Module SSA


The SSA intermediate language: abstract syntax and semantics. SSA is the Single Static Assignment form of RTL. SSA code is organized as instructions and phi-blocks, functions and programs. Instructions are those of RTL, phi-blocks are sequences of phi-instructions. All instructions manipulates pseudo-registers, as in RTL.

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Integers.
Require Import Values.
Require Import Events.
Require Import Memory.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Op.
Require Import Registers.
Require Import Floats.

Require Import Kildall.
Require Import Utils.
Require Import Relation_Operators.
Require Import Classical.
Require Import Relations.Relation_Definitions.
Require Import DLib.
Require Import Dom.

Unset Allow StrictProp.

Abstract syntax


Instructions are organized as a control-flow graph: a function is a finite map from nodes (abstract program points) to instructions, and each instruction lists explicitly the nodes of its successors.

Definition node := positive.

Variant instruction: Type :=
  | Inop: node -> instruction
No operation -- just branch to the successor.
  | Iop: operation -> list reg -> reg -> node -> instruction
Iop op args dst succ performs the arithmetic operation op over the values of registers args, stores the result in dst, and branches to succ.
  | Iload: memory_chunk -> addressing -> list reg -> reg -> node -> instruction
Iload chunk addr args dst succ loads a chunk quantity from the address determined by the addressing mode addr and the values of the args registers, stores the quantity just read into dst, and branches to succ.
  | Istore: memory_chunk -> addressing -> list reg -> reg -> node -> instruction
Istore chunk addr args src succ stores the value of register src in the chunk quantity at the the address determined by the addressing mode addr and the values of the args registers, then branches to succ.
  | Icall: signature -> reg + ident -> list reg -> reg -> node -> instruction
Icall sig fn args dst succ invokes the function determined by fn (either a function pointer found in a register or a function name), giving it the values of registers args as arguments. It stores the return value in dst and branches to succ.
  | Itailcall: signature -> reg + ident -> list reg -> instruction
Itailcall sig fn args performs a function invocation in tail-call position.
  | Ibuiltin: external_function -> list (builtin_arg reg) -> (builtin_res reg) -> node -> instruction
Ibuiltin ef args dst succ calls the built-in function identified by ef, giving it the values of args as arguments. It stores the return value in dst and branches to succ.
  | Icond: condition -> list reg -> node -> node -> instruction
Icond cond args ifso ifnot evaluates the boolean condition cond over the values of registers args. If the condition is true, it transitions to ifso. If the condition is false, it transitions to ifnot.
  | Ijumptable: reg -> list node -> instruction
Ijumptable arg tbl transitions to the node that is the n-th element of the list tbl, where n is the signed integer value of register arg.
  | Ireturn: option reg -> instruction.
Ireturn terminates the execution of the current function (it has no successor). It returns the value of the given register, or Vundef if none is given.

Variant phiinstruction: Type :=
| Iphi: list reg -> reg -> phiinstruction.
Iphi args dst copies the value of the k-th register of args, where k is the index of the predecessor of the current point.

Phi blocks are lists of phi-instructions
Definition phiblock:= list phiinstruction.

A function description comprises a control-flow graph (CFG) fn_code (a partial finite mapping from nodes to instructions) and a phi-block graph fn_phicode (a partial finite mapping from nodes to phi-blocks. Consequently, whenever a abstract program point is a junction point holding a phi-block, the phi-block can be retrieved in fn_phicode, while its regular instruction is stored in fn_code. As in Cminor, fn_sig is the function signature and fn_stacksize the number of bytes for its stack-allocated activation record. fn_params is the list of registers that are bound to the values of arguments at call time. fn_entrypoint is the node of the first instruction of the function in the CFG. fn_ext_params is a list of all SSA registers that are used in the function but not defined in it. It includes all the function's parameters. fn_dom_dest is a dominance test between two CFG nodes for the function.

Definition code: Type := PTree.t instruction.

Definition phicode: Type := PTree.t phiblock.

Record function: Type := mkfunction {
  fn_sig: signature;
  fn_params: list reg;
  fn_stacksize: Z;
  
  fn_code: code;
  fn_phicode: phicode;

  fn_entrypoint: node;

  fn_ext_params : list reg;
  fn_dom_test: node -> node -> bool
}.


Definition fundef := AST.fundef function.
Definition program := AST.program fundef unit.

Definition funsig (fd: fundef) :=
  match fd with
  | Internal f => fn_sig f
  | External ef => ef_sig ef
  end.

