Module CSE

Common subexpression elimination over RTL. This optimization proceeds by value numbering over extended basic blocks.

Require Import Coqlib.
Require Import Maps.
Require Import Ast.
Require Import Integers.
Require Import Pointers.
Require Import Floats.
Require Import Values.
Require Import Mem.
Require Import Globalenvs.
Require Import Op.
Require Import Registers.
Require Import RTL.
Require Import Kildall.

Value numbering

The idea behind value numbering algorithms is to associate abstract identifiers (``value numbers'') to the contents of registers at various program points, and record equations between these identifiers. For instance, consider the instruction r1 = add(r2, r3) and assume that r2 and r3 are mapped to abstract identifiers x and y respectively at the program point just before this instruction. At the program point just after, we can add the equation z = add(x, y) and associate r1 with z, where z is a fresh abstract identifier. However, if we already knew an equation u = add(x, y), we can preferably add no equation and just associate r1 with u. If there exists a register r4 mapped with u at this point, we can then replace the instruction r1 = add(r2, r3) by a move instruction r1 = r4, therefore eliminating a common subexpression and reusing the result of an earlier addition. Abstract identifiers / value numbers are represented by positive integers. Equations are of the form valnum = rhs, where the right-hand sides rhs are either arithmetic operations or memory loads.

Definition valnum := positive.

Inductive rhs : Type :=
  | Op: operation -> list valnum -> rhs.

Definition eq_valnum: forall (x y: valnum), {x=y}+{x<>y} := peq.

Definition eq_list_valnum (x y: list valnum) : {x=y}+{x<>y}.
  revert y; induction x; intros; case y; intros.
  left; auto.
  right; congruence.
  right; congruence.
  case (eq_valnum a v); intros.
  case (IHx l); intros.
  left; congruence.
  right; congruence.
  right; congruence.

Definition eq_rhs (x y: rhs) : {x=y}+{x<>y}.
  generalize Int.eq_dec; intro.
  generalize Float.eq_dec; intro.
  generalize eq_operation; intro.
  generalize eq_addressing; intro.
  assert (forall (x y: memory_chunk), {x=y}+{x<>y}). decide equality.
  generalize eq_valnum; intro.
  generalize eq_list_valnum; intro.
  decide equality.

A value numbering is a collection of equations between value numbers plus a partial map from registers to value numbers. Additionally, we maintain the next unused value number, so as to easily generate fresh value numbers.

Record numbering : Type := mknumbering {
  num_next: valnum;
  num_eqs: list (valnum * rhs);
  num_reg: PTree.t valnum

Definition empty_numbering :=
  mknumbering 1%positive nil (PTree.empty valnum).

valnum_reg n r returns the value number for the contents of register r. If none exists, a fresh value number is returned and associated with register r. The possibly updated numbering is also returned. valnum_regs is similar, but for a list of registers.

Definition valnum_reg (n: numbering) (r: reg) : numbering * valnum :=
  match PTree.get r n.(num_reg) with
  | Some v => (n, v)
  | None => (mknumbering (Psucc n.(num_next))
                           (PTree.set r n.(num_next) n.(num_reg)),

Fixpoint valnum_regs (n: numbering) (rl: list reg)
                     {struct rl} : numbering * list valnum :=
  match rl with
  | nil =>
      (n, nil)
  | r1 :: rs =>
      let (n1, v1) := valnum_reg n r1 in
      let (ns, vs) := valnum_regs n1 rs in
      (ns, v1 :: vs)

find_valnum_rhs rhs eqs searches the list of equations eqs for an equation of the form vn = rhs for some value number vn. If found, Some vn is returned, otherwise None is returned.

Fixpoint find_valnum_rhs (r: rhs) (eqs: list (valnum * rhs))
                         {struct eqs} : option valnum :=
  match eqs with
  | nil => None
  | (v, r') :: eqs1 =>
      if eq_rhs r r' then Some v else find_valnum_rhs r eqs1

add_rhs n rd rhs updates the value numbering n to reflect the computation of the operation or load represented by rhs and the storing of the result in register rd. If an equation vn = rhs is known, register rd is set to vn. Otherwise, a fresh value number vn is generated and associated with rd, and the equation vn = rhs is added.

Definition add_rhs (n: numbering) (rd: reg) (rh: rhs) : numbering :=
  match find_valnum_rhs rh n.(num_eqs) with
  | Some vres =>
      mknumbering n.(num_next) n.(num_eqs)
                  (PTree.set rd vres n.(num_reg))
  | None =>
      mknumbering (Psucc n.(num_next))
                  ((n.(num_next), rh) :: n.(num_eqs))
                  (PTree.set rd n.(num_next) n.(num_reg))

add_op n rd op rs specializes add_rhs for the case of an arithmetic operation. The right-hand side corresponding to op and the value numbers for the argument registers rs is built and added to n as described in add_rhs. If op is a move instruction, we simply assign the value number of the source register to the destination register, since we know that the source and destination registers have exactly the same value. This enables more common subexpressions to be recognized. For instance:
     z = add(x, y);  u = x; v = add(u, y);
Since u and x have the same value number, the second add is recognized as computing the same result as the first add, and therefore u and z have the same value number.

Definition add_op (n: numbering) (rd: reg) (op: operation) (rs: list reg) :=
  match is_move_operation op rs with
  | Some r =>
      let (n1, v) := valnum_reg n r in
      mknumbering n1.(num_next) n1.(num_eqs) (PTree.set rd v n1.(num_reg))
  | None =>
      let (n1, vs) := valnum_regs n rs in
      add_rhs n1 rd (Op op vs)

add_unknown n rd returns a numbering where rd is mapped to a fresh value number, and no equations are added. This is useful to model instructions with unpredictable results such as Ialloc.

