Reading 12: Grammars & Parsing

Productions can use operators to combine terminals and nonterminals on the righthand side. The three most important operators in a production expression are:

Repetition, represented by *:

x ::= y*        // x matches zero or more y

Concatenation, represented not by a symbol, but just a space:

x ::= y z       // x matches y followed by z 

Union, also called alternation, represented by |:

x ::= y | z     // x matches either y or z 

By convention, postfix operators like * have highest precedence, which means they are applied first. Concatenation is applied next. Alternation | has lowest precedence, which means it is applied last. Parentheses can be used to override precedence:

m ::= a (b|c) d      // m matches a, followed by either b or c, followed by d
x ::= (y z | a b)*   // x matches zero or more yz or ab pairs

Let’s use these operators to generalize our url grammar to match some other hostnames, such as http://stanford.edu/ and http://google.com/.

url ::= 'http://' hostname '/'
hostname ::= 'mit.edu' | 'stanford.edu' | 'google.com'

The first rule of this grammar demonstrates concatenation. The url nonterminal matches strings that start with the literal string http://, followed by a match to the hostname nonterminal, followed by the literal string /.

The hostname rule demonstrates union. A hostname can match one of the three literal strings, mit.edu or stanford.edu or google.com.

So this grammar represents the set of three strings, http://mit.edu/, http://google.com/, and http://stanford.edu/.

Let’s take it one step further by allowing any lowercase word in place of mit, stanford, google, com and edu:

url ::= 'http://' hostname '/'
hostname ::= word '.' word
word ::= letter*
letter ::= ('a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' 
                | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' 
                | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z')

The new word rule matches a string of zero or more lowercase letters, so the overall grammar can now match http://alibaba.com/ and http://zyxw.edu/ as well. Unfortunately word can also match an empty string, so this url grammar also matches http://./, which is not a legal URL. Here’s a verbose way to fix that, which requires word to match at least one letter.

word ::= letter letter*

We’ll see a simpler way to write word in the next section.

You can also use additional operators which are just syntactic sugar (i.e., they’re equivalent to combinations of the main three operators).

0 or 1 occurrence is represented by ?:

x ::= y?       // an x is a y or is the empty string
               //    equivalent to x ::=   | y
               //                        ^--- note the empty string here

1 or more occurrences is represented by +:

x ::= y+       // an x is one or more y
               //    equivalent to  x ::= y y*

An exact number of occurrences is represented by {n}, and a range of occurrences by {n,m}, {n,}, or {,m}:

x ::= y{3}     // an x is three y
               //    equivalent to x ::= y y y 

x ::= y{1,3}   // an x is between one and three y
               //    equivalent to x ::= y | y y | y y y

x ::= y{,4}    // an x is at most four y
               //    equivalent to x ::=   | y | y y | y y y | y y y y
               //                        ^--- note the empty string here

x ::= y{2,}    // an x is two or more y
               //    equivalent to x ::= y y y*

A character class [...] represents the set of single-character strings matching any of the characters listed in the square brackets:

x ::= [aeiou]     // equivalent to  x ::= 'a' | 'e' | 'i' | 'o' | 'u'

Ranges of characters can be described compactly using -:

x ::= [a-ckx-z]   // equivalent to  x ::= 'a' | 'b' | 'c' | 'k' | 'x' | 'y' | 'z'

An inverted character class [^...] represents the set of single-character strings matching a character not listed in the brackets:

x ::= [^a-c]  // equivalent to  x ::= 'd' | 'e' | 'f' | ... | '0' | '1' | ... | '!' | '@'
              //                          | ... (all other possible characters)

These additional operators allow the word production to be expressed more compactly:

url ::= 'http://' hostname '/'
hostname ::= word '.' word
word ::= [a-z]+

How else do we need to generalize? Hostnames can have more than two components, and there can be an optional port number:

http://didit.csail.mit.edu:4949/

To handle this kind of string, the grammar is now:

url ::= 'http://' hostname (':' port)? '/' 
hostname ::= word '.' hostname | word '.' word
port ::= [0-9]+
word ::= [a-z]+

Notice how hostname is now defined recursively in terms of itself. Which part of the hostname definition is the base case, and which part is the recursive step? What kinds of hostnames are allowed?

Using the repetition operator, we could also write hostname without recursion, like this:

hostname ::= (word '.')+ word

Recursion can sometimes be eliminated from a grammar using operators like this, but not always.

