Inspired by only a partial understanding of PEG parsing I decided to build one. The result may not be a great general-purpose PEG parser generator — there are already many of those (e.g. TatSu is written in Python and generates Python code) — but it was a good way to learn about PEG, and it furthers my goal of replacing CPython’s parser with one built from a PEG grammar.

[This is part 2 of my PEG series. See the Series Overview for the rest.]

In this section I lay the groundwork for understanding how the generated parser works, by showing a simple hand-written parser.

(By the way, as an experiment, I’m not sprinkling links all over my writings. If there’s something you don’t understand, just Google it. :-)

The most common way of PEG parsing uses a recursive descent parser with unlimited backtracking. Take the toy grammar from last week’s article:

statement: assignment | expr | if_statement
expr: expr '+' term | expr '-' term | term
term: term '*' atom | term '/' atom | atom
atom: NAME | NUMBER | '(' expr ')'
assignment: target '=' expr
target: NAME
if_statement: 'if' expr ':' statement

A super-abstract recursive descent parser for this language would define a function for each symbol that tries to call the functions corresponding to the alternatives. For example, for statement we’d have this function:

def statement():
if assignment():
return True
if expr():
return True
if if_statement():
return True
return False

Of course this is too simplistic: it leaves out essential details about the parser’s input and output.

Let’s start with the input side. Classic parsers use a separate tokenizer which breaks the input (a text file or string) into a series of tokens, such as keywords, identifiers (names), numbers and operators. PEG parsers (like other modern parsers such as ANTLR) often unify tokenizing and parsing, but for my project I chose to keep the separate tokenizer.

Tokenizing Python is complicated enough that I don’t want to reimplement it using PEG’s formalism. For example, you have to keep track of indentation (this requires a stack inside the tokenizer), and the handling of newlines in Python is interesting (they are significant except inside matching brackets). The many types of string quotes also cause some complexity. In short, I have no beef with Python’s existing tokenizer, so I want to keep it. (Aside: CPython has two tokenizers —an internal one used by the parser, written in C, and the standard library one, which is a faithful reimplementation in pure Python. This is helpful for my project.)

Classic tokenizers typically have a simple interface whereby you call a function, e.g. get_token(), which returns the next token in the input, consuming the input a few characters at a time. The tokenize module simplifies this even further: its basic API is a generator which yields one token at a time. Each token is a TypeInfo object which has several fields, the most important ones of which indicate the type of the token (e.g. NAME, NUMBER, STRING), and its string value, meaning the string of characters comprising the token (e.g. abc, 42, or "hello world"). There are also additional fields that give the coordinates of the token in the input file, which is useful for error reporting.

A special token type isENDMARKER, which indicates that the end of the input file has been reached. The generator terminates if you ignore this and try to get the next token.

But I digress. How do we implement unlimited backtracking? Backtracking requires you to be able to remember a position in the source code and re-parse from that point. The tokenizer API doesn’t allow us to reset its input pointer, but it’s easy to capture the stream of tokens in an array and replay it from there, so that’s what we do. (You could also do this using itertools.tee(), but based on warnings in the docs that’s probably less efficient in our case.)

I suppose you could just first tokenize the entire input into a Python list and then use that as the parser input, but that would mean if there’s an invalid token near the end of the file (such as a string with a missing closing quote) and there’s also a syntax error earlier in the file, you would get an error message about the bad token first. I would find that a poor user experience, since the syntax error could actually be the root cause for the bad string. So my design tokenizes on demand, and the list becomes a lazy list.

The basic API is very simple. The Tokenizer object encapsulates the array of tokens and the position in that array. It has three basic methods:

  • get_token() returns the next token, advancing the position in the array (reading another token from the source if we’re at the end of the array);
  • mark() returns the current position in the array;
  • reset(pos) sets the position in the array (the argument must be something you got from mark()).

We add one convenience function, peek_token() which returns the next token without advancing the position.

Here, then, is the core of the Tokenizer class:

class Tokenizer:    def __init__(self, tokengen):
"""Call with tokenize.generate_tokens(...)."""
self.tokengen = tokengen
self.tokens = []
self.pos = 0
def mark(self):
return self.pos
def reset(self, pos):
self.pos = pos
def get_token(self):
token = self.peek_token()
self.pos += 1
return token
def peek_token(self):
if self.pos == len(self.tokens):
return self.tokens[self.pos]

Now, there are various things still missing (and the names of the methods and instance variables should really start with an underscore), but this will do as a sketch of the Tokenizer API.

The parser also needs to become a class, so that statement(), expr() and so on can become methods. The tokenizer becomes an instance variable, but we don’t want the parsing methods to call get_token() directly — instead, we give the Parser class an expect() method which can succeed or fail just like a parsing method. The argument to expect() is the expected token — either a string (like "+") or a token type (like NAME). I’ll get to the return type after discussing the parser’s output.

