6

Grammar, which knows how to control even kings. Molière

This chapter marks the first major milestone of the book. Many of us have cobbled together a mishmash of regular expressions and substring operations to extract some sense out of a pile of text. The code was probably riddled with bugs and a beast to maintain. Writing a real parser—one with decent error-handling, a coherent internal structure, and the ability to robustly chew through a sophisticated syntax—is considered a rare, impressive skill. In this chapter, you will attain it.

It’s easier than you think, partially because we front-loaded a lot of the hard work in the last chapter. You already know your way around a formal grammar. You’re familiar with syntax trees, and we have some Java classes to represent them. The only remaining piece is parsing—transmogrifying a sequence of tokens into one of those syntax trees.

Some CS textbooks make a big deal out of parsers. In the 60s, computer scientists—reasonably fed up with programming in assembly language—started designing more sophisticated, human-friendly languages like FORTRAN and ALGOL. Alas, they weren’t very machine-friendly, for the primitive machines at the time.

They designed languages that they honestly weren’t even sure how to write compilers for, and then did ground-breaking work inventing parsing and compiling techniques that could handle these new big languages on those old tiny machines.

Classic compiler books read like fawning hagiographies of these pioneers and their tools. The cover of “Compilers: Principles, Techniques, and Tools” literally has a dragon labeled “complexity of compiler design” being slain by a knight bearing a sword and shield branded “LALR parser generator” and “syntax directed translation”. They laid it on thick.

A little self-congratulation is well-deserved, but the truth is you don’t need to know most of that stuff to bang out a high quality parser for a modern machine. As always, I encourage you to broaden your education and take it in later, but this book omits the trophy case.

6 . 1 Ambiguity and the Parsing Game

In the last chapter, I said you can “play” a context free grammar like a game in order to generate strings. Now, we play that game in reverse. Given a string—a series of tokens—we map those tokens to terminals in the grammar to figure out which rules could have generated that string.

The “could have” part is interesting. It’s entirely possible to create a grammar that is ambiguous, where different choices of productions can lead to the same string. When you’re using the grammar to generate strings, that doesn’t matter much. Once you have the string, who cares how you got to it?

When parsing, ambiguity means the parser may misunderstand the user’s code. Here’s the Lox expression grammar we put together in the last chapter:

expression → literal
           | unary
           | binary
           | grouping ;

literal    → NUMBER | STRING | "false" | "true" | "nil" ;
grouping   → "(" expression ")" ;
unary      → ( "-" | "!" ) expression ;
binary     → expression operator expression ;
operator   → "==" | "!=" | "<" | "<=" | ">" | ">="
           | "+"  | "-"  | "*" | "/" ;

This is a valid string in that grammar:

But there are two ways we could have generated it. One way is:

1. Starting at expression, pick binary.
2. For the left-hand expression, pick NUMBER, and use 6.
3. For the operator, pick "/".
4. For the right-hand expression, pick binary again.
5. In that nested binary expression, pick 3 - 1.

Another is:

1. Starting at expression, pick binary.
2. For the left-hand expression, pick binary again.
3. In that nested binary expression, pick 6 / 3.
4. Back at the outer binary, for the operator, pick "-".
5. For the right-hand expression, pick NUMBER, and use 1.

Those produce the same strings, but not the same syntax trees:

In other words, the grammar allows seeing the expression as (6 / 3) - 1 or 6 / (3 - 1). That in turn affects the result of evaluating it. The way mathematicians have solved this ambiguity since blackboards were first invented is by defining rules for precedence and associativity.

• Precedence determines which operator is evaluated first in an expression containing a mixture of different operators. Precedence rules tell us that we evaluate the / before the - in the above example. Operators with higher precedence are evaluated before operators with lower precedence. Equivalently, higher precedence operators are said to “bind tighter”.

• Associativity determines which operator is evaluated first in a series of the same operator. When an operator is left-associative (think “left-to-right”), operators on the left evaluate before those on the right. Since - is left-associative, this expression:

  is equivalent to:

  Assignment, on the other hand, is right-associative. This:

  is equivalent to:

Without well-defined precedence and associativity, an expression that uses multiple operators is ambiguous—it can be parsed into different syntax trees, which could in turn evaluate to different results. For Lox, we follow the same precedence rules as C, going from highest to lowest:

How do we stuff these restrictions into our context-free grammar? Right now, when we have an expression that contains subexpressions, like a binary operator, we allow any expression in there:

binary → expression operator expression ;

The expression nonterminal allows us to pick any kind of expression as an operand, regardless of the operator we picked. The rules of precedence limit that. For example, an operand of a * expression cannot be a + expression, since the latter has lower precedence.