Another thing to observe is that this grammar allows port numbers that are not technically legal, since port numbers can only range from 0 to 65535 (2¹⁶-1). We could write a more complex definition of port that would match only these integers, but that’s not typically done in a grammar. Instead, the constraint 0 ≤ port ≤ 65535 would be specified in the program that uses the grammar.

There are more things we should do to go further:

• generalizing http to support the additional protocols that URLs can have
• generalizing the / at the end to a slash-separated path
• allowing hostnames with the full set of legal characters instead of just a-z

root    ::= integer ('-' integer)+
integer ::= [0-9]+

root   ::= (A B)+
A      ::= [Aa]
B      ::= [Bb]

https://websis.mit.edu/
ftp://ftp.athena.mit.edu/

ptth://web.mit.edu/
mailto:bitdiddle@mit.edu

url ::= protocol '://' hostname (':' port)? '/' 
protocol ::= TODO
hostname ::= word '.' hostname | word '.' word
port ::= [0-9]+
word ::= [a-z]+

Now let’s look at grammars for some file formats. We’ll be using two different markup languages that represent typographic style in text. Here they are:

Markdown:

This is _italic_.

(To learn about Markdown, see the Markdown syntax documentation or Markdown on Wikipedia.)

HTML:

Here is an <i>italic</i> word.

(To learn about HTML, see the HTML Living Standard, or HTML on Wikipedia.)

For simplicity, our example HTML and Markdown grammars will only specify italics, but other text styles are of course possible. Also for simplicity, we will assume the plain text between the formatting delimiters isn’t allowed to use any formatting punctuation, like _ or <.

Here’s the grammar for our simplified version of Markdown:

markdown ::= ( normal | italic ) *
italic ::= '_' normal '_'
normal ::= text
text ::= [^_]*

Here’s the grammar for our simplified version of HTML:

html ::= ( normal | italic ) *
italic ::= '<i>' html '</i>'
normal ::= text
text ::= [^<>]*

Regexes are widely used in programming, and you should have them in your toolbox.

In TypeScript/JavaScript, you can use regexes for manipulating strings (see string.split, string.match, RegExp). They’re built-in as a first-class feature of other modern scripting languages like Python and Ruby, and you can use them in many text editors for find and replace.

In some languages (including TypeScript/JavaScript), regexes have their own literal syntax delimited by forward slashes, /..../. In others (including Python and Java), there is no special syntactic support for regexes, so you just use a quoted string.

Here are some examples of using regexes in TypeScript.

Replace all runs of spaces in a string s with a single space:

const singleSpacedString = s.replace(/ +/g, " ");

Notice the g character in the example above. By default, a regex will only match the first instance in a string. Adding a g character after the regex makes this a “global” pattern which forces it to match all instances in the string.

Match a URL:

if (s.match(/http:\/\/([a-z]+\.)+[a-z]+(:[0-9]+)?\//)) {
    // then s is a url
}

Notice the backslashes in the example above. We want to match a literal period ., so we have to first escape it as \. to protect it from being interpreted as the regex match-any-character operator. Furthermore, we want to match literal forward slashes /, so we also have to escape them as \/ to prevent them from prematurely signalling the end of the regex literal. The frequent necessity for backslash escapes makes regexes less readable.

Extract parts of a date like "2020-03-18":

const s = "2020-03-18";
const regex = /(?<year>\d{4})-(?<month>\d{2})-(?<day>\d{2})/;
const m = s.match(regex);
if (m) {
  assert(m.groups);
  const year = m.groups.year;
  const month = m.groups.month;
  const day = m.groups.day;
  // m.groups.name is the part of s that matched (?<name>...)
}

This example uses named capturing groups like (?<year>...) to extract parts of the matched string and assign them names. The (?<name>...) syntax matches the regex ... inside the parentheses, and then assigns name to the string that match. Note that ? here does not mean 0 or 1 repetition. In this context, right after an open parenthesis, the ? signals that these parentheses have special meaning, not just grouping.

