[clang][docs][dataflow] Added an introduction to dataflow analysis

[gribozavr]

Summary

This documentation supports the dataflow analysis framework (see "[RFC]
A dataflow analysis framework for Clang AST" on cfe-dev).

Since the implementation of the framework has not been committed yet,
right now the doc describes dataflow analysis in general.

Since this is the first markdown document in clang/docs, I added support
for Markdown to clang/docs/conf.py in the same way as it is done in
llvm/docs.

[xazax.hun]

Thanks for working on this! Publishing documentation alongside the feature is a great practice. I liked this documentation a lot. I have strong opinion about the ambiguous use of path condition in this document, otherwise I was only nitpicking.

clang/docs/DataFlowAnalysisIntro.md

25

I know, everyone has their favorite set of resources. Let me share mine: https://cs.au.dk/~amoeller/spa/
I think this one might have a bit more content with more examples. Feel free to leave it as is.

50

Nit: I wonder if representing the value of x with an empty set is a good choice. One could argue that x has an indeterminate value that could be represented with the set of all integers using top. Others could argue for a bottom value. Those concepts are not yet introduced. Maybe initializing x in the source code would side step this problem.

72–73

Technically, I think this would be a join-semilattice. To have a lattice, we would need both join and meet.

87–88

If we want to be practical, I wonder if we want to give some guidance how to implement that efficiently. E.g. we could cite a paper like this: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.106.4911&rep=rep1&type=pdf

112–113

I wonder if we want to cite widening and narrowing in this context to give people some pointers where to look for more info.

183

I think it would make sense to include the whole program state in the comments. E.g. // x is ⊥, n is ⊤.
This could showcase how we constrain the value of n to 42 in one of the branches.

280

While I agree that this is the general formula, people sometimes do small variations, e.g.:

out =  join(transfer(basic_block,in_1), transfer(basic_block,in_2), ..., transfer(basic_block,in_n))

This is less efficient as we end up invoking the transfer function more often, but it can be more precise. E.g. with some ad-hoc notation:

Given the branches: x: {1}, x: {2}, x: {3}, x: {4}
join(in...) : x : {⊤}
transfer("x/2", ...) == x : {⊤}

vs.
Given the branches: x: {1}, x: {2}, x: {3}, x: {4}
transfer("x/2", ...) ==x : {0}, x : {1}, x : {1}, x: {2} == outs
join(outs) == x: {0, 1, 2}

289

Only if our transfer functions are monotonic :)

300

I think many people find it confusing, what is the relationship between symbolic execution and abstract interpretation? I found this answer helpful: https://cstheory.stackexchange.com/questions/19708/symbolic-execution-is-a-case-of-abstract-interpretation
I wonder if it might be worth citing.

366

Here, I think we want to include ptr in the comments describing the program state.

382–383

One could argue that you also cannot read from the pointee, otherwise you have an input-output parameter. There is an example clarifying this below, but I wonder if we want to get the definition right from the beginning.

470

While I do agree we cannot explain everything, like what borrowing or escaping means in this document, I wonder if we want to add some citations.

479–480

I wonder if the distinction between normal states and failure states is useful. In general, we can combine arbitrary number of lattices and propagate all the information in a single pass. I.e., we could have multiple "normal" or "failure" states.

There are multiple ways to combine lattices, we can put them on top of each other, or next to each other introducing new top/bottom values, or we can take their products.

571

It might be a good idea to reference liveness analysis which is often used to answer many related questions:

• Finding dead stores
• Finding uninitialized variable
• Finding a good point to deallocate memory
• Finding out if it would be safe to move an object
• ...

809

Here, I think the term path condition becomes ambiguous. Traditionally, path condition is a path sensitive property, in the code snippet below:

if (b1)
{ ... }

if (b1 && b2)
{ ... }

We would reach the second branch on two paths (depending on whether we took the first one). On one of the paths, the path condition will have b1, on the other, it will have !b1.

I think here you mean the non-path-sensitive property. I'd suggest doing a strict distinction and use path condition for path sensitive properties and flow condition for flow sensitive properties.

[gribozavr]

Address review comments.

[gribozavr2]

clang/docs/DataFlowAnalysisIntro.md

25

Forgot about this book -- I also like it a lot, added!

50

Since there is a discussion of these topics below, I agree that avoiding the question at this point is better. Added initialization.

72–73

Corrected.

87–88

Thanks, added.

112–113

I think it should be a separate section, with a slightly different example. To show the value of widening we need a lattice that can abstract the infinite set {0; 1; 2; ...} more precisely than "⊤" (e.g., "non-negative"). There would be also the usual handwaving about figuring out how to make those types of conclusions in the general case. I think adding all that here would break the flow and make the explanation less crisp. Adding a separate section on widening would be good one day, when the dataflow framework gets the feature. I don't think it is necessary right now.

183

So far we have been focusing on figuring out the values of x the local variable, not n the parameter, so I'd rather not deviate. Also, comparing n to 42 wouldn't necessarily lead to constraining the set of values here (for example, in the framework we do it in the flow condition).

280

This section is only attempting to give the readers an intuition of why dataflow works, hence it must be straightforward, not necessarily exhaustive or rigorous. I tried to add your suggestion here, but it breaks the flow. I tried to add it at the end of the section, but it looks out of place. So, sorry, I couldn't fit it here. If you have a specific wording suggestion that works here, I would be happy to apply it.

I think one would need to add a section like "variations on dataflow equations" to properly introduce the idea. It also seems to me that this specific variation is also a special case of a more general idea of deferring join operations to improve precision; that is, instead of doing a join everywhere where classic dataflow prescribes it, we can instead keep exploring separate paths, and only do a join at some later point in the CFG. Similarly we can also unroll loops in a precise way up to a certain number of iterations, and attempt to join/widen only at after that. These are of course important ideas, but they don't help with what this document is trying to achieve.

289

That's an important condition, added.

300

So far I have avoided using the term "abstract interpretation" in this document, or going into more deep topics. However I read this answer and I like it too, so I added a link to it here.

366

Added.

382–383

Corrected.

470

I reworded to avoid using the term "borrow".

479–480

I wonder if the distinction between normal states and failure states is useful.

I'm not sure I understand -- this distinction is useful in this particular approach to solve this problem, since it helps solve the problem? Or are you objecting to the term "failure"?

Of course, in general, an analysis does not need to have failure/normal states, and like you said, if we track information about multiple candidate output parameters at the same time, each can be in either a normal or failure state at every program point independently of other parameters. However, this document provides an example of a solution for this particular problem; the goal is not to solve the problem, but to give the reader an intuition of how dataflow ideas can be applied to solve real problems.

