# Parsers. Xiaokang Qiu Purdue University. August 31, 2018 ECE 468

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1 Parsers Xiaokang Qiu Purdue University ECE 468 August 31, 2018

2 What is a parser A parser has two jobs: 1) Determine whether a string (program) is valid (think: grammatically correct) 2) Determine the structure of a program (think: diagramming a sentence) e.g. I saw the man with the binoculars.

3 Agenda How do we define a language? How do we define the set of strings that are grammatically correct Context free grammars How do we recognize strings in the language? How can we tell (easily) whether a program is a valid string in the language How can we determine the structure of a program? LL parsers and LR parsers

4 Languages A language is a (possibly infinite) set of strings Regular expressions describe regular languages Fundamental drawback: can only use finite state to recognize whether a string is in the language Consider this valid piece of C code: { { { int x; } } } Need to make sure that there are the same number of { as } How would you write a regular expression to capture that?

5 Languages Key problem: programming language syntax is recursive If statements can be nested inside while loops which can themselves be nested inside if statements which can be nested inside for loops which can be nested inside switch statements... Nesting can be arbitrarily deep New formalism for specifying these kinds of recursive languages: Context-free Grammars

6 Terminology Grammar G = (Vt, Vn, S, P) Vt is the set of terminals Vn is the set of non-terminals S is the start symbol P is the set of productions Each production takes the form: Vn λ (Vn Vt)+ Grammar is context-free (why?) A simple grammar: G = ({a, b}, {S, A, B}, S, {S AB, A Aa, A a, B Bb, B b})

7 Simple grammar Start symbol Non-terminals S A B A A a A a B B b B b Terminals Production

8 Generating strings S A B A A a A a B B b B b Given a start rule, productions tell us how to rewrite a non-terminal into a different set of symbols Some productions may rewrite to λ. That just removes the nonterminal To derive the string a a b b b we can do the following rewrites: S A B A a B a a B a a B b a a B b b a a b b b

9 Terminology Strings are composed of symbols A A a a B b b A a is a string We will use Greek letters to represent strings composed of both terminals and non-terminals L(G) is the language produced by the grammar G All strings consisting of only terminals that can be produced by G In our example, L(G) = a + b + The language of a context-free grammar is a context-free language All regular languages are context-free, but not vice versa

10 Why is this useful? statement statement ; statement statement if_stmt ; statement while_loop ; statement id = lit ; statement id = id + id ; if_stmt if ( cond_expr ) then statement while_loop while ( cond_expr ) statment cond_expr id < lit

11 Programming language syntax Programming language syntax is defined with CFGs Constructs in language become non-terminals May use auxiliary non-terminals to make it easier to define constructs if_stmt if ( cond_expr ) then statement else_part else_part else statement else_part λ Tokens in language become terminals

12 Parse trees Tree which shows how a string was produced by a language Interior nodes of tree: non-terminals Children: the terminals and nonterminals generated by applying a production rule Leaf nodes: terminals

13 Leftmost derivation Rewriting of a given string starts with the leftmost symbol Exercise: do a leftmost derivation of the input program f(v + v) using the following grammar: E Prefix (E) E v Tail Prefix f Prefix λ Tail + E Tail λ What does the parse tree look like?

14 Rightmost derivation Rewrite using the rightmost non-terminal, instead of the left What is the rightmost derivation of this string? f(v + v) E Prefix (E) E v Tail Prefix f Prefix λ Tail + E Tail λ

15 Simple conversions A B C D E F? D G* A B A C D E D E F D λ D G D

16 Top-down vs. Bottom-up parsers Top-down parsers expand the parse tree in pre-order Identify parent nodes before the children (a+b) Bottom-up parsers expand the parse tree in post-order Identify children before the parents (a+b) Notation: LL(1): Top-down derivation with 1 symbol lookahead LL(k): Top-down derivation with k symbols lookahead LR(1): Bottom-up derivation with 1 symbol lookahead

17 Top-down parsing

18 Top-down parsing Idea: we know sentence has to start with initial symbol Build up partial derivations by predicting what rules are used to expand non-terminals Often called predictive parsers If partial derivation has terminal characters, match them from the input stream

19 A simple example S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$

20 A simple example S A B c \$ A x a A A y a A special end of input symbol A c B b B λ A sentence in the grammar: x a c c \$

21 A simple example S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: S

22 A simple example S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: A B c \$ Predict rule

23 A simple example Choose based on first set of rules S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: x a A B c \$ Predict rule based on next token

24 A simple example S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: x a A B c \$ Match token

25 A simple example S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: x a A B c \$ Match token

26 A simple example Choose based on first set of rules S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: x a c B c \$ Predict rule based on next token