In my first sketch of the parser, the parsing functions just returned True or False. That’s fine for theoretical computer science (where the question a parser answers is “is this a valid string in the language?”) but not when you’re building a parser — instead, we want the parser to create an AST. So let’s just arrange it so that each parsing method returns a Node object on success, or None on failure.

The Node class can be super simple:

class Node:    def __init__(self, type, children):
self.type = type
self.children = children

Here, type indicates what kind of AST node this is (e.g. an "add" node or an "if" node), and children is a list of nodes and tokens (instances of TokenInfo). This is enough for a compiler to generate code or do other analysis such as linting or static type checking, although in the future I’d like to change the way we represent the AST.

To fit into this scheme, the expect() method returns a TokenInfo object on success, and None on failure. To support backtracking, I wrap the tokenizer’s mark() and reset() methods (no API change here). Here then is the infrastructure for the Parser class:

class Parser:    def __init__(self, tokenizer):
self.tokenizer = tokenizer
def mark(self):
return self.tokenizer.mark()
def reset(self, pos):
def expect(self, arg):
token = self.tokenizer.peek_token()
if token.type == arg or token.string == arg:
return self.tokenizer.get_token()
return None

Again, I’ve left out some details, but this works.

At this point I need to introduce an important requirement for parsing methods: a parsing method either returns a Node, positioning the tokenizer after the last token of the grammar rule it recognized; or it returns None, and then it leaves the tokenizer position unchanged. If a parsing method reads several tokens and then decides to fail, it must restore the tokenizer’s position. That’s what mark() and reset() are for. Note that expect() also follows this rule.

So here’s a sketch of the actual parser. Note that I am using Python 3.8’s walrus operator (:=):

class ToyParser(Parser):    def statement(self):
if a := self.assignment():
return a
if e := self.expr():
return e
if i := self.if_statement():
return i
return None
def expr(self):
if t := self.term():
pos = self.mark()
if op := self.expect("+"):
if e := self.expr():
return Node("add", [t, e])
if op := self.expect("-"):
if e := self.expr():
return Node("sub", [t, e])
return t
return None
def term(self):
# Very similar...
def atom(self):
if token := self.expect(NAME):
return token
if token := self.expect(NUMBER):
return token
pos = self.mark()
if self.expect("("):
if e := self.expr():
if self.expect(")"):
return e
return None

I’ve left some parsing methods as exercises for the reader — this is really more to give a flavor of what such a parser looks like, and eventually we’ll generate code like this automatically from the grammar. Constants like NAME and NUMBER are imported from the token module in the standard library. (This ties us further to Python tokenization; there are ways around this that we should explore if we want to make a more general PEG parser generator.)

Also note that I cheated a bit: expr is left-recursive, but I made the parser right-recursive, because recursive-descent parsers don’t work with left-recursive grammar rules. There’s a fix for this, but it’s still the topic of some academic research and I’d like to present it separately. Just realize that this version doesn’t correspond 100% with the toy grammar.

The key things I want you to get at this point are:

  • Grammar rules correspond to parser methods, and when a grammar rule references another grammar rule, its parsing method calls the other rule’s parsing method.
  • When multiple items make up an alternative, the parsing method calls the corresponding methods one after the other.
  • When a grammar rule references a token, its parsing method calls expect().
  • When a parsing method successfully recognizes its grammar rule at the given input position, it returns a corresponding AST node; when it fails to recognize its grammar rule, it returns None.
  • Parsing methods must explicitly reset the tokenizer position when they abandon a parse after having consumed one or more tokens (directly, or indirectly by calling another parsing method that succeeded). This applies when abandoning one alternative to try the next, and also when abandoning the parse altogether.

If all parsing methods abide by these rules, it’s not necessary to use mark() and reset() around a single parsing method. You can prove this using induction.

As an aside, it’s tempting to try to get rid of the explicit mark() and reset() calls by using a context manager and a with statement, but this doesn’t work: the reset() call shouldn’t be called upon success! As a further fix you could try to use exceptions for control flow, so the context manager knows whether to reset the tokenizer (I think TatSu does something like this). For example, you could arrange for this to work:

    def statement(self):
with self.alt():
return self.assignment()
with self.alt():
return self.expr()
with self.alt():
return self.if_statement()
raise ParsingFailure

In particular, the little ladder of if statements in atom() for recognizing a parenthesized expression could become:

        with self.alt():
e = self.expr()
return e

But I find this too “magical” — when reading such code you must stay aware that each parsing method (and expect()) may raise an exception, and that this exception is caught and ignored by the context manager in the with statement. That’s pretty unusual, although definitely supported (by returning true from __exit__). Also, my ultimate goal is to generate C, not Python, and in C there’s no with statement to alter the control flow.

Anyway, here are some topics for future installments:

  • generating parsing code from the grammar;
  • packrat parsing (memoization);
  • EBNF features like (x | y), [x y ...], x*, x+;
  • tracing (for debugging the parser or grammar);
  • PEG features like lookahead and “cut”;
  • how to handling left recursive rules;
  • generating C code.

License for this article and the code shown: CC BY-NC-SA 4.0

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