Definition successors_instr (i: instruction) : list node :=
  match i with
  | Inop s => s :: nil
  | Iop op args res s => s :: nil
  | Iload chunk addr args dst s => s :: nil
  | Istore chunk addr args src s => s :: nil
  | Icall sig ros args res s => s :: nil
  | Itailcall sig ros args => nil
  | Ibuiltin ef args res s => s :: nil
  | Icond cond args ifso ifnot => ifso :: ifnot :: nil
  | Ijumptable arg tbl => tbl
  | Ireturn optarg => nil
  end.

Notation preds :=
  (fun f pc => (make_predecessors (fn_code f) successors_instr) !!! pc).

Inductive join_point (jp: node) (f:function) : Prop :=
  | jp_cons : forall l,
      forall (Hpreds: (make_predecessors (fn_code f) successors_instr) ! jp = Some l)
             (Hl: (length l > 1)%nat),
        join_point jp f.

Fixpoint get_index_acc (l:list node) (a:node) acc : option nat :=
  match l with
    | nil => None
    | x::m => if (peq x a)
      then Some acc
      else get_index_acc m a (acc+1)%nat
  end.

Definition get_index (l:list node) (a:node) :=
  get_index_acc l a O.

Definition index_pred (predsf: PTree.t (list node)):=
  fun (i:node) (j:node) =>
    let predsj := predsf !!! j in
      match predsj with
        | nil => None
        | _ => get_index predsj i
      end.

Use and definition of registers

Section UDEF.

assigned_code_spec code pc r holds if the register r is assigned at point pc in the code code
Inductive assigned_code_spec (code:code) (pc:node) : reg -> Prop :=
| AIop: forall op args dst succ,
  code!pc = Some (Iop op args dst succ) ->
  assigned_code_spec code pc dst
| AIload: forall chunk addr args dst succ,
  code!pc = Some (Iload chunk addr args dst succ) ->
  assigned_code_spec code pc dst
| AIcall: forall sig fn args dst succ,
  code!pc = Some (Icall sig fn args dst succ) ->
  assigned_code_spec code pc dst
| AIbuiltin: forall fn args dst succ,
    code!pc = Some (Ibuiltin fn args (BR dst) succ) ->
    assigned_code_spec code pc dst.

assigned_phi_spec phicode pc r holds if the register r is assigned at point pc in the phi-code phicode
Variant assigned_phi_spec (phicode: phicode) (pc: node): reg -> Prop :=
  APhi: forall phiinstr dst,
    (phicode!pc) = Some phiinstr ->
    (exists args, List.In (Iphi args dst) phiinstr) ->
    assigned_phi_spec phicode pc dst.

Variable f : function.
  
use_code f r pc holds whenever register r is used at pc in the regular code of f
Inductive use_code : reg -> node -> Prop :=
| UIop: forall pc arg op args dst s,
  (fn_code f) !pc = Some (Iop op args dst s) -> In arg args -> use_code arg pc
| UIload: forall pc arg chk adr args dst s,
  (fn_code f) !pc = Some (Iload chk adr args dst s) -> In arg args -> use_code arg pc
| UIcond: forall pc arg cond args s s',
  (fn_code f) !pc = Some (Icond cond args s s') -> In arg args -> use_code arg pc
| UIbuiltin: forall pc arg ef args dst s,
  (fn_code f) !pc = Some (Ibuiltin ef args dst s) -> In arg (params_of_builtin_args args) -> use_code arg pc
| UIstore: forall pc arg chk adr args src s,
  (fn_code f) !pc = Some (Istore chk adr args src s) -> In arg (src::args) -> use_code arg pc
| UIcall: forall pc arg sig r args dst s,
  (fn_code f) !pc = Some (Icall sig (inl ident r) args dst s) -> In arg (r::args) -> use_code arg pc
| UItailcall: forall pc arg sig r args,
  (fn_code f) !pc = Some (Itailcall sig (inl ident r) args) -> In arg (r::args) -> use_code arg pc
| UIcall2: forall pc arg sig id args dst s,
  (fn_code f) !pc = Some (Icall sig (inr reg id) args dst s) -> In arg args -> use_code arg pc
| UItailcall2: forall pc arg sig id args,
  (fn_code f) !pc = Some (Itailcall sig (inr reg id) args) -> In arg args -> use_code arg pc
| UIjump: forall pc arg tbl,
  (fn_code f) !pc = Some (Ijumptable arg tbl) -> use_code arg pc
| UIret: forall pc arg,
  (fn_code f) !pc = Some (Ireturn (Some arg)) -> use_code arg pc.

use_phicode f r pc holds whenever register r is used at pc in the phi-code of f
Variant use_phicode : reg -> node -> Prop :=
| upc_intro : forall pc pred k arg args dst phib
  (PHIB: (fn_phicode f) ! pc = Some phib)
  (ASSIG : In (Iphi args dst) phib)
  (KARG : nth_error args k = Some arg)
  (KPRED : index_pred (make_predecessors (fn_code f) successors_instr) pred pc = Some k),
  use_phicode arg pred.