Named capturing groups can be retrieved by the groups property after a successful match. If this regex were matched against "2025-03-18", for example, then m.groups.year would return "2025", m.groups.month would return "03", and m.groups.day would return "18".

const input = "The <b>Good</b>, the <i>Bad</i>, and the <strong>Ugly</strong>";
const regex = /TODO/g;
const output = input.replace(regex, "");

const input = "77 Rose Court Ln";
const regex = /[0-9]+ .* (Rd|St|Ave|Ln)/;
const m = input.match(regex);
if (m) {
    ...
}

const houseNumber = m.groups.houseNumber; // should be "77"
const streetName = m.groups.streetName; // should be "Rose Court"
const streetType = m.groups.streetType; // should be "Ln"

In general, a language that can be expressed with our system of grammars is called context-free. Not all context-free languages are also regular; that is, some grammars can’t be reduced to single nonrecursive productions. Our HTML grammar is context-free but not regular.

The grammars for most programming languages are also context-free. In general, any language with nested structure (like nesting parentheses or braces) is context-free but not regular. That description applies to the TypeScript grammar, shown here in part:

statement ::= 
  '{' statement* '}'
| 'if' '(' expression ')' statement ('else' statement)?
| 'for' '(' forinit? ';' expression? ';' forupdate? ')' statement
| 'while' '(' expression ')' statement
| 'do' statement 'while' '(' expression ')' ';'
| 'try' '{' statement* '}' ( catches | catches? 'finally' '{' statement* '}' )
| 'switch' '(' expression ')' '{' switchgroups '}'
| 'return' expression? ';'
| 'throw' expression ';' 
| 'break' label? ';'
| 'continue' label? ';'
| expression ';' 
| label ':' statement
| ';'

So far, we’ve discussed:

• grammars, which describe groups of strings using a production of the form nonterminal ::= expression of terminals, nonterminals, and operators
  • grammars make use of operators; the three most important ones are a * b (repetition), a b (concatenation), and a | b (union)
  • grammars help differentiate between meaningful strings and nonmeaningful strings; for example, a properly formatted URL will match our url grammar, but a file-system path will not
• parse trees, which diagram how a concrete string matches a grammar
  • parse trees will come in handy when designing software to automatically match strings to a grammar in useful ways, which we will see in the next section
• regular expressions, or regexes, a compact version of regular grammars
  • regexes have a rich history in computer science; they are widely used throughout the full stack and across many languages
  • TypeScript/JavaScript (like many languages) has built-in support for regexes, which includes named capturing groups for meaningfully handling matches
  • regex101.com is a great site to experiment with regular expressions

The last part of this reading introduces the idea of a parser generator, which automates the process of parsing with a grammar.

Here is what our HTML grammar looks like as a ParserLib grammar:

html ::= ( italic | normal ) * ;
italic ::= '<i>' html '</i>' ;
normal ::= text ; 
text ::= [^<>]+ ;  /* represents a string of one or more characters that are not < or > */

Let’s break it down.

Each ParserLib rule consists of a name, followed by a ::=, followed by its definition, terminated by a semicolon. The ParserLib grammar can also include TypeScript-style comments, both single line and multiline.

By convention, we use lowercase for nonterminals: html, normal, italic, text. The ParserLib library is case-insensitive with respect to nonterminal names; it canonicalizes names to all-lowercase, so even if you don’t write all your names into lowercase, you will see them as lowercase when you print your grammar.

Terminals are quoted strings, like '<i>', or more generally regular expressions, like [^<>]+. Unlike normal regular expressions, however, ParserLib syntax requires literal characters to be either quoted or enclosed in character-class brackets. Thus the regex (alpha|beta|[c-z])* is written as ('alpha'|'beta'|[c-z])* in ParserLib syntax.

html ::= ( italic | normal ) * ;

This rule shows that ParserLib rules can have the alternation operator |, repetition operators like * (and also + and ?, even though they’re not shown in this rule), and parentheses for grouping, in the same way we’ve been using in the grammars section.

Whitespace in a grammar is not significant (outside of quoted strings or [...] character classes). Here the operators have been surrounded by spaces to make them more visible. Writing the rule as html::=(italic|normal)*; would have the same effect, just with less readability.

The html nonterminal also happens to be the root of this grammar (also called starting symbol). The root is the nonterminal that the whole input needs to match. It’s good practice to put the rule for the root nonterminal first in the grammar, so that a human reader can take a top-down view, but it isn’t essential. When we create a parser from the grammar, we tell ParserLib which nonterminal the parse should use as the root.

italic ::= '<i>' html '</i>' ;
normal ::= text ;
text ::= [^<>]+ ;

Note that the text production uses the inverted character class [^<>], discussed in the grammars section, to represent any character except < and >.