Definition add_unknown (n: numbering) (rd: reg) :=
  mknumbering (Psucc n.(num_next))
              (PTree.set rd n.(num_next) n.(num_reg)).

Definition reg_valnum (n: numbering) (vn: valnum) : option reg :=
    (fun (res: option reg) (r: reg) (v: valnum) =>
       if peq v vn then Some r else res)
    n.(num_reg) None.

Definition find_rhs (n: numbering) (rh: rhs) : option reg :=
  match find_valnum_rhs rh n.(num_eqs) with
  | None => None
  | Some vres => reg_valnum n vres

Definition find_op
    (n: numbering) (op: operation) (rs: list reg) : option reg :=
  let (n1, vl) := valnum_regs n rs in
  find_rhs n1 (Op op vl).

The static analysis

We now define a notion of satisfiability of a numbering. This semantic notion plays a central role in the correctness proof (see CSEproof), but is defined here because we need it to define the ordering over numberings used in the static analysis. A numbering is satisfiable in a given register environment and memory state if there exists a valuation, mapping value numbers to actual values, that validates both the equations and the association of registers to value numbers.

Definition equation_holds
     (valuation: valnum -> val)
     (ge: genv) (sp: option pointer)
     (vres: valnum) (rh: rhs) : Prop :=
  match rh with
  | Op op vl =>
      eval_operation ge sp op ( valuation vl) =
      Some (valuation vres)

Definition numbering_holds
   (valuation: valnum -> val)
   (ge: genv) sp (rs: regset) (n: numbering) : Prop :=
     (forall vn rh,
       In (vn, rh) n.(num_eqs) ->
       equation_holds valuation ge sp vn rh)
  /\ (forall r vn,
       PTree.get r n.(num_reg) = Some vn -> rs#r = valuation vn).

Definition numbering_satisfiable
   (ge: genv) (sp: option pointer) (rs: regset) (n: numbering) : Prop :=
  exists valuation, numbering_holds valuation ge sp rs n.

Lemma empty_numbering_satisfiable:
  forall ge sp rs, numbering_satisfiable ge sp rs empty_numbering.
  intros; red.
  exists (fun (vn: valnum) => Vundef); split; simpl; intros; try done.
  by rewrite PTree.gempty in H.

We now equip the type numbering with a partial order and a greatest element. The partial order is based on entailment: n1 is greater than n2 if n1 is satisfiable whenever n2 is. The greatest element is, of course, the empty numbering (no equations).

Module Numbering.
  Definition t := numbering.
  Definition ge (n1 n2: numbering) : Prop :=
    forall ge sp rs,
    numbering_satisfiable ge sp rs n2 ->
    numbering_satisfiable ge sp rs n1.
  Definition top := empty_numbering.
  Lemma top_ge: forall x, ge top x.
    intros; red; intros. unfold top. apply empty_numbering_satisfiable.
  Lemma refl_ge: forall x, ge x x.
    intros; red; auto.
End Numbering.

We reuse the solver for forward dataflow inequations based on propagation over extended basic blocks defined in library Kildall.

Module Solver := BBlock_solver(Numbering).

The transfer function for the dataflow analysis returns the numbering ``after'' execution of the instruction at pc, as a function of the numbering ``before''. For Iop and Iload instructions, we add equations or reuse existing value numbers as described for add_op and add_load. For Istore instructions, we forget all equations involving memory loads. For Icall instructions, we could simply associate a fresh, unconstrained by equations value number to the result register. However, it is often undesirable to eliminate common subexpressions across a function call (there is a risk of increasing too much the register pressure across the call), so we just forget all equations and start afresh with an empty numbering. Finally, the remaining instructions modify neither registers nor the memory, so we keep the numbering unchanged.

Definition transfer (f: function) (pc: node) (before: numbering) :=
  match f.(fn_code)!pc with
  | None => before
  | Some i =>
      match i with
      | Inop s =>
      | Iop op args res s =>
          add_op before res op args
      | Iload chunk addr args dst s =>
          add_unknown before dst
      | Istore chunk addr args src s =>
      | Iatomic atom args dst s =>
          add_unknown before dst
      | Ifence s =>
      | Icall sig ros args res s =>
      | Icond cond args ifso ifnot =>
      | Ireturn optarg =>
      | Ithreadcreate fn args s =>

The static analysis solves the dataflow inequations implied by the transfer function using the ``extended basic block'' solver, which produces sub-optimal solutions quickly. The result is a mapping from program points to numberings. In the unlikely case where the solver fails to find a solution, we simply associate empty numberings to all program points, which is semantically correct and effectively deactivates the CSE optimization.

Definition analyze (f: RTL.function): PMap.t numbering :=
  match Solver.fixpoint (successors f) (transfer f) f.(fn_entrypoint) with
  | None => PMap.init empty_numbering
  | Some res => res

Code transformation

The code transformation is performed instruction by instruction. Iload instructions and non-trivial Iop instructions are turned into move instructions if their result is already available in a register, as indicated by the numbering inferred at that program point.

Definition transf_instr (n: numbering) (instr: instruction) :=
  match instr with
  | Iop op args res s =>
      if is_trivial_op op then instr else
        match find_op n op args with
        | None => instr
        | Some r => Iop Omove (r :: nil) res s
  | _ =>

Definition transf_code (approxs: PMap.t numbering) (instrs: code) : code := (fun pc instr => transf_instr approxs!!pc instr) instrs.

Definition transf_function (f: function) : function :=
  let approxs := analyze f in
        (transf_code approxs f.(fn_code))

Definition transf_fundef (f: fundef) : fundef :=
  Ast.transf_fundef transf_function f.

Definition transf_program (p: program) : program :=
  transform_program transf_fundef p.