571

Good point, added a reference.

809

Replaced with "flow condition".

[xazax.hun]

Thanks!

clang/docs/DataFlowAnalysisIntro.md

112–113

Sounds good.

183

I see. Yeah, if the framework does not do that (yet), I see how it might be confusing. Let's leave this as is.

280

Agreed, we cannot introduce everything in an introductory document. I mainly wanted to avoid creating the impression that there is one definitive way to data flow analysis and this is it. Maybe one sentence at the end mentioning there are several variations that we do not discuss here could be sufficient.

479–480

Yeah, I found the term failure a bit confusing. It is a regular analysis state, the analysis itself did not fail. I think I understand that this is named failure after the fact that it is blocking the transformation.

Probably my main problem was that it was not clear whether the document suggested having normal and failure states is a general design principle to all analysis or specific to the problem we want to solve. I think in this particular example the idea of combining lattices in a certain way to build larger lattices is the general design principle (combining something that one could think of 3 separate analysis: escape, unsafe reads, and field sensitive initialized variable into one) and the classification of states into normal and failure is specific to the problem.

I think the placement of this section makes it pretty clear that this is just an example and not describing general methods. Probably I took a break before reading this section and read it with the wrong context in mind :)

Feel free to leave this as it is.

[gribozavr2]

clang/docs/DataFlowAnalysisIntro.md

280

Added a long parenthetical:

(Note that there are other ways to write this equation that produce higher
precision analysis results. The trick is to keep exploring the execution paths
separately and delay joining until later. Hoowever, we won't discuss those
variations here.)

479–480

Sounds good :)

[MyDeveloperDay]

Did you add somewhere what dependency we need in order to build the documentation now as mine is having problems with "recommonmark"

/usr/bin/sphinx-build -n ./docs ./html
Running Sphinx v3.4.3

Configuration error:
There is a programmable error in your configuration file:

Traceback (most recent call last):
  File "/usr/lib/python3.6/site-packages/sphinx/config.py", line 326, in eval_config_file
    execfile_(filename, namespace)
  File "/usr/lib/python3.6/site-packages/sphinx/util/pycompat.py", line 88, in execfile_
    exec(code, _globals)
  File "/buildareas/llvm3/llvm-project/clang/docs/conf.py", line 41, in <module>
    import recommonmark
ModuleNotFoundError: No module named 'recommonmark'

[gribozavr2]

Did you add somewhere what dependency we need in order to build the documentation now as mine is having problems with "recommonmark"

I didn't change the documentation because recommonmark is not a new dependency for llvm-project, llvm/docs was depending on it before this change.