27 A simple example S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: x a c B c \$ Match token

28 A simple example Choose based on follow set S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: x a c λ c \$ Predict rule based on next token

29 A simple example S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: x a c c \$ Match token

30 A simple example S A B c \$ A x a A A y a A A c B b B λ A sentence in the grammar: x a c c \$ Current derivation: x a c c \$ Match token

31 First and follow sets First(α): the set of terminals (and/or λ) that begin all strings that can be derived from α First(A) = {x, y, λ} First(xaA) = {x} First (AB) = {x, y, b} Follow(A): the set of terminals (and/or \$, but no λs) that can appear immediately after A in some partial derivation Follow(A) = {b} S A B \$ A x a A A y a A A λ B b

32 First and follow sets First(α) = {a Vt α * aβ} {λ if α * λ} Follow(A) = {a Vt S +... Aa...} {\$ if S +...A \$} S: start symbol a: a terminal symbol A: a non-terminal symbol α,β: a string composed of terminals and non-terminals (typically, α is the RHS of a production : derived in 1 step * : derived in 0 or more steps + :derived in 1 or more steps

33 Computing first sets Terminal: First(a) = {a} Non-terminal: First(A) Look at all productions for A A X1X2... Xk First(A) (First(X1) - λ) If λ First(X1), First(A) (First(X2) - λ) If λ is in First(Xi) for all i, then λ First(A) Computing First(α): similar procedure to computing First(A)

34 Exercise What are the first sets for all the non-terminals in following grammar: S A B \$ A x a A A y a A A λ B b B A

35 Computing follow sets Follow(S) = {} To compute Follow(A): Find productions which have A on rhs. Three rules: 1.X α A β: Follow(A) (First(β) - λ) 2.X α A β: If λ First(β), Follow(A) Follow(X) 3.X α A: Follow(A) Follow(X) Note: Follow(X) never has λ in it.

36 Exercise What are the follow sets for S A B \$ A x a A A y a A A λ B b B A

37 Towards parser generators Key problem: as we read the source program, we need to decide what productions to use Step 1: find the tokens that can tell which production P (of the form A X1X2... Xm) applies If next token is in Predict(P), then we should choose this production

38 Parse tables Step 2: build a parse table Given some non-terminal Vn (the non-terminal we are currently processing) and a terminal Vt (the lookahead symbol), the parse table tells us which production P to use (or that we have an error) More formally: T: Vn Vt P {Error}

39 Building the parse table Start: T[A][t] = //initialize all fields to error foreach A: foreach P with A on its lhs: foreach t in Predict(P): T[A][t] = P Exercise: build parse table for our toy grammar 1.S A B \$ 2.A x a A 3.A y a A 4.A λ 5.B b

40 Stack-based parser for LL(1) Given the parse table, a stack-based algorithm is much simpler to generate than a recursive descent parser Basic algorithm: 1.Push the RHS of a production onto the stack 2.Pop a symbol, if it is a terminal, match it 3.If it is a non-terminal, take its production according to the parse table and go to 1 Note: always start with start state

41 1. S A B \$ An example How would a stack-based parser parse: x a y a b 2. A x a A 3. A y a A 4. A λ 5. B b Parse stack Remaining input Parser action S x a y a b \$ predict 1 A B \$ x a y a b \$ predict 2 x a A B \$ x a y a b \$ match(x) a A B \$ a y a b \$ match(a) A B \$ y a b \$ predict 3 y a A B \$ y a b \$ match(y) a A B \$ a b \$ match(a) A B \$ b \$ predict 4 B \$ b \$ predict 5 b \$ b \$ match(b) \$ \$ Done!

42 Non-LL(1) grammars Not all grammars are LL(1)! Consider <stmt> if <expr> then <stmt list> endif <stmt> if <expr> then <stmt list> else <stmt list> endif This is not LL(1) (why?) We can turn this in to <stmt> if <expr> then <stmt list> <if suffix> <if suffix> endif <if suffix> else <stmt list> endif

43 Left recursion Left recursion is a problem for LL(1) parsers LHS is also the first symbol of the RHS Consider: E E + T What would happen with the stack-based algorithm?

44 Removing left recursion E E + T E T E E1 Etail E1 T Etail + T Etail Etail λ E + T + T + T E1 Etail

45 LL(k) parsers Can look ahead more than one symbol at a time k-symbol lookahead requires extending first and follow sets 2-symbol lookahead can distinguish between more rules: A ax ay More lookahead leads to more powerful parsers What are the downsides?

46 Are all grammars LL(k)? No! Consider the following grammar: S E E (E + E) E (E E) E x y z When parsing E, how do we know whether to use rule 2 or 3? Potentially unbounded number of characters before the distinguishing + or is found No amount of lookahead will help!

