==========================
Convergence And Uniformity
==========================

.. contents::
   :local:

Introduction
============

Some parallel environments execute threads in groups that allow
communication within the group using special primitives called
*convergent* operations. The outcome of a convergent operation is
sensitive to the set of threads that executes it "together", i.e.,
convergently.

A value is said to be *uniform* across a set of threads if it is the
same across those threads, and *divergent* otherwise. Correspondingly,
a branch is said to be a uniform branch if its condition is uniform,
and it is a divergent branch otherwise.

Whether threads are *converged* or not depends on the paths they take
through the control flow graph. Threads take different outgoing edges
at a *divergent branch*. Divergent branches constrain
program transforms such as changing the CFG or moving a convergent
operation to a different point of the CFG. Performing these
transformations across a divergent branch can change the sets of
threads that execute convergent operations convergently. While these
constraints are out of scope for this document, the described
*uniformity analysis* allows these transformations to identify
uniform branches where these constraints do not hold.

Convergence and
uniformity are inter-dependent: When threads diverge at a divergent
branch, they may later *reconverge* at a common program point.
Subsequent operations are performed convergently, but the inputs may
be non-uniform, thus producing divergent outputs.

Uniformity is also useful by itself on targets that execute threads in
groups with shared execution resources (e.g. waves, warps, or
subgroups):

- Uniform outputs can potentially be computed or stored on shared
  resources.
- These targets must "linearize" a divergent branch to ensure that
  each side of the branch is followed by the corresponding threads in
  the same group. But linearization is unnecessary at uniform
  branches, since the whole group of threads follows either one side
  of the branch or the other.

This document presents a definition of convergence that is reasonable
for real targets and is compatible with the currently implicit
semantics of convergent operations in LLVM IR. This is accompanied by
a *uniformity analysis* that extends previous work on divergence analysis
[DivergenceSPMD]_ to cover irreducible control-flow.

.. [DivergenceSPMD] Julian Rosemann, Simon Moll, and Sebastian
   Hack. 2021. An Abstract Interpretation for SPMD Divergence on
   Reducible Control Flow Graphs. Proc. ACM Program. Lang. 5, POPL,
   Article 31 (January 2021), 35 pages.
   https://doi.org/10.1145/3434312

Terminology
===========

Cycles
   Described in :ref:`cycle-terminology`.

Closed path
   Described in :ref:`cycle-closed-path`.

Disjoint paths
   Two paths in a CFG are said to be disjoint if the only nodes common
   to both are the start node or the end node, or both.

Join node
   A join node of a branch is a node reachable along disjoint paths
   starting from that branch.

Diverged path
   A diverged path is a path that starts from a divergent branch and
   either reaches a join node of the branch or reaches the end of the
   function without passing through any join node of the branch.

Threads and Dynamic Instances
=============================

Each occurrence of an instruction in the program source is called a
*static instance*. When a thread executes a program, each execution of
a static instance produces a distinct *dynamic instance* of that
instruction.

Each thread produces a unique sequence of dynamic instances:

- The sequence is generated along branch decisions and loop
  traversals.
- Starts with a dynamic instance of a "first" instruction.
- Continues with dynamic instances of successive "next"
  instructions.

Threads are independent; some targets may choose to execute them in
groups in order to share resources when possible.

.. figure:: convergence-natural-loop.png
   :name: convergence-natural-loop

.. table::
   :name: convergence-thread-example
   :align: left

   +----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
   |          |        | 1   | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   |      |
   +----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
   | Thread 1 | Entry1 | H1  | B1  | L1  | H3  |     | L3  |     |     |     | Exit |
   +----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
   | Thread 2 | Entry1 | H2  |     | L2  | H4  | B2  | L4  | H5  | B3  | L5  | Exit |
   +----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+

In the above table, each row is a different thread, listing the
dynamic instances produced by that thread from left to right. Each
thread executes the same program that starts with an ``Entry`` node
and ends with an ``Exit`` node, but different threads may take
different paths through the control flow of the program. The columns
are numbered merely for convenience, and empty cells have no special
meaning. Dynamic instances listed in the same column are converged.

.. _convergence-definition:

Convergence
===========

*Converged-with* is a transitive symmetric relation over dynamic
instances produced by *different threads* for the *same static
instance*. Informally, two threads that produce converged dynamic
instances are said to be *converged*, and they are said to execute
that static instance *convergently*, at that point in the execution.

