When testing a program, one not only needs to cover its several behaviors; one also needs to *check* whether the result is as expected. In this chapter, we introduce a technique that allows us to *mine* function specifications from a set of given executions, resulting in abstract and formal *descriptions* of what the function expects and what it delivers.

These so-called *dynamic invariants* produce pre- and post-conditions over function arguments and variables from a set of executions. They are useful in a variety of contexts:

- Dynamic invariants provide important information for symbolic fuzzing, such as types and ranges of function arguments.
- Dynamic invariants provide pre- and postconditions for formal program proofs and verification.
- Dynamic invariants provide numerous assertions that can check whether function behavior has changed
- Checks provided by dynamic invariants can be very useful as
*oracles*for checking the effects of generated tests

Traditionally, dynamic invariants are dependent on the executions they are derived from. However, when paired with comprehensive test generators, they quickly become very precise, as we show in this chapter.

**Prerequisites**

- You should be familiar with tracing program executions, as in the chapter on coverage.
- Later in this section, we access the internal
*abstract syntax tree*representations of Python programs and transform them, as in the chapter on information flow.

When implementing a function or program, one usually works against a *specification* – a set of documented requirements to be satisfied by the code. Such specifications can come in natural language. A formal specification, however, allows the computer to check whether the specification is satisfied.

In the introduction to testing, we have seen how *preconditions* and *postconditions* can describe what a function does. Consider the following (simple) square root function:

In [3]:

```
def any_sqrt(x):
assert x >= 0 # Precondition
...
assert result * result == x # Postcondition
return result
```

The assertion `assert p`

checks the condition `p`

; if it does not hold, execution is aborted. Here, the actual body is not yet written; we use the assertions as a specification of what `any_sqrt()`

*expects*, and what it *delivers*.

The topmost assertion is the *precondition*, stating the requirements on the function arguments. The assertion at the end is the *postcondition*, stating the properties of the function result (including its relationship with the original arguments). Using these pre- and postconditions as a specification, we can now go and implement a square root function that satisfies them. Once implemented, we can have the assertions check at runtime whether `any_sqrt()`

works as expected; a symbolic or concolic test generator will even specifically try to find inputs where the assertions do *not* hold. (An assertion can be seen as a conditional branch towards aborting the execution, and any technique that tries to cover all code branches will also try to invalidate as many assertions as possible.)

*retrofit* existing code with "specifications" that properly describe their behavior, allowing developers to simply *check* them rather than having to write them from scratch? This is what we do in this chapter.

Before we go into *mining* specifications, let us first discuss why it could be useful to *have* them. As a motivating example, consider the full implementation of a square root function from the introduction to testing:

In [5]:

```
def my_sqrt(x):
"""Computes the square root of x, using the Newton-Raphson method"""
approx = None
guess = x / 2
while approx != guess:
approx = guess
guess = (approx + x / approx) / 2
return approx
```

`my_sqrt()`

does not come with any functionality that would check types or values. Hence, it is easy for callers to make mistakes when calling `my_sqrt()`

:

In [7]:

```
with ExpectError():
my_sqrt("foo")
```

In [8]:

```
with ExpectError():
x = my_sqrt(0.0)
```

In [9]:

```
with ExpectTimeout(1):
x = my_sqrt(-1.0)
```

*annotating* functions with information that prevents errors like the above ones. The idea is to provide a *specification* of expected properties – a specification that can then be checked at runtime or statically.

\todo{Introduce the concept of *contract*.}

For our Python code, one of the most important "specifications" we need is *types*. Python being a "dynamically" typed language means that all data types are determined at run time; the code itself does not explicitly state whether a variable is an integer, a string, an array, a dictionary – or whatever.

*writer* of Python code, omitting explicit type declarations may save time (and allows for some fun hacks). It is not sure whether a lack of types helps in *reading* and *understanding* code for humans. For a *computer* trying to analyze code, the lack of explicit types is detrimental. If, say, a constraint solver, sees `if x:`

and cannot know whether `x`

is supposed to be a number or a string, this introduces an *ambiguity*. Such ambiguities may multiply over the entire analysis in a combinatorial explosion – or in the analysis yielding an overly inaccurate result.

*annotations* to function arguments (actually, to all variables) and return values. We can, for instance, state that `my_sqrt()`

is a function that accepts a floating-point value and returns one:

In [10]:

```
def my_sqrt_with_type_annotations(x: float) -> float:
"""Computes the square root of x, using the Newton-Raphson method"""
return my_sqrt(x)
```

`my_sqrt_typed()`

with a string as an argument and get the exact same result as above. However, one can make use of special *typechecking* modules that would check types – *dynamically* at runtime or *statically* by analyzing the code without having to execute it.

Type annotations can also be checked *statically* – that is, without even running the code. Let us create a simple Python file consisting of the above `my_sqrt_typed()`

definition and a bad invocation.

In [16]:

```
f = tempfile.NamedTemporaryFile(mode='w', suffix='.py')
f.name
```

Out[16]:

'/var/folders/n2/xd9445p97rb3xh7m1dfx8_4h0006ts/T/tmpatyxqdg2.py'

In [17]:

```
f.write(inspect.getsource(my_sqrt))
f.write('\n')
f.write(inspect.getsource(my_sqrt_with_type_annotations))
f.write('\n')
f.write("print(my_sqrt_with_type_annotations('123'))\n")
f.flush()
```

These are the contents of our newly created Python file:

In [19]:

```
print_file(f.name)
```

`mypy`

produces on the above file:

In [21]:

```
result = subprocess.run(["mypy", "--strict", f.name], universal_newlines=True, stdout=subprocess.PIPE)
del f # Delete temporary file
```

In [22]:

```
print(result.stdout)
```

`mypy`

complains about untyped function definitions such as `my_sqrt()`

; most important, however, it finds that the call to `my_sqrt_with_type_annotations()`

in the last line has the wrong type.

`mypy`

, we can achieve the same type safety with Python as in statically typed languages – provided that we as programmers also produce the necessary type annotations. Is there a simple way to obtain these?

Our first task will be to mine type annotations (as part of the code) from *values* we observe at run time. These type annotations would be *mined* from actual function executions, *learning* from (normal) runs what the expected argument and return types should be. By observing a series of calls such as these, we could infer that both `x`

and the return value are of type `float`

:

In [23]:

```
y = my_sqrt(25.0)
y
```

Out[23]:

5.0

In [24]:

```
y = my_sqrt(2.0)
y
```

Out[24]:

1.414213562373095

How can we mine types from executions? The answer is simple:

- We
*observe*a function during execution - We track the
*types*of its arguments - We include these types as
*annotations*into the code.

To do so, we can make use of Python's tracing facility we already observed in the chapter on coverage. With every call to a function, we retrieve the arguments, their values, and their types.

To observe argument types at runtime, we define a *tracer function* that tracks the execution of `my_sqrt()`

, checking its arguments and return values. The `Tracker`

class is set to trace functions in a `with`

block as follows:

```
with Tracker() as tracker:
function_to_be_tracked(...)
info = tracker.collected_information()
```

As in the chapter on coverage, we use the `sys.settrace()`

function to trace individual functions during execution. We turn on tracking when the `with`

block starts; at this point, the `__enter__()`

method is called. When execution of the `with`

block ends, `__exit()__`

is called.

In [26]:

```
class Tracker:
def __init__(self, log=False):
self._log = log
self.reset()
def reset(self):
self._calls = {}
self._stack = []
def traceit(self):
"""Placeholder to be overloaded in subclasses"""
pass
# Start of `with` block
def __enter__(self):
self.original_trace_function = sys.gettrace()
sys.settrace(self.traceit)
return self
# End of `with` block
def __exit__(self, exc_type, exc_value, tb):
sys.settrace(self.original_trace_function)
```

`traceit()`

method does nothing yet; this is done in specialized subclasses. The `CallTracker`

class implements a `traceit()`

function that checks for function calls and returns:

In [27]:

```
class CallTracker(Tracker):
def traceit(self, frame, event, arg):
"""Tracking function: Record all calls and all args"""
if event == "call":
self.trace_call(frame, event, arg)
elif event == "return":
self.trace_return(frame, event, arg)
return self.traceit
```

`trace_call()`

is called when a function is called; it retrieves the function name and current arguments, and saves them on a stack.

In [28]:

```
class CallTracker(CallTracker):
def trace_call(self, frame, event, arg):
"""Save current function name and args on the stack"""
code = frame.f_code
function_name = code.co_name
arguments = get_arguments(frame)
self._stack.append((function_name, arguments))
if self._log:
print(simple_call_string(function_name, arguments))
```

In [29]:

```
def get_arguments(frame):
"""Return call arguments in the given frame"""
# When called, all arguments are local variables
local_variables = dict(frame.f_locals) # explicit copy
arguments = [(var, frame.f_locals[var]) for var in local_variables]
arguments.reverse() # Want same order as call
return arguments
```

`trace_return()`

is called. We now also have the return value. We log the whole call with arguments and return value (if desired) and save it in our list of calls.

In [30]:

```
class CallTracker(CallTracker):
def trace_return(self, frame, event, arg):
"""Get return value and store complete call with arguments and return value"""
code = frame.f_code
function_name = code.co_name
return_value = arg
# TODO: Could call get_arguments() here to also retrieve _final_ values of argument variables
called_function_name, called_arguments = self._stack.pop()
assert function_name == called_function_name
if self._log:
print(simple_call_string(function_name, called_arguments), "returns", return_value)
self.add_call(function_name, called_arguments, return_value)
```

`simple_call_string()`

is a helper for logging that prints out calls in a user-friendly manner.

In [31]:

```
def simple_call_string(function_name, argument_list, return_value=None):
"""Return function_name(arg[0], arg[1], ...) as a string"""
call = function_name + "(" + \
", ".join([var + "=" + repr(value)
for (var, value) in argument_list]) + ")"
if return_value is not None:
call += " = " + repr(return_value)
return call
```

`add_call()`

saves the calls in a list; each function name has its own list.

In [32]:

```
class CallTracker(CallTracker):
def add_call(self, function_name, arguments, return_value=None):
"""Add given call to list of calls"""
if function_name not in self._calls:
self._calls[function_name] = []
self._calls[function_name].append((arguments, return_value))
```

`calls()`

, we can retrieve the list of calls, either for a given function, or for all functions.

In [33]:

```
class CallTracker(CallTracker):
def calls(self, function_name=None):
"""Return list of calls for function_name,
or a mapping function_name -> calls for all functions tracked"""
if function_name is None:
return self._calls
return self._calls[function_name]
```

In [34]:

```
with CallTracker(log=True) as tracker:
y = my_sqrt(25)
y = my_sqrt(2.0)
```

After execution, we can retrieve the individual calls:

In [35]:

```
calls = tracker.calls('my_sqrt')
calls
```

Out[35]:

[([('x', 25)], 5.0), ([('x', 2.0)], 1.414213562373095)]

`argument_list`

, `return_value`

), where `argument_list`

is a list of pairs (`parameter_name`

, `value`

).

In [36]:

```
my_sqrt_argument_list, my_sqrt_return_value = calls[0]
simple_call_string('my_sqrt', my_sqrt_argument_list, my_sqrt_return_value)
```

Out[36]:

'my_sqrt(x=25) = 5.0'

If the function does not return a value, `return_value`

is `None`

.

In [37]:

```
def hello(name):
print("Hello,", name)
```

In [38]:

```
with CallTracker() as tracker:
hello("world")
```

Hello, world

In [39]:

```
hello_calls = tracker.calls('hello')
hello_calls
```

Out[39]:

[([('name', 'world')], None)]

In [40]:

```
hello_argument_list, hello_return_value = hello_calls[0]
simple_call_string('hello', hello_argument_list, hello_return_value)
```

Out[40]:

"hello(name='world')"

Despite what you may have read or heard, Python actually *is* a typed language. It is just that it is *dynamically typed* – types are used and checked only at runtime (rather than declared in the code, where they can be *statically checked* at compile time). We can thus retrieve types of all values within Python:

In [41]:

```
type(4)
```

Out[41]:

int

In [42]:

```
type(2.0)
```

Out[42]:

float

In [43]:

```
type([4])
```

Out[43]:

list

We can retrieve the type of the first argument to `my_sqrt()`

:

In [44]:

```
parameter, value = my_sqrt_argument_list[0]
parameter, type(value)
```

Out[44]:

('x', int)

as well as the type of the return value:

In [45]:

```
type(my_sqrt_return_value)
```

Out[45]:

float

`my_sqrt()`

is a function taking (among others) integers and floats and returning floats. We could declare `my_sqrt()`

as:

In [46]:

```
def my_sqrt_annotated(x: float) -> float:
return my_sqrt(x)
```

`my_sqrt()`

actually pass a number. A dynamic type checker could run such checks at runtime. And of course, any symbolic interpretation will greatly profit from the additional annotations.

In [47]:

```
my_sqrt_annotated.__annotations__
```

Out[47]:

{'x': float, 'return': float}

This is how run-time checkers access the annotations to check against.

Our plan is to annotate functions automatically, based on the types we have seen. To do so, we need a few modules that allow us to convert a function into a tree representation (called *abstract syntax trees*, or ASTs) and back; we already have seen these in the chapters on concolic and symbolic testing.

`inspect.getsource()`

. (Note that this does not work for functions defined in other notebooks.)

In [49]:

```
my_sqrt_source = inspect.getsource(my_sqrt)
my_sqrt_source
```

Out[49]:

'def my_sqrt(x):\n """Computes the square root of x, using the Newton-Raphson method"""\n approx = None\n guess = x / 2\n while approx != guess:\n approx = guess\n guess = (approx + x / approx) / 2\n return approx\n'

`print_content(s, suffix)`

formats and highlights the string `s`

as if it were a file with ending `suffix`

. We can thus view (and highlight) the source as if it were a Python file:

In [51]:

```
print_content(my_sqrt_source, '.py')
```

Parsing this gives us an abstract syntax tree (AST) – a representation of the program in tree form.

In [52]:

```
my_sqrt_ast = ast.parse(my_sqrt_source)
```

`ast.dump()`

(textual output) and `showast.show_ast()`

(graphical output with showast) allow us to inspect the structure of the tree. We see that the function starts as a `FunctionDef`

with name and arguments, followed by a body, which is a list of statements of type `Expr`

(the docstring), type `Assign`

(assignments), `While`

(while loop with its own body), and finally `Return`

.

In [53]:

```
print(ast.dump(my_sqrt_ast, indent=4))
```

Too much text for you? This graphical representation may make things simpler.

In [55]:

```
if rich_output():
import showast
showast.show_ast(my_sqrt_ast)
```

`ast.unparse()`

converts such a tree back into the more familiar textual Python code representation. Comments are gone, and there may be more parentheses than before, but the result has the same semantics:

In [56]:

```
print_content(ast.unparse(my_sqrt_ast), '.py')
```

Let us now go and transform these trees to add type annotations. We start with a helper function `parse_type(name)`

which parses a type name into an AST.

In [57]:

```
def parse_type(name):
class ValueVisitor(ast.NodeVisitor):
def visit_Expr(self, node):
self.value_node = node.value
tree = ast.parse(name)
name_visitor = ValueVisitor()
name_visitor.visit(tree)
return name_visitor.value_node
```

In [58]:

```
print(ast.dump(parse_type('int')))
```

Name(id='int', ctx=Load())

In [59]:

```
print(ast.dump(parse_type('[object]')))
```

List(elts=[Name(id='object', ctx=Load())], ctx=Load())

We now define a helper function that actually adds type annotations to a function AST. The `TypeTransformer`

class builds on the Python standard library `ast.NodeTransformer`

infrastructure. It would be called as

```
TypeTransformer({'x': 'int'}, 'float').visit(ast)
```

to annotate the arguments of `my_sqrt()`

: `x`

with `int`

, and the return type with `float`

. The returned AST can then be unparsed, compiled or analyzed.

In [60]:

```
class TypeTransformer(ast.NodeTransformer):
def __init__(self, argument_types, return_type=None):
self.argument_types = argument_types
self.return_type = return_type
super().__init__()
```

`TypeTransformer`

is the method `visit_FunctionDef()`

, which is called for every function definition in the AST. Its argument `node`

is the subtree of the function definition to be transformed. Our implementation accesses the individual arguments and invokes `annotate_args()`

on them; it also sets the return type in the `returns`

attribute of the node.

In [61]:

```
class TypeTransformer(TypeTransformer):
def visit_FunctionDef(self, node):
"""Add annotation to function"""
# Set argument types
new_args = []
for arg in node.args.args:
new_args.append(self.annotate_arg(arg))
new_arguments = ast.arguments(
node.args.posonlyargs,
new_args,
node.args.vararg,
node.args.kwonlyargs,
node.args.kw_defaults,
node.args.kwarg,
node.args.defaults
)
# Set return type
if self.return_type is not None:
node.returns = parse_type(self.return_type)
return ast.copy_location(
ast.FunctionDef(node.name, new_arguments,
node.body, node.decorator_list,
node.returns), node)
```

Each argument gets its own annotation, taken from the types originally passed to the class:

In [62]:

```
class TypeTransformer(TypeTransformer):
def annotate_arg(self, arg):
"""Add annotation to single function argument"""
arg_name = arg.arg
if arg_name in self.argument_types:
arg.annotation = parse_type(self.argument_types[arg_name])
return arg
```

`my_sqrt()`

with types for the arguments and return types:

In [63]:

```
new_ast = TypeTransformer({'x': 'int'}, 'float').visit(my_sqrt_ast)
```

When we unparse the new AST, we see that the annotations actually are present:

In [64]:

```
print_content(ast.unparse(new_ast), '.py')
```

Similarly, we can annotate the `hello()`

function from above:

In [65]:

```
hello_source = inspect.getsource(hello)
```

In [66]:

```
hello_ast = ast.parse(hello_source)
```

In [67]:

```
new_ast = TypeTransformer({'name': 'str'}, 'None').visit(hello_ast)
```

In [68]:

```
print_content(ast.unparse(new_ast), '.py')
```

def hello(name: str) -> None: print('Hello,', name)

Let us now annotate functions with types mined at runtime. We start with a simple function `type_string()`

that determines the appropriate type of a given value (as a string):

In [69]:

```
def type_string(value):
return type(value).__name__
```

In [70]:

```
type_string(4)
```

Out[70]:

'int'

In [71]:

```
type_string([])
```

Out[71]:

'list'

`type_string()`

does not examine element types; hence, the type of `[3]`

is simply `list`

instead of, say, `list[int]`

. For now, `list`

will do fine.

In [72]:

```
type_string([3])
```

Out[72]:

'list'

`type_string()`

will be used to infer the types of argument values found at runtime, as returned by `CallTracker.calls()`

:

In [74]:

```
tracker.calls()
```

Out[74]:

{'my_sqrt': [([('x', 25.0)], 5.0), ([('x', 2.0)], 1.414213562373095)]}

The function `annotate_types()`

takes such a list of calls and annotates each function listed:

In [75]:

```
def annotate_types(calls):
annotated_functions = {}
for function_name in calls:
try:
annotated_functions[function_name] = annotate_function_with_types(function_name, calls[function_name])
except KeyError:
continue
return annotated_functions
```

`annotate_function_ast_with_types()`

:

In [76]:

```
def annotate_function_with_types(function_name, function_calls):
function = globals()[function_name] # May raise KeyError for internal functions
function_code = inspect.getsource(function)
function_ast = ast.parse(function_code)
return annotate_function_ast_with_types(function_ast, function_calls)
```

`annotate_function_ast_with_types()`

invokes the `TypeTransformer`

with the calls seen, and for each call, iterate over the arguments, determine their types, and annotate the AST with these. The universal type `Any`

is used when we encounter type conflicts, which we will discuss below.

In [78]:

```
def annotate_function_ast_with_types(function_ast, function_calls):
parameter_types = {}
return_type = None
for calls_seen in function_calls:
args, return_value = calls_seen
if return_value is not None:
if return_type is not None and return_type != type_string(return_value):
return_type = 'Any'
else:
return_type = type_string(return_value)
for parameter, value in args:
try:
different_type = parameter_types[parameter] != type_string(value)
except KeyError:
different_type = False
if different_type:
parameter_types[parameter] = 'Any'
else:
parameter_types[parameter] = type_string(value)
annotated_function_ast = TypeTransformer(parameter_types, return_type).visit(function_ast)
return annotated_function_ast
```

Here is `my_sqrt()`

annotated with the types recorded usign the tracker, above.

In [79]:

```
print_content(ast.unparse(annotate_types(tracker.calls())['my_sqrt']), '.py')
```

Let us bring all of this together in a single class `TypeAnnotator`

that first tracks calls of functions and then allows accesing the AST (and the source code form) of the tracked functions annotated with types. The method `typed_functions()`

returns the annotated functions as a string; `typed_functions_ast()`

returns their AST.

In [80]:

```
class TypeTracker(CallTracker):
pass
```

In [81]:

```
class TypeAnnotator(TypeTracker):
def typed_functions_ast(self, function_name=None):
if function_name is None:
return annotate_types(self.calls())
return annotate_function_with_types(function_name,
self.calls(function_name))
def typed_functions(self, function_name=None):
if function_name is None:
functions = ''
for f_name in self.calls():
try:
f_text = ast.unparse(self.typed_functions_ast(f_name))
except KeyError:
f_text = ''
functions += f_text
return functions
return ast.unparse(self.typed_functions_ast(function_name))
```

Here is how to use `TypeAnnotator`

. We first track a series of calls:

In [82]:

```
with TypeAnnotator() as annotator:
y = my_sqrt(25.0)
y = my_sqrt(2.0)
```

After tracking, we can immediately retrieve an annotated version of the functions tracked:

In [83]:

```
print_content(annotator.typed_functions(), '.py')
```

In [84]:

```
with TypeAnnotator() as annotator:
hello('type annotations')
y = my_sqrt(1.0)
```

Hello, type annotations

In [85]:

```
print_content(annotator.typed_functions(), '.py')
```

Let us now resolve the role of the magic `Any`

type in `annotate_function_ast_with_types()`

. If we see multiple types for the same argument, we set its type to `Any`

. For `my_sqrt()`

, this makes sense, as its arguments can be integers as well as floats:

In [87]:

```
print_content(ast.unparse(annotate_types(tracker.calls())['my_sqrt']), '.py')
```

`sum3()`

can be called with floating-point numbers as arguments, resulting in the parameters getting a `float`

type:

In [88]:

```
def sum3(a, b, c):
return a + b + c
```

In [89]:

```
with TypeAnnotator() as annotator:
y = sum3(1.0, 2.0, 3.0)
y
```

Out[89]:

6.0

In [90]:

```
print_content(annotator.typed_functions(), '.py')
```

def sum3(a: float, b: float, c: float) -> float: return a + b + c

If we call `sum3()`

with integers, though, the arguments get an `int`

type:

In [91]:

```
with TypeAnnotator() as annotator:
y = sum3(1, 2, 3)
y
```

Out[91]:

6

In [92]:

```
print_content(annotator.typed_functions(), '.py')
```

def sum3(a: int, b: int, c: int) -> int: return a + b + c

And we can also call `sum3()`

with strings, giving the arguments a `str`

type:

In [93]:

```
with TypeAnnotator() as annotator:
y = sum3("one", "two", "three")
y
```

Out[93]:

'onetwothree'

In [94]:

```
print_content(annotator.typed_functions(), '.py')
```

def sum3(a: str, b: str, c: str) -> str: return a + b + c

`TypeAnnotator()`

will assign an `Any`

type to both arguments and return values:

In [95]:

```
with TypeAnnotator() as annotator:
y = sum3(1, 2, 3)
y = sum3("one", "two", "three")
```

In [96]:

```
typed_sum3_def = annotator.typed_functions('sum3')
```

In [97]:

```
print_content(typed_sum3_def, '.py')
```

def sum3(a: Any, b: Any, c: Any) -> Any: return a + b + c

`Any`

makes it explicit that an object can, indeed, have any type; it will not be type-checked at runtime or statically. To some extent, this defeats the power of type checking; but it also preserves some of the type flexibility that many Python programmers enjoy. Besides `Any`

, the `typing`

module supports several additional ways to define ambiguous types; we will keep this in mind for a later exercise.

Besides basic data types. we can check several further properties from arguments. We can, for instance, whether an argument can be negative, zero, or positive; or that one argument should be smaller than the second; or that the result should be the sum of two arguments – properties that cannot be expressed in a (Python) type.

Such properties are called *invariants*, as they hold across all invocations of a function. Specifically, invariants come as *pre*- and *postconditions* – conditions that always hold at the beginning and at the end of a function. (There are also *data* and *object* invariants that express always-holding properties over the state of data or objects, but we do not consider these in this book.)

The classical means to specify pre- and postconditions is via *assertions*, which we have introduced in the chapter on testing. A precondition checks whether the arguments to a function satisfy the expected properties; a postcondition does the same for the result. We can express and check both using assertions as follows:

In [98]:

```
def my_sqrt_with_invariants(x):
assert x >= 0 # Precondition
...
assert result * result == x # Postcondition
return result
```

A nicer way, however, is to syntactically separate invariants from the function at hand. Using appropriate decorators, we could specify pre- and postconditions as follows:

```
@precondition lambda x: x >= 0
@postcondition lambda return_value, x: return_value * return_value == x
def my_sqrt_with_invariants(x):
# normal code without assertions
...
```

The decorators `@precondition`

and `@postcondition`

would run the given functions (specified as anonymous `lambda`

functions) before and after the decorated function, respectively. If the functions return `False`

, the condition is violated. `@precondition`

gets the function arguments as arguments; `@postcondition`

additionally gets the return value as first argument.

In [100]:

```
def condition(precondition=None, postcondition=None):
def decorator(func):
@functools.wraps(func) # preserves name, docstring, etc
def wrapper(*args, **kwargs):
if precondition is not None:
assert precondition(*args, **kwargs), "Precondition violated"
retval = func(*args, **kwargs) # call original function or method
if postcondition is not None:
assert postcondition(retval, *args, **kwargs), "Postcondition violated"
return retval
return wrapper
return decorator
def precondition(check):
return condition(precondition=check)
def postcondition(check):
return condition(postcondition=check)
```

With these, we can now start decorating `my_sqrt()`

:

In [101]:

```
@precondition(lambda x: x > 0)
def my_sqrt_with_precondition(x):
return my_sqrt(x)
```

This catches arguments violating the precondition:

In [102]:

```
with ExpectError():
my_sqrt_with_precondition(-1.0)
```

Likewise, we can provide a postcondition:

In [103]:

```
EPSILON = 1e-5
```

In [104]:

```
@postcondition(lambda ret, x: ret * ret - x < EPSILON)
def my_sqrt_with_postcondition(x):
return my_sqrt(x)
```

In [105]:

```
y = my_sqrt_with_postcondition(2.0)
y
```

Out[105]:

1.414213562373095

If we have a buggy implementation of $\sqrt{x}$, this gets caught quickly:

In [106]:

```
@postcondition(lambda ret, x: ret * ret - x < EPSILON)
def buggy_my_sqrt_with_postcondition(x):
return my_sqrt(x) + 0.1
```

In [107]:

```
with ExpectError():
y = buggy_my_sqrt_with_postcondition(2.0)
```

*mine* some of them.

To *mine* invariants, we can use the same tracking functionality as before; instead of saving values for individual variables, though, we now check whether the values satisfy specific *properties* or not. For instance, if all values of `x`

seen satisfy the condition `x > 0`

, then we make `x > 0`

an invariant of the function. If we see positive, zero, and negative values of `x`

, though, then there is no property of `x`

left to talk about.

The general idea is thus:

- Check all variable values observed against a set of predefined properties; and
- Keep only those properties that hold for all runs observed.

What precisely do we mean by properties? Here is a small collection of value properties that would frequently be used in invariants. All these properties would be evaluated with the *metavariables* `X`

, `Y`

, and `Z`

(actually, any upper-case identifier) being replaced with the names of function parameters:

In [108]:

```
INVARIANT_PROPERTIES = [
"X < 0",
"X <= 0",
"X > 0",
"X >= 0",
"X == 0",
"X != 0",
]
```

`my_sqrt(x)`

is called as, say `my_sqrt(5.0)`

, we see that `x = 5.0`

holds. The above properties would then all be checked for `x`

. Only the properties `X > 0`

, `X >= 0`

, and `X != 0`

hold for the call seen; and hence `x > 0`

, `x >= 0`

, and `x != 0`

would make potential preconditions for `my_sqrt(x)`

.

We can check for many more properties such as relations between two arguments:

In [109]:

```
INVARIANT_PROPERTIES += [
"X == Y",
"X > Y",
"X < Y",
"X >= Y",
"X <= Y",
]
```

`X`

, only one of these will hold:

In [110]:

```
INVARIANT_PROPERTIES += [
"isinstance(X, bool)",
"isinstance(X, int)",
"isinstance(X, float)",
"isinstance(X, list)",
"isinstance(X, dict)",
]
```

We can check for arithmetic properties:

In [111]:

```
INVARIANT_PROPERTIES += [
"X == Y + Z",
"X == Y * Z",
"X == Y - Z",
"X == Y / Z",
]
```

Here's relations over three values, a Python special:

In [112]:

```
INVARIANT_PROPERTIES += [
"X < Y < Z",
"X <= Y <= Z",
"X > Y > Z",
"X >= Y >= Z",
]
```

Finally, we can also check for list or string properties. Again, this is just a tiny selection.

In [113]:

```
INVARIANT_PROPERTIES += [
"X == len(Y)",
"X == sum(Y)",
"X.startswith(Y)",
]
```

Let us first introduce a few *helper functions* before we can get to the actual mining. `metavars()`

extracts the set of meta-variables (`X`

, `Y`

, `Z`

, etc.) from a property. To this end, we parse the property as a Python expression and then visit the identifiers.

In [114]:

```
def metavars(prop):
metavar_list = []
class ArgVisitor(ast.NodeVisitor):
def visit_Name(self, node):
if node.id.isupper():
metavar_list.append(node.id)
ArgVisitor().visit(ast.parse(prop))
return metavar_list
```

In [115]:

```
assert metavars("X < 0") == ['X']
```

In [116]:

```
assert metavars("X.startswith(Y)") == ['X', 'Y']
```

In [117]:

```
assert metavars("isinstance(X, str)") == ['X']
```

To produce a property as invariant, we need to be able to *instantiate* it with variable names. The instantiation of `X > 0`

with `X`

being instantiated to `a`

, for instance, gets us `a > 0`

. To this end, the function `instantiate_prop()`

takes a property and a collection of variable names and instantiates the meta-variables left-to-right with the corresponding variables names in the collection.

In [118]:

```
def instantiate_prop_ast(prop, var_names):
class NameTransformer(ast.NodeTransformer):
def visit_Name(self, node):
if node.id not in mapping:
return node
return ast.Name(id=mapping[node.id], ctx=ast.Load())
meta_variables = metavars(prop)
assert len(meta_variables) == len(var_names)
mapping = {}
for i in range(0, len(meta_variables)):
mapping[meta_variables[i]] = var_names[i]
prop_ast = ast.parse(prop, mode='eval')
new_ast = NameTransformer().visit(prop_ast)
return new_ast
```

In [119]:

```
def instantiate_prop(prop, var_names):
prop_ast = instantiate_prop_ast(prop, var_names)
prop_text = ast.unparse(prop_ast).strip()
while prop_text.startswith('(') and prop_text.endswith(')'):
prop_text = prop_text[1:-1]
return prop_text
```

In [120]:

```
assert instantiate_prop("X > Y", ['a', 'b']) == 'a > b'
```

In [121]:

```
assert instantiate_prop("X.startswith(Y)", ['x', 'y']) == 'x.startswith(y)'
```

To actually *evaluate* properties, we do not need to instantiate them. Instead, we simply convert them into a boolean function, using `lambda`

:

In [122]:

```
def prop_function_text(prop):
return "lambda " + ", ".join(metavars(prop)) + ": " + prop
def prop_function(prop):
return eval(prop_function_text(prop))
```

Here is a simple example:

In [123]:

```
prop_function_text("X > Y")
```

Out[123]:

'lambda X, Y: X > Y'

In [124]:

```
p = prop_function("X > Y")
p(100, 1)
```

Out[124]:

True

In [125]:

```
p(1, 100)
```

Out[125]:

False

To extract invariants from an execution, we need to check them on all possible instantiations of arguments. If the function to be checked has two arguments `a`

and `b`

, we instantiate the property `X < Y`

both as `a < b`

and `b < a`

and check each of them.

To get all combinations, we use the Python `permutations()`

function:

In [127]:

```
for combination in itertools.permutations([1.0, 2.0, 3.0], 2):
print(combination)
```

(1.0, 2.0) (1.0, 3.0) (2.0, 1.0) (2.0, 3.0) (3.0, 1.0) (3.0, 2.0)

`true_property_instantiations()`

takes a property and a list of tuples (`var_name`

, `value`

). It then produces all instantiations of the property with the given values and returns those that evaluate to True.

In [128]:

```
def true_property_instantiations(prop, vars_and_values, log=False):
instantiations = set()
p = prop_function(prop)
len_metavars = len(metavars(prop))
for combination in itertools.permutations(vars_and_values, len_metavars):
args = [value for var_name, value in combination]
var_names = [var_name for var_name, value in combination]
try:
result = p(*args)
except:
result = None
if log:
print(prop, combination, result)
if result:
instantiations.add((prop, tuple(var_names)))
return instantiations
```

Here is an example. If `x == -1`

and `y == 1`

, the property `X < Y`

holds for `x < y`

, but not for `y < x`

:

In [129]:

```
invs = true_property_instantiations("X < Y", [('x', -1), ('y', 1)], log=True)
invs
```

X < Y (('x', -1), ('y', 1)) True X < Y (('y', 1), ('x', -1)) False

Out[129]:

{('X < Y', ('x', 'y'))}

The instantiation retrieves the short form:

In [130]:

```
for prop, var_names in invs:
print(instantiate_prop(prop, var_names))
```

x < y

Likewise, with values for `x`

and `y`

as above, the property `X < 0`

only holds for `x`

, but not for `y`

:

In [131]:

```
invs = true_property_instantiations("X < 0", [('x', -1), ('y', 1)], log=True)
```

X < 0 (('x', -1),) True X < 0 (('y', 1),) False

In [132]:

```
for prop, var_names in invs:
print(instantiate_prop(prop, var_names))
```

x < 0

Let us now run the above invariant extraction on function arguments and return values as observed during a function execution. To this end, we extend the `CallTracker`

class into an `InvariantTracker`

class, which automatically computes invariants for all functions and all calls observed during tracking.

`InvariantTracker`

uses the properties as defined above; however, one can specify alternate sets of properties.

In [133]:

```
class InvariantTracker(CallTracker):
def __init__(self, props=None, **kwargs):
if props is None:
props = INVARIANT_PROPERTIES
self.props = props
super().__init__(**kwargs)
```

`InvariantTracker`

is the `invariants()`

method. This iterates over the calls observed and checks which properties hold. Only the intersection of properties – that is, the set of properties that hold for all calls – is preserved, and eventually returned. The special variable `return_value`

is set to hold the return value.

In [134]:

```
RETURN_VALUE = 'return_value'
```

In [135]:

```
class InvariantTracker(InvariantTracker):
def invariants(self, function_name=None):
if function_name is None:
return {function_name: self.invariants(function_name) for function_name in self.calls()}
invariants = None
for variables, return_value in self.calls(function_name):
vars_and_values = variables + [(RETURN_VALUE, return_value)]
s = set()
for prop in self.props:
s |= true_property_instantiations(prop, vars_and_values, self._log)
if invariants is None:
invariants = s
else:
invariants &= s
return invariants
```

Here's an example of how to use `invariants()`

. We run the tracker on a small set of calls.

In [136]:

```
with InvariantTracker() as tracker:
y = my_sqrt(25.0)
y = my_sqrt(10.0)
tracker.calls()
```

Out[136]:

{'my_sqrt': [([('x', 25.0)], 5.0), ([('x', 10.0)], 3.162277660168379)]}

`invariants()`

method produces a set of properties that hold for the observed runs, together with their instantiations over function arguments.

In [137]:

```
invs = tracker.invariants('my_sqrt')
invs
```

Out[137]:

{('X != 0', ('return_value',)), ('X != 0', ('x',)), ('X < Y', ('return_value', 'x')), ('X <= Y', ('return_value', 'x')), ('X > 0', ('return_value',)), ('X > 0', ('x',)), ('X > Y', ('x', 'return_value')), ('X >= 0', ('return_value',)), ('X >= 0', ('x',)), ('X >= Y', ('x', 'return_value')), ('isinstance(X, float)', ('return_value',)), ('isinstance(X, float)', ('x',))}

As before, the actual instantiations are easier to read:

In [138]:

```
def pretty_invariants(invariants):
props = []
for (prop, var_names) in invariants:
props.append(instantiate_prop(prop, var_names))
return sorted(props)
```

In [139]:

```
pretty_invariants(invs)
```

Out[139]:

['isinstance(return_value, float)', 'isinstance(x, float)', 'return_value != 0', 'return_value < x', 'return_value <= x', 'return_value > 0', 'return_value >= 0', 'x != 0', 'x > 0', 'x > return_value', 'x >= 0', 'x >= return_value']

`x`

and the return value have a `float`

type. We also see that both are always greater than zero. These are properties that may make useful pre- and postconditions, notably for symbolic analysis.

*not* universally hold, namely `return_value <= x`

, as the following example shows:

In [140]:

```
my_sqrt(0.01)
```

Out[140]:

0.1

`x = 0.1`

, though, the invariant `return_value <= x`

is eliminated:

In [141]:

```
with InvariantTracker() as tracker:
y = my_sqrt(25.0)
y = my_sqrt(10.0)
y = my_sqrt(0.01)
pretty_invariants(tracker.invariants('my_sqrt'))
```

Out[141]:

['isinstance(return_value, float)', 'isinstance(x, float)', 'return_value != 0', 'return_value > 0', 'return_value >= 0', 'x != 0', 'x > 0', 'x >= 0']

`sum3()`

. We see that all types are well-defined; the properties that all arguments are non-zero, however, is specific to the calls observed.

In [142]:

```
with InvariantTracker() as tracker:
y = sum3(1, 2, 3)
y = sum3(-4, -5, -6)
pretty_invariants(tracker.invariants('sum3'))
```

Out[142]:

['a != 0', 'b != 0', 'c != 0', 'isinstance(a, int)', 'isinstance(b, int)', 'isinstance(c, int)', 'isinstance(return_value, int)', 'return_value != 0']

`sum3()`

with strings instead, we get different invariants. Notably, we obtain the postcondition that the return value starts with the value of `a`

– a universal postcondition if strings are used.

In [143]:

```
with InvariantTracker() as tracker:
y = sum3('a', 'b', 'c')
y = sum3('f', 'e', 'd')
pretty_invariants(tracker.invariants('sum3'))
```

Out[143]:

['a != 0', 'a < return_value', 'a <= return_value', 'b != 0', 'c != 0', 'return_value != 0', 'return_value > a', 'return_value >= a', 'return_value.startswith(a)']

`sum3()`

with both strings and numbers (and zeros, too), there are no properties left that would hold across all calls. That's the price of flexibility.

In [144]:

```
with InvariantTracker() as tracker:
y = sum3('a', 'b', 'c')
y = sum3('c', 'b', 'a')
y = sum3(-4, -5, -6)
y = sum3(0, 0, 0)
pretty_invariants(tracker.invariants('sum3'))
```

Out[144]:

[]

As with types, above, we would like to have some functionality where we can add the mined invariants as annotations to existing functions. To this end, we introduce the `InvariantAnnotator`

class, extending `InvariantTracker`

.

`params()`

returns a comma-separated list of parameter names as observed during calls.

In [145]:

```
class InvariantAnnotator(InvariantTracker):
def params(self, function_name):
arguments, return_value = self.calls(function_name)[0]
return ", ".join(arg_name for (arg_name, arg_value) in arguments)
```

In [146]:

```
with InvariantAnnotator() as annotator:
y = my_sqrt(25.0)
y = sum3(1, 2, 3)
```

In [147]:

```
annotator.params('my_sqrt')
```

Out[147]:

'x'

In [148]:

```
annotator.params('sum3')
```

Out[148]:

'c, b, a'

`preconditions()`

returns the preconditions from the mined invariants (i.e., those propertes that do not depend on the return value) as a string with annotations:

In [149]:

```
class InvariantAnnotator(InvariantAnnotator):
def preconditions(self, function_name):
conditions = []
for inv in pretty_invariants(self.invariants(function_name)):
if inv.find(RETURN_VALUE) >= 0:
continue # Postcondition
cond = "@precondition(lambda " + self.params(function_name) + ": " + inv + ")"
conditions.append(cond)
return conditions
```

In [150]:

```
with InvariantAnnotator() as annotator:
y = my_sqrt(25.0)
y = my_sqrt(0.01)
y = sum3(1, 2, 3)
```

In [151]:

```
annotator.preconditions('my_sqrt')
```

Out[151]:

['@precondition(lambda x: isinstance(x, float))', '@precondition(lambda x: x != 0)', '@precondition(lambda x: x > 0)', '@precondition(lambda x: x >= 0)']

`postconditions()`

does the same for postconditions:

In [152]:

```
class InvariantAnnotator(InvariantAnnotator):
def postconditions(self, function_name):
conditions = []
for inv in pretty_invariants(self.invariants(function_name)):
if inv.find(RETURN_VALUE) < 0:
continue # Precondition
cond = ("@postcondition(lambda " +
RETURN_VALUE + ", " + self.params(function_name) + ": " + inv + ")")
conditions.append(cond)
return conditions
```

In [153]:

```
with InvariantAnnotator() as annotator:
y = my_sqrt(25.0)
y = my_sqrt(0.01)
y = sum3(1, 2, 3)
```

In [154]:

```
annotator.postconditions('my_sqrt')
```

Out[154]:

['@postcondition(lambda return_value, x: isinstance(return_value, float))', '@postcondition(lambda return_value, x: return_value != 0)', '@postcondition(lambda return_value, x: return_value > 0)', '@postcondition(lambda return_value, x: return_value >= 0)']

With these, we can take a function and add both pre- and postconditions as annotations:

In [155]:

```
class InvariantAnnotator(InvariantAnnotator):
def functions_with_invariants(self):
functions = ""
for function_name in self.invariants():
try:
function = self.function_with_invariants(function_name)
except KeyError:
continue
functions += function
return functions
def function_with_invariants(self, function_name):
function = globals()[function_name] # Can throw KeyError
source = inspect.getsource(function)
return "\n".join(self.preconditions(function_name) +
self.postconditions(function_name)) + '\n' + source
```

Here comes `function_with_invariants()`

in all its glory:

In [156]:

```
with InvariantAnnotator() as annotator:
y = my_sqrt(25.0)
y = my_sqrt(0.01)
y = sum3(1, 2, 3)
```

In [157]:

```
print_content(annotator.function_with_invariants('my_sqrt'), '.py')
```

Here's another example. `list_length()`

recursively computes the length of a Python function. Let us see whether we can mine its invariants:

In [158]:

```
def list_length(L):
if L == []:
length = 0
else:
length = 1 + list_length(L[1:])
return length
```

In [159]:

```
with InvariantAnnotator() as annotator:
length = list_length([1, 2, 3])
print_content(annotator.functions_with_invariants(), '.py')
```

`len(L)`

is that `X == len(Y)`

is part of the list of properties to be checked.

The next example is a very simple function:

In [160]:

```
def sum2(a, b):
return a + b
```

In [161]:

```
with InvariantAnnotator() as annotator:
sum2(31, 45)
sum2(0, 0)
sum2(-1, -5)
```

`a`

, `b`

, and the return value as `return_value == a + b`

in all its variations.

In [162]:

```
print_content(annotator.functions_with_invariants(), '.py')
```

`None`

, and we can only mine preconditions. (Well, we get a "postcondition" that the return value is non-zero, which holds for `None`

).

In [163]:

```
def print_sum(a, b):
print(a + b)
```

In [164]:

```
with InvariantAnnotator() as annotator:
print_sum(31, 45)
print_sum(0, 0)
print_sum(-1, -5)
```

76 0 -6

In [165]:

```
print_content(annotator.functions_with_invariants(), '.py')
```

A function with invariants, as above, can be fed into the Python interpreter, such that all pre- and postconditions are checked. We create a function `my_sqrt_annotated()`

which includes all the invariants mined above.

In [166]:

```
with InvariantAnnotator() as annotator:
y = my_sqrt(25.0)
y = my_sqrt(0.01)
```

In [167]:

```
my_sqrt_def = annotator.functions_with_invariants()
my_sqrt_def = my_sqrt_def.replace('my_sqrt', 'my_sqrt_annotated')
```

In [168]:

```
print_content(my_sqrt_def, '.py')
```

In [169]:

```
exec(my_sqrt_def)
```

In [170]:

```
with ExpectError():
my_sqrt_annotated(-1.0)
```

This is in contrast to the original version, which just hangs on negative values:

In [171]:

```
with ExpectTimeout(1):
my_sqrt(-1.0)
```

*regressions* are caught as violations of the postconditions. Let us illustrate this by simply inverting the result, and return $-2$ as square root of 4.

In [172]:

```
my_sqrt_def = my_sqrt_def.replace('my_sqrt_annotated', 'my_sqrt_negative')
my_sqrt_def = my_sqrt_def.replace('return approx', 'return -approx')
```

In [173]:

```
print_content(my_sqrt_def, '.py')
```

In [174]:

```
exec(my_sqrt_def)
```

*is* a square root of 4, since $(-2)^2 = 4$ holds. Yet, such a change may be unexpected by callers of `my_sqrt()`

, and hence, this would be caught with the first call:

In [175]:

```
with ExpectError():
my_sqrt_negative(2.0) # type: ignore
```

*oracles* during testing. In particular, once we have mined them for a set of functions, we can check them again and again with test generators – especially after code changes. The more checks we have, and the more specific they are, the more likely it is we can detect unwanted effects of changes.

Mined specifications can only be as good as the executions they were mined from. If we only see a single call to, say, `sum2()`

as defined above, we will be faced with several mined pre- and postconditions that *overspecialize* towards the values seen:

In [176]:

```
with InvariantAnnotator() as annotator:
y = sum2(2, 2)
print_content(annotator.functions_with_invariants(), '.py')
```

`a == b`

, for instance, only holds for the single call observed; the same holds for the mined postcondition `return_value == a * b`

. Yet, `sum2()`

can obviously be successfully called with other values that do not satisfy these conditions.

*learn from more and more diverse runs*. If we have a few more calls of `sum2()`

, we see how the set of invariants quickly gets smaller:

In [177]:

```
with InvariantAnnotator() as annotator:
length = sum2(1, 2)
length = sum2(-1, -2)
length = sum2(0, 0)
print_content(annotator.functions_with_invariants(), '.py')
```

`sum2()`

will easily resolve the problem.

In [179]:

```
SUM2_EBNF_GRAMMAR: Grammar = {
"<start>": ["<sum2>"],
"<sum2>": ["sum2(<int>, <int>)"],
"<int>": ["<_int>"],
"<_int>": ["(-)?<leaddigit><digit>*", "0"],
"<leaddigit>": crange('1', '9'),
"<digit>": crange('0', '9')
}
```

In [180]:

```
assert is_valid_grammar(SUM2_EBNF_GRAMMAR)
```

In [181]:

```
sum2_grammar = convert_ebnf_grammar(SUM2_EBNF_GRAMMAR)
```

In [182]:

```
sum2_fuzzer = GrammarFuzzer(sum2_grammar)
[sum2_fuzzer.fuzz() for i in range(10)]
```

Out[182]:

['sum2(60, 3)', 'sum2(-4, 0)', 'sum2(-579, 34)', 'sum2(3, 0)', 'sum2(-8, 0)', 'sum2(0, 8)', 'sum2(3, -9)', 'sum2(0, 0)', 'sum2(0, 5)', 'sum2(-3181, 0)']

In [183]:

```
with InvariantAnnotator() as annotator:
for i in range(10):
eval(sum2_fuzzer.fuzz())
print_content(annotator.function_with_invariants('sum2'), '.py')
```

`sqrt()`

with positive numbers only, already assuming its precondition. In some way, one thus needs a specification (a model, a grammar) to mine another specification – a chicken-and-egg problem.

*infinite source of executions* to learn invariants from. In each of these executions, all functions would be called with values that satisfy the (implicit) precondition, allowing us to mine invariants for these functions. This holds, because at the system level, invalid inputs must be rejected by the system in the first place. The meaningful precondition at the system level, ensuring that only valid inputs get through, thus gets broken down into a multitude of meaningful preconditions (and subsequent postconditions) at the function level.

This chapter provides two classes that automatically extract specifications from a function and a set of inputs:

`TypeAnnotator`

for*types*, and`InvariantAnnotator`

for*pre-*and*postconditions*.

Both work by *observing* a function and its invocations within a `with`

clause. Here is an example for the type annotator:

In [184]:

```
def sum(a, b):
return a + b
```

In [185]:

```
with TypeAnnotator() as type_annotator:
sum(1, 2)
sum(-4, -5)
sum(0, 0)
```

`typed_functions()`

method will return a representation of `sum2()`

annotated with types observed during execution.

In [186]:

```
print(type_annotator.typed_functions())
```

def sum(a: int, b: int) -> int: return a + b

The invariant annotator works similarly:

In [187]:

```
with InvariantAnnotator() as inv_annotator:
sum(1, 2)
sum(-4, -5)
sum(0, 0)
```

`functions_with_invariants()`

method will return a representation of `sum2()`

annotated with inferred pre- and postconditions that all hold for the observed values.

In [188]:

```
print(inv_annotator.functions_with_invariants())
```

*oracles* (to detect deviations from a given set of runs) as well as for all kinds of *symbolic code analyses*. The chapter gives details on how to customize the properties checked for.

- Type annotations and explicit invariants allow for
*checking*arguments and results for expected data types and other properties. - One can automatically
*mine*data types and invariants by observing arguments and results at runtime. - The quality of mined invariants depends on the diversity of values observed during executions; this variety can be increased by generating tests.

This chapter concludes the part on semantic fuzzing techniques. In the next part, we will explore domain-specific fuzzing techniques from configurations and APIs to graphical user interfaces.

The DAIKON dynamic invariant detector can be considered the mother of function specification miners. Continuously maintained and extended for more than 20 years, it mines likely invariants in the style of this chapter for a variety of languages, including C, C++, C#, Eiffel, F#, Java, Perl, and Visual Basic. On top of the functionality discussed above, it holds a rich catalog of patterns for likely invariants, supports data invariants, can eliminate invariants that are implied by others, and determines statistical confidence to disregard unlikely invariants. The corresponding paper \cite{Ernst2001} is one of the seminal and most-cited papers of Software Engineering. A multitude of works have been published based on DAIKON and detecting invariants; see this curated list for details.

As it comes to adding type annotations to existing code, the blog post "The state of type hints in Python" gives a great overview on how Python type hints can be used and checked. To add type annotations, there are two important tools available that also implement our above approach:

- MonkeyType implements the above approach of tracing executions and annotating Python 3 arguments, returns, and variables with type hints.
- PyAnnotate does a similar job, focusing on code in Python 2. It does not produce Python 3-style annotations, but instead produces annotations as comments that can be processed by static type checkers.

These tools have been created by engineers at Facebook and Dropbox, respectively, assisting them in checking millions of lines of code for type issues.

Our code for mining types and invariants is in no way complete. There are dozens of ways to extend our implementations, some of which we discuss in exercises.

The Python `typing`

module allows expressing that an argument can have multiple types. For `my_sqrt(x)`

, this allows expressing that `x`

can be an `int`

or a `float`

:

In [190]:

```
def my_sqrt_with_union_type(x: Union[int, float]) -> float: # type: ignore
...
```

`TypeAnnotator`

such that it supports union types for arguments and return values. Use `Optional[X]`

as a shorthand for `Union[X, None]`

.

In Python, one cannot only annotate arguments with types, but actually also local and global variables – for instance, `approx`

and `guess`

in our `my_sqrt()`

implementation:

In [191]:

```
def my_sqrt_with_local_types(x: Union[int, float]) -> float:
"""Computes the square root of x, using the Newton-Raphson method"""
approx: Optional[float] = None
guess: float = x / 2
while approx != guess:
approx = guess
guess = (approx + x / approx) / 2
return approx
```

`TypeAnnotator`

such that it also annotates local variables with types. Search the function AST for assignments, determine the type of the assigned value, and make it an annotation on the left-hand side.

Our implementation of invariant checkers does not make it clear for the user which pre-/postcondition failed.

In [192]:

```
@precondition(lambda s: len(s) > 0)
def remove_first_char(s):
return s[1:]
```

In [193]:

```
with ExpectError():
remove_first_char('')
```

`doc`

keyword argument which is printed if the invariant is violated:

In [194]:

```
def verbose_condition(precondition=None, postcondition=None, doc='Unknown'):
def decorator(func):
@functools.wraps(func) # preserves name, docstring, etc
def wrapper(*args, **kwargs):
if precondition is not None:
assert precondition(*args, **kwargs), "Precondition violated: " + doc
retval = func(*args, **kwargs) # call original function or method
if postcondition is not None:
assert postcondition(retval, *args, **kwargs), "Postcondition violated: " + doc
return retval
return wrapper
return decorator
```

In [195]:

```
def verbose_precondition(check, **kwargs): # type: ignore
return verbose_condition(precondition=check, doc=kwargs.get('doc', 'Unknown'))
```

In [196]:

```
def verbose_postcondition(check, **kwargs): # type: ignore
return verbose_condition(postcondition=check, doc=kwargs.get('doc', 'Unknown'))
```

In [197]:

```
@verbose_precondition(lambda s: len(s) > 0, doc="len(s) > 0") # type: ignore
def remove_first_char(s):
return s[1:]
remove_first_char('abc')
```

Out[197]:

'bc'

In [198]:

```
with ExpectError():
remove_first_char('')
```

`InvariantAnnotator`

such that it includes the conditions in the generated pre- and postconditions.

If the value of an argument changes during function execution, this can easily confuse our implementation: The values are tracked at the beginning of the function, but checked only when it returns. Extend the `InvariantAnnotator`

and the infrastructure it uses such that

- it saves argument values both at the beginning and at the end of a function invocation;
- postconditions can be expressed over both
*initial*values of arguments as well as the*final*values of arguments; - the mined postconditions refer to both these values as well.

Several mined invariant are actually *implied* by others: If `x > 0`

holds, then this implies `x >= 0`

and `x != 0`

. Extend the `InvariantAnnotator`

such that implications between properties are explicitly encoded, and such that implied properties are no longer listed as invariants. See \cite{Ernst2001} for ideas.

Postconditions may also refer to the values of local variables. Consider extending `InvariantAnnotator`

and its infrastructure such that the values of local variables at the end of the execution are also recorded and made part of the invariant inference mechanism.

After mining a first set of invariants, have a concolic fuzzer generate tests that systematically attempt to invalidate pre- and postconditions. How far can you generalize?

The larger the set of properties to be checked, the more potential invariants can be discovered. Create a *grammar* that systematically produces a large set of properties. See \cite{Ernst2001} for possible patterns.

Rather than producing invariants as annotations for pre- and postconditions, insert them as `assert`

statements into the function code, as in:

```
def my_sqrt(x):
'Computes the square root of x, using the Newton-Raphson method'
assert isinstance(x, int), 'violated precondition'
assert (x > 0), 'violated precondition'
approx = None
guess = (x / 2)
while (approx != guess):
approx = guess
guess = ((approx + (x / approx)) / 2)
return_value = approx
assert (return_value < x), 'violated postcondition'
assert isinstance(return_value, float), 'violated postcondition'
return approx
```

Such a formulation may make it easier for test generators and symbolic analysis to access and interpret pre- and postconditions.