clang/docs/DataFlowAnalysisIntro.md

+# Data flow analysis: an informal introduction
+
+## Abstract
+
+This document introduces data flow analysis in an informal way. The goal is to
+give the reader an intuitive understanding of how it works, and show how it
+applies to a range of refactoring and bug finding problems.
+
+Data flow analysis is a well-established technique; it is described in many
+papers, books, and videos. If you would like a more formal, or a more thorough
+explanation of the concepts mentioned in this document, please refer to the
+following resources:
+
+*   [The Lattice article in Wikipedia](https://en.wikipedia.org/wiki/Lattice_\(order\)).
+*   Videos on the PacketPrep YouTube channel that introduce lattices and the
+    necessary background information:
+    [#20](https://www.youtube.com/watch?v=73j_FXBXGm8),
+    [#21](https://www.youtube.com/watch?v=b5sDjo9tfE8),
+    [#22](https://www.youtube.com/watch?v=saOG7Uooeho),
+    [#23](https://www.youtube.com/watch?v=3EAYX-wZH0g),
+    [#24](https://www.youtube.com/watch?v=KRkHwQtW6Cc),
+    [#25](https://www.youtube.com/watch?v=7Gwzsc4rAgw).
+*   [Introduction to Dataflow Analysis](https://www.youtube.com/watch?v=OROXJ9-wUQE)
+*   [Introduction to abstract interpretation](http://www.cs.tau.ac.il/~msagiv/courses/asv/absint-1.pdf).
+*   [Introduction to symbolic execution](https://www.cs.umd.edu/~mwh/se-tutorial/symbolic-exec.pdf).
+*   [Static Program Analysis by Anders Møller and Michael I. Schwartzbach](https://cs.au.dk/~amoeller/spa/).
+*   [EXE: automatically generating inputs of death](https://css.csail.mit.edu/6.858/2020/readings/exe.pdf)
+    (a paper that successfully applies symbolic execution to real-world
+    software).
+
+## Data flow analysis
+
+### The purpose of data flow analysis
+
+Data flow analysis is a static analysis technique that proves facts about a
+program or its fragment. It can make conclusions about all paths through the
+program, while taking control flow into account and scaling to large programs.
+The basic idea is propagating facts about the program through the edges of the
+control flow graph (CFG) until a fixpoint is reached.
+
+### Sample problem and an ad-hoc solution
+
+We would like to explain data flow analysis while discussing an example. Let's
+imagine that we want to track possible values of an integer variable in our
+program. Here is how a human could annotate the code:
+
+```c++
+void Example(int n) {
+  int x = 0;
+  // x is {0}
+  if (n > 0) {
+    x = 5;
+    // x is {5}
+  } else {
+    x = 42;
+    // x is {42}
+  }
+  // x is {5; 42}
+  print(x);
+}
+```
+
+We use sets of integers to represent possible values of `x`. Local variables
+have unambiguous values between statements, so we annotate program points
+between statements with sets of possible values.
+
+Here is how we arrived at these annotations. Assigning a constant to `x` allows
+us to make a conclusion that `x` can only have one value. When control flow from
+the "then" and "else" branches joins, `x` can have either value.
+
+Abstract algebra provides a nice formalism that models this kind of structure,
+namely, a lattice. A join-semilattice is a partially ordered set, in which every
+two elements have a least upper bound (called a *join*).
+
+```
+join(a, b) ⩾ a   and   join(a, b) ⩾ b   and   join(x, x) = x
+```
+
+For this problem we will use the lattice of subsets of integers, with set
+inclusion relation as ordering and set union as a join.
+
+Lattices are often represented visually as Hasse diagrams. Here is a Hasse
+diagram for our lattice that tracks subsets of integers:
+
+![Hasse diagram for a lattice of integer sets](DataFlowAnalysisIntroImages/IntegerSetsInfiniteLattice.svg)
+
+Computing the join in the lattice corresponds to finding the lowest common
+ancestor (LCA) between two nodes in its Hasse diagram. There is a vast amount of
+literature on efficiently implementing LCA queries for a DAG, however Efficient
+Implementation of Lattice Operations (1989)
+([CiteSeerX](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.106.4911),
+[doi](https://doi.org/10.1145%2F59287.59293)) describes a scheme that
+particularly well-suited for programmatic implementation.
+
+### Too much information and "top" values
+
+Let's try to find the possible sets of values of `x` in a function that modifies
+`x` in a loop:
+
+```c++
+void ExampleOfInfiniteSets() {
+  int x = 0; // x is {0}
+  while (condition()) {
+    x += 1;  // x is {0; 1; 2; …}
+  }
+  print(x);  // x is {0; 1; 2; …}
+}
+```
+
+We have an issue: `x` can have any value greater than zero; that's an infinite
+set of values, if the program operated on mathematical integers. In C++ `int` is
+limited by `INT_MAX` so technically we have a set `{0; 1; …; INT_MAX}` which is
+still really big.
+
+To make our analysis practical to compute, we have to limit the amount of
+information that we track. In this case, we can, for example, arbitrarily limit
+the size of sets to 3 elements. If at a certain program point `x` has more than
+3 possible values, we stop tracking specific values at that program point.
+Instead, we denote possible values of `x` with the symbol `⊤` (pronounced "top"
+according to a convention in abstract algebra).
+
+```c++
+void ExampleOfTopWithALoop() {
+  int x = 0;  // x is {0}
+  while (condition()) {
+    x += 1;   // x is ⊤
+  }
+  print(x);   // x is ⊤
+}
+```
+
+The statement "at this program point, `x`'s possible values are `⊤`" is
+understood as "at this program point `x` can have any value because we have too
+much information, or the information is conflicting".
+
+Note that we can get more than 3 possible values even without a loop:
+
+```c++
+void ExampleOfTopWithoutLoops(int n) {
+  int x = 0;  // x is {0}
+  switch(n) {
+    case 0:  x = 1; break; // x is {1}
+    case 1:  x = 9; break; // x is {9}
+    case 2:  x = 7; break; // x is {7}
+    default: x = 3; break; // x is {3}
+  }
+  // x is ⊤
+}
+```
+
+### Uninitialized variables and "bottom" values
+
+When `x` is declared but not initialized, it has no possible values. We
+represent this fact symbolically as `⊥` (pronounced "bottom").
+
+```c++
+void ExampleOfBottom() {
+  int x;    // x is ⊥
+  x = 42;   // x is {42}
+  print(x);
+}
+```
+
+Note that using values read from uninitialized variables is undefined behaviour
+in C++. Generally, compilers and static analysis tools can assume undefined
+behavior does not happen. We must model uninitialized variables only when we are
+implementing a checker that specifically is trying to find uninitialized reads.
+In this example we show how to model uninitialized variables only to demonstrate
+the concept of "bottom", and how it applies to possible value analysis. We
+describe an analysis that finds uninitialized reads in a section below.
+
+### A practical lattice that tracks sets of concrete values
+
+Taking into account all corner cases covered above, we can put together a
+lattice that we can use in practice to track possible values of integer
+variables. This lattice represents sets of integers with 1, 2, or 3 elements, as
+well as top and bottom. Here is a Hasse diagram for it:
+
+![Hasse diagram for a lattice of integer sets](DataFlowAnalysisIntroImages/IntegerSetsFiniteLattice.svg)
+
+### Formalization
+
+Let's consider a slightly more complex example, and think about how we can
+compute the sets of possible values algorithmically.
+
+```c++
+void Example(int n) {
+  int x;          // x is ⊥
+  if (n > 0) {
+    if (n == 42) {
+       x = 44;    // x is {44}
+    } else {
+       x = 5;     // x is {5}
+    }
+    print(x);     // x is {44; 5}
+  } else {
+    x = n;        // x is ⊤
+  }
+  print(x);       // x is ⊤
+}
+```
+
+As humans, we understand the control flow from the program text. We used our
+understanding of control flow to find program points where two flows join.
+Formally, control flow is represented by a CFG (control flow graph):