*Convergence order* is a strict partial order over dynamic instances
that is defined as the transitive closure of:

1. If dynamic instance ``P`` is executed strictly before ``Q`` in the
   same thread, then ``P`` is *convergence-before* ``Q``.
2. If dynamic instance ``P`` is executed strictly before ``Q1`` in the
   same thread, and ``Q1`` is *converged-with* ``Q2``, then ``P`` is
   *convergence-before* ``Q2``.
3. If dynamic instance ``P1`` is *converged-with* ``P2``, and ``P2``
   is executed strictly before ``Q`` in the same thread, then ``P1``
   is *convergence-before* ``Q``.

.. table::
   :name: convergence-order-example
   :align: left

   +----------+-------+-----+-----+-----+-----+-----+-----+-----+------+
   |          | 1     | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9    |
   +----------+-------+-----+-----+-----+-----+-----+-----+-----+------+
   | Thread 1 | Entry | ... |     |     |     | S2  | T   | ... | Exit |
   +----------+-------+-----+-----+-----+-----+-----+-----+-----+------+
   | Thread 2 | Entry | ... |     | Q2  | R   | S1  |     | ... | Exit |
   +----------+-------+-----+-----+-----+-----+-----+-----+-----+------+
   | Thread 3 | Entry | ... | P   | Q1  |     |     |     | ... |      |
   +----------+-------+-----+-----+-----+-----+-----+-----+-----+------+

The above table shows partial sequences of dynamic instances from
different threads. Dynamic instances in the same column are assumed
to be converged (i.e., related to each other in the converged-with
relation). The resulting convergence order includes the edges ``P ->
Q2``, ``Q1 -> R``, ``P -> R``, ``P -> T``, etc.

The fact that *convergence-before* is a strict partial order is a
constraint on the *converged-with* relation. It is trivially satisfied
if different dynamic instances are never converged. It is also
trivially satisfied for all known implementations for which
convergence plays some role. Aside from the strict partial convergence
order, there are currently no additional constraints on the
*converged-with* relation imposed in LLVM IR.

.. _convergence-note-convergence:

.. note::

   1. The ``convergent`` attribute on convergent operations does
      constrain changes to ``converged-with``, but it is expressed in
      terms of control flow and does not explicitly deal with thread
      convergence.

   2. The convergence-before relation is not
      directly observable. Program transforms are in general free to
      change the order of instructions, even though that obviously
      changes the convergence-before relation.

   3. Converged dynamic instances need not be executed at the same
      time or even on the same resource. Converged dynamic instances
      of a convergent operation may appear to do so but that is an
      implementation detail. The fact that ``P`` is convergence-before
      ``Q`` does not automatically imply that ``P`` happens-before
      ``Q`` in a memory model sense.

   4. **Future work:** Providing convergence-related guarantees to
      compiler frontends enables some powerful optimization techniques
      that can be used by programmers or by high-level program
      transforms. Constraints on the ``converged-with`` relation may
      be added eventually as part of the definition of LLVM
      IR, so that guarantees can be made that frontends can rely on.
      For a proposal on how this might work, see `D85603
      <https://reviews.llvm.org/D85603>`_.

.. _convergence-maximal:

Maximal Convergence
-------------------

This section defines a constraint that may be used to
produce a *maximal converged-with* relation without violating the
strict *convergence-before* order. This maximal converged-with
relation is reasonable for real targets and is compatible with
convergent operations.

The maximal converged-with relation is defined in terms of cycle
headers, which are not unique to a given CFG. Each cycle hierarchy for
the same CFG results in a different maximal converged-with relation.

   **Maximal converged-with:**

   Dynamic instances ``X1`` and ``X2`` produced by different threads
   for the same static instance ``X`` are converged in the maximal
   converged-with relation if and only if for every cycle ``C`` with
   header ``H`` that contains ``X``:

   - every dynamic instance ``H1`` of ``H`` that precedes ``X1`` in
     the respective thread is convergence-before ``X2``, and,
   - every dynamic instance ``H2`` of ``H`` that precedes ``X2`` in
     the respective thread is convergence-before ``X1``,
   - without assuming that ``X1`` is converged with ``X2``.

.. note::

   For brevity, the rest of the document restricts the term
   *converged* to mean "related under the maximal converged-with
   relation for the given cycle hierarchy".

