machine Inappropriateness of FSA Constituents: only terminals Recursion: do not allow A => B => A RTN (Recursive Transition Network) FSA with augmentation of recursion arc: terminal or nonterminal if arc is nonterminal: call to a subtransition network & return upon traversal 42
41 Nonregular Constructs REs can denote only a fixed number of repetitions or an unspecified number of repetitions of one given construct E.g. a*b* A nonregular construct: L = {a n b n n 1} 43
42 NonContextFree Constructs CFGs can denote only a fixed number of repetitions or an unspecified number of repetitions of one or two (paired) given constructs E.g. a n b n Some noncontextfree constructs: L 1 = {wcw w is in (a b)*} declaration/use of identifiers L 2 = {a n b m c n d m n 1 and m 1} #formal arguments/#actual arguments L 3 = {a n b n c n n 0} e.g., b: Backspace, c: under score 44
43 ContextFree Constructs FA (RE) cannot keep counts CFGs can keep count of two items but not three Similar contextfree constructs: L 1 = {wcw R w is in (a b)*, R: reverse order} L 2 = {a n b m c m d n n 1 and m 1} L 2 = {a n b n c m d m n 1 and m 1} L 3 = {a n b n n 1} 45
44 CFG Parsers 46
45 Types of CFG Parsers Universal: can parse any CFG grammar CYK, Earley CYK: Exhaustively matching subranges of input tokens against grammar rules, from smaller ranges to larger ranges Earley: Exhaustively enumerating possible expectations from lefttoright, according to current input token and grammar Nonuniversal: e.g., recursive descent parser Universal (to all grammars) is NOT always efficient 47
46 Types of CFG Parsers Practical Parsers: [ what is a good parser? ] Simple: simple program structure Lefttoright (or righttoleft) scan middleout or island driven is often not preferred Topdown or Bottom up matching Efficient: efficient for good/bad inputs Parse normal syntax quickly Detect errors immediately on next token Deterministic: No alternative choices during parsing given next token Small lookahead buffer (also contribute to efficiency) 48
47 Types of CFG Parsers Top Down: Matching from start symbol down to terminal tokens Bottom Up: Matching input tokens with reducible rules from terminal up to start symbol 49
48 Efficient CFG Parsers Top Down: LL Parsers Matching from start symbol down to terminal tokens, lefttoright, according to a leftmost derivation sequence Bottom Up: LR Parsers Matching input tokens with reducible rules, lefttoright, from terminal up to start symbol, in a reverse order of rightmost derivation sequence 50
49 Efficient CFG Parsers Efficient & Deterministic Parsing only possible for some subclasses of grammars with special parsing algorithms Top Down: Parsing LL Grammars with LL Parsers Bottom Up: Parsing LR Grammars with LR Parsers LR grammar is a larger class of grammars than LL 51
50 Parsing Table Construction for Parsing Table: Efficient Parsers A precomputed table (according to the grammar), indicating the appropriate action(s) to take in any predefined state when some input token(s) is/are under examination Lookahead symbol(s): the input symbol(s) under examination for determining next action(s) id + * State0 State1 State2 action1 action2 Good parsers do not change their codes when the grammar is revised. Table driven. action3 action4 num action5 52
51 Parsing Table Construction for Efficient Parsers Parsing Table Construction: Decide a predefined number of lookaheads to use for predicting next state Define and enumerate all the unique states for the parsing method Decide the actions to take in all states with all possible lookahead(s) 53
52 Parsing Table Construction for Efficient Parsers XParser: you can invent any parser and call it the XParser But its parsing algorithm may not handle all grammars deterministically, thus efficiently. XGrammar: Any grammar whose parsing table for the X parsing method/xparser has no conflicting actions in all states NonX Grammar: has more than one action to take under any state 54