For the multiplication operands, we need a nonterminal that means “any kind of expression of higher precedence than *”. Something like:

multiplication → higherThanMultiply "*" higherThanMultiply ;

Since * and / have the same precedence and the level above them is unary operators, a better approximation is:

multiplication → unary ( "*" | "/" ) unary ;

Except that’s not quite right. We broke associativity. This multiplication rule doesn’t allow 1 * 2 * 3. To support associativity, we make one side permit expressions at the same level. Which side we choose determines if the operator is left- or right-associative. Since multiplication and division are left-associative, it’s:

multiplication → multiplication ( "*" | "/" ) unary ;

This is correct, but the fact that the first nonterminal in the body of the rule is the same as the head of the rule means this production is left-recursive. Some parsing techniques, including the one we’re going to use, have trouble with left recursion. Instead, we’ll use this other style:

multiplication → unary ( ( "/" | "*" ) unary )* ;

At the grammar level, this sidesteps left recursion by saying a multiplication expression is a flat sequence of multiplications and divisions. This mirrors the code we’ll use to parse a sequence of multiplications.

If we rejigger all of the binary operator rules in the same way, we get:

expression     → equality ;
equality       → comparison ( ( "!=" | "==" ) comparison )* ;
comparison     → addition ( ( ">" | ">=" | "<" | "<=" ) addition )* ;
addition       → multiplication ( ( "-" | "+" ) multiplication )* ;
multiplication → unary ( ( "/" | "*" ) unary )* ;
unary          → ( "!" | "-" ) unary
               | primary ;
primary        → NUMBER | STRING | "false" | "true" | "nil"
               | "(" expression ")" ;

Instead of a single binary rule, there are now four separate rules for each binary operator precedence level. The main expression rule is no longer a flat series of | branches for each kind of expression. Instead, it is simply an alias for the lowest-precedence expression form, equality, because that includes all higher-precedence expressions too.

Each binary operator’s operands use the next-higher precedence level. After the binary operators, we go to unary, the rule for unary operator expressions, since those bind tighter than binary ones. For unary, we do use a recursive rule because unary operators are right-associative, which means instead of left recursion, we have right recursion. The unary nonterminal is at the end of the body for unary, not the beginning, and our parser won’t have any trouble with that.

Finally, the unary rule alternately bubbles up to primary in cases where it doesn’t match a unary operator. “Primary” is the time-honored name for the highest level of precedence including the atomic expressions like literals.

Our grammar grew a bit, but it’s unambiguous now. We’re ready to make a parser for it.

6 . 2 Recursive Descent Parsing

There is a whole pack of parsing techniques whose names mostly seem to be combinations of “L” and “R”—LL(k), LR(1), LALR—along with more exotic beasts like parser combinators, Earley parsers, the shunting yard algorithm, and packrat parsing. For our first interpreter, one technique is more than sufficient: recursive descent.

Recursive descent is the simplest way to build a parser, and doesn’t require using complex parser generator tools like Yacc, Bison or ANTLR. All you need is straightforward hand-written code. Don’t be fooled by its simplicity, though. Recursive descent parsers are fast, robust, and can support sophisticated error-handling. In fact, GCC, V8 (the JavaScript VM in Chrome), Roslyn (the C# compiler written in C#) and many other heavyweight production language implementations use recursive descent. It kicks ass.

It is considered a top-down parser because it starts from the top or outermost grammar rule (here expression) and works its way down into the nested subexpressions before finally reaching the leaves of the syntax tree. This is in contrast with bottom-up parsers like LR that start with primary expressions and compose them into larger and larger chunks of syntax.

A recursive descent parser is a literal translation of the grammar’s rules straight into imperative code. Each rule becomes a function. The body of the rule translates to code roughly like:

It’s called “recursive descent” because when a grammar rule refers to itself—directly or indirectly—that translates to recursive method calls.