+
+![CFG for the code above](DataFlowAnalysisIntroImages/CFGExample.svg)
+
+We can compute sets of possible values by propagating them through the CFG of
+the function:
+
+*   When `x` is declared but not initialized, its possible values are `{}`. The
+    empty set plays the role of `⊥` in this lattice.
+
+*   When `x` is assigned a concrete value, its possible set of values contains
+    just that specific value.
+
+*   When `x` is assigned some unknown value, it can have any value. We represent
+    this fact as `⊤`.
+
+*   When two control flow paths join, we compute the set union of incoming
+    values (limiting the number of elements to 3, representig larger sets as
+    `⊤`).
+
+The sets of possible values are influenced by:
+
+*   Statements, for example, assignments.
+
+*   Joins in control flow, for example, ones that appear at the end of "if"
+    statements.
+
+**Effects of statements** are modeled by what is formally known as a transfer
+function. A transfer function takes two arguments: the statement, and the state
+of `x` at the previous program point. It produces the state of `x` at the next
+program point. For example, the transfer function for assignment ignores the
+state at the previous program point:
+
+```c++
+// GIVEN: x is {42; 44}
+x = 0;
+// CONCLUSION: x is {0}
+```
+
+The transfer function for `+` performs arithmetic on every set member:
+
+```c++
+// GIVEN: x is {42, 44}
+x = x + 100;
+// CONCLUSION: x is {142, 144}
+```
+
+**Effects of control flow** are modeled by joining the knowledge from all
+possible previous program points.
+
+```c++
+if (...) {
+  ...
+  // GIVEN: x is {42}
+} else {
+  ...
+  // GIVEN: x is {44}
+}
+// CONCLUSION: x is {42; 44}
+```
+
+```c++
+// GIVEN: x is {42}
+while (...) {
+  ...
+  // GIVEN: x is {44}
+}
+// CONCLUSION: {42; 44}
+```
+
+The predicate that we marked "given" is usually called a precondition, and the
+conclusion is called a postcondition.
+
+In terms of the CFG, we join the information from all predecessor basic blocks.
+
+![Modeling the effects of a CFG basic block](DataFlowAnalysisIntroImages/CFGJoinRule.svg)
+
+Putting it all together, to model the effects of a basic block we compute:
+
+```
+out = transfer(basic_block, join(in_1, in_2, ..., in_n))
+```
+
+(Note that there are other ways to write this equation that produce higher
+precision analysis results. The trick is to keep exploring the execution paths
+separately and delay joining until later. Hoowever, we won't discuss those
+variations here.)
+
+To make a conclusion about all paths through the program, we repeat this
+computation on all basic blocks until we reach a fixpoint. In other words, we
+keep propagating information through the CFG until the computed sets of values
+stop changing.
+
+If the lattice has a finite height and transfer functions are monotonic the
+algorithm is guaranteed to terminate.  Each iteration of the algorithm can
+change computed values only to larger values from the lattice. In the worst
+case, all computed values become `⊤`, which is not very useful, but at least the
+analysis terminates at that point, because it can't change any of the values.
+
+Fixpoint iteration can be optimised by only reprocessing basic blocks which had
+one of their inputs changed on the previous iteration. This is typically
+implemented using a worklist queue. With this optimisation the time complexity
+becomes `O(m * |L|)`, where `m` is the number of basic blocks in the CFG and
+`|L|` is the size of lattice used by the analysis.
+
+## Symbolic execution: a very short informal introduction
+
+### Symbolic values
+
+In the previous example where we tried to figure out what values a variable can
+have, the analysis had to be seeded with a concrete value. What if there are no
+assignments of concrete values in the program? We can still deduce some
+interesting information by representing unknown input values symbolically, and
+computing results as symbolic expressions:
+
+```c++
+void PrintAbs(int x) {
+  int result;
+  if (x >= 0) {
+    result = x;   // result is {x}
+  } else {
+    result = -x;  // result is {-x}
+  }
+  print(result);  // result is {x; -x}
+}
+```
+
+We can't say what specific value gets printed, but we know that it is either `x`
+or `-x`.
+
+Dataflow analysis is an istance of abstract interpretation, and does not dictate
+how exactly the lattice and transfer functions should be designed, beyond the
+necessary conditions for the analysis to converge. Nevertheless, we can use
+symbolic execution ideas to guide our design of the lattice and transfer
+functions: lattice values can be symbolic expressions, and transfer functions
+can construct more complex symbolic expressions from symbolic expressions that
+represent arguments. See [this StackOverflow
+discussion](https://cstheory.stackexchange.com/questions/19708/symbolic-execution-is-a-case-of-abstract-interpretation)
+for a further comparison of abstract interpretation and symbolic execution.
+
+### Flow condition
+
+A human can say about the previous example that the function returns `x` when
+`x >= 0`, and `-x` when `x < 0`. We can make this conclusion programmatically by
+tracking a flow condition. A flow condition is a predicate written in terms of
+the program state that is true at a specific program point regardless of the
+execution path that led to this statement. For example, the flow condition for
+the program point right before evaluating `result = x` is `x >= 0`.
+
+If we enhance the lattice to be a set of pairs of values and predicates, the
+dataflow analysis computes the following values:
+
+```c++
+void PrintAbs(int x) {
+  int result;
+  if (x >= 0) {
+    // Flow condition: x >= 0.
+    result = x;   // result is {x if x >= 0}
+  } else {
+    // Flow condition: x < 0.
+    result = -x;  // result is {-x if x < 0}
+  }
+  print(result);  // result is {x if x >= 0; -x if x < 0}
+}
+```
+
+Of course, in a program with loops, symbolic expressions for flow conditions can
+grow unbounded. A practical static analysis system must control this growth to
+keep the symbolic representations manageable and ensure that the data flow
+analysis terminates. For example, it can use a constraint solver to prune
+impossible flow conditions, and/or it can abstract them, losing precision, after
+their symbolic representations grow beyond some threshold. This is similar to
+how we had to limit the sizes of computed sets of possible values to 3 elements.
+
+### Symbolic pointers
+
+This approach proves to be particularly useful for modeling pointer values,
+since we don't care about specific addresses but just want to give a unique
+identifier to a memory location.
+
+```c++
+void ExampleOfSymbolicPointers(bool b) {
+  int x = 0;     // x is {0}
+  int* ptr = &x; // x is {0}      ptr is {&x}
+  if (b) {
+    *ptr = 42;   // x is {42}     ptr is {&x}
+  }
+  print(x);      // x is {0; 42}  ptr is {&x}
+}
+```
+
+## Example: finding output parameters
+
+Let's explore how data flow analysis can help with a problem that is hard to
+solve with other tools in Clang.
+
+### Problem description
+
+Output parameters are function parameters of pointer or reference type whose
+pointee is completely overwritten by the function, and not read before it is
+overwritten. They are common in pre-C++11 code due to the absence of move
+semantics. In modern C++ output parameters are non-idiomatic, and return values
+are used instead.
+
+Imagine that we would like to refactor output parameters to return values to
+modernize old code. The first step is to identify refactoring candidates through
+static analysis.
+
+For example, in the following code snippet the pointer `c` is an output
+parameter:
+
+```c++
+struct Customer {
+  int account_id;
+  std::string name;
+}
+
+void GetCustomer(Customer *c) {
+  c->account_id = ...;
+  if (...) {
+    c->name = ...;
+  } else {
+    c->name = ...;
+  }
+}
+```
+
+We would like to refactor this code into:
+
+```c++
+Customer GetCustomer() {
+  Customer c;
+  c.account_id = ...;
+  if (...) {
+    c.name = ...;
+  } else {
+    c.name = ...;
+  }
+  return c;
+}
+```
+
+However, in the function below the parameter `c` is not an output parameter
+because its field `name` is not overwritten on every path through the function.
+
+```c++
+void GetCustomer(Customer *c) {
+  c->account_id = ...;
+  if (...) {
+    c->name = ...;
+  }
+}
+```
+
+The code also cannot read the value of the parameter before overwriting it:
+
+```c++
+void GetCustomer(Customer *c) {
+  use(c->account_id);
+  c->name = ...;
+  c->account_id = ...;
+}
+```
+
+Functions that escape the pointer also block the refactoring:
+
+```c++
+Customer* kGlobalCustomer;
+
+void GetCustomer(Customer *c) {
+  c->name = ...;
+  c->account_id = ...;
+  kGlobalCustomer = c;
+}
+```
+
+To identify a candidate function for refactoring, we need to do the following:
+
+*   Find a function with a non-const pointer or reference parameter.
+
+*   Find the definition of that function.
+
+*   Prove that the function completely overwrites the pointee on all paths
+    before returning.
+
+*   Prove that the function reads the pointee only after overwriting it.
+
+*   Prove that the function does not persist the pointer in a data structure
+    that is live after the function returns.
+
+There are also requirements that all usage sites of the candidate function must
+satisfy, for example, that function arguments do not alias, that users are not
+taking the address of the function, and so on. Let's consider verifying usage
+site conditions to be a separate static analysis problem.
+
+### Lattice design
+
+To analyze the function body we can use a lattice which consists of normal
+states and failure states. A normal state describes program points where we are
+sure that no behaviors that block the refactoring have occurred. Normal states
+keep track of all parameter's member fields that are known to be overwritten on
+every path from function entry to the corresponding program point. Failure
+states accumulate observed violations (unsafe reads and pointer escapes) that
+block the refactoring.
+
+In the partial order of the lattice failure states compare greater than normal
+states, which guarantees that they "win" when joined with normal states. Order
+between failure states is determined by inclusion relation on the set of
+accumulated violations (lattice's `⩽` is `⊆` on the set of violations). Order
+between normal states is determined by reversed inclusion relation on the set of
+overwritten parameter's member fields (lattice's `⩽` is `⊇` on the set of
+overwritten fields).
+
+![Lattice for data flow analysis that identifies output parameters](DataFlowAnalysisIntroImages/OutputParameterIdentificationLattice.svg)
+
+To determine whether a statement reads or writes a field we can implement
+symbolic evaluation of `DeclRefExpr`s, `LValueToRValue` casts, pointer
+dereference operator and `MemberExpr`s.
+
+### Using data flow results to identify output parameters
+
+Let's take a look at how we use data flow analysis to identify an output
+parameter. The refactoring can be safely done when the data flow algorithm
+computes a normal state with all of the fields proven to be overwritten in the
+exit basic block of the function.
+
+```c++
+struct Customer {
+  int account_id;
+  std::string name;
+};
+
+void GetCustomer(Customer* c) {
+  // Overwritten: {}
+  c->account_id = ...; // Overwritten: {c->account_id}
+  if (...) {
+    c->name = ...;     // Overwritten: {c->account_id, c->name}
+  } else {
+    c->name = ...;     // Overwritten: {c->account_id, c->name}
+  }
+  // Overwritten: {c->account_id, c->name}
+}
+```
+
+When the data flow algorithm computes a normal state, but not all fields are
+proven to be overwritten we can't perform the refactoring.
+
+```c++
+void target(bool b, Customer* c) {
+  // Overwritten: {}
+  if (b) {
+    c->account_id = 42;     // Overwritten: {c->account_id}
+  } else {
+    c->name = "Konrad";  // Overwritten: {c->name}
+  }
+  // Overwritten: {}
+}
+```
+
+Similarly, when the data flow algorithm computes a failure state, we also can't
+perform the refactoring.
+
+```c++
+Customer* kGlobalCustomer;
+
+void GetCustomer(Customer* c) {
+  // Overwritten: {}
+  c->account_id = ...;    // Overwritten: {c->account_id}
+  if (...) {
+    print(c->name);       // Unsafe read
+  } else {
+    kGlobalCustomer = c;  // Pointer escape
+  }
+  // Unsafe read, Pointer escape
+}
+```
+
+## Example: finding dead stores
+
+Let's say we want to find redundant stores, because they indicate potential
+bugs.
+
+```c++
+x = GetX();
+x = GetY();
+```
+
+The first store to `x` is never read, probably there is a bug.
+
+The implementation of dead store analysis is very similar to output parameter
+analysis: we need to track stores and loads, and find stores that were never
+read.
+
+[Liveness analysis](https://en.wikipedia.org/wiki/Live_variable_analysis) is a
+generalization of this idea, which is often used to answer many related
+questions, for example:
+
+* finding dead stores,
+* finding uninitialized variables,
+* finding a good point to deallocate memory,
+* finding out if it would be safe to move an object.
+
+## Example: definitive initialization
+
+Definitive initialization proves that variables are known to be initialized when
+read. If we find a variable which is read when not initialized then we generate
+a warning.
+
+```c++
+void Init() {
+  int x;    // x is uninitialized
+  if (cond()) {
+    x = 10; // x is initialized
+  } else {
+    x = 20; // x is initialized
+  }
+  print(x); // x is initialized
+}
+```
+
+```c++
+void Uninit() {
+  int x;    // x is uninitialized
+  if (cond()) {
+    x = 10; // x is initialized
+  }
+  print(x); // x is maybe uninitialized, x is being read, report a bug.
+}
+```
+
+For this purpose we can use lattice in a form of a mapping from variable
+declarations to initialization states; each initialization state is represented
+by the followingn lattice:
+
+![Lattice for definitive initialization analysis](DataFlowAnalysisIntroImages/DefinitiveInitializationLattice.svg)
+
+A lattice element could also capture the source locations of the branches that
+lead us to the corresponding program point. Diagnostics would use this
+information to show a sample buggy code path to the user.
+
+## Example: refactoring raw pointers to `unique_ptr`
+
+Modern idiomatic C++ uses smart pointers to express memory ownership, however in
+pre-C++11 code one can often find raw pointers that own heap memory blocks.
+
+Imagine that we would like to refactor raw pointers that own memory to
+`unique_ptr`. There are multiple ways to design a data flow analysis for this
+problem; let's look at one way to do it.
+
+For example, we would like to refactor the following code that uses raw
+pointers:
+
+```c++
+void UniqueOwnership1() {
+  int *pi = new int;
+  if (...) {
+    Borrow(pi);
+    delete pi;
+  } else {
+    TakeOwnership(pi);
+  }
+}
+```
+
+into code that uses `unique_ptr`:
+
+```c++
+void UniqueOwnership1() {
+  auto pi = std::make_unique<int>();
+  if (...) {
+    Borrow(pi.get());
+  } else {
+    TakeOwnership(pi.release());
+  }
+}
+```
+
+This problem can be solved with a lattice in form of map from value declarations
+to pointer states:
+
+![Lattice that identifies candidates for unique_ptr refactoring](DataFlowAnalysisIntroImages/UniquePtrLattice.svg)
+
+We can perform the refactoring if at the exit of a function `pi` is
+`Compatible`.
+
+```c++
+void UniqueOwnership1() {
+  int *pi;             // pi is Compatible
+  pi = new int;        // pi is Defined
+  if (...) {
+    Borrow(pi);        // pi is Defined
+    delete pi;         // pi is Compatible
+  } else {
+    TakeOwnership(pi); // pi is Compatible
+  }
+  // pi is Compatible
+}
+```
+
+Let's look at an example where the raw pointer owns two different memory blocks:
+
+```c++
+void UniqueOwnership2() {
+  int *pi = new int;  // pi is Defined
+  Borrow(pi);
+  delete pi;          // pi is Compatible
+  if (smth) {
+    pi = new int;     // pi is Defined
+    Borrow(pi);
+    delete pi;        // pi is Compatible
+  }
+  // pi is Compatible
+}
+```
+
+It can be refactored to use `unique_ptr` like this:
+
+```c++
+void UniqueOwnership2() {
+  auto pi = make_unique<int>();
+  Borrow(pi);
+  if (smth) {
+    pi = make_unique<int>();
+    Borrow(pi);
+  }
+}
+```
+
+In the following example, the raw pointer is used to access the heap object
+after the ownership has been transferred.
+
+```c++
+void UniqueOwnership3() {
+  int *pi = new int; // pi is Defined
+  if (...) {
+    Borrow(pi);
+    delete pi;       // pi is Compatible
+  } else {
+    vector<unique_ptr<int>> v = {std::unique_ptr(pi)}; // pi is Compatible
+    print(*pi);
+    use(v);
+  }
+  // pi is Compatible
+}
+```
+
+We can refactor this code to use `unique_ptr`, however we would have to
+introduce a non-owning pointer variable, since we can't use the moved-from
+`unique_ptr` to access the object:
+
+```c++
+void UniqueOwnership3() {
+  std::unique_ptr<int> pi = std::make_unique<int>();
+  if (...) {
+    Borrow(pi);
+  } else {
+    int *pi_non_owning = pi.get();
+    vector<unique_ptr<int>> v = {std::move(pi)};
+    print(*pi_non_owning);
+    use(v);
+  }
+}
+```
+
+If the original code didn't call `delete` at the very end of the function, then
+our refactoring may change the point at which we run the destructor and release
+memory. Specifically, if there is some user code after `delete`, then extending
+the lifetime of the object until the end of the function may hold locks for
+longer than necessary, introduce memory overhead etc.
+
+One solution is to always replace `delete` with a call to `reset()`, and then
+perform another analysis that removes unnecessary `reset()` calls.
+
+```c++
+void AddedMemoryOverhead() {
+  HugeObject *ho = new HugeObject();
+  use(ho);
+  delete ho; // Release the large amount of memory quickly.
+  LongRunningFunction();
+}
+```
+
+This analysis will refuse to refactor code that mixes borrowed pointer values
+and unique ownership. In the following code, `GetPtr()` returns a borrowed
+pointer, which is assigned to `pi`. Then, `pi` is used to hold a uniquely-owned
+pointer. We don't distinguish between these two assignments, and we want each
+assignment to be paired with a corresponding sink; otherwise, we transition the
+pointer to a `Conflicting` state, like in this example.
+
+```c++
+void ConflictingOwnership() {
+  int *pi;           // pi is Compatible
+  pi = GetPtr();     // pi is Defined
+  Borrow(pi);        // pi is Defined
+
+  pi = new int;      // pi is Conflicting
+  Borrow(pi);
+  delete pi;
+  // pi is Conflicting
+}
+```
+
+We could still handle this case by finding a maximal range in the code where
+`pi` could be in the Compatible state, and only refactoring that part.
+
+```c++
+void ConflictingOwnership() {
+  int *pi;
+  pi = GetPtr();
+  Borrow(pi);
+
+  std::unique_ptr<int> pi_unique = std::make_unique<int>();
+  Borrow(pi_unique.get());
+}
+```
+
+## Example: finding redundant branch conditions
+
+In the code below `b1` should not be checked in both the outer and inner "if"
+statements. It is likely there is a bug in this code.
+
+```c++
+int F(bool b1, bool b2) {
+  if (b1) {
+    f();
+    if (b1 && b2) {  // Check `b1` again -- unnecessary!
+      g();
+    }
+  }
+}
+```
+
+A checker that finds this pattern syntactically is already implemented in
+ClangTidy using AST matchers (`bugprone-redundant-branch-condition`).
+
+To implement it using the data flow analysis framework, we can produce a warning
+if any part of the branch condition is implied by the flow condition.
+
+```c++
+int F(bool b1, bool b2) {
+  // Flow condition: true.
+  if (b1) {
+    // Flow condition: b1.
+    f();
+    if (b1 && b2) { // `b1` is implied by the flow condition.
+      g();
+    }
+  }
+}
+```
+
+One way to check this implication is to use a SAT solver. Without a SAT solver,
+we could keep the flow condition in the CNF form and then it would be easy to
+check the implication.
+
+## Example: finding unchecked `std::optional` unwraps
+
+Calling `optional::value()` is only valid if `optional::has_value()` is true. We
+want to show that when `x.value()` is executed, the flow condition implies
+`x.has_value()`.
+
+In the example below `x.value()` is accessed safely because it is guarded by the
+`x.has_value()` check.
+
+```c++
+void Example(std::optional<int> &x) {
+  if (x.has_value()) {
+    use(x.value());
+  }
+}
+```
+
+While entering the if branch we deduce that `x.has_value()` is implied by the
+flow condition.
+
+```c++
+void Example(std::optional<int> x) {
+  // Flow condition: true.
+  if (x.has_value()) {
+    // Flow condition: x.has_value() == true.
+    use(x.value());
+  }
+  // Flow condition: true.
+}
+```
+
+We also need to prove that `x` is not modified between check and value access.
+The modification of `x` may be very subtle:
+
+```c++
+void F(std::optional<int> &x);
+
+void Example(std::optional<int> &x) {
+  if (x.has_value()) {
+    // Flow condition: x.has_value() == true.
+    unknown_function(x); // may change x.
+    // Flow condition: true.
+    use(x.value());
+  }
+}
+```
+
+## Example: finding dead code behind A/B experiment flags
+
+Finding dead code is a classic application of data flow analysis.
+
+Unused flags for A/B experiment hide dead code. However, this flavor of dead
+code is invisible to the compiler because the flag can be turned on at any
+moment.
+
+We could make a tool that deletes experiment flags. The user tells us which flag
+they want to delete, and we assume that the it's value is a given constant.
+
+For example, the user could use the tool to remove `example_flag` from this
+code:
+
+```c++
+DEFINE_FLAG(std::string, example_flag, "", "A sample flag.");
+
+void Example() {
+  bool x = GetFlag(FLAGS_example_flag).empty();
+  f();
+  if (x) {
+    g();
+  } else {
+    h();
+  }
+}
+```
+
+The tool would simplify the code to:
+
+```c++
+void Example() {
+  f();
+  g();
+}
+```
+
+We can solve this problem with a classic constant propagation lattice combined
+with symbolic evaluation.
+
+## Example: finding inefficient usages of associative containers
+
+Real-world code often accidentally performs repeated lookups in associative
+containers:
+
+```c++
+map<int, Employee> xs;
+xs[42]->name = "...";
+xs[42]->title = "...";
+```
+
+To find the above inefficiency we can use the available expressions analysis to
+understand that `m[42]` is evaluated twice.
+
+```c++
+map<int, Employee> xs;
+Employee &e = xs[42];
+e->name = "...";
+e->title = "...";
+```
+
+We can also track the `m.contains()` check in the flow condition to find
+redundant checks, like in the example below.
+
+```c++
+std::map<int, Employee> xs;
+if (!xs.contains(42)) {
+  xs.insert({42, someEmployee});
+}
+```
+
+## Example: refactoring types that implicitly convert to each other
+
+Refactoring one strong type to another is difficult, but the compiler can help:
+once you refactor one reference to the type, the compiler will flag other places
+where this information flows with type mismatch errors. Unfortunately this
+strategy does not work when you are refactoring types that implicitly convert to
+each other, for example, replacing `int32_t` with `int64_t`.
+
+Imagine that we want to change user IDs from 32 to 64-bit integers. In other
+words, we need to find all integers tainted with user IDs. We can use data flow
+analysis to implement taint analysis.
+
+```c++
+void UseUser(int32_t user_id) {
+  int32_t id = user_id;
+  // Variable `id` is tainted with a user ID.
+  ...
+}
+```
+
+Taint analysis is very well suited to this problem because the program rarely
+branches on user IDs, and almost certainly does not perform any computation
+(like arithmetic).