Consider the grammar shown below.

expr ::= sum ;
sum ::= primary ('+' primary)* ;
primary ::= constant | '(' sum ')' ;
constant ::= [0-9]+ ;

This grammar will accept an expression like 42+2+5, but will reject a similar expression that has any spaces between the constants and the + signs. We could modify the grammar to allow whitespace around the plus sign by modifying the production rule for sum like this:

sum ::= primary (whitespace* '+' whitespace* primary)* ;
whitespace ::= [ \t\r\n]+ ;

However, this can become cumbersome very quickly once the grammar becomes more complicated. ParserLib allows a shorthand to indicate that certain kinds of characters should be skipped.

// the IntegerExpression grammar
@skip whitespace {
    expr ::= sum ;
    sum ::= primary ('+' primary)* ;
    primary ::= constant | '(' sum ')' ;
}
whitespace ::= [ \t\r\n]+ ;
constant ::= [0-9]+ ;

The @skip whitespace notation indicates that zero or more matches to the whitespace nonterminal should be automatically ignored, before and after each terminal, nonterminal, and character class on the righthand side of expr, sum, and primary. So from the point of view of grammar matching, these three rules effectively become:

expr ::= whitespace* sum whitespace* ;
sum ::= whitespace* primary whitespace* (whitespace* '+' whitespace* primary whitespace*)* whitespace* ;
primary ::= whitespace* constant whitespace* | whitespace* '('  whitespace* sum  whitespace* ')' whitespace* ;

Several things are important to note about @skip. First, there is nothing special about whitespace. The @skip directive works with any nonterminal defined in the grammar – so you could @skip punctuation or @skip spacesAndComments instead, by defining appropriate rules for those nonterminals.

Second, note how the definition of constant was intentionally placed outside the @skip block. This is because we want to accept expressions with extra space around a constant, like 42 + 2, but we want to reject spaces within constants, like 4 2 + 2. Putting the constant rule inside the @skip block would cause it to effectively become constant ::= (whitespace* [0-9] whitespace*)+, so it would accept constants with spaces inside them. Putting the rule for primary inside the @skip block – so that its use of constant can be surrounded by whitespace – but keeping the rule for constant outside, provides the desired behavior.

@skip spaces {
  semester ::= season year ;
  season ::= 'Fall' | 'Spring' ;
  year ::= [0-9] [0-9] ;
}
spaces ::= ' '+ ;

@skip spaces {
  semester ::= season year ;
}
season ::= 'Fall' | 'Spring' ;
year ::= [0-9] [0-9] ;
spaces ::= ' '+ ;

@skip space {
  semester ::= season year ;
}
season ::= 'Fall' | 'Spring' ;
year ::= [0-9] [0-9] ;
space ::= ' ' ;

The rest of this reading will use as a running example the IntegerExpression grammar that we defined in the previous section. In order to embed the grammar in TypeScript with its newlines intact, we will use backquotes (also called a template literal):

const grammar: string = `
  @skip whitespace {
      expr ::= sum;
      sum ::= primary ('+' primary)*;
      primary ::= constant | '(' sum ')';
  }
  constant ::= [0-9]+;
  whitespace ::= [ \\t\\r\\n]+;  // <-- note that backslashes must be escaped here, because this regex is inside a string
`;

The ParserLib parser generator tool converts a grammar string like this into a parser. In order to do this, you need to follow three steps. First, you need to import the ParserLib library:

import { Parser, ParseTree, compile } from "parserlib";

The second step is to define an enum type that contains all the nonterminals used by your grammar. This will tell ParserLib which definitions to expect in the grammar and will allow it to check for any missing ones.

enum IntegerGrammar {
  Expr, Sum, Primary, Constant, Whitespace
}

Note that ParserLib itself is case insensitive, but by convention, the names of enum values are capitalized.

From within your code, you can create a Parser by calling its compile factory function.

const parser: Parser<IntegerGrammar> = compile(grammar, IntegerGrammar, IntegerGrammar.Expr);

The compile method takes a grammar string, the nonterminal enumeration, and the nonterminal to use as the root nonterminal of the grammar. In this case, the root nonterminal we want is expr, so we pass IntegerGrammar.Expr.

Assuming you don’t have any syntax errors in your grammar, the result will be a Parser object that can be used to parse text. Notice that the Parser is a generic type that is parameterized by the IntegerGrammar enum that you defined earlier.