6 . 2 . 1 The parser class

Each grammar rule becomes a method inside this new class:

package com.craftinginterpreters.lox;

import java.util.List;                                 

import static com.craftinginterpreters.lox.TokenType.*;

class Parser {                                         
  private final List<Token> tokens;                    
  private int current = 0;                             

  Parser(List<Token> tokens) {                         
    this.tokens = tokens;                              
  }                                                    
}                                                      

lox/Parser.java, create new file

Like the scanner, it consumes a sequence, only now we’re working at the level of entire tokens. It takes in a list of tokens and uses current to point to the next token eagerly waiting to be used.

We’re going to run straight through the expression grammar now and translate each rule to Java code. The first rule, expression, simply expands to the equality rule, so that’s straightforward:

  private Expr expression() {
    return equality();       
  }                          

lox/Parser.java, add after Parser()

Each method for parsing a grammar rule produces a syntax tree for that rule and returns it to the caller. When the body of the rule contains a nonterminal—a reference to another rule—we call that rule’s method.

The rule for equality is a little more complex:

equality → comparison ( ( "!=" | "==" ) comparison )* ;

In Java, that becomes:

  private Expr equality() {                         
    Expr expr = comparison();

    while (match(BANG_EQUAL, EQUAL_EQUAL)) {        
      Token operator = previous();                  
      Expr right = comparison();                    
      expr = new Expr.Binary(expr, operator, right);
    }                                               

    return expr;                                    
  }                                                 

lox/Parser.java, add after expression()

Let’s step through it. The left comparison nonterminal in the body is translated to the first call to comparison() and we store that in a local variable.

Then, the ( ... )* loop in the rule is mapped to a while loop. We need to know when to exit that loop. We can see that inside the rule, we must first find either a != or == token. So, if we don’t see one of those, we must be done with the sequence of equality operators. We express that check using a handy match() method:

  private boolean match(TokenType... types) {
    for (TokenType type : types) {           
      if (check(type)) {                     
        advance();                           
        return true;                         
      }                                      
    }

    return false;                            
  }                                          

lox/Parser.java, add after equality()

This checks to see if the current token is any of the given types. If so, it consumes the token and returns true. Otherwise, it returns false and leaves the token as the current one.

The match() method is defined in terms of two more fundamental operations:

  private boolean check(TokenType type) {
    if (isAtEnd()) return false;         
    return peek().type == type;          
  }                                      

lox/Parser.java, add after match()

This returns true if the current token is of the given type. Unlike match(), it doesn’t consume the token, it only looks at it.

  private Token advance() {   
    if (!isAtEnd()) current++;
    return previous();        
  }                           

lox/Parser.java, add after check()

This consumes the current token and returns it, similar to how our scanner’s advance() method did with characters.

These methods bottom out on the last handful of primitive operations:

  private boolean isAtEnd() {      
    return peek().type == EOF;     
  }

  private Token peek() {           
    return tokens.get(current);    
  }                                

  private Token previous() {       
    return tokens.get(current - 1);
  }                                

lox/Parser.java, add after advance()

isAtEnd() checks if we’ve run out of tokens to parse. peek() returns the current token we have yet to consume and previous() returns the most recently consumed token. The latter makes it easier to use match() and then access the just-matched token.

That’s most of the parsing infrastructure we need. Where were we? Right, so if we are inside the while loop in equality(), then the parser knows it found a != or == operator and must be parsing an equality expression.

It grabs the token that was matched for the operator so we can track which kind of binary expression this is. Then it calls comparison() again to parse the right-hand operand. It combines the operator and the two operands into a new Expr.Binary syntax tree node, and then loops around. Each time, it stores the expression back in the same expr local variable. As it zips through a sequence of equality expressions, that creates a left-associative nested tree of binary operator nodes.

The parser falls out of the loop once it hits a token that’s not an equality operator. Finally, it returns the expression. Note that if it doesn’t encounter a single equality operator, then it never enters the loop. In that case, the equality() method effectively calls and returns comparison(). In that way, this method matches an equality operator or anything of higher precedence.