53 Parsing Table Construction for Efficient Parsers k: The number of lookahead symbols used by a parser to determine the next action A larger number of lookahead symbols tends to make it less possible to have conflicting actions But may result in a much larger table that grows exponentially with the number of lookaheads Does not guarantee unambiguous for some grammars (inherently ambiguous) X(k) Parser: X Parser that uses k lookahead symbols to determine the next action X(k) Grammar: any grammar deterministically parsable with X(k) Parser 55
54 Types of Grammars Capable of Efficient Parsing LL(k) Grammars Grammars that can be deterministically parsed using an LL(k) parsing algorithm e.g., LL(1) grammar LR(k) Grammars Grammars that can be deterministically parsed using an LR(k) parsing algorithm e.g., SLR(1) grammar, LR(1) grammar, LALR(1) grammar 56
55 TopDown CFG Parsers Recursive Descent Parser vs. NonRecursive LL(1) Parser 57
56 TopDown Parsing Construct a parse tree from the root to the leaves using leftmost derivation S c A B A a b a B d input: cad S S S S c A B c A B c A B c A B a b a a d 58
57 Predictive Parsing A topdown parsing without backtracking there is only one alternative production to choose at each derivation step stmt if expr then stmt else stmt while expr do stmt begin stmt_list end 59
58 LL(k) Parsing The first L stands for scanning the input from left to right The second L stands for producing a leftmost derivation The k stands for the number of input symbols for lookahead used to choose alternative productions at each derivation step 60
59 LL(1) Parsing Use one input symbol of lookahead Same as Recursivedescent parsing But, Nonrecursive predictive parsing 61
60 Recursive Descent Parsing (more) The parser consists of a set of (possibly recursive) procedures Each procedure is associated with a nonterminal of the grammar The sequence of procedures called in processing the input implicitly defines a parse tree for the input 62
61 An Example type simple id array [ simple ] of type simple integer char num dotdot num 63
62 An Example array [ num dotdot num ] of integer type array [ simple ] of type num dotdot num simple integer 64
63 An Example procedure type; begin if lookahead is in { integer, char, num } then simple else if lookahead = id then match(id) else if lookahead = array then begin match(array); match('['); simple; match(']'); match(of); type end else error end; 65
64 An Example procedure match(t : token); begin if lookahead = t then lookahead := nexttoken else error end; 66
65 An Example procedure simple; begin if lookahead = integer then match(integer) else if lookahead = char then match(char) else if lookahead = num then begin match(num); match(dotdot); match(num) end else error end; 67
66 LL(k) Constraint: Left Recursion A grammar is left recursive if it has a nonterminal A such that A + A A A A R R R A A A A * A R R R R 68
67 Direct/Immediate Left Recursion A A 1 A 2... A m n is equivalent to A A i j (i=1,m ; j=1,n) A 1 A' 2 A'... n A' A' 1 A' 2 A'... m A' ( n ) ( m )* 69
68 An Example E E + T T T T * F F F ( E ) id E T E' E' + T E' T F T' T' * F T' F ( E ) id 70
69 Indirect Left Recursion G0: S A a b A A c S d Problem: Indirect LeftRecursion: S A a S d a Scan rules topdown Do not start with symbols defined earlier (=> substitute them if any) Resolve direct recursion SolutionStep1: Indirect to Direct LeftRecursion: A A c A a d b d SolutionStep2: Direct LeftRecursion to RightRecursion: S A a b A b d A' A' 71 A' ca' a d A'