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+       refY="0"
+       refX="0"
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+       style="overflow:visible"
+       inkscape:isstock="true">
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+       y="2.6458333"
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+       height="10.583333"
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+       x="76.561562"
+       style="font-size:4.93889px;line-height:1;font-family:sans-serif;text-align:center;word-spacing:0px;text-anchor:middle;stroke-width:0.264583"
+       xml:space="preserve"><tspan
+         style="font-size:4.93889px;text-align:center;text-anchor:middle;stroke-width:0.264583"
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+       style="fill:none;stroke:#000000;stroke-width:0.264999;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
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+       width="52.916668"
+       height="26.458334"
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+    <text
+       xml:space="preserve"
+       style="font-size:4.9389px;line-height:1.1;font-family:sans-serif;text-align:start;word-spacing:0px;text-anchor:start;stroke-width:0.264583"
+       x="54.303524"
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+       id="text839-6"><tspan
+         sodipodi:role="line"
+         id="tspan837-3"
+         x="54.303524"
+         y="30.288889"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;text-anchor:start;stroke-width:0.264583">// Pre: x is ⊥</tspan><tspan
+         id="tspan876"
+         sodipodi:role="line"
+         x="54.303524"
+         y="35.772423"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;text-anchor:start;stroke-width:0.264583">int x;</tspan><tspan
+         id="tspan878"
+         sodipodi:role="line"
+         x="54.303524"
+         y="41.255955"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;text-anchor:start;stroke-width:0.264583">if (n &gt; 0)</tspan><tspan
+         id="tspan874"
+         sodipodi:role="line"
+         x="54.303524"
+         y="47.197174"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;text-anchor:start;stroke-width:0.264583">// Post: x is ⊥</tspan></text>
+    <rect
+       y="92.604164"
+       x="2.6458302"
+       height="21.166666"
+       width="52.916668"
+       id="rect12-8-6"
+       style="fill:none;stroke:#000000;stroke-width:0.264999;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1" />
+    <text
+       id="text839-6-0"
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+       style="font-size:4.9389px;line-height:1.1;font-family:sans-serif;text-align:start;word-spacing:0px;text-anchor:start;stroke-width:0.264583"
+       xml:space="preserve"><tspan
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583"
+         y="99.405327"
+         x="6.154747"
+         id="tspan837-3-1"
+         sodipodi:role="line">// Pre: x is ⊥</tspan><tspan
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583"
+         y="104.88886"
+         x="6.154747"
+         sodipodi:role="line"
+         id="tspan878-2">x = n;</tspan><tspan
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+         y="110.37239"
+         x="6.154747"
+         sodipodi:role="line"
+         id="tspan874-3">// Post: x is ⊤</tspan></text>
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+       height="21.166666"
+       width="52.916668"
+       id="rect12-8-3"
+       style="fill:none;stroke:#000000;stroke-width:0.264999;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1" />
+    <text
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+       style="font-size:4.9389px;line-height:1.1;font-family:sans-serif;text-align:start;word-spacing:0px;text-anchor:start;stroke-width:0.264583"
+       xml:space="preserve"><tspan
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583"
+         y="67.426483"
+         x="125.74102"
+         id="tspan837-3-7"
+         sodipodi:role="line">// Pre: x is ⊥</tspan><tspan
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583"
+         y="72.910019"
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+         sodipodi:role="line"
+         id="tspan878-1">if (n == 42)</tspan><tspan
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583"
+         y="78.851234"
+         x="125.74102"
+         sodipodi:role="line"
+         id="tspan874-8">// Post: x is ⊥</tspan></text>
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+       style="fill:none;stroke:#000000;stroke-width:0.264999;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
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+       height="21.16667"
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+       x="88.860504"
+       y="99.025513"
+       id="text839-6-3-7"><tspan
+         sodipodi:role="line"
+         id="tspan837-3-7-6"
+         x="88.860504"
+         y="99.025513"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583">// Pre: x is ⊥</tspan><tspan
+         id="tspan878-1-1"
+         sodipodi:role="line"
+         x="88.860504"
+         y="104.50905"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583">x = 5;</tspan><tspan
+         id="tspan874-8-5"
+         sodipodi:role="line"
+         x="88.860504"
+         y="109.99258"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583">// Post: x is {5}</tspan></text>
+    <rect
+       y="92.604164"
+       x="153.45834"
+       height="21.166677"
+       width="58.208328"
+       id="rect12-8-3-3-3"
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+    <text
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+       style="font-size:4.9389px;line-height:1.1;font-family:sans-serif;text-align:start;word-spacing:0px;text-anchor:start;stroke-width:0.264583"
+       xml:space="preserve"><tspan
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583"
+         y="99.025513"
+         x="156.17146"
+         id="tspan837-3-7-6-0"
+         sodipodi:role="line">// Pre: x is ⊥</tspan><tspan
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583"
+         y="104.50905"
+         x="156.17146"
+         sodipodi:role="line"
+         id="tspan878-1-1-7">x = 44;</tspan><tspan
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583"
+         y="109.99258"
+         x="156.17146"
+         sodipodi:role="line"
+         id="tspan874-8-5-9">// Post: x is {44}</tspan></text>
+    <rect
+       style="fill:none;stroke:#000000;stroke-width:0.264999;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
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+       width="68.791672"
+       height="21.166672"
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+       xml:space="preserve"
+       style="font-size:4.9389px;line-height:1.1;font-family:sans-serif;text-align:start;word-spacing:0px;text-anchor:start;stroke-width:0.264583"
+       x="117.2491"
+       y="130.77551"
+       id="text839-6-3-8"><tspan
+         sodipodi:role="line"
+         id="tspan837-3-7-7"
+         x="117.2491"
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+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583">// Pre: x is {5; 44}</tspan><tspan
+         id="tspan878-1-2"
+         sodipodi:role="line"
+         x="117.2491"
+         y="136.25905"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583">print(x);</tspan><tspan
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+         sodipodi:role="line"
+         x="117.2491"
+         y="141.74258"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583">// Post: x is {5; 44}</tspan></text>
+    <rect
+       style="fill:none;stroke:#000000;stroke-width:0.264999;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
+       id="rect12-8-6-3"
+       width="52.916668"
+       height="21.166666"
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+    <text
+       xml:space="preserve"
+       style="font-size:4.9389px;line-height:1.1;font-family:sans-serif;text-align:start;word-spacing:0px;text-anchor:start;stroke-width:0.264583"
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+         sodipodi:role="line"
+         id="tspan837-3-1-7"
+         x="53.695339"
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+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583">// Pre: x is ⊤</tspan><tspan
+         id="tspan878-2-0"
+         sodipodi:role="line"
+         x="53.695339"
+         y="168.49834"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583">print(x);</tspan><tspan
+         id="tspan874-3-8"
+         sodipodi:role="line"
+         x="53.695339"
+         y="173.98187"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:monospace;-inkscape-font-specification:monospace;text-align:start;text-anchor:start;stroke-width:0.264583">// Post: x is ⊤</tspan></text>
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+    <text
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+       style="font-size:4.9389px;line-height:1;font-family:sans-serif;text-align:center;word-spacing:0px;text-anchor:middle;stroke-width:0.264583"
+       x="119.0625"
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+       id="text839-1"><tspan
+         sodipodi:role="line"
+         id="tspan837-4"
+         x="119.0625"
+         y="55.562504"
+         style="font-size:4.9389px;text-align:center;text-anchor:middle;stroke-width:0.264583">True</tspan></text>
+    <text
+       id="text839-1-4"
+       y="87.3125"
+       x="179.91667"
+       style="font-size:4.9389px;line-height:1;font-family:sans-serif;text-align:center;word-spacing:0px;text-anchor:middle;stroke-width:0.264583"
+       xml:space="preserve"><tspan
+         style="font-size:4.9389px;text-align:center;text-anchor:middle;stroke-width:0.264583"
+         y="87.3125"
+         x="179.91667"
+         id="tspan837-4-2"
+         sodipodi:role="line">True</tspan></text>
+    <text
+       id="text839-1-5"
+       y="87.3125"
+       x="113.77083"
+       style="font-size:4.9389px;line-height:1;font-family:sans-serif;text-align:center;word-spacing:0px;text-anchor:middle;stroke-width:0.264583"
+       xml:space="preserve"><tspan
+         style="font-size:4.9389px;text-align:center;text-anchor:middle;stroke-width:0.264583"
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+</svg>

