Linearization of the control-flow graph: translation from LTL to LTLin

Require Import Coqlib.
Require Import Maps.
Require Import Ordered.
Require Import FSets.
Require FSetAVL.
Require Import AST.
Require Import Values.
Require Import Globalenvs.
Require Import Errors.
Require Import Op.
Require Import Locations.
Require Import LTL.
Require Import LTLin.
Require Import Kildall.
Require Import Lattice.

Open Scope error_monad_scope.

To translate from LTL to LTLin, we must lay out the nodes of the LTL control-flow graph in some linear order, and insert explicit branches and conditional branches to make sure that each node jumps to its successors as prescribed by the LTL control-flow graph. However, branches are not necessary if the fall-through behaviour of LTLin instructions already implements the desired flow of control. For instance, consider the two LTL instructions

    L1: Lop op args res L2
    L2: ...

If the instructions L1 and L2 are laid out consecutively in the LTLin code, we can generate the following LTLin code:

    L1: Lop op args res
    L2: ...

However, if this is not possible, an explicit Lgoto is needed:

    L1: Lop op args res
        Lgoto L2
        ...
    L2: ...

The main challenge in code linearization is therefore to pick a ``good'' order for the nodes that exploits well the fall-through behavior. Many clever trace picking heuristics have been developed for this purpose. In this file, we present linearization in a way that clearly separates the heuristic part (choosing an order for the basic blocks) from the actual code transformation parts. We proceed in two passes:

Choosing an order for the nodes. This returns an enumeration of CFG nodes stating that they must be laid out in the order shown in the list.
Generate LTLin code where each node branches explicitly to its successors, except if one of these successors is the immediately following instruction.

The beauty of this approach is that correct code is generated under surprisingly weak hypotheses on the enumeration of CFG nodes: it suffices that every reachable instruction occurs exactly once in the enumeration. We therefore follow an approach based on validation a posteriori: a piece of untrusted Caml code implements the node enumeration heuristics, and the resulting enumeration is checked for correctness by Coq functions that are proved to be sound.

Determination of the order of basic blocks

We first compute a mapping from CFG nodes to booleans, indicating whether a CFG instruction is reachable or not. This computation is a trivial forward dataflow analysis where the transfer function is the identity: the successors of a reachable instruction are reachable, by the very definition of reachability.

Module DS := Dataflow_Solver(LBoolean)(NodeSetForward).

Definition reachable_aux (f: LTL.function) : option (PMap.t bool) :=
  DS.fixpoint
    (successors f)
    (fun pc r => r)
    ((f.(fn_entrypoint), true) :: nil).

Definition reachable (f: LTL.function) : PMap.t bool :=
  match reachable_aux f with
  | None => PMap.init true
  | Some rs => rs
  end.

We then enumerate the nodes of reachable instructions. This task is performed by external, untrusted Caml code.

Parameter enumerate_aux: LTL.function -> PMap.t bool -> list node.

Now comes the a posteriori validation of a node enumeration.

Module Nodeset := FSetAVL.Make(OrderedPositive).

Build a Nodeset.t from a list of nodes, checking that the list contains no duplicates.

Fixpoint nodeset_of_list (l: list node) (s: Nodeset.t)
                         {struct l}: res Nodeset.t :=
  match l with
  | nil => OK s
  | hd :: tl =>
      if Nodeset.mem hd s
      then Error (msg "Linearize: duplicates in enumeration")
      else nodeset_of_list tl (Nodeset.add hd s)
  end.

Definition check_reachable_aux
     (reach: PMap.t bool) (s: Nodeset.t)
     (ok: bool) (pc: node) (i: LTL.instruction) : bool :=
  if reach!!pc then ok && Nodeset.mem pc s else ok.

Definition check_reachable
     (f: LTL.function) (reach: PMap.t bool) (s: Nodeset.t) : bool :=
  PTree.fold (check_reachable_aux reach s) f.(LTL.fn_code) true.

Definition enumerate (f: LTL.function) : res (list node) :=
  let reach := reachable f in
  let enum := enumerate_aux f reach in
  do s <- nodeset_of_list enum Nodeset.empty;
  if check_reachable f reach s
  then OK enum
  else Error (msg "Linearize: wrong enumeration").

Translation from LTL to LTLin

We now flatten the structure of the CFG graph, laying out LTL instructions consecutively in the order computed by enumerate, and inserting branches to the labels of sucessors if necessary. Whether to insert a branch or not is determined by the starts_with function below. For LTL conditional branches Lcond cond args s1 s2, we have two possible translations:

      Lcond cond args s1;       or     Lcond (not cond) args s2;
      Lgoto s2                         Lgoto s1

We favour the first translation if s2 is the label of the next instruction, and the second if s1 is the label of the next instruction, thus avoiding the insertion of a redundant Lgoto instruction.

Fixpoint starts_with (lbl: label) (k: code) {struct k} : bool :=
  match k with
  | Llabel lbl' :: k' => if peq lbl lbl' then true else starts_with lbl k'
  | _ => false
  end.

Definition add_branch (s: label) (k: code) : code :=
  if starts_with s k then k else Lgoto s :: k.

Definition linearize_instr (b: LTL.instruction) (k: code) : code :=
  match b with
  | LTL.Lnop s =>
      add_branch s k
  | LTL.Lop op args res s =>
      Lop op args res :: add_branch s k
  | LTL.Lload chunk addr args dst s =>
      Lload chunk addr args dst :: add_branch s k
  | LTL.Lstore chunk addr args src s =>
      Lstore chunk addr args src :: add_branch s k
  | LTL.Lcall sig ros args res s =>
      Lcall sig ros args res :: add_branch s k
  | LTL.Ltailcall sig ros args =>
      Ltailcall sig ros args :: k
  | LTL.Lbuiltin ef args res s =>
      Lbuiltin ef args res :: add_branch s k
  | LTL.Lcond cond args s1 s2 =>
      if starts_with s1 k then
        Lcond (negate_condition cond) args s2 :: add_branch s1 k
      else
        Lcond cond args s1 :: add_branch s2 k
  | LTL.Ljumptable arg tbl =>
      Ljumptable arg tbl :: k
  | LTL.Lreturn or =>
      Lreturn or :: k
  end.

Linearize a function body according to an enumeration of its nodes.

Fixpoint linearize_body (f: LTL.function) (enum: list node)
                        {struct enum} : code :=
  match enum with
  | nil => nil
  | pc :: rem =>
      match f.(LTL.fn_code)!pc with
      | None => linearize_body f rem
      | Some b => Llabel pc :: linearize_instr b (linearize_body f rem)
      end
  end.

Entry points for code linearization

Definition transf_function (f: LTL.function) : res LTLin.function :=
  do enum <- enumerate f;
  OK (mkfunction
       (LTL.fn_sig f)
       (LTL.fn_params f)
       (LTL.fn_stacksize f)
       (add_branch (LTL.fn_entrypoint f) (linearize_body f enum))).

Definition transf_fundef (f: LTL.fundef) : res LTLin.fundef :=
  AST.transf_partial_fundef transf_function f.

Definition transf_program (p: LTL.program) : res LTLin.program :=
  transform_partial_program transf_fundef p.

Module Linearize

Determination of the order of basic blocks

Translation from LTL to LTLin

Entry points for code linearization