OPERATORS = [
[("<=", "left"), ("<", "left")],
[("+", "any"), ("-", "left")],
[("*", "any"), ("/", "left")],
[("^", "right")],
[("(", "left")],
]A list of operators in groups of increasing precedence (tighter binding power). It doesn’t matter exactly what the precedence is, just that each entry within a group has the same precedence (+ and -, etc) and that each group is lower precedence than the next.
OPERATORS = [
[("<=", "left"), ("<", "left")],
[("+", "any"), ("-", "left")],
[("*", "any"), ("/", "left")],
[("^", "right")],
[("(", "left")],
]Pull out the operator names from each entry.
OPERATOR_NAMES = [info[0] for group in OPERATORS for info in group]
class ParseError(Exception):
passTurn an input string into a list of tokens.
def tokenize(source: str) -> list:
if not source.isascii():
raise ParseError("Only ASCII characters are supported")
result: list = []We use an index to keep track of where we are in the input string.
idx: int = 0
def at_end() -> bool:
nonlocal idx
return idx >= len(source)Assuming we have characters left (not at_end()), pull out one character
from the input string and advance the position.
def advance() -> str:
nonlocal idx
result = source[idx]
idx += 1
return resultAssuming we have characters left (not at_end()), look at the next one
available without advancing the position.
def peek() -> str:
nonlocal idx
return source[idx]
while not at_end():
c = advance()Skip all whitespace between tokens
if c.isspace():
continueSkip line comments
if c == "#":
while not at_end() and advance() != "\n":
pass
continueRead a number and append it as an int
if c.isdigit():
num = int(c)
while not at_end() and peek().isdigit():
num = num * 10 + int(advance())
result.append(num)
continueFor operators that share a prefix, we have to disambiguate by looking ahead.
if c == "<":
if not at_end() and peek() == "=":
advance()
result.append("<=")
else:
result.append("<")
continueRead a single-character operator and append it as a string
if c in OPERATOR_NAMES or c in "(),":
result.append(c)
continueRead a variable name and append it as a string
if c.isalpha():
var = c
while not at_end() and peek().isalpha():
var += advance()
result.append(var)
continue
raise ParseError(f"Unexpected character: {c}")
return resultPull out the precedence for each entry by using its group’s index in
OPERATORS. This gives operators * and / precendece 1, for example.
OPERATOR_PREC = {
info[0]: idx for (idx, group) in enumerate(OPERATORS) for info in group
}Pull out the associativity for each entry.
OPERATOR_ASSOC = {info[0]: info[1] for group in OPERATORS for info in group}Precedence climbing is composed of a recursive function parse_ that also contains a loop. The loop is responsible for reading all tokens at or above the given precedence level. When it sees a token below the minimum required precedence, it drops down a precedence level by returning. Otherwise, it uses the current operator’s precedence as the minimum precedence for the next tokens (by recursively calling).
This parser treats the tokens list as a mutable stream of tokens by doing
pop(0) repeatedly. In a real parser, use a deque or stream and don’t
actually pop from the front of a list.
def parse_(tokens: list, min_prec: int):Atomic groups such as integers and parenthesized sub-expressions make up the left hand side of an expression.
def atom():
if not tokens:
raise ParseError("Unexpected end of input")
token = tokens.pop(0)
if token == "-":We only support unary negation, not unary addition.
return ["negate", atom()]
if token == "(":Parse a parenthesized expression.
result = parse_(tokens, 0)
if not tokens or tokens.pop(0) != ")":
raise ParseError("Expected closing parenthesis")
return result
if token in OPERATOR_NAMES:
raise ParseError(f"Unexpected operator: {token}")
if isinstance(token, int):
return token
if isinstance(token, str) and token.isalpha():Return a variable name.
return token
raise ParseError(f"Unexpected token: {token}")
def comma_separated():
if not tokens:
raise ParseError("Expected closing parenthesis in function call")
if tokens[0] == ")":Empty argument list
return []At least one argument
result = [parse_(tokens, 0)]If there are more, each argument starts with a comma
while tokens and tokens[0] == ",":
tokens.pop(0)
result.append(parse_(tokens, 0))
return result
lhs = atom()The main precedence climbing loop.