Moving on to the next rule…

comparison → addition ( ( ">" | ">=" | "<" | "<=" ) addition )* ;

Translated to Java:

  private Expr comparison() {                                
    Expr expr = addition();

    while (match(GREATER, GREATER_EQUAL, LESS, LESS_EQUAL)) {
      Token operator = previous();                           
      Expr right = addition();                               
      expr = new Expr.Binary(expr, operator, right);         
    }                                                        

    return expr;                                             
  }                                                          

lox/Parser.java, add after equality()

The grammar rule is virtually identical to equality and so is the corresponding code. The only differences are the token types for the operators we match, and the method we call for the operands, now addition() instead of comparison(). The remaining two binary operator rules follow the same pattern:

  private Expr addition() {                         
    Expr expr = multiplication();

    while (match(MINUS, PLUS)) {                    
      Token operator = previous();                  
      Expr right = multiplication();                
      expr = new Expr.Binary(expr, operator, right);
    }                                               

    return expr;                                    
  }                                                 

  private Expr multiplication() {                   
    Expr expr = unary();                            

    while (match(SLASH, STAR)) {                    
      Token operator = previous();                  
      Expr right = unary();                         
      expr = new Expr.Binary(expr, operator, right);
    }                                               

    return expr;                                    
  }                                                 

lox/Parser.java, add after comparison()

That’s all of the binary operators, parsed with the correct precedence and associativity. We’re crawling up the precedence hierarchy and now we’ve reached the unary operators:

unary → ( "!" | "-" ) unary
      | primary ;

The code for this is a little different:

  private Expr unary() {                     
    if (match(BANG, MINUS)) {                
      Token operator = previous();           
      Expr right = unary();                  
      return new Expr.Unary(operator, right);
    }

    return primary();                        
  }                                          

lox/Parser.java, add after multiplication()

Again, we look at the current token to see how to parse. If it’s a ! or -, we must have a unary expression. In that case, we grab the token, and then recursively call unary() again to parse the operand. Wrap that all up in a unary expression syntax tree and we’re done.

Otherwise, we must have reached the highest level of precedence, primary expressions.

primary → NUMBER | STRING | "false" | "true" | "nil"
        | "(" expression ")" ;

Most of the cases for the rule are single terminals, so it’s pretty straightforward:

  private Expr primary() {                                 
    if (match(FALSE)) return new Expr.Literal(false);      
    if (match(TRUE)) return new Expr.Literal(true);        
    if (match(NIL)) return new Expr.Literal(null);

    if (match(NUMBER, STRING)) {                           
      return new Expr.Literal(previous().literal);         
    }                                                      

    if (match(LEFT_PAREN)) {                               
      Expr expr = expression();                            
      consume(RIGHT_PAREN, "Expect ')' after expression.");
      return new Expr.Grouping(expr);                      
    }                                                      
  }                                                        

lox/Parser.java, add after unary()

The interesting branch is the one for handling parentheses. After we match an opening ( and parse the expression inside it, we must find a ) token. If we don’t, that’s an error.

6 . 3 Syntax Errors

A parser really has two jobs:

1. Given a valid sequence of tokens, produce a corresponding syntax tree.

2. Given an invalid sequence of tokens, detect any errors and tell the user about their mistakes.

Don’t underestimate how important the second job is! In modern IDEs and editors, the parser is constantly reparsing code—often while the user is still editing it—in order to syntax highlight and support things like auto-complete. That means it will encounter code in incomplete, half-wrong states all the time.

When the user doesn’t realize the syntax is wrong, it is up to the parser to help guide them back onto the right path. The way it reports errors is a large part of your language’s user interface. Good syntax error handling is hard. By definition, the code isn’t in a well-defined state, so there’s no infallible way to know what the user meant to write. The parser can’t read your mind.

There are a couple of hard requirements for when the parser runs into a syntax error:

• It must detect and report the error. If it doesn’t detect the error and passes the resulting malformed syntax tree on to the interpreter, all manner of horrors may be summoned.

• It must not crash or hang. Syntax errors are a fact of life and language tools have to be robust in the face of them. Segfaulting or getting stuck in an infinite loop isn’t allowed. While the source may not be valid code, it’s still a valid input to the parser because users use the parser to learn what syntax is allowed.

Those are the table stakes if you want to get in the parser game at all, but you really want to raise the ante beyond that. A decent parser should:

• Be fast. Computers are thousands of times faster than they were when parser technology was first invented. The days of needing to optimize your parser so that it could get through an entire source file during a coffee break are over. But programmer expectations have risen as quickly, if not faster. They expect their editors to reparse files in milliseconds after every keystroke.