Maximal convergence can now be demonstrated in the earlier example as follows:

.. table::
   :align: left

   +----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
   |          |        | 1   | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   |      |
   +----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
   | Thread 1 | Entry1 | H1  | B1  | L1  | H3  |     | L3  |     |     |     | Exit |
   +----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
   | Thread 2 | Entry2 | H2  |     | L2  | H4  | B2  | L4  | H5  | B3  | L5  | Exit |
   +----------+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+

- ``Entry1`` and ``Entry2`` are converged.
- ``H1`` and ``H2`` are converged.
- ``B1`` and ``B2`` are not converged due to ``H4`` which is not
  convergence-before ``B1``.
- ``H3`` and ``H4`` are converged.
- ``H3`` is not converged with ``H5`` due to ``H4`` which is not
  convergence-before ``H3``.
- ``L1`` and ``L2`` are converged.
- ``L3`` and ``L4`` are converged.
- ``L3`` is not converged with ``L5`` due to ``H5`` which is not
  convergence-before ``L3``.

.. _convergence-cycle-headers:

Dependence on Cycles Headers
----------------------------

Contradictions in convergence order are possible only between two
nodes that are inside some cycle. The dynamic instances of such nodes
may be interleaved in the same thread, and this interleaving may be
different for different threads.

When a thread executes a node ``X`` once and then executes it again,
it must have followed a closed path in the CFG that includes ``X``.
Such a path must pass through the header of at least one cycle --- the
smallest cycle that includes the entire closed path. In a given
thread, two dynamic instances of ``X`` are either separated by the
execution of at least one cycle header, or ``X`` itself is a cycle
header.

In reducible cycles (natural loops), each execution of the header is
equivalent to the start of a new iteration of the cycle. But this
analogy breaks down in the presence of explicit constraints on the
converged-with relation, such as those described in :ref:`future
work<convergence-note-convergence>`. Instead, cycle headers should be
treated as implicit *points of convergence* in a maximal
converged-with relation.

Consider a sequence of nested cycles ``C1``, ``C2``, ..., ``Ck`` such
that ``C1`` is the outermost cycle and ``Ck`` is the innermost cycle,
with headers ``H1``, ``H2``, ..., ``Hk`` respectively. When a thread
enters the cycle ``Ck``, any of the following is possible:

1. The thread directly entered cycle ``Ck`` without having executed
   any of the headers ``H1`` to ``Hk``.

2. The thread executed some or all of the nested headers one or more
   times.

The maximal converged-with relation captures the following intuition
about cycles:

1. When two threads enter a top-level cycle ``C1``, they execute
   converged dynamic instances of every node that is a :ref:`child
   <cycle-parent-block>` of ``C1``.

2. When two threads enter a nested cycle ``Ck``, they execute
   converged dynamic instances of every node that is a child of
   ``Ck``, until either thread exits ``Ck``, if and only if they
   executed converged dynamic instances of the last nested header that
   either thread encountered.

   Note that when a thread exits a nested cycle ``Ck``, it must follow
   a closed path outside ``Ck`` to reenter it. This requires executing
   the header of some outer cycle, as described earlier.

Consider two dynamic instances ``X1`` and ``X2`` produced by threads ``T1``
and ``T2`` for a node ``X`` that is a child of nested cycle ``Ck``.
Maximal convergence relates ``X1`` and ``X2`` as follows:

1. If neither thread executed any header from ``H1`` to ``Hk``, then
   ``X1`` and ``X2`` are converged.

2. Otherwise, if there are no converged dynamic instances ``Q1`` and
   ``Q2`` of any header ``Q`` from ``H1`` to ``Hk`` (where ``Q`` is
   possibly the same as ``X``), such that ``Q1`` precedes ``X1`` and
   ``Q2`` precedes ``X2`` in the respective threads, then ``X1`` and
   ``X2`` are not converged.

3. Otherwise, consider the pair ``Q1`` and ``Q2`` of converged dynamic
   instances of a header ``Q`` from ``H1`` to ``Hk`` that occur most
   recently before ``X1`` and ``X2`` in the respective threads. Then
   ``X1`` and ``X2`` are converged if and only if there is no dynamic
   instance of any header from ``H1`` to ``Hk`` that occurs between
   ``Q1`` and ``X1`` in thread ``T1``, or between ``Q2`` and ``X2`` in
   thread ``T2``. In other words, ``Q1`` and ``Q2`` represent the last
   point of convergence, with no other header being executed before
   executing ``X``.