70 Indirect Left Recursion Input. Grammar G with no cycles or production. Output. An equivalent grammar with no left recursion. 1. Arrange the nonterminals in some order A 1, A 2,..., A n 2. for i := 1 to n do begin // Step1: Substitute 1 st symbols of Ai for j := 1 to i  1 do begin // which are previous Aj s replace each production of the form A i A j ( j < i ) by the production A i k where A j k are all the current A j productions; end eliminate direct left recursion among A i productions // Step2 end 72
71 Left Factoring Two alternatives of a nonterminal A have a nontrivial common prefix if, and A 1 2 A A' A'
72 An Example S i E t S i E t S e S a E b S i E t S S' a S' es E b 74
73 TopDown Parsing: as Stack Matching Construct a parse tree from the root to the leaves using leftmost derivation S c A B A a b a B d input: cad S S S S c A B c A B c A B c A B a b a a d 76
74 Nonrecursive Predictive Parsing General State a b c x y z Input Stack X Non Recursive: Stack + Driver Program (instead of Recursive procedures) Parsing program (parser/driver) Parsing table Output M[X,a]= {X > Y 1 Y 2 Y k } Predictive: precomputed parsing actions 77
75 Nonrecursive Predictive Parsing Expand Nonterminal a b c x y z Input Stack Y 1 Non Recursive: Stack + Driver Program (instead of Recursive procedures) Y 2 Y k Parsing program (parser/driver) Parsing table Output M[X,a]= {X > Y 1 Y 2 Y k } Predictive: precomputed parsing actions 78
76 Nonrecursive Predictive Parsing Match Terminal a b c x y z Input Stack Y 1 =a Non Recursive: Stack + Driver Program (instead of Recursive procedures) Y 2 Y k Parsing program (parser/driver) Parsing table Output M[X,a]= {X > Y 1 Y 2 Y k } Predictive: precomputed parsing actions 79
77 Nonrecursive Predictive Parsing  Error Recovery a b c x y z Input Stack Y 1 =a Non Recursive: Stack + Driver Program (instead of Recursive procedures) Y 2 Y k =c Parsing program (parser/driver) Parsing table Output M[X,a]= {X > Y 1 Y 2 Y k } Predictive: precomputed parsing actions 80
78 Nonrecursive Predictive Parsing  Error Recovery a b c x y z Input Stack Y 1 =a Non Recursive: Stack + Driver Program (instead of Recursive procedures) Y 2 Y k =c Parsing program (parser/driver) Parsing table Output M[X,a]= {X > Y 1 Y 2 Y k } Predictive: precomputed parsing actions 81
79 Stack Operations Match when the top stack symbol is a terminal and it matches the input symbol, pop the top stack symbol and advance the input pointer Expand when the top stack symbol is a nonterminal, replace this symbol by the right hand side of one of its productions Leftmost RHS symbol at TopofStack 83
80 An Example type simple id array [ simple ] of type simple integer char num dotdot num 84
81 An Example Action Stack Input E type array [ num dotdot num ] of integer M type of ] simple [ array array [ num dotdot num ] of integer M type of ] simple [ [ num dotdot num ] of integer E type of ] simple num dotdot num ] of integer M type of ] num dotdot num num dotdot num ] of integer M type of ] num dotdot dotdot num ] of integer M type of ] num num ] of integer M type of ] ] of integer M type of of integer E type integer E simple integer M integer integer 85
82 Parsing program push $S onto the stack, where S is the start symbol set ip to point to the first symbol of w$; // try to match S$ with w$ repeat let X be the top stack symbol and a the symbol pointed to by ip; if X is a terminal or $ then if X = a then pop X from the stack and advance ip else error // or error_recovery() else // X is a nonterminal if M[X, a] = X Y 1 Y 2... Y k then pop X from and push Y k... Y 2 Y 1 onto the stack else error // or error_recovery() until X = $ 86
83 Parser Driven by a Parsing Table: Nonrecursive Descent a b c d X X Y1 Y2 Yk X Z1 Z2 Zm Y1 Y1 1 Y1 2 Z1 Z1 1 Z1 2 X() { // WITHOUT εproduction: X ε if (LA= a ) then Y1(); Y2(); Yk(); else if (LA= b ) Z1(); Z2(); ; Zm(); a in FirstSet( Y1 Y2 Yk ) else ERROR(); // no X ε b in FirstSet( Z1 Z2 Zm ) // else RETURN; if X exists } // Recursive decent procedure for matching X 87
