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Our subset of Python is now complex enough that the meaning of programs is non-obvious. What if a formal parameter has the same name as a built-in function? Can two functions share names without confusion? To resolve such questions, we must describe environments in more detail.

An environment in which an expression is evaluated consists of a sequence of frames, depicted as boxes. Each frame contains bindings, each of which associates a name with its corresponding value. There is a single global frame. Assignment and import statements add entries to the first frame of the current environment. So far, our environment consists only of the global frame.

1 from math import pi
2 tau = 2 * pi
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Global
pi 3.1416
tau 6.2832

This environment diagram shows the bindings of the current environment, along with the values to which names are bound. The environment diagrams in this text are interactive: you can step through the lines of the small program on the left to see the state of the environment evolve on the right. You can also click on the "Edit code in Online Python Tutor" link to load the example into the Online Python Tutor, a tool created by Philip Guo for generating these environment diagrams. You are encouraged to create examples yourself and study the resulting environment diagrams.

Functions appear in environment diagrams as well. An import statement binds a name to a built-in function. A def statement binds a name to a user-defined function created by the definition. The resulting environment after importing mul and defining square appears below:

1 from operator import mul
2 def square(x):
3     return mul(x, x)
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Global
mul
 
square
 
func mul(...)
func square(x)

Each function is a line that starts with func, followed by the function name and formal parameters. Built-in functions such as mul do not have formal parameter names, and so ... is always used instead.

The name of a function is repeated twice, once in the frame and again as part of the function itself. The name appearing in the function is called the intrinsic name. The name in a frame is a bound name. There is a difference between the two: different names may refer to the same function, but that function itself has only one intrinsic name.

The name bound to a function in a frame is the one used during evaluation. The intrinsic name of a function does not play a role in evaluation. Step through the example below using the Forward button to see that once the name max is bound to the value 3, it can no longer be used as a function.

1 f = max
2 max = 3
3 result = f(2, 3, 4)
4 max(1, 2)  # Causes an error
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Global
f
 
max 3
func max(...)

The error message TypeError: 'int' object is not callable is reporting that the name max (currently bound to the number 3) is an integer and not a function. Therefore, it cannot be used as the operator in a call expression.

Function Signatures. Functions differ in the number of arguments that they are allowed to take. To track these requirements, we draw each function in a way that shows the function name and its formal parameters. The user-defined function square takes only x; providing more or fewer arguments will result in an error. A description of the formal parameters of a function is called the function's signature.

The function max can take an arbitrary number of arguments. It is rendered as max(...). Regardless of the number of arguments taken, all built-in functions will be rendered as <name>(...), because these primitive functions were never explicitly defined.