A register is used either in the code or in the phicode of a function
Variant use : reg -> node -> Prop :=
| u_code : forall x pc, use_code x pc -> use x pc
| u_phicode : forall x pc, use_phicode x pc -> use x pc.

Special definition point for function parameters and registers that are used in the function without having been defined anywhere in the function
Variant ext_params (x: reg) : Prop :=
| ext_params_params :
  In x (fn_params f) -> ext_params x
| ext_params_undef :
  (exists pc, use x pc) ->
  (forall pc, ~ assigned_phi_spec (fn_phicode f) pc x) ->
  (forall pc, ~ assigned_code_spec (fn_code f) pc x) ->
  ext_params x.
Hint Constructors ext_params: core.

def r pc holds if register r is defined at node pc
Variant def : reg -> node -> Prop :=
| def_params : forall x,
  ext_params x -> def x (fn_entrypoint f)
| def_code : forall x pc, assigned_code_spec (fn_code f) pc x -> def x pc
| def_phicode : forall x pc, assigned_phi_spec (fn_phicode f) pc x -> def x pc.

End UDEF.

Hint Constructors ext_params def assigned_code_spec assigned_phi_spec: core.

Formalization of Dominators

Section DOMINATORS.
  
  Variable f : function.

  Definition entry := (fn_entrypoint f).
  Notation code := (fn_code f).
  
cfg i j holds if j is a successor of i in the code of f
  Variant _cfg (i j:node) : Prop :=
  | CFG : forall ins
    (HCFG_ins: code !i = Some ins)
    (HCFG_in : In j (successors_instr ins)),
    _cfg i j.
  
  Definition exit (pc: node) : Prop :=
  match code ! pc with
  | Some (Itailcall _ _ _) => True
  | Some (Ireturn _) => True
  | Some (Ijumptable _ succs) => succs = nil
  | _ => False
  end.

  Definition cfg := _cfg.
  
  Definition dom := dom cfg exit entry.

End DOMINATORS.

Notation SSApath := (fun f => Dom.path (cfg f) (exit f) (entry f)).

Well-formed SSA functions


Every variable is assigned as most once
Definition unique_def_spec (f : function) :=
  (forall (r:reg) (pc pc':node),
    (assigned_code_spec (f.(fn_code)) pc r ->
      assigned_code_spec (f.(fn_code)) pc' r ->
      pc = pc')
    /\
    (assigned_phi_spec (f.(fn_phicode)) pc r ->
      assigned_phi_spec (f.(fn_phicode)) pc' r ->
      pc = pc')
    /\
    ((assigned_code_spec (f.(fn_code)) pc r ->
      ~ assigned_phi_spec (f.(fn_phicode)) pc' r)
    /\ (assigned_phi_spec (f.(fn_phicode)) pc r ->
      ~ assigned_code_spec (f.(fn_code)) pc' r)))
  /\
  (forall pc phiinstr,
    (f.(fn_phicode))!pc = Some phiinstr ->
    ( (NoDup phiinstr)
      /\ (forall r args args',
          In (Iphi args r) phiinstr -> In (Iphi args' r) phiinstr -> args = args'))
  ).


All phi-instruction have the right numbers of phi-arguments
Definition block_nb_args (f: function) : Prop :=
  forall pc block args x,
    (fn_phicode f) ! pc = Some block ->
    In (Iphi args x) block ->
    (length (preds f pc)) = (length args).

Definition successors (f: function) : PTree.t (list positive) :=
  PTree.map1 successors_instr (fn_code f).

Notation succs := (fun f pc => (successors f) !!! pc).

Well-formed SSA functions

Notation reached := (fun f => (reached (cfg f) (entry f))).
Notation sdom := (fun f => (sdom (cfg f) (exit f) (entry f))).