**Example:**

.. figure:: convergence-both-diverged-nested.png
   :name: convergence-both-diverged-nested

The above figure shows two nested irreducible cycles with headers
``R`` and ``S``. The nodes ``Entry`` and ``Q`` have divergent
branches. The table below shows the convergence between three threads
taking different paths through the CFG. Dynamic instances listed in
the same column are converged.

   .. table::
      :align: left

      +---------+-------+-----+-----+-----+-----+-----+-----+-----+------+
      |         | 1     | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 10   |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+------+
      | Thread1 | Entry | P1  | Q1  | S1  | P3  | Q3  | R1  | S2  | Exit |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+------+
      | Thread2 | Entry | P2  | Q2  |     |     |     | R2  | S3  | Exit |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+------+
      | Thread3 | Entry |     |     |     |     |     | R3  | S4  | Exit |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+------+

- ``P2`` and ``P3`` are not converged due to ``S1``
- ``Q2`` and ``Q3`` are not converged due to ``S1``
- ``S1`` and ``S3`` are not converged due to ``R2``
- ``S1`` and ``S4`` are not converged due to ``R3``

Informally, ``T1`` and ``T2`` execute the inner cycle a different
number of times, without executing the header of the outer cycle. All
threads converge in the outer cycle when they first execute the header
of the outer cycle.

.. _convergence-uniformity:

Uniformity
==========

1. The output of two converged dynamic instances is uniform if and
   only if it compares equal for those two dynamic instances.
2. The output of a static instance ``X`` is uniform *for a given set
   of threads* if and only if it is uniform for every pair of
   converged dynamic instances of ``X`` produced by those threads.

A non-uniform value is said to be *divergent*.

For a set ``S`` of threads, the uniformity of each output of a static
instance is determined as follows:

1. The semantics of the instruction may specify the output to be
   uniform.
2. Otherwise, the output is divergent if the static instance is not
   :ref:`m-converged <convergence-m-converged>`.
3. Otherwise, if the static instance is m-converged:

   1. If it is a PHI node, its output is uniform if and only
      if for every pair of converged dynamic instances produced by all
      threads in ``S``:

      a. Both instances choose the same output from converged
         dynamic instances, and,
      b. That output is uniform for all threads in ``S``.
   2. Otherwise, the output is uniform if and only if the input
      operands are uniform for all threads in ``S``.

Divergent Cycle Exits
---------------------

When a divergent branch occurs inside a cycle, it is possible that a
diverged path continues to an exit of the cycle. This is called a
divergent cycle exit. If the cycle is irreducible, the diverged path
may re-enter and eventually reach a join within the cycle. Such a join
should be examined for the :ref:`diverged entry
<convergence-diverged-entry>` criterion.

Nodes along the diverged path that lie outside the cycle experience
*temporal divergence*, when two threads executing convergently inside
the cycle produce uniform values, but exit the cycle along the same
divergent path after executing the header a different number of times
(informally, on different iterations of the cycle). For a node ``N``
inside the cycle the outputs may be uniform for the two threads, but
any use ``U`` outside the cycle receives a value from non-converged
dynamic instances of ``N``. An output of ``U`` may be divergent,
depending on the semantics of the instruction.

Static Uniformity Analysis
==========================

Irreducible control flow results in different cycle hierarchies
depending on the choice of headers during depth-first traversal. As a
result, a static analysis cannot always determine the convergence of
nodes in irreducible cycles, and any uniformity analysis is limited to
those static instances whose convergence is independent of the cycle
hierarchy:

.. _convergence-m-converged:

  **m-converged static instances:**

  A static instance ``X`` is *m-converged* for a given CFG if and only
  if the maximal converged-with relation for its dynamic instances is
  the same in every cycle hierarchy that can be constructed for that CFG.

  .. note::

   In other words, two dynamic instances ``X1`` and ``X2`` of an
   m-converged static instance ``X`` are converged in some cycle
   hierarchy if and only if they are also converged in every other
   cycle hierarchy for the same CFG.

   As noted earlier, for brevity, we restrict the term *converged* to
   mean "related under the maximal converged-with relation for a given
   cycle hierarchy".


Each node ``X`` in a given CFG is reported to be m-converged if and
only if:

1. ``X`` is a :ref:`top-level<cycle-toplevel-block>` node, in which
   case, there are no cycle headers to influence the convergence of
   ``X``.

2. Otherwise, if ``X`` is inside a cycle, then every cycle that
   contains ``X`` satisfies the following necessary conditions:

   a. Every divergent branch inside the cycle satisfies the
      :ref:`diverged entry criterion<convergence-diverged-entry>`, and,
   b. There are no :ref:`diverged paths reaching the
      cycle<convergence-diverged-outside>` from a divergent branch
      outside it.