47 In real languages? Consider the if-then-else problem if x then y else z Problem: else is optional if a then if b then c else d Which if does the else belong to? This is analogous to a bracket language : [ i ] j (i j) S [ S C S λ C ] C λ [ [ ] can be parsed: SSλC or SSCλ (it s ambiguous!)

48 Solving the if-then-else problem The ambiguity exists at the language level. To fix, we need to define the semantics properly ] matches nearest unmatched [ This is the rule C uses for if-then-else What if we try this? S [ S S S1 S1 [ S1 ] S1 λ This grammar is still not LL(1) (or LL(k) for any k!)

49 Two possible fixes If there is an ambiguity, prioritize one production over another e.g., if C is on the stack, always match ] before matching λ S [ S C S λ C ] C λ Another option: change the language! e.g., all if-statements need to be closed with an endif S S E E if S E other else S endif endif

50 Parsing if-then-else What if we don t want to change the language? C does not require { } to delimit single-statement blocks To parse if-then-else, we need to be able to look ahead at the entire rhs of a production before deciding which production to use In other words, we need to determine how many ] to match before we start matching [ s LR parsers can do this!

51 Bottom-up parsing

52 LR Parsers Parser which does a Left-to-right, Right-most derivation Rather than parse top-down, like LL parsers do, parse bottom-up, starting from leaves Basic idea: put tokens on a stack until an entire production is found Issues: Recognizing the endpoint of a production Finding the length of a production (RHS) Finding the corresponding nonterminal (the LHS of the production)

53 LR Parsers Basic idea: shift tokens onto the stack. At any step, keep the set of productions that could generate the read-in tokens reduce the RHS of recognized productions to the corresponding non-terminal on the LHS of the production. Replace the RHS tokens on the stack with the LHS non-terminal.

54 Data structures At each state, given the next token, A goto table defines the successor state An action table defines whether to shift put the next state and token on the stack reduce an RHS is found; process the production terminate parsing is complete

55 Simple example 1.P S 2.S x ; S 3.S e State Symbol x ; e P S Action Shift Shift Shift Reduce 3 Reduce 2 Accept

56 Parsing using an LR(0) parser Basic idea: parser keeps track, simultaneously, of all possible productions that could be matched given what it s seen so far. When it sees a full production, match it. Maintain a parse stack that tells you what state you re in Start in state 0 In each state, look up in action table whether to: shift: consume a token off the input; look for next state in goto table; push next state onto stack reduce: match a production; pop off as many symbols from state stack as seen in production; look up where to go according to non-terminal we just matched; push next state onto stack accept: terminate parse

57 Example Parse x ; x ; e Step Parse Stack Reading Input Parser Action 1 0 x ; x ; e Shift x ; x ; e Shift x ; x ; e Shift x ; x ; e Shift x ; x ; e Shift x ; x ; e Reduce 3 (goto 4) x ; x ; S Reduce 2 (goto 4) x ; S Reduce 2 (goto 4) S Accept

58 LR(k) parsers LR(0) parsers No lookahead Predict which action to take by looking only at the symbols currently on the stack LR(k) parsers Can look ahead k symbols Most powerful class of deterministic bottom-up parsers LR(1) and variants are the most common parsers

59 Terminology for LR parsers Configuration: a production augmented with a A X1... Xi Xi+1... Xj The marks the point to which the production has been recognized. In this case, we have recognized X1... Xi Configuration set: all the configurations that can apply at a given point during the parse: A B CD A B GH T B Z Idea: every configuration in a configuration set is a production that we could be in the process of matching

60 Configuration closure set Include all the configurations necessary to recognize the next symbol after the For each configuration in set: If next symbol is terminal, no new configuration added If next symbol is non-terminal X, for each production of the form X α, add configuration X α S E \$ E E + T T T ID (E) closure0({s E \$}) = { S E \$ E E + T E T T ID } T (E)

61 Successor configuration set Starting with the initial configuration set s0 = closure0({s α \$}) an LR(0) parser will find the successor given the next symbol X X can be either a terminal (the next token from the scanner) or a non-terminal (the result of applying a reduction) Determining the successor s = go_to0(s, X): For each configuration in s of the form A β X γ add A β X γ to t s = closure0(t)

62 CFSM CFSM = Characteristic Finite State Machine Nodes are configuration sets (starting from s0) Arcs are go_to relationships S S \$ S ID

63 Building the goto table We can just read this off from the CFSM Symbol State ID \$ S

64 Building the action table Given the configuration set s: We shift if the next token matches a terminal after the in some configuration A α a β s and a Vt, else error We reduce production P if the is at the end of a production B α s where production P is B α Extra actions: shift if goto table transitions between states on a nonterminal accept if we have matched the goal production

