parsing nested functions in python

Here is a solution that uses pyparsing, and as such will be much easier to expand:

from pyparsing import *

first a convenience function (use the second tag function and print the parse tree to see why)

def tag(name):
    """This version converts ["expr", 4] => 4
       comment in the version below to see the original parse tree
    """
    def tagfn(tokens):
        tklist = tokens.asList()
        if name == 'expr' and len(tklist) == 1:
            # LL1 artifact removal
            return tklist
        return tuple([name] + tklist)
    return tagfn

# def tag(name):
#     return lambda tokens: tuple([name] + tokens.asList())

Our lexer needs ot recognize left and right parenthesis, integers, and names. This is how you define them with pyparsing:

LPAR = Suppress("(")
RPAR = Suppress(")")
integer = Word(nums).setParseAction(lambda s,l,t: [int(t[0])])
name = Word(alphas)

our parser has function calls, which take zero or more expressions as parameters. A function call is also an expression, so to deal with the circularity we have to forward declare expr and fncall:

expr = Forward()
fncall = Forward()

expr << (integer | fncall).setParseAction(tag('expr'))
fnparams = delimitedList(expr)

fncall << (name + Group(LPAR + Optional(fnparams, default=[]) + RPAR)).setParseAction(tag('fncall'))

Now we can parse our string (we can add spaces and more or less than two parameters to functions as well):

line = "add(multiply(add(2,3),add(4,5)),1)"
res = fncall.parseString(line)

to see what is returned you can print it, this is called the parse-tree (or, since our tag function has simplified it, an abstract syntax tree):

import pprint
pprint.pprint(list(res))

which outputs:

[('fncall',
  'add',
  [('fncall',
    'multiply',
    [('fncall', 'add', [2, 3]), ('fncall', 'add', [4, 5])]),
   1])]

with the commented out tag function it would be (which is just more work to deal with for no added benefit):

[('fncall',
  'add',
  [('expr',
    ('fncall',
     'multiply',
     [('expr', ('fncall', 'add', [('expr', 2), ('expr', 3)])),
      ('expr', ('fncall', 'add', [('expr', 4), ('expr', 5)]))])),
   ('expr', 1)])]

Now define the functions that are available to our program:

FUNCTIONS = {
    'add': lambda *args: sum(args, 0),
    'multiply': lambda *args: reduce(lambda a, b: a*b, args, 1),
}

# print FUNCTIONS['multiply'](1,2,3,4)   # test that it works ;-)

Our parser is now very simple to write:

def parse(ast):
    if not ast:  # will not happen in our program, but it's good practice to exit early on no input
        return

    if isinstance(ast, tuple) and ast[0] == 'fncall':
        # ast is here ('fncall', <name-of-function>, [list-of-arguments])
        fn_name = ast[1]          # get the function name
        fn_args = parse(ast[2])   # parse each parameter (see elif below)
        return FUNCTIONS[fn_name](*fn_args)  # find and apply the function to its arguments
    elif isinstance(ast, list):
        # this is called when we hit a parameter list
        return [parse(item) for item in ast]
    elif isinstance(ast, int):
        return ast

Now call the parser on the result of the lexing phase:

>>> print parse(res[0])  # the outermost item is an expression
46

You can use a generator function to build a very simple parser:

import re, operator
line, f = "add(multiply(add(2,3),add(4,5)),1)", {'add':operator.add, 'multiply':operator.mul}
def parse(d):
   n = next(d, None)
   if n is not None and n != ')':
     if n == '(':
       yield iter(parse(d))
     else:
       yield n
     yield from parse(d)

parsed = parse(iter(re.findall('\(|\)|\w+', line)))
def _eval(d):
   _r = []
   n = next(d, None)
   while n is not None:
      if n.isdigit():
        _r.append(int(n))
      else:
        _r.append(f[n](*_eval(next(d))))
      n = next(d, None)
   return _r

print(_eval(parsed)[0])

Output:

46