84 Parser Driven by a Parsing Table: Nonrecursive Descent a b c d X X Y1 Y2 Yk X Z1 Z2 Zm X Y1 Y1 1 Y1 2 Z1 Z1 1 Z1 2 a in FirstSet( Y1 Y2 Yk ) b in FirstSet( Z1 Z2 Zm ) d in FollowSet(X) (S => * X d ) X() { // WITH εproduction: X ε if (LA= a ) then else if (LA= b ) Y1(); Y2(); Yk(); Z1(); Z2(); ; Zm(); // else ERROR(); // no X ε else if (LA=??) RETURN; // if X exists } // Recursive decent procedure for matching X 88
85 First Sets The first set of a string is the set of terminals that begin the strings derived from. If *, then is also in the first set of. Used simply to flag whether can be null for computing First Set Not for matching any real input when parsing FIRST( ) = {a * a }+{, if * } FIRST( ) includes { }: means that * 89
86 Compute First Sets If X is terminal, then FIRST(X) is {X} If X is nonterminal and X is a production, then add to FIRST(X) If X is nonterminal and X Y 1 Y 2... Y k is a production, then add a to FIRST(X) if for some i, a is in FIRST(Y i ) and is in all of FIRST(Y 1 ),..., FIRST(Y i1 ). If is in FIRST(Y j ) for all j, then add to FIRST(X) 90
87 Follow Sets What to do with matching null: A? TD Recursive Descent Parsing: assumes success LL: more predictive => Follow Set of A The follow set of a nonterminal A is the set of terminals that can appear immediately to the right of A in some sentential form, namely, S * A a a is in the follow set of A. 91
88 Compute Follow Sets Initialization: Place $ in FOLLOW(S), where S is the start symbol and $ is the input right end marker. If there is a production A B, then everything in FIRST( ) except for is placed in FOLLOW(B) is not considered a visible input to follow any symbol If there is a production A B or A B where FIRST( ) contains (i.e., * ), then everything in FOLLOW(A) is in FOLLOW(B) S * A a implies S * B a YES: every symbol that can follow A will also follow B NO!: every symbol that can follow B will also follow A 92
89 An Example E T E' E' +TE' T F T' T' *FT' F ( E ) id FIRST(E) = FIRST(T) = FIRST(F) = { (, id } FIRST(E') = { +, } FIRST(T') = { *, } FOLLOW(E) = FOLLOW(E') = { ), $ } FOLLOW(T) = FOLLOW(T') = { +, ), $ } FOLLOW(F) = { +, *, ), $ } 93
90 Constructing Parsing Table Input. Grammar G. Output. Parsing Table M. Method. 1. For each production A of the grammar, do steps 2 and For each terminal a in FIRST( ), add A to M[A, a]. 3. If is in FIRST( ) [A * ], add A to M[A, b] for each terminal b [including $ ] in FOLLOW(A).  If is in FIRST( ) and $ is in FOLLOW(A), add A to M[A, $]. 4. Make each undefined entry of M be error. 94
91 LL(1) Parsing Table Construction A B C a in First( ) A b in Follow(A) c not in First( ) or Follow(A) A ( * ) error When to apply A? including A A() { // WITH/WITHOUT εproductions: A ( * ) if (LA= a in First(Y1 Y2 Yk)) then Y1(); Y2(); Yk(); else if (LA= b in Follow(A) & εin First(Z1 Z2... )) Z1(); Z2(); ; Zm(); // Nullable else ERROR(); } // Recursive version of LL(1) parser 95
92 An Example id + * ( ) $ E E TE' E TE' E' E' +TE' E' E' T T FT' T FT' T' T' T' *FT' T' T' F F id F (E) 96
93 An Example Stack Input Output $E id + id * id$ $E'T id + id * id$ E TE' $E'T'F id + id * id$ T FT' $E'T'id id + id * id$ F id $E'T' + id * id$ $E' + id * id$ T' $E'T+ + id * id$ E' +TE' $E'T id * id$ $E'T'F id * id$ T FT' $E'T'id id * id$ F id $E'T' * id$ $E'T'F* * id$ T' *FT' $E'T'F id$ $E'T'id id$ F id $E'T' $ $E' $ T' $ $ E' 97
94 LL(1) Grammars A grammar is an LL(1) grammar if its predictive parsing table has no multiplydefined entries 98
95 A Counter Example S ietss' a S' es E b e FOLLOW(S ) a b e i t $ S S a S ietss' S' S' S' S' es E E b Disambiguation: matching closest then e FIRST(e S) 99