• Report as many distinct errors as there are. Aborting after the first error is easy to implement, but it’s annoying for users if every time they fix what they think is the one error in a file, a new one appears. They want to see them all.

• Minimize cascaded errors. Once a single error is found, the parser no longer really knows what’s going on. It tries to get itself back on track and keep going, but if it gets confused, it may report a slew of ghost errors that don’t indicate other real problems in the code. When the first error is fixed, they disappear, because they merely represent the parser’s own confusion. These are annoying because they can scare the user into thinking their code is in a worse state than it is.

The last two points are in tension. We want to report as many separate errors as we can, but we don’t want to report ones that are merely side effects of an earlier one.

The way a parser responds to an error and keeps going to look for later errors is called “error recovery”. It was a hot research topic in the 60s. Back then, you’d hand a stack of punch cards to the secretary and come back the next day to see if the compiler succeeded. With an iteration loop that slow, you really wanted to find every single error in your code in one pass.

Today, when parsers complete before you’ve even finished typing, it’s less of an issue. Simple, fast error recovery is fine.

6 . 3 . 1 Panic mode error recovery

Of all the recovery techniques devised in yesteryear, the one that best stood the test of time is called—somewhat alarmingly—“panic mode”. As soon as the parser detects an error, it enters panic mode. It knows at least one token doesn’t make sense given its current state in the middle of some stack of grammar productions.

Before it can get back to parsing, it needs to get its state and the sequence of forthcoming tokens aligned such that the next token does match the rule being parsed. This process is called synchronization.

To do that, we select some rule in the grammar that will mark the synchronization point. The parser fixes its parsing state by jumping out of any nested productions until it gets back to that rule. Then it synchronizes the token stream by discarding tokens until it reaches one that can appear at that point in the rule.

Any additional real syntax errors hiding in those discarded tokens aren’t reported, but it also means that any mistaken cascaded errors that are side effects of the initial error aren’t falsely reported either, which is a decent trade-off.

The traditional place in the grammar to synchronize is between statements. We don’t have those yet, so we won’t actually synchronize in this chapter, but we’ll get the machinery in place for later.

6 . 3 . 2 Entering panic mode

Back before we went on this side trek about error recovery, we were writing the code to parse a parenthesized expression. After parsing the expression, it looks for the closing ) by calling consume(). Here, finally, is that method:

  private Token consume(TokenType type, String message) {
    if (check(type)) return advance();

    throw error(peek(), message);                        
  }                                                      

lox/Parser.java, add after match()

It’s similar to match() in that it checks to see if the next token is of the expected type. If so, it consumes it and everything is groovy. If some other token is there, then we’ve hit an error. We report it by calling this:

  private ParseError error(Token token, String message) {
    Lox.error(token, message);                           
    return new ParseError();                             
  }                                                      

lox/Parser.java, add after previous()

First, that shows the error to the user by calling:

  static void error(Token token, String message) {              
    if (token.type == TokenType.EOF) {                          
      report(token.line, " at end", message);                   
    } else {                                                    
      report(token.line, " at '" + token.lexeme + "'", message);
    }                                                           
  }                                                             

lox/Lox.java, add after report()

This reports an error at a given token. It shows the token’s location and the token itself. This will come in handy later since we use tokens throughout the interpreter to track locations in code.

After this is called, the user knows about the syntax error, but what does the parser do next? Back in error(), it creates and returns a ParseError, an instance of:

class Parser {                                               

  private static class ParseError extends RuntimeException {}

  private final List<Token> tokens;                          

lox/Parser.java, nest inside class Parser

This is a simple sentinel class we use to unwind the parser. The error() method returns it instead of throwing because we want to let the caller decide whether to unwind or not.

Some parse errors occur in places where the parser isn’t likely to get into a weird state and we don’t need to synchronize. In those places, we simply report the error and keep on truckin’. For example, Lox limits the number of arguments you can pass to a function. If you pass too many, the parser needs to report that error, but it can and should simply keep on parsing the extra arguments instead of freaking out and going into panic mode.

In our case, though, the syntax error is nasty enough that we want to panic and synchronize. Discarding tokens is pretty easy, but how do we synchronize the parser’s own state?

