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diff --git a/docs/chapter06/README.md b/docs/chapter06/README.md new file mode 100644 index 00000000..0f3ef945 --- /dev/null +++ b/docs/chapter06/README.md @@ -0,0 +1,203 @@ +# Chapter 6: Peepholes for If + +[Previous: Chapter 5](../chapter05/README.md) | +[Next: Chapter 7](../chapter07/README.md) + +# Table of Contents + +1. [Type System Revision](#type-system-revision) +2. [Peephole of `if`](#peephole-of-if) +3. [`Region` and `Phi`](#region-and-phi) +4. [Discussion](#discussion) +5. [Expanding Peephole Beyond Constant Expressions](#expanding-peephole-beyond-constant-expressions) +6. [Dominators](#dominators) + +You can also read [this chapter](https://github.com/SeaOfNodes/Simple/tree/linear-chapter06) in a linear Git revision history on the [linear](https://github.com/SeaOfNodes/Simple/tree/linear) branch and [compare](https://github.com/SeaOfNodes/Simple/compare/linear-chapter05...linear-chapter06) it to the previous chapter. + +In this chapter we do not add new language features. Our goal is to add peephole optimization to `if` statements. + + +## Type System Revision + +We now need to extend the type system to introduce another lattice structure, in addition to the Integer sub-lattice. +As before we denote "Top" by T and "Bottom" by ⊥. + +* "Ctrl" represents a live control, whereas "~Ctrl" represents a dead control. +* "INT" refers to an integer value, note that integer values have their own sub-lattice. + +| | ⊥ | Ctrl | ~Ctrl | T | INT | +|-------|---|------|-------|-------|-----| +| T | ⊥ | Ctrl | ~Ctrl | T | INT | +| Ctrl | ⊥ | Ctrl | Ctrl | Ctrl | ⊥ | +| ~Ctrl | ⊥ | Ctrl | ~Ctrl | ~Ctrl | ⊥ | +| ⊥ | ⊥ | ⊥ | ⊥ | ⊥ | ⊥ | + +* "Top" meets any type is that type, +* "Bottom" meets any type is "Bottom", +* "Top", "Bottom", "Ctrl", "~Ctrl" are classed as simple types, +* "Ctrl" meets "~Ctrl" is "Ctrl" +* Unless covered above, a simple type meets non-simple results in "Bottom" + +The entire revised Lattice is shown below: + + + + +## Peephole of `if` + +* Our general strategy is to continue parsing the entire `if` statement even if we know that one branch is dead. +* This approach creates dead nodes which get cleaned up by dead code + elimination. The benefits are that we do not introduce special logic during + parsing, the normal graph building machinery takes care of cleaning up dead + nodes. +* When we peephole an `If` node we test if the predicate is a constant. If so, + then one branch or the other is dead, depending on if the constant is `0` or not. +* When we add the `Proj` nodes, in this scenario, we already know one of the projections is dead. The peephole of the + relevant `Proj` node replaces the `Proj` with a `Constant` of type "~Ctrl" indicating dead control. +* At this point our dead code elimination would kill the `If` node since it has no uses + - but we cannot do that because we need to continue the parse. The + important observation is that *we __can__ kill the `If` node after __both__ + the `Proj` nodes have been created*, because any subsequent control flow can + only see the `Proj` nodes. +* To ensure this, we add a dummy user to `If` before creating the first `Proj` and remove it immediately after, allowing the + second `Proj` to trigger a dead code elimination of the `If`. +* The live `Proj` gets replaced by the parent of `If`, whereas the dead `Proj` gets replaced by "~Ctrl". +* The parsing then continues as normal. +* The other changes are when we reach the merge point with a dead control. +* Again keeping with our strategy we merge as normal, including creating `Phi` nodes. +* The `Region` may have one of its inputs as "~Ctrl". This will be seen + by the `Phi` which then optimizes itself. If the `Phi` has just one live + input, the `Phi` peephole replaces itself with the remaining input. +* Finally, at the end, when the `Region` node is peepholed, we see that it has only one live input and no Phi uses + and can be replaced with its one live input. + +## `Region` and `Phi` + +One of the invariants we maintain is that for each control input to a +`Region` every `Phi` has a 1-to-1 relationship with a data input. Thus, if a +`Region` loses a control input, every `Phi`'s corresponding data input must be +deleted. Conversely, we cannot collapse a Region until it has no dependent +`Phi`s. + +When processing `If` we do not remove control