96 LL(1) Grammars or Not?? A grammar G is LL(1) iff whenever A are two distinct productions of G, the following conditions hold: For no terminal a do both and derive strings beginning with a. or M[A, first( )&first( )] entries will have conflicting actions At most one of and can derive the empty string or M[A, follow(a)] entries have conflicting actions If *, then does not derive any string beginning with a terminal in FOLLOW(A). or M[A, first( )&follow(a)] entries have conflicting actions 100
97 NonLL(1) Grammar: Ambiguous According to LL(1) Parsing Table Construction a in First( ) & First( ) b in Follow(A) a in First( ) & Follow(A) A A A ( * ) A (/ * ) (but * a ) A A ( * ) A ( * ) B C When will A & A appear in the same table cell?? 101
98 LL(1) Grammars or Not?? If G is leftrecursive or ambiguous, then M will have at least one multiplydefined entry => nonll(1) E.g., X Xa b => FIRST(X) = {b} (and, of course, FIRST(b) = {b}) => M[X,b] includes both {X Xa} and {X b} Ambiguous G and G with leftrecursive productions can not be LL(1). No LL(1) grammar can be ambiguous 102
99 Error Recovery for LL Parsers 103
100 Syntactic Errors Empty entries in a parsing table: Syntactic error is encountered when the lookahead symbol corresponding to this entry is in input buffer Error Recovery information can be encoded in such entries to take appropriate actions upon error Error Detection: (1) Stacktop = x && x!= input (a) (2) Stacktop = A && M[A, a] = empty (error) 104
101 Error Recovery Strategies Panic mode: skip tokens until a token in a set of synchronizing tokens appears INS(eration) type of errors sync at delimiters, keywords,, that have clear functions Phrase Level Recovery local INS(eration), DEL(eation), SUB(stitution) types of errors Error Production define error patterns in grammar Global Correction [Grammar Correction] minimum distance correction 105
102 Error Recovery Panic Mode Panic mode: skip tokens until a token in a set of synchronizing tokens appears Commonly used Synchronizing tokens: SUB(A,ip): use FOLLOW(A) as sync set for A (pop A) use the FIRST set of a higher construct as sync set for a lower construct INS(ip): use FIRST(A) as sync set for A *ip= : use the production deriving as the default DEL(ip): If a terminal on stack cannot be matched, pop the terminal 106
103 Error Recovery Panic Mode Action Stack Input SUB(A,ip) INS(ip) DEL(ip) X A *ip Follow(A) A A *ip First(A) A x *ip x X X A A x *ip Follow(A) *ip First(A) x *ip 107
104 Error Recovery Actions Using Follow & First Sets to Sync Expanding nonterminal A: M[A,a] = error (blank): Skip a in input = delete all such a (until sync with sync symbol, b) /* panic */ M[A,b] = sync (at FOLLOW(A)) Pop A from stack = b is a sync symbol following A M[A,b] = A (sync at FIRST(A)) Expand A (same as normal parsing action) Matching terminal x : (*sp= x )!= a Pop(x) from stack = missing input token x 108
105 FOLLOW(X) An is Example used to Expand productions or Sync (on errors) FOLLOW(E)=FOLLOW(E )={),$} id + * ( ) $ E E TE' E TE' sync sync E' E' +TE' E' E' T T FT' sync T FT' sync sync T' T' T' *FT' T' T' F F id sync sync F (E) sync sync FOLLOW(F)={+,*,),$} FIRST(X) is used to Expand non productions or Sync (on errors) 109
106 An Example Stack Input Output $E ) id * + id$ error, skip ) $E id * + id$ id is in FIRST(E) $E'T id * + id$ E TE' $E'T'F id * + id$ T FT' $E'T'id id * + id$ F id $E'T' * + id$ $E'T'F* * + id$ T' *FT' $E'T'F + id$ error, M[F,+]=synch / FOLLOW(F) $E'T' + id$ F popped $E' + id$ T' $E'T+ + id$ E' +TE' $E'T id$ $E'T'F id$ T FT' $E'T'id id$ F id $E'T' $ $E' $ T' $ $ E' 110
107 Parse Tree  Error Recovered ) id * + id => id * F + id E ) T E F T + T E id * F T F T ε ε id ε 111
Parsing. Roadmap. > Contextfree grammars > Derivations and precedence > Topdown parsing > Leftrecursion > Lookahead > Tabledriven parsing
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