Record wf_ssa_function (f:function) : Prop := {
  fn_ssa : unique_def_spec f;
  
  fn_ssa_params : forall x, In x (fn_params f) ->
    (forall pc, ~ assigned_code_spec (fn_code f) pc x) /\
    (forall pc, ~ assigned_phi_spec (fn_phicode f) pc x);
    
  fn_strict : forall x u d, use f x u -> def f x d -> dom f d u;
      
  fn_use_def_code : forall x pc,
    use_code f x pc ->
    assigned_code_spec (fn_code f) pc x -> False;

  fn_wf_block: block_nb_args f;
  
  fn_normalized: forall jp pc,
                   (join_point jp f) ->
                   In jp (succs f pc) -> (fn_code f) ! pc = Some (Inop jp);
  
  fn_phicode_inv: forall jp,
    join_point jp f <->
    f.(fn_phicode) ! jp <> None;
    
  fn_code_reached: forall pc ins, (fn_code f) ! pc = Some ins -> reached f pc;

  fn_code_closed:
    forall pc pc' instr, (fn_code f) ! pc = Some instr ->
    In pc' (successors_instr instr) ->
    exists instr', (fn_code f) ! pc' = Some instr';

  fn_entry : exists s, (fn_code f) ! (fn_entrypoint f) = Some (Inop s);
  fn_entry_pred: forall pc, ~ cfg f pc (fn_entrypoint f);

  fn_ext_params_complete: forall r,
    ext_params f r -> In r (fn_ext_params f);

  fn_dom_test_correct : forall n d,
    reached f n -> fn_dom_test f d n = true -> dom f d n
}.

Require Import KildallComp.

Lemma no_assigned_phi_spec_fn_entrypoint: forall f,
  wf_ssa_function f -> forall x,
    ~ assigned_phi_spec (fn_phicode f) (fn_entrypoint f) x.
Proof.
  red; intros.
  inv H0.
  assert (join_point (fn_entrypoint f) f).
  { rewrite fn_phicode_inv; auto.
    congruence.
  }
  inv H0.
  destruct l.
  simpl in Hl; apply False_ind; omega.
  exploit @make_predecessors_some; eauto. intros [pins Hpins].
  assert (In (fn_entrypoint f) (successors_instr pins)).
  {
    eapply @make_predecessors_correct2; eauto.
    unfold Kildall.successors_list.
    unfold make_preds.
    rewrite Hpreds.
    left; auto.
  }
  elim (fn_entry_pred _ H p).
  econstructor; eauto.
Qed.

Well-formed SSA function definitions
Inductive wf_ssa_fundef: fundef -> Prop :=
  | wf_ssa_fundef_external: forall ef,
      wf_ssa_fundef (External ef)
  | wf_ssa_function_internal: forall f,
      wf_ssa_function f ->
      wf_ssa_fundef (Internal f).

Well-formed SSA programs
Definition wf_ssa_program (p: program) : Prop :=
  forall f id, In (id, Gfun f) (prog_defs p) -> wf_ssa_fundef f.

Operational semantics


Definition genv := Genv.t fundef unit.

The dynamic semantics of SSA is given in small-step style for basic instructions, as a set of transitions between states, and in a big step style for phi-blocks. A state captures the current point in the execution. Three kinds of states appear in the transitions: In all three kinds of states, the cs parameter represents the call stack. It is a list of frames Stackframe res f sp pc rs. Each frame represents a function call in progress. res is the pseudo-register that will receive the result of the call. f is the calling function. sp is its stack pointer. pc is the program point for the instruction that follows the call. rs is the state of registers in the calling function.

Definition regset := Regmap.t val.

Variant stackframe : Type :=
  | Stackframe:
      forall (res: reg) (* where to store the result *)
             (f: function) (* calling function *)
             (sp: val) (* stack pointer in calling function *)
             (pc: node) (* program point in calling function *)
             (rs: regset), (* register state in calling function *)
      stackframe.

Variant state : Type :=
  | State:
      forall (stack: list stackframe) (* call stack *)
             (f: function) (* current function *)
             (sp: val) (* stack pointer *)
             (pc: node) (* current program point in c *)
             (rs: regset) (* register state *)
             (m: mem), (* memory state *)
      state
  | Callstate:
      forall (stack: list stackframe) (* call stack *)
             (f: fundef) (* function to call *)
             (args: list val) (* arguments to the call *)
             (m: mem), (* memory state *)
      state
  | Returnstate:
      forall (stack: list stackframe) (* call stack *)
             (v: val) (* return value for the call *)
             (m: mem), (* memory state *)
      state.

Section RELSEM.

Variable ge: genv.