clang/docs/DataFlowAnalysisIntroImages/CFGJoinRule.svg

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+    <marker
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+    <text
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+       style="font-size:4.9389px;line-height:1.25;font-family:sans-serif;word-spacing:0px;stroke-width:0.264583"
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+         id="tspan30-8"
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+    <text
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+           id="tspan70"
+           style="font-size:65%;text-align:center;baseline-shift:sub;text-anchor:middle">1</tspan></tspan></text>
+    <text
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+       style="font-size:4.9389px;line-height:1.25;font-family:sans-serif;word-spacing:0px;stroke-width:0.264583"
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+         style="font-size:4.9389px;text-align:center;text-anchor:middle;stroke-width:0.264583"
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+   style="font-style:italic;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Italic';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal">in</tspan><tspan
+   id="tspan152"
+   style="font-size:65%;baseline-shift:sub">2</tspan>, …, <tspan
+   id="tspan160"
+   style="font-style:italic;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Italic';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal">in</tspan><tspan
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+       style="font-size:4.9389px;line-height:1.25;font-family:sans-serif;text-align:start;word-spacing:0px;text-anchor:start;stroke-width:0.264583"
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+   style="font-style:italic;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Italic';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;text-anchor:start;stroke-width:0.264583"
+   id="tspan148-3">out<tspan
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+   id="tspan201"
+   style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;text-anchor:start">(<tspan
+   id="tspan205"
+   style="font-style:italic;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Italic';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;text-anchor:start">basic_block</tspan>, </tspan>join</tspan>(<tspan
+   style="font-style:italic;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.9389px;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Italic';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;text-anchor:start;stroke-width:0.264583"
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+   style="font-size:3.2103px;text-align:start;baseline-shift:sub;text-anchor:start;stroke-width:0.264583"
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+       d="M 68.791666,6.6145827 50.270832,23.8125"
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clang/docs/DataFlowAnalysisIntroImages/DefinitiveInitializationLattice.svg

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clang/docs/DataFlowAnalysisIntroImages/IntegerSetsFiniteLattice.svg

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+       id="path900"
+       d="M 92.604167,26.458333 60.854166,15.875"
+       style="fill:none;stroke:#000000;stroke-width:0.265;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#marker1174)" />
+  </g>
+</svg>

clang/docs/conf.py

 # Add any Sphinx extension module names here, as strings. They can be extensions
 # coming with Sphinx (named 'sphinx.ext.*') or your custom ones.
 extensions = ['sphinx.ext.todo', 'sphinx.ext.mathjax']
 
 # Add any paths that contain templates here, relative to this directory.
 templates_path = ['_templates']
 
 # The suffix of source filenames.
-source_suffix = '.rst'
+source_suffix = {
+    '.rst': 'restructuredtext',
+}
+
+try:
+  import recommonmark
+except ImportError:
+  # manpages do not use any .md sources
+  if not tags.has('builder-man'):
+    raise
+else:
+  import sphinx
+  if sphinx.version_info >= (3, 0):
+    # This requires 0.5 or later.
+    extensions.append('recommonmark')
+  else:
+    source_parsers = {'.md': 'recommonmark.parser.CommonMarkParser'}
+  source_suffix['.md'] = 'markdown'
 
 # The encoding of source files.
 #source_encoding = 'utf-8-sig'
 
 # The master toctree document.
 master_doc = 'index'
 
 # General information about the project.

clang/docs/index.rst

.. toctree::
    Toolchain
    LanguageExtensions
    ClangCommandLineReference
    AttributeReference
    DiagnosticsReference
    CrossCompilation
    ClangStaticAnalyzer
    ThreadSafetyAnalysis
+   DataFlowAnalysisIntro
    AddressSanitizer
    ThreadSanitizer
    MemorySanitizer
    UndefinedBehaviorSanitizer
    DataFlowSanitizer
    LeakSanitizer
    SanitizerCoverage
    SanitizerStats