Now that you’ve generated the parser object, you are ready to parse your own text. The parser has a method called parse that takes in the text to be parsed as a string, and returns a ParseTree. Calling it produces a parse tree:

const parseTree: ParseTree<IntegerGrammar> = parser.parse("54+(2+ 89)");

Note that the ParseTree is also a generic type that is parameterized by the enum type IntegerGrammar.

For debugging, we can then print this tree out:

console.log(parseTree.toString());

You can also try calling the function visualizeAsUrl(tree) which will provide you with a URL which you can copy and paste to your browser to view the visualization.

So we’ve used the parser to turn a stream of characters into a parse tree, which shows how the grammar matches the stream. Next we want to do something useful with this parse tree: translating it into a value of a recursive abstract data type.

Like the Parser itself, the ParseTree is parameterized by the type NT, an enum type that lists all the nonterminals in the grammar, like the IntegerGrammar enumeration we defined earlier.

The first step is to learn how to traverse the parse tree. The ParseTree object has four methods that you need to be most familiar with. Three of them are fundamental observers, represented as getter properties:

interface ParseTree<NT> {

  /**
   * The nonterminal corresponding to this node in the parse tree.
   */
  name: NT;

  /**
   * The children of this node, in order, excluding @skipped subtrees
   */
  children: ReadonlyArray<ParseTree<NT>>;

  /**
   * The substring of the original string that this subtree matched
   */
  text: string;

Additionally, you can query the ParseTree for all children that match a particular production rule:

  /**
   * Get the children that correspond to a particular production rule 
   * @param name Name of the nonterminal corresponding to the desired production rule.
   * @returns children that represent matches of name's production rule.
   */
  public childrenByName(name: NT): ReadonlyArray<ParseTree<NT>>;
}

A good way to visit the nodes in a parse tree is to write a recursive function. For example, the recursive function below prints all nodes in the parse tree with proper indentation.

/**
 * Traverse a parse tree, indenting to make it easier to read.
 * @param node   parse tree to print.
 * @param indent indentation to use.
 */
function printNodes(node: ParseTree<IntegerGrammar>, indent: string): void {
    console.log(indent + node.name + ":" + node.text);
    for (const child of node.children){
        printNodes(child, indent + "  ");
    }
}

Running this function on the parse tree for 54+(2+ 89) produces the output below. For reference, the grammar and the visualized parse tree are shown at right.

Expr:54+(2+ 89)
  Sum:54+(2+ 89)
    Primary:54
      Constant:54
    Primary:(2+ 89)
      Sum:2+ 89
        Primary:2
          Constant:2
        Primary:89
          Constant:89

// the IntegerExpression grammar
@skip whitespace {
    expr ::= sum ;
    sum ::= primary ('+' primary)* ;
    primary ::= constant | '(' sum ')' ;
}
whitespace ::= [ \t\r\n]+ ;
constant ::= [0-9]+ ;

reading exercises

Parse trees

Which of the following statements are true of a ParserLib parse tree, from careful examination of the example output just above?

the root node of the tree corresponds to the root nonterminal of the grammar(missing answer)

(missing explanation)

each internal node of the tree is named by a nonterminal in the grammar(missing answer)

(missing explanation)

terminals do not have their own separate nodes that can be retrieved by children(missing answer)

(missing explanation)

only a grammar with recursive productions can generate a parse tree(missing answer)

(missing explanation)

a node’s immediate children must correspond to nonterminals mentioned in the node’s production (missing answer)

(missing explanation)

@skip N means that the skipped nonterminal N never appears among the children of nodes whose rules are inside the @skip block(missing answer)

(missing explanation)

check explain

Snapshot diagram of a ParseTree

Let’s see how a Parser­Lib parse tree looks as a snapshot diagram, because it’s closer to how we will think about it and work with it in TypeScript code.

The partial snapshot diagram at the right corresponds to part of the Integer­Grammar parse tree for "2+ 89", also shown.

The snapshot diagram shows the three correspondingly-colored nodes along the left branch of the parse tree: sum, primary, and constant.

Fill in the gray rectangles in the snapshot diagram.

What should appear at location A?

Constant,Primary,Sum,“2”,“+”,“89”(missing answer)

(missing explanation)

What should appear at location B?