inputs to a `Region`, instead the +dead control input is simply set to `Constant(~Ctrl)`. The peephole logic in +a `Phi` notices this and replaces itself with the live input. + +## Discussion + +With the changes above, our peephole optimization is able to fold dead code when +an `if` statement has a condition that evaluates to `true` or `false` at compile time. + +Let's look at a simple example below: + +```java +if( true ) return 2; +return 1; +``` + +Before peephole: + + + +After peephole: + + + +Another example: + +```java +int a=1; +if( true ) + a=2; +else + a=3; +return a; +``` + +Before peephole: + + + +After peephole: + + + +## Expanding Peephole Beyond Constant Expressions + +While our peephole of the `CFG` collapses dead code when the expression is +a compile-time constant, it does not do as well as it could with the following example: + +```java +int a = 0; +int b = 1; +if( arg ) { + a = 2; + if( arg ) { b = 2; } + else b = 3; +} +return a+b; +``` + +In this example, `arg` is defined externally, and we have to assume it is not a constant. +However, the code repeats the `if( arg )` condition in the inner `if` statement. +We know that if the outer `if( arg )` is true, then the *inner `if` must also evaluate to true*. + +Our peephole at present cannot see this: + + + +What we would like is for our peephole to see that the inner `if` is fully determined by the +outer `if` and thus, the code can be simplified as follows: + +```java +int a = 0; +int b = 1; +if ( arg ) { + a = 2; + b = 2; +} +return a+b; +``` + +That is, the final outcome is either `4` if `arg` happens to be true, or `1` if `arg` happens to be false. + +## Dominators + +The key to implementing this peephole is the observation that the inner `if` is completely determined by the +outer `if`. In compiler parlance, we say that the outer `if` dominates the inner `if`. + +There is a generic way to determine this type of domination; it involves constructing a dominator +tree. For details of how to do this, please refer to the paper `A Simple, Fast Dominance Algorithm`.[^1] + +However, constructing a dominator tree is an expensive operation; one that we would not like to +do in the middle of parsing. Fortunately, since we do not have loops yet, we do not +have back edges in our control flow graph. Our control flow graph is essentially hierarchical, tree like. +Thus, a simple form of dominator identification is possible. + +The main idea is that we can assign a `depth` to each control node, this represents the node's depth +in the hierarchy. We can compute this depth incrementally as we need to. The nodes that are deeper in the hierarchy +have a greater depth than the ones above. + +Secondly we know that for our control nodes, except for `Region`s, there is only one incoming control edge, +which means that most control nodes have a single immediate dominating control node. `Region` nodes are different +as they can have one or more control edges; but, we know that for `if` statements, a `Region` node has exactly two +control inputs, one from each branch of the `if`. To find the immediate dominator of a `Region` node in the +limited scenario of an `if`, we can follow the immediate dominator of each branch until we find the common ancestor +node that dominates both the branches. + +If this ancestor node is also an `if` and its predicate is the same as the current `if`, then we know that +the outcome of the present `if` is fully determined. + +The code that does this check is in the [`compute` method of the `if`](https://github.com/SeaOfNodes/Simple/blob/main/chapter06/src/main/java/com/seaofnodes/simple/node/IfNode.java#L38-L45). + +The `idom` method in `Node` provides a default implementation of the depth based immediate dominator calculation. +`Region` [overrides this](https://github.com/SeaOfNodes/Simple/blob/main/chapter06/src/main/java/com/seaofnodes/simple/node/RegionNode.java#L57-L74) to implement the more complex search described above. + +It turns out that to simplify the example above, we need a [further peephole in our `Add` node](https://github.com/SeaOfNodes/Simple/blob/main/chapter06/src/main/java/com/seaofnodes/simple/node/AddNode.java#L64-L67). + +With all these changes, the outcome of the peephole is just what we would like to see! + + + +[^1]: Cooper, K. D. Cooper, Harvey, T. J., and Kennedy, K. (2001). + A Simple, Fast Dominance Algorithm. |