.. note::

   A reducible cycle :ref:`trivially satisfies
   <convergence-reducible-cycle>` the above conditions. In particular,
   if the whole CFG is reducible, then all nodes in the CFG are
   m-converged.

The uniformity of each output of a static instance
is determined using the criteria
:ref:`described earlier <convergence-uniformity>`. The discovery of
divergent outputs may cause their uses (including branches) to also
become divergent. The analysis propagates this divergence until a
fixed point is reached.

The convergence inferred using these criteria is a safe subset of the
maximal converged-with relation for any cycle hierarchy. In
particular, it is sufficient to determine if a static instance is
m-converged for a given cycle hierarchy ``T``, even if that fact is
not detected when examining some other cycle hierarchy ``T'``.

This property allows compiler transforms to use the uniformity
analysis without being affected by DFS choices made in the underlying
cycle analysis. When two transforms use different instances of the
uniformity analysis for the same CFG, a "divergent value" result in
one analysis instance cannot contradict a "uniform value" result in
the other.

Generic transforms such as SimplifyCFG, CSE, and loop transforms
commonly change the program in ways that change the maximal
converged-with relations. This also means that a value that was
previously uniform can become divergent after such a transform.
Uniformity has to be recomputed after such transforms.

Divergent Branch inside a Cycle
-------------------------------

.. figure:: convergence-divergent-inside.png
   :name: convergence-divergent-inside

The above figure shows a divergent branch ``Q`` inside an irreducible
cyclic region. When two threads diverge at ``Q``, the convergence of
dynamic instances within the cyclic region depends on the cycle
hierarchy chosen:

1. In an implementation that detects a single cycle ``C`` with header
   ``P``, convergence inside the cycle is determined by ``P``.

2. In an implementation that detects two nested cycles with headers
   ``R`` and ``S``, convergence inside those cycles is determined by
   their respective headers.

.. _convergence-diverged-entry:

A conservative approach would be to simply report all nodes inside
irreducible cycles as having divergent outputs. But it is desirable to
recognize m-converged nodes in the CFG in order to maximize
uniformity. This section describes one such pattern of nodes derived
from *closed paths*, which are a property of the CFG and do not depend
on the cycle hierarchy.

  **Diverged Entry Criterion:**

  The dynamic instances of all the nodes in a closed path ``P`` are
  m-converged only if for every divergent branch ``B`` and its
  join node ``J`` that lie on ``P``, there is no entry to ``P`` which
  lies on a diverged path from ``B`` to ``J``.

.. figure:: convergence-closed-path.png
   :name: convergence-closed-path

Consider the closed path ``P -> Q -> R -> S`` in the above figure.
``P`` and ``R`` are :ref:`entries to the closed
path<cycle-closed-path>`. ``Q`` is a divergent branch and ``S`` is a
join for that branch, with diverged paths ``Q -> R -> S`` and ``Q ->
S``.

- If a diverged entry ``R`` exists, then in some cycle hierarchy,
  ``R`` is the header of the smallest cycle ``C`` containing the
  closed path and a :ref:`child cycle<cycle-definition>` ``C'``
  exists in the set ``C - R``, containing both branch ``Q`` and join
  ``S``. When threads diverge at ``Q``, one subset ``M`` continues
  inside cycle ``C'``, while the complement ``N`` exits ``C'`` and
  reaches ``R``. Dynamic instances of ``S`` executed by threads in set
  ``M`` are not converged with those executed in set ``N`` due to the
  presence of ``R``. Informally, threads that diverge at ``Q``
  reconverge in the same iteration of the outer cycle ``C``, but they
  may have executed the inner cycle ``C'`` differently.

  .. table::
     :align: left

     +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
     |         | 1     | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10  | 11   |
     +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
     | Thread1 | Entry | P1  | Q1  |     |     |     | R1  | S1  | P3  | ... | Exit |
     +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
     | Thread2 | Entry | P2  | Q2  | S2  | P4  | Q4  | R2  | S4  |     |     | Exit |
     +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+

  In the table above, ``S2`` is not converged with ``S1`` due to ``R1``.

|

- If ``R`` does not exist, or if any node other than ``R`` is the
  header of ``C``, then no such child cycle ``C'`` is detected.
  Threads that diverge at ``Q`` execute converged dynamic instances of
  ``S`` since they do not encounter the cycle header on any path from
  ``Q`` to ``S``. Informally, threads that diverge at ``Q``
  reconverge at ``S`` in the same iteration of ``C``.