Constant,Primary,Sum,“2”,“+”,“89”(missing answer)

(missing explanation)

What should appear at all the locations marked C?

IntegerGrammar,ReadonlyArray<ParseTree<IntegerGrammar>>,ParseTree<IntegerGrammar>(missing answer)

(missing explanation)

check explain

We need to convert the parse tree into a recursive data type. Here’s the definition of the recursive data type that we’re going to use to represent integer arithmetic expressions:

IntegerExpression = Constant(n: number)
                  + Plus(left: IntegerExpression, right: IntegerExpression)

If this syntax is mysterious, review recursive data type definitions.

When a recursive data type represents a language this way, it is often called an abstract syntax tree. An IntegerExpression value captures the important features of the expression – its grouping and the integers in it – while omitting unnecessary details of the sequence of characters that created it.

By contrast, the parse tree that we just generated with the IntegerExpression parser is a concrete syntax tree. It’s called concrete, rather than abstract, because it contains more details about how the expression is represented in actual characters. For example, the strings 2+2, ((2)+(2)), and 0002+0002 would each produce a different concrete syntax tree, but these trees would all correspond to the same abstract IntegerExpression value: Plus(Constant(2), Constant(2)).

Now, we can create a recursive function that walks the ParseTree to produce an IntegerExpression as follows:

/**
 * Convert a parse tree into an abstract syntax tree.
 * 
 * @param parseTree constructed according to the IntegerExpression grammar
 * @returns abstract syntax tree corresponding to parseTree
 */
function makeAbstractSyntaxTree(parseTree: ParseTree<IntegerGrammar>): IntegerExpression {
    if (parseTree.name === IntegerGrammar.Expr) {
        // expr ::= sum;
        return makeAbstractSyntaxTree(parseTree.children[0] ?? assert.fail('missing child'));

    } else if (parseTree.name === IntegerGrammar.Sum) {
        // sum ::= primary ('+' primary)*;
        const children = parseTree.childrenByName(IntegerGrammar.Primary);
        const subexprs = children.map(makeAbstractSyntaxTree);
        const expression: IntegerExpression = subexprs.reduce((result, subexpr) => new Plus(result, subexpr));
        return expression;

    } else if (parseTree.name === IntegerGrammar.Primary) {
        // primary ::= constant | '(' sum ')';
        const child: ParseTree<IntegerGrammar> = parseTree.children[0] ?? assert.fail('missing child');
        // check which alternative (constant or sum) was actually matched
        switch (child.name) {
        case IntegerGrammar.Constant:
            return makeAbstractSyntaxTree(child);
        case IntegerGrammar.Sum:
            return makeAbstractSyntaxTree(child); // for this parser, we do the same thing either way
        default:
            assert.fail(`Primary node unexpected child ${IntegerGrammar[child.name]}`);
        }

    } else if (parseTree.name === IntegerGrammar.Constant) {
        // constant ::= [0-9]+;
        const n: number = parseInt(parseTree.text);
        return new Constant(n);

    } else {
        assert.fail(`cannot make AST for ${IntegerGrammar[parseTree.name]} node`);
    }
}

The function follows the structure of the grammar, handling each rule from the grammar in turn: expr, sum, primary, and constant. The only rule we don’t need to handle here is whitespace, because the grammar uses whitespace only in a @skip block, and skipped subtrees are not returned by children so they should never appear in the traversal. Note, however, that skipped whitespace will still appear in the text field of a matched nonterminal.

Note that this code is tied very closely to the grammar. If you change the rules of the grammar in a significant way, this code will likely need to change too.

For sum nodes, the code uses map and reduce to avoid writing loops (and notice that parseTree.childrenByName is identical to performing a filter on parseTree.children).

To handle primary nodes, this code uses a switch statement. Any time you use switch, remember: the execution of a switch statement starts at the matching case and falls through to subsequent cases. Our code here is careful to return (or raise an error) in every case, never falling through.