while tokens and (token := tokens[0]) in OPERATOR_NAMES:
op_prec = OPERATOR_PREC[token]
if op_prec < min_prec:Drop a precedence level by returning.
return lhs
tokens.pop(0)Special-case function application: the function arguments can re-start
the precedence climbing at precedence 0.
if token == "(":
args = comma_separated()
if not tokens or tokens.pop(0) != ")":
raise ParseError("Expected closing parenthesis in function call")
lhs = [lhs, *args]
continueFor left-associative operators such as - and /, bump up the
minimum precedence by one. Don’t bump for any-associative operators
such as + or right-associative operators such as ^.
next_prec = op_prec + 1 if OPERATOR_ASSOC[token] == "left" else op_prec
rhs = parse_(tokens, next_prec)This is where you could build an AST node instead of making a list.
lhs = [token, lhs, rhs]
return lhs
def parse(tokens: list):Start off with minimum precedence of 0.
result = parse_(tokens, 0)
if tokens:If we have tokens left-over, that’s a syntax error.
raise ParseError("Unexpected tokens: " + " ".join(map(str, tokens)))
return resultAlso include a bunch of tests that push the entirety of the example over 200 lines but are illustrative.
import re
import unittest
class TokenizerTests(unittest.TestCase):
def test_empty(self):
self.assertEqual(tokenize(""), [])
def test_digit(self):
self.assertEqual(tokenize("1"), [1])
def test_number(self):
self.assertEqual(tokenize("123"), [123])
def test_add(self):
self.assertEqual(tokenize("+"), ["+"])
def test_sub(self):
self.assertEqual(tokenize("-"), ["-"])
def test_sub(self):
self.assertEqual(tokenize("-"), ["-"])
def test_mul(self):
self.assertEqual(tokenize("*"), ["*"])
def test_div(self):
self.assertEqual(tokenize("/"), ["/"])
def test_pow(self):
self.assertEqual(tokenize("^"), ["^"])
def test_comma(self):
self.assertEqual(tokenize(","), [","])
def test_less(self):
self.assertEqual(tokenize("<"), ["<"])
def test_less_than_one(self):
self.assertEqual(tokenize("<1"), ["<", 1])
def test_less_equal_one(self):
self.assertEqual(tokenize("<=1"), ["<=", 1])
def test_skip_whitespace(self):
self.assertEqual(tokenize("1 2"), [1, 2])
def test_skip_line_comment(self):
self.assertEqual(tokenize("1# 2\n3"), [1, 3])
def test_unrecognized_operator(self):
with self.assertRaises(ParseError):
tokenize("%")
def test_var(self):
self.assertEqual(tokenize("a"), ["a"])
self.assertEqual(tokenize("abc"), ["abc"])
class ParseTests(unittest.TestCase):
def assert_parse_error(self, message):
return self.assertRaisesRegex(ParseError, re.escape(message))
def test_empty(self):
with self.assert_parse_error("Unexpected end of input"):
parse([])
def test_const(self) -> None:
self.assertEqual(parse([3]), 3)
def test_var(self) -> None:
self.assertEqual(parse(["abc"]), "abc")
def test_const_leftover_raises(self) -> None:
with self.assert_parse_error("Unexpected tokens: 4 5"):
parse([3, 4, 5])
def test_const_paren(self) -> None:
self.assertEqual(parse(["(", 3, ")"]), 3)
self.assertEqual(parse(["(", "(", "(", 3, ")", ")", ")"]), 3)
def test_const_paren_missing(self) -> None:
with self.assert_parse_error("Expected closing parenthesis"):
parse(["(", 3])
with self.assert_parse_error("Unexpected tokens: )"):
parse([3, ")"])
def test_call_fun(self) -> None:
self.assertEqual(parse(["f", "(", "x", ")"]), ["f", "x"])
def test_call_fun_no_arguments(self) -> None:
self.assertEqual(parse(["f", "(", ")"]), ["f"])
def test_call_fun_missing_closing_paren(self) -> None:
with self.assert_parse_error("Expected closing parenthesis in function call"):
parse(["f", "("])
with self.assert_parse_error("Expected closing parenthesis in function call"):
parse(["f", "(", "x"])
def test_call_fun_need_comma(self) -> None:
with self.assert_parse_error("Expected closing parenthesis in function call"):
parse(["f", "(", "x", "y", ")"])
def test_call_fun_double_comma(self) -> None:
with self.assert_parse_error("Unexpected token: ,"):
parse(["f", "(", "x", ",", ",", "y", ")"])
def test_call_fun_more_than_one_argument(self) -> None:
self.assertEqual(parse(["f", "(", "x", ",", "y", ")"]), ["f", "x", "y"])
def test_negate_const(self) -> None:
with self.assert_parse_error("Unexpected end of input"):
parse(["-"])
self.assertEqual(parse(["-", 3]), ["negate", 3])
def test_add(self) -> None:
self.assertEqual(parse([1, "+", 2]), ["+", 1, 2])
def test_add_negate(self) -> None:
self.assertEqual(parse([1, "+", "-", 2]), ["+", 1, ["negate", 2]])
def test_mul_negate(self) -> None:
self.assertEqual(parse([1, "*", "-", 2]), ["*", 1, ["negate", 2]])
def test_sub_negate(self) -> None:
self.assertEqual(parse([1, "-", "-", 2]), ["-", 1, ["negate", 2]])
def test_begin_add_raises(self) -> None:
with self.assert_parse_error("Unexpected operator: +"):
parse(["+"])
with self.assert_parse_error("Unexpected operator: +"):
parse(["+", 2])
def test_double_add_raises(self) -> None:
with self.assert_parse_error("Unexpected operator: +"):
parse([1, "+", "+", 2])
def test_add_add(self) -> None:
self.assertEqual(parse([1, "+", 2, "+", 3]), ["+", 1, ["+", 2, 3]])
def test_add_mul(self) -> None:
self.assertEqual(parse([1, "+", 2, "*", 3]), ["+", 1, ["*", 2, 3]])
self.assertEqual(parse(["(", 1, "+", 2, ")", "*", 3]), ["*", ["+", 1, 2], 3])
def test_mul_add(self) -> None:
self.assertEqual(parse([1, "*", 2, "+", 3]), ["+", ["*", 1, 2], 3])
def test_sub(self) -> None:
self.assertEqual(parse([1, "-", 2]), ["-", 1, 2])
def test_sub_sub(self) -> None:
self.assertEqual(parse([1, "-", 2, "-", 3]), ["-", ["-", 1, 2], 3])
def test_add_mul(self) -> None:
self.assertEqual(parse([1, "+", 2, "*", 3]), ["+", 1, ["*", 2, 3]])
self.assertEqual(parse([1, "*", 2, "+", 3]), ["+", ["*", 1, 2], 3])
class EndToEndTests(unittest.TestCase):
def parse(self, source: str) -> list:
return parse(tokenize(source))
def test_int(self):
self.assertEqual(self.parse("123"), 123)
def test_add(self):
self.assertEqual(self.parse("3+4"), ["+", 3, 4])
def test_add_call(self):
self.assertEqual(self.parse("1+f(2)*3"), ["+", 1, ["*", ["f", 2], 3]])
self.assertEqual(self.parse("1*f(2)+3"), ["+", ["*", 1, ["f", 2]], 3])
def test_call_call(self):
self.assertEqual(self.parse("f(1)(2)"), [["f", 1], 2])
def test_call0(self):
self.assertEqual(self.parse("f()"), ["f"])
def test_call1(self):
self.assertEqual(self.parse("f(x)"), ["f", "x"])
def test_call2(self):
self.assertEqual(self.parse("f(x, y)"), ["f", "x", "y"])
def test_call3(self):
self.assertEqual(self.parse("f(x, y, z)"), ["f", "x", "y", "z"])
def test_call_expression_argument(self):
self.assertEqual(self.parse("f(1+2, 3*4)"), ["f", ["+", 1, 2], ["*", 3, 4]])
def test_call_nested(self):
self.assertEqual(self.parse("f(g(x), h(y))"), ["f", ["g", "x"], ["h", "y"]])
if __name__ == "__main__":
unittest.main()