6 . 3 . 3 Synchronizing a recursive descent parser

With recursive descent, the parser’s state—which rules it is in the middle of recognizing—is not stored explicitly in fields. Instead, we use Java’s own call stack to track what the parser is doing. Each rule in the process of being parsed is a callframe on the stack. In order to reset that state, we need to clear out those callframes.

The natural way to do that in Java is exceptions. When we want to synchronize, we throw that ParseError object. Higher up in the method for the grammar rule we are synchronizing to, we’ll catch it. Since we are synchronizing on statement boundaries, we’ll catch the exception there. After the exception is caught, the parser is in the right state. All that’s left is to synchronize the tokens.

We want to discard tokens until we’re right at the beginning of the next statement. That boundary is pretty easy to spot—it’s one of the main reasons we picked it. After a semicolon, we’re probably finished with a statement. Most statements start with a keyword—for, if, return, var, etc. When the next token is any of those, we’re probably about to start a statement.

This method encapsulates that logic:

  private void synchronize() {                 
    advance();

    while (!isAtEnd()) {                       
      if (previous().type == SEMICOLON) return;

      switch (peek().type) {                   
        case CLASS:                            
        case FUN:                              
        case VAR:                              
        case FOR:                              
        case IF:                               
        case WHILE:                            
        case PRINT:                            
        case RETURN:                           
          return;                              
      }                                        

      advance();                               
    }                                          
  }                                            

lox/Parser.java, add after error()

It discards tokens until it thinks it found a statement boundary. After catching a ParseError, we’ll call this and then we are hopefully back in sync. When it works well, we have discarded tokens that would have likely caused cascaded errors anyway and now we can parse the rest of the file starting at the next statement.

Alas, we don’t get to see this method in action, since we don’t have statements yet. We’ll get to that in a couple of chapters. For now, if an error occurs, we’ll panic and unwind all the way to the top and stop parsing. Since we can only parse a single expression anyway, that’s no big loss.

6 . 4 Wiring up the Parser

We are mostly done parsing expressions now. There is one other place where we need to add a little error handling. As the parser descends through the parsing methods for each grammar rule, it eventually hits primary(). If none of the cases in there match, it means we are sitting on a token that can’t start an expression. We need to handle that error too:

    if (match(LEFT_PAREN)) {                               
      Expr expr = expression();                            
      consume(RIGHT_PAREN, "Expect ')' after expression.");
      return new Expr.Grouping(expr);                      
    }                                                      

    throw error(peek(), "Expect expression.");             

  }                                                        

lox/Parser.java, in primary()

With that, all that remains in the parser is to define an initial method to kick it off. It’s called, naturally enough, parse():

  Expr parse() {                
    try {                       
      return expression();      
    } catch (ParseError error) {
      return null;              
    }                           
  }                             

lox/Parser.java, add after Parser()

We’ll revisit this method later when we add statements to the language. For now, it parses a single expression and returns it. We also have some temporary code to exit out of panic mode. Syntax error recovery is the parser’s job, so we don’t want the ParseError exception to escape into the rest of the interpreter.

When a syntax error does occur, this method returns null. That’s OK. The parser promises not to crash or hang on invalid syntax, but it doesn’t promise to return a usable syntax tree if an error is found. As soon as the parser reports an error, hadError gets set, and subsequent phases are skipped.

Finally, we can hook up our brand new parser to the main Lox class and try it out. We still don’t have an interpreter so, for now, we’ll parse to a syntax tree and then use the AstPrinter class from the last chapter to display it.

Delete the old code to print the scanned tokens and replace it with this:

    List<Token> tokens = scanner.scanTokens();             

    Parser parser = new Parser(tokens);                    
    Expr expression = parser.parse();

    // Stop if there was a syntax error.                   
    if (hadError) return;                                  

    System.out.println(new AstPrinter().print(expression));

  }                                                        

lox/Lox.java, in run(), replace 5 lines

Congratulations, you have crossed the threshold! That really is all there is to hand-writing a parser. We’ll extend the grammar in later chapters with assignment, statements, and other stuff, but none of that is any more complex than the binary operators we tackled here.

Fire up the interpreter and type in some expressions. See how it handles precedence and associativity correctly? Not bad for less than 200 lines of code.

if (flags & FLAG_MASK == SOME_FLAG) { ... } // Wrong.
if ((flags & FLAG_MASK) == SOME_FLAG) { ... } // Right.