Plus(Plus(Constant(1), Constant(2)), 
     Plus(Constant(3), Constant(4)))

@skip spaces { 
  date ::= monthname day ',' year | day '/' monthnum '/' year ;
}
monthname ::= 'January' | 'February' ;
monthnum ::= [0-9]+ ; 
day ::= [0-9]+ ;
year ::= [0-9]+ ;
spaces ::= ' '+ ;

@skip spaces { 
  list ::= int (',' list)? ;
}
int ::= '-'? [0-9]+ ;
spaces ::= ' '+ ;

enum ListGrammar { List, Int, Spaces }
function makeList(parseTree: ParseTree<ListGrammar>): Array<number> {
    switch (parseTree.name) {
    case ListGrammar.List:
        const list: Array<number> = [];
        const intChild: ParseTree<ListGrammar>|undefined = parseTree.children[0];
        const listChild: ParseTree<ListGrammar>|undefined = parseTree.children[1];
        assert(intChild);
        list.push(parseInt(▶▶A◀◀));
        if (▶▶B◀◀) {
            list = list.concat(▶▶C◀◀);
        }
        return list;
    default:
        throw new Error("should never get here");
    }
}

@skip spaces {
  semester ::= season year ;
  season ::= 'Fall' | 'Spring' ;
  year ::= [0-9] [0-9] ;
}
spaces ::= ' '+ ;

// nonterminals in the grammar
enum SemesterGrammar { Semester, Season, Year, Spaces }

// abstract data type representing a season of the year
enum Season { Fall, Spring, Winter, Summer }

/**
 * @param node must be a match to the season rule
 * @returns corresponding Season value
 */
function convertToSeason(node: ParseTree<SemesterGrammar>): Season {
  assert.equal(node.name, SemesterGrammar.Season);
  return ▶▶A◀◀ ? Season.Fall : Season.Spring;
}

@skip spaces {
  semester ::= season year ;
}
season ::= 'Fall' | 'Spring' ;
year ::= [0-9] [0-9] ;
spaces ::= ' '+ ;

  return ▶▶A◀◀ ? Season.Fall : Season.Spring;

Several things can go wrong when parsing.

Your grammar may have a syntax error in it. In this case, compile will throw a ParseError.

The string you are trying to parse may not be parseable with your given grammar. This might happen because your grammar is incorrect, or because your string is incorrect. Either way, the problem will be signaled by the parse method throwing an ParseError.

The ParseError error contains some information about the possible location of the error, although parse errors are sometimes inherently difficult to localize, since the parser cannot know what string you intended to write, so you may need to search a little to find the true location of the error.

ParserLib works by generating a top-down recursive descent parser. These kind of parsers have a few limitations in terms of the grammars that they can parse. There are two in particular that are worth pointing out.

Left recursion. A recursive descent parser can go into an infinite loop if the grammar involves left recursion. This is a case where a definition for a nonterminal involves that nonterminal as its leftmost symbol. For example, the grammar below includes left recursion because one of the possible definitions of sum is sum '+' number which has sum as its leftmost symbol.

sum ::= number | sum '+' number ;
number ::= [0-9]+ ;

This left-recursive definition is problematic because the recursive descent parser needs to try matching all alternatives for sum, both number and sum '+' number. But trying to match sum '+' number immediately requires matching sum as its first step. The parser keeps trying the rule recursively but never makes any progress through the string being parsed – not reducing to a smaller subproblem as correct recursion requires to terminate. By contrast, a rule like expr = number | '(' expr ')', which is recursive but not left-recursive, is able to make progress through the string by matching ( before recursively applying the expr rule again.

Left recursion can also happen indirectly. For example, changing the grammar above to the one below does not address the problem because the definition of sum still indirectly involves a symbol that has sum as its first symbol.

sum ::= number | thing number ;
thing ::= sum '+' ;
number ::= [0-9]+ ;

If you give any of these grammars to ParserLib and then try to use them to parse a symbol, ParserLib will fail with a ParseError listing the offending nonterminal.

There are some general techniques to eliminate left recursion; for our purposes, the simplest approach will be to replace left recursion with repetition (*), so the grammar above becomes:

sum ::= (number '+')* number ;
number ::= [0-9]+ ;

Greediness. This is an issue that you may not run into in this class, but it is a limitation of ParserLib you should be aware of. The ParserLib parsers are greedy in that at every point they try to match a maximal string for any rule they are currently considering. For example, consider the following grammar:

g ::= ab threeb ;
ab ::= 'a'*'b'* ;
threeb ::= 'bbb' ;

The string 'aaaabbb' clearly should match g, but a greedy parser cannot parse it because it will try to parse a maximal substring that matches the ab symbol, and then it will find that it cannot parse threeb because it has already consumed the entire string. Unlike left recursion, which is easy to fix, this is a more fundamental limitation of the type of parser implemented by ParserLib.