  .. table::
     :align: left

     +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+------+
     |         | 1     | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10   |
     +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+------+
     | Thread1 | Entry | P1  | Q1  | R1  | S1  | P3  | Q3  | R3  | S3  | Exit |
     +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+------+
     | Thread2 | Entry | P2  | Q2  |     | S2  | P4  | Q4  | R2  | S4  | Exit |
     +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+------+

|

  .. note::

     In general, the cycle ``C`` in the above statements is not
     expected to be the same cycle for different headers. Cycles and
     their headers are tightly coupled; for different headers in the
     same outermost cycle, the child cycles detected may be different.
     The property relevant to the above examples is that for every
     closed path, there is a cycle ``C`` that contains the path and
     whose header is on that path.

The diverged entry criterion must be checked for every closed path
passing through a divergent branch ``B`` and its join ``J``. Since
:ref:`every closed path passes through the header of some
cycle<cycle-closed-path-header>`, this amounts to checking every cycle
``C`` that contains ``B`` and ``J``. When the header of ``C``
dominates the join ``J``, there can be no entry to any path from the
header to ``J``, which includes any diverged path from ``B`` to ``J``.
This is also true for any closed paths passing through the header of
an outer cycle that contains ``C``.

Thus, the diverged entry criterion can be conservatively simplified
as follows:

  For a divergent branch ``B`` and its join node ``J``, the nodes in a
  cycle ``C`` that contains both ``B`` and ``J`` are m-converged only
  if:

  - ``B`` strictly dominates ``J``, or,
  - The header ``H`` of ``C`` strictly dominates ``J``, or,
  - Recursively, there is cycle ``C'`` inside ``C`` that satisfies the
    same condition.

When ``J`` is the same as ``H`` or ``B``, the trivial dominance is
insufficient to make any statement about entries to diverged paths.

.. _convergence-diverged-outside:

Diverged Paths reaching a Cycle
-------------------------------

.. figure:: convergence-divergent-outside.png
   :name: convergence-divergent-outside

The figure shows two cycle hierarchies with a divergent branch in
``Entry`` instead of ``Q``. For two threads that enter the closed path
``P -> Q -> R -> S`` at ``P`` and ``R`` respectively, the convergence
of dynamic instances generated along the path depends on whether ``P``
or ``R`` is the header.

-  Convergence when ``P`` is the header.

   .. table::
      :align: left

      +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
      |         | 1     | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10  | 11  | 12  | 13   |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
      | Thread1 | Entry |     |     |     | P1  | Q1  | R1  | S1  | P3  | Q3  |     | S3  | Exit |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
      | Thread2 | Entry |     | R2  | S2  | P2  | Q2  |     | S2  | P4  | Q4  | R3  | S4  | Exit |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+

   |

-  Convergence when ``R`` is the header.

   .. table::
      :align: left

      +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
      |         | 1     | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10  | 11  | 12   |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
      | Thread1 | Entry |     | P1  | Q1  | R1  | S1  | P3  | Q3  | S3  |     |     | Exit |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+
      | Thread2 | Entry |     |     |     | R2  | S2  | P2  | Q2  | S2  | P4  | ... | Exit |
      +---------+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------+

   |

Thus, when diverged paths reach different entries of an irreducible
cycle from outside the cycle, the static analysis conservatively
reports every node in the cycle as not m-converged.

.. _convergence-reducible-cycle:

Reducible Cycle
---------------

If ``C`` is a reducible cycle with header ``H``, then in any DFS,
``H`` :ref:`must be the header of some cycle<cycle-reducible-headers>`
``C'`` that contains ``C``. Independent of the DFS, there is no entry
to the subgraph ``C`` other than ``H`` itself. Thus, we have the
following:

1. The diverged entry criterion is trivially satisfied for a divergent
   branch and its join, where both are inside subgraph ``C``.
2. When diverged paths reach the subgraph ``C`` from outside, their
   convergence is always determined by the same header ``H``.

Clearly, this can be determined only in a cycle hierarchy ``T`` where
``C`` is detected as a reducible cycle. No such conclusion can be made
in a different cycle hierarchy ``T'`` where ``C`` is part of a larger
cycle ``C'`` with the same header, but this does not contradict the
conclusion in ``T``.