Definition find_function
      (ros: reg + ident) (rs: regset) : option fundef :=
      match ros with
        | inl r => Genv.find_funct ge (rs # r)
        | inr symb =>
          match Genv.find_symbol ge symb with
            | None => None
            | Some b => Genv.find_funct_ptr ge b
          end
      end.

Definition of the effect of a phi-block on a register set, when the control flow comes from the k-th predecessor of the current program point. Phi-blocks are given a parallel semantics
Fixpoint phi_store k phib (rs:regset) :=
  match phib with
    | nil => rs
    | (Iphi args dst)::phib =>
      match nth_error args k with
        | None => (phi_store k phib rs)
        | Some arg => (phi_store k phib rs)# dst <- (rs # arg)
      end
  end.

            
The transitions are presented as an inductive predicate step ge st1 t st2, where ge is the global environment, st1 the initial state, st2 the final state, and t the trace of system calls performed during this transition.

Fixpoint init_regs (vl: list val) (rl: list reg) {struct rl} : regset :=
  match rl, vl with
  | r1 :: rs, v1 :: vs => PMap.set r1 v1 (init_regs vs rs)
  | _, _ => PMap.init Vundef
  end.

Inductive step: state -> trace -> state -> Prop :=
| exec_Inop_njp:
  forall s f sp pc rs m pc',
    (fn_code f)!pc = Some(Inop pc') ->
    ~ join_point pc' f ->
    step (State s f sp pc rs m)
    E0 (State s f sp pc' rs m)
| exec_Inop_jp:
  forall s f sp pc rs m pc' phib k,
    (fn_code f)!pc = Some(Inop pc') ->
    join_point pc' f ->
    (fn_phicode f)!pc' = Some phib ->
    index_pred (make_predecessors (fn_code f) successors_instr) pc pc' = Some k ->
    step (State s f sp pc rs m)
    E0 (State s f sp pc' (phi_store k phib rs) m)
| exec_Iop:
  forall s f sp pc rs m op args res pc' v,
    (fn_code f)!pc = Some(Iop op args res pc') ->
    eval_operation ge sp op rs## args m = Some v ->
    step (State s f sp pc rs m)
    E0 (State s f sp pc' (rs#res <- v) m)
| exec_Iload:
  forall s f sp pc rs m chunk addr args dst pc' a v,
    (fn_code f)!pc = Some(Iload chunk addr args dst pc') ->
    eval_addressing ge sp addr rs## args = Some a ->
    Mem.loadv chunk m a = Some v ->
    step (State s f sp pc rs m)
    E0 (State s f sp pc' (rs# dst <- v) m)
| exec_Istore:
  forall s f sp pc rs m chunk addr args src pc' a m',
    (fn_code f)!pc = Some(Istore chunk addr args src pc') ->
    eval_addressing ge sp addr rs## args = Some a ->
    Mem.storev chunk m a rs# src = Some m' ->
    step (State s f sp pc rs m)
    E0 (State s f sp pc' rs m')
| exec_Icall:
  forall s f sp pc rs m sig ros args res pc' fd,
    (fn_code f)!pc = Some(Icall sig ros args res pc') ->
    find_function ros rs = Some fd ->
    funsig fd = sig ->
    step (State s f sp pc rs m)
    E0 (Callstate (Stackframe res f sp pc' rs :: s) fd rs## args m)
| exec_Itailcall:
  forall s f stk pc rs m sig ros args fd m',
    (fn_code f)!pc = Some(Itailcall sig ros args) ->
    find_function ros rs = Some fd ->
    funsig fd = sig ->
    Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
    step (State s f (Vptr stk (Ptrofs.zero)) pc rs m)
    E0 (Callstate s fd rs## args m')
| exec_Ibuiltin:
  forall s f sp pc rs m ef args vargs res vres pc' t m',
    (fn_code f)!pc = Some(Ibuiltin ef args res pc') ->
    eval_builtin_args ge (fun r => rs# r) sp m args vargs ->
    external_call ef ge vargs m t vres m' ->
    step (State s f sp pc rs m)
    t (State s f sp pc' (regmap_setres res vres rs) m')
| exec_Icond_true:
  forall s f sp pc rs m cond args ifso ifnot,
    (fn_code f)!pc = Some(Icond cond args ifso ifnot) ->
    eval_condition cond rs## args m = Some true ->
    step (State s f sp pc rs m)
    E0 (State s f sp ifso rs m)
| exec_Icond_false:
  forall s f sp pc rs m cond args ifso ifnot,
    (fn_code f)!pc = Some(Icond cond args ifso ifnot) ->
    eval_condