A static type analyzer for Python code

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The Typegraph


This document gives an overview of pytype’s typegraph and explains some of the grittier details. Because it is a significant part of the hottest paths in pytype, the typegraph has a variety of optimizations and caching to improve performance.

The Control Flow Graph

As pytype analyzes a program, it builds a control flow graph (CFG) that represents how the parts of the program work together. Each Node in the CFG roughly correlates with a single statement in the program.

For example:

if some_val:  # 1
  x = 5  # 2
  y = 6  # 3
  x = "a"  # 4
z = x.upper() + str(y)  # 5

This program has a CFG that looks like:

     / \
(2)<+   +>(4)
 |         |
 v         |

Note how the two branches of the if-else statement are represented by two paths starting at Node 1 and coming together at Node 5.

A Variable tracks the type information for a variable in the program being analyzed. This includes simple variables (e.g. x in x = 5), function arguments (a and b in def f(a, b)), and functions, classes and modules.

A Binding associates a Variable with a value at a particular Node. In the example above, the Variable for x is bound to the value 5 at Node 2 (Binding(5, Node 2)) and to "a" at Node 4 (Binding("a", Node 4)). Meanwhile, y has only a single binding: Binding(6, Node 3). The Bindings that a Variable is connected to represent the possible values for that Variable.

Building up the CFG in this way allows pytype to perform type checking. When pytype reaches Node 5 (z = x.upper() + str(y)), it queries the CFG to find what x may be. Depending on the value of some_val, x could an int or a str. Since int doesn’t have a method called upper, pytype reports an attribute-error, even though str does have an upper method.

However, pytype is limited in what it knows. Looking at the example again, we know that y won’t be defined if some_val is False, which would make str(y) fail. But pytype can’t know for sure if some_val will evaluate to True or False. Since there’s a path through the CFG where y is defined (y = 6 if some_val == True), pytype won’t report an error for str(y). If we change the condition to if False, so that pytype knows unambiguously that only the code under else: will be executed, then pytype will report a name-error on str(y) because there is no path through the CFG where y is defined.

Code Components

The CFG, described above, is the core of type checking for pytype. By building a graph of the paths of execution through a program, pytype knows which objects are in scope and the types of those objects at any point in the program. The source code for the CFG lives in the typegraph/ directory. It’s written in C++ for performance reasons.

(Note: there is also a Python implementation, and, which is no longer used and is not guaranteed to match the semantics of the C++ implementation.)


A CFG maps the paths of execution through a program; similarly, a typegraph maps the flow of types through a program – it’s a CFG that’s been enhanced with type information. (In pytype, we use the two terms interchangeably.) typegraph.h covers the classes that are used to build the typegraph. Some of them were already mentioned earlier in Basic Concepts, namely Variables, Bindings and Nodes (called CFGNodes here).

First is CFGNode, the building block of the CFG. A CFGNode corresponds to one or more opcodes in the Python program being analyzed. As mentioned previously, each Binding associates a Variable with a value at a particular CFGNode.

CFGNodes may also have conditions. Recall the example program and CFG discussed in Basic Concepts:

if some_val:  # 1
  x = 5  # 2
  y = 6  # 3
  x = "a"  # 4
z = x.upper() + str(y)  # 5
     / \
(2)<+   +>(4)
 |         |
 v         |

The condition of the if-statement is some_val, or more explicitly some_val == True. Pytype understands that the path 1 -> 2 -> 3 -> 5 can only be taken when some_val evaluates to True, while 1 -> 4 -> 5 is taken when some_val evaluates to False. This is accomplished by adding the condition to the list of goals in a query. (Queries and goals are discussed in the [Solver] section.)

CFGNodes may be associated with Variables by one or more Bindings. A simple case, x = 5, was explained above. In more complex cases, such as y = m*x + b, the Binding for ("m*x+b", n0) is derived from several other Bindings. This is represented by an Origin that explains how the Binding is constructed. An Origin consists of two parts: the CFGNode of the Binding (n0, in this case) and a Source Set, which contains all the Bindings that are used to construct the Binding. In y = m*x + b, the Source Set is {m, x, b}. The Solver uses Source Sets when deciding if a Binding is visible at a particular node: for a Binding to be visible, all of the Bindings in its Source Sets must also be visible.

Pytype often needs to know what Bindings are visible at a given node. “Visible” means there is a path in the CFG to the given node from the node where the Binding originates. Consider the example typegraph above: when checking if x.upper() is valid, pytype wants to know: “is x visible as a str at node 5?” That Binding is set in node 4, and there’s a path from node 4 to node 5 if some_val is False. Pytype will then ask: “is some_val False at node 4?” Since this is a limited example, pytype doesn’t know, but it assumes that some_val may be False. (And it will also assume some_val may be True! This means “is x a str” and “is x an int” are both considered possible – which is safer when we don’t know the value of some_val.) Therefore yes, x is visible as a str at node 5.

Binding Data

What data does a Binding actually bind?

In an expression like x = a(), pytype will add a Binding to x stating that x is bound to whatever value a returns. This value will be an instance of (a subclass of) abstract.BaseValue, the type that represents values in the Python program being analyzed.

But the typegraph is implemented in C++. That means the typegraph works with PyObject pointers. And to store those, the Binding class can hold onto a void pointer:

class Binding {
  void* data;

In, adding data turns a PyObject* into a void*, and pulling data out of a binding casts the void* back into a PyObject*. Simple enough!

But there’s a subtle bug here. Python objects are tracked by a reference counting garbage collector, and when nothing has a reference to a PyObject, that PyObject will be cleaned up. It’s easy to ensure the Binding increments the reference count when data is added, of course. Conversely, it must be ensured that if the Binding is cleaned up, the reference count is decremented. As an additional twist, a single PyObject* may be held by multiple Bindings, possibly across multiple Programs.

The solution is BindingData. Instead of a void*, a Binding holds a std::shared_ptr to a DataType:

class DataType;  // A fictional opaque type for data that's added to Bindings.
typedef std::shared_ptr<DataType> BindingData;
class Binding {
  BindingData data;

When a PyObject* value is added to a Binding, it’s transformed into a BindingData pointer with a cleanup function that decrements the reference count. This approach ensures that each PyObject* has a correct reference count, that no data objects will be deleted before the Binding is cleaned up, and that deleting a Binding (or a whole Program) will correctly decrememnt the reference count. This avoids potential memory leaks caused by PyObjects not being cleaned up.

NOTE: Why use DataType instead of PyObject? This decouples the typegraph implementation from Python, which is a cleaner design and makes testing easier. Regular C++ values can be used for testing instead of having to construct PyObjects.


The main purpose of the Program is to organize the CFG. Almost all of its functionality is devoted to creating CFGNodes, Variables and Bindings. It also holds the Solver that queries the CFG. Only one Program is alive at any time, except during tests. While Program and the typegraph as a whole are designed to be thread-safe, the rest of pytype is not guaranteed to be.

The advantage of having the Program maintain ownership of all CFG objects is we can guarantee that all parts of the CFG will stay alive as long as the Program is alive. It provides a single, centralized location for accessing and modifying the CFG and the solver.

Default Data

Sometimes a Binding needs to be created without any particular data to be stored in it. By default, the Program will fill in the Binding with a nullptr, which is a sane enough default for C++ code. But pytype expects Bindings to contain valid data, and since Bindings are immutable, it’s useful for the Program to allow the default Binding data to be set to a more meaningful value.

That said, the default data is rarely used, because Bindings are almost always created with particular data in mind. In fact, a Binding is only created without specific data when pytype hits a complexity limit. In particular, default data is used when:

  1. there are too many possible variable combinations, as seen in abstract_utils.get_views()
  2. there are too many Bindings on one Variable, as seen in Variable.FindOrAddBinding()

In both cases, default data ensures the Bindings have valid data, allowing pytype to continue analysis in the face of complexity.

Pytype sets the default data to abstract.Unsolvable.

Sets in the Typegraph

The typegraph makes good use of sets to keep track of things like which nodes are blocked by some condition, or the set of Bindings that form the source set of another Binding. The sets are implemented by giving each CFGNode and Binding an ID (which are unique between objects of the same type, but not globally unique) and ordering them by their ID by overloading <. There is also a helper function, pointer_less, which compares pointers by dereferencing them. Hence, CFGNodeSet is defined as std::set<const CFGNode*, pointer_less<CFGNode>, and similarly for SourceSet.

Originally, these sets did not use pointer_less and just compared the pointers. This is undefined behavior.

std::set or std::unordered_set?

The typegraph uses std::set to increase determinism: any source of nondeterminism or randomness in pytype has a chance to leak into the pickled type stub files, which is problematic. Additionally, the typegraph does not and cannot guarantee that the items in a set won’t be modified, which violates the std::unordered_set invariants. Finally, iterating over and comparing sets is more efficient for std::set.


A basic operation when analyzing the CFG is checking if there is a path between two nodes. This is used when checking if a binding’s origin is visible at a given node, for example. Queries always progress from a child node to the parent, in the opposite direction of the edges of the CFG. To make these lookups faster, the Program class uses a backwards reachability cache that’s handled by the ReachabilityAnalyzer class. Backwards reachability means that the reachability analysis proceeds from child to parent, in the opposite direction of the directed edges of the graph.

The ReachabilityAnalyzer tracks the list of adjacent nodes for each CFG node. For node i, reachable[i][j] indicates whether j is reachable from i. When an edge is added to the reachability cache, the cache updates every node to see if connections are possible.

For example, consider three nodes A -> B -> C. The cache would be initialised as:

reachable[A] = [True, False, False]
reachable[B] = [False, True, False]
reacahble[C] = [False, False, True]

(A node can always be reached from itself.) Since there’s a connection from B to C, the backwards edge C -> B is added to the cache:

reachable[A] = [True, False, False]
reachable[B] = [False, True, False]
reacahble[C] = [False, True, True]

Then the backwards edge B -> A is added:

reachable[A] = [True, False, False]
reachable[B] = [True, True, False]
reacahble[C] = [True, True, True]

The cache now shows that A can only reach itself, B can reach itself and A, and C can reach all three nodes.


Every CFG node has an ID number of type size_t, which these docs will assume is 64 bits. A naive implementation would make reachable[n] a list of the IDs of the nodes reachable from node n, but at 8 bytes per node and potentially thousands of nodes, that will get too expensive quickly. Instead, nodes are split into buckets based on their ID. Each bucket tracks 64 nodes, so the top 58 bits of the ID determine the node’s bucket and the bottom 6 bits determine the node’s index within the bucket.

These buckets are implemented using bit vectors. Since a bucket covers 64 nodes, it’s represented by an int64_t. A node’s reachable list is then a std::vector<int64_t>, and the reachability cache that tracks every node’s reachable list is a std::vector<std::vector<int64_t>>. All together, node j is reachable from node i if reachable[i][bucket(j)] & bit(j) == 1.

For example, consider a CFG with 100 nodes, which have IDs from 0 to 99. There will be 100 entries in the reachability cache, one for each node, such that reachable[n] corresponds to the nodes that are backwards reachable from node n. reachable[n] is a std::vector<int64_t> with two elements, the first tracking nodes 0 - 63 and the second tracking nodes 64 - 99, with room to track another 28 nodes.

Let’s check if node 75 is backwards reachable from node 30: is_reachable(30, 75).

  1. Find node 75’s bucket: bucket = 75 >> 6 = 1. (This is equivalent to 75 / 64.)
  2. Find node 75’s bit: bit = 1 << (75 & 0x3F) = 1 << 11.
  3. Check node 30’s reachability: reachable[30][bucket] & bit.

Adding a new node to the reachability cache is accomplished by adding another entry to reachable. There is a catch: the cache must check if a new bucket is needed to track the new node. If one is, then every node’s reachable list is extended with one more bucket. Finally, reachable[n][bucket(n)] = bit[n] is set, indicating that node n is reachable from itself.

Adding an edge between two nodes is only slightly more complex. Because the cache tracks reachability, adding an edge may update every node. Remember the A -> B -> C example previously: add_edge(B, A) updated both B and C, because B is reachable from C. For add_edge(src, dst), the cache checks if src is reachable from each node i, and if so, bitwise-ORs reachable[i] and reachable[dst] together. Because src is reachable from itself, this will also update reachable[src] when i == src.

The Solver Algorithm

This section is intended as a starting point to help you understand what the solver is doing. The code itself is somewhat complex, but the comments, this guide and the solver tests should clarify it.

A Simple Example

Let’s start with a simple example. The CFG is just three nodes:

n0 -> n1 -> n2

There are also two bindings. For clarity, they’re named after the variable and value that they bind together.

  1. Binding x=5: x = 5 at n0.
  2. Binding y=7: y = x + 2 at n1.

A query for this very simple CFG would be, for example, “is y = x + 2 visible at n2?” For this query, the list of goals is [y=7], and the start node is n2. The tuple of the list of goals and the start node forms the state of the solver: ([y=7], n2) is the initial state for this query.

To answer this, the solver starts at n2. There are no relevant bindings here, so no goals are resolved. It then checks which node it should move to next. Since y=7 is bound at n1, and n1 can be reached moving backwards from n2, the solver chooses to move to n1. The new solver state is ([y=7], n1).

At n1, the solver finds binding y=7. This fulfills a goal! It removes y=7 from the goal list, then checks y=7’s source set for new goals. The source set of a binding is the list of bindings that are used to construct a new binding. The value bound to y is x + 2, which contains the binding x=5, so that binding is in the source set of y=7. It then looks for a new node to move to, same as before. In this case, it’s looking for the origin node for x=5, which is n0. Since there’s a path between n0 and n1, the new state is ([x=5], n0).

Finally, at n0, the solver finds x=5. That goal is satisfied and therefore removed from the goal set. Since x=5 has no bindings in its source set, the solver stops here – the query has been solved and the solver can answer true.

A More Complex Example

Let’s consider a more complex example with more CFG features. Consider this sample program:

a = 1          x0
b = 2          | \
c = True       |  \
if c:          x1 |
  d = a + 2    |  |
else:          |  x2
  d = a + b    | /
e = d + a      x3

In this example, there are 6 bindings (a=1, b=2, c=True, d=a2, d=ab, e=da) across four nodes. Additionally, x1 and x2 both have conditions, which are the bindings c=True and c=False.

The example query will be: “Is e = d + a visible from x3?”

State 1: ([e=da], x3). The goal is immediately fulfilled. But the binding has two possible constructions: d=a2 + a=1 or d=ab + a=1. The solver now has multiple states to consider: ([d=ab, a=1], x2) and ([d=a2, a=1], x1).

State 2.1: ([d=ab, a=1], x2). This node fulfills the first goal, creating a new goal set of ([b=2, a=1]). Also, x2 has the condition c=False, so that’s added to the goal set. All three bindings have the same origin of x0, so that’s the next node to visit: ([b=2, a=1, c=False], x0).

State 2.2: ([b=2, a=1, c=False], x0). This node fulfills a=1 and b=2, leaving just c=False. Since that binding has no source set, there are no more nodes to consider, and the solver returns false for this branch.

State 3.1: ([d=a2, a=1]), x1). The goal d=a2 is fulfilled, and its source set (just a=1, since 2 is a constant) is checked for new goals. Since a=1 is already a goal, the goal set doesn’t change. The node condition (c=True) is also added to the goal set. The new state is ([a=1, c=True], x0).

State 3.2: ([a=1, c=True], x0). At this node, the solver finds both goals are fulfilled. Since neither one has a source set, the solver finds no new goals, which means this branch succeeds.

The queries can get more complex from here. One case to consider: it’s possible to construct a goal set like ([d=a2, d=ab]). The solver will detect this contradiction – there’s no way to satisfy two different bindings on one variable – and reject the state.

Shortcircuiting and the solver cache

The solver employs two small optimizations for speading up queries.

The first trick is shortcircuiting. Queries with multiple bindings take longer to evaluate, so the solver first checks if the query is at all possible by breaking the initial goals into their own, individual queries.

Consider the CFG used in the previous example. A query like ([d=a2, ...], x2) will fail because there’s no path from x2 to where d=a2 is bound in x1. Or, similarly, ([c=False, ...], x3), which will fail because c is only bound to True in the CFG. Larger queries with these bindings at these nodes will always fail, so the solver saves time by shortcircuiting them.

Note that shortcircuiting only stops queries that are easily contradicted in this way. More complex queries that fail due to the interplay between bindings – such as ([d=ab, d=a2], x3) – will not be shortcircuited.

In addition, each state the solver encounters is cached, including all the states created during shortcircuiting, with the solution that was found. Note that the cache does not persist between solver instances, so it is only useful if a solver is queried multiple times. It is also helpful for preventing infinite recursion; new states are considered to be solvable, on the reasoning that if the state couldn’t be solved in an ensuing state then the cache entry would be updated.