65 Action table 0 Shift State 1 Reduce 2 2 Shift 3 Accept

66 Conflicts in action table For LR(0) grammars, the action table entries are unique: from each state, can only shift or reduce But other grammars may have conflicts Reduce/reduce conflicts: multiple reductions possible from the given configuration Shift/reduce conflicts: we can either shift or reduce from the given configuration

67 Shift/reduce conflict Consider the following grammar: S A y A x xx This leads to the following configuration set (after shifting one x : A x x A x Can shift or reduce here

68 Shift/reduce example (2) Consider the following grammar: S A y A λ x This leads to the following initial configuration set: S A y A x A λ Can shift or reduce here

69 Lookahead Can resolve reduce/reduce conflicts and shift/reduce conflicts by employing lookahead Looking ahead one (or more) tokens allows us to determine whether to shift or reduce (cf how we resolved ambiguity in LL(1) parsers by looking ahead one token)

70 Parsers with lookahead Adding lookahead creates an LR(1) parser Built using similar techniques as LR(0) parsers, but uses lookahead to distinguish states LR(1) machines can be much larger than LR(0) machines, but resolve many shift/reduce and reduce/reduce conflicts Other types of LR parsers are SLR(1) and LALR(1) Differ in how they resolve ambiguities yacc and bison produce LALR(1) parsers

71 LR(1) parsing Configurations in LR(1) look similar to LR(0), but they are extended to include a lookahead symbol A X1... Xi Xi+1... Xj, l (where l Vt λ) If two configurations differ only in their lookahead component, we combine them A X1... Xi Xi+1... Xj, {l1... lm}

72 Building configuration sets To close a configuration B α A β, l Add all configurations of the form A γ, u where u First(βl) Intuition: the lookahead symbol for any configuration is the terminal we expect to see after the configuration has been matched The parse could apply the production for A, and the lookahead after we apply the production should match the next token that would be produced by B

73 Example closure1({s E \$, {λ}}) = S E \$, {λ} S E \$ E E + T T T ID (E) E E + T, {\$} E T, {\$} T ID, {\$} T (E), {\$} E E + T, {+} E T, {+} T ID, {+} T (E), {+}

74 Building goto and action tables The function goto1(configuration-set, symbol) is analogous to goto0(configuration-set, symbol) for LR(0) Build goto table in the same way as for LR(0) Key difference: the action table. action[s][x] = reduce when is at end of configuration and x lookahead set of configuration A α, {... x...} s shift when is before x A β x γ s

75 Action and goto tables begin end ; SimpleStmt \$ <program> <stmts> 0 S / 1 1 S / 4 R4 S / 5 S / 2 2 S / 3 3 A 4 S / 4 R4 S / 5 S / 7 5 S / 6 6 S / 4 R4 S / 5 S / 10 7 S / 8 8 S / 9 9 S / 4 R4 S / 6 S / R2 11 R3

76 <program> begin <stmts> end \$ <stmts> <stmts> SimpleStmt ; <stmts> begin <stmts> end ; <stmts> Example <stmts> λ Parse: begin SimpleStmt ; SimpleStmt ; end \$ Step Parse Stack Reading Input Parser Action 1 0 begin S ; S ; end \$ Shift begin S ; S ; end \$ Shift begin S ; S ; end \$ Shift begin S ; S ; end \$ Shift begin S ; S ; end \$ Shift begin S ; S ; end \$ Reduce 4 (goto 10) begin S ; S ; stmt end \$ Reduce 2 (goto 10) begin S ; stmt end \$ Reduce 2 (goto 2) begin stmt end \$ Shift begin stmt end \$ Accept

77 Problems with LR(1) parsers LR(1) parsers are very powerful... But the table size is much larger than LR(0) as much as a factor of Vt (why?) Example: Algol 60 (a simple language) includes several thousand states! Storage efficient representations of tables are an important issue

78 Solutions to the size problem Different parser schemes SLR (simple LR): build an CFSM for a language, then add lookahead wherever necessary (i.e., add lookahead to resolve shift/reduce conflicts) What should the lookahead symbol be? To decide whether to reduce using production A α, use Follow(A) LALR: merge LR states when they only differ by lookahead symbols

79 Dealing with semantic actions When a construct (corresponding to a production in a grammar) is recognized, a typical parser will invoke a semantic action In a compiler, this action generates an intermediate representation of the program construct In an interpreter, this action might be to perform the action specified by the construct. Thus, if a+b is recognized, the value of a and b would be added and placed in a temporary variable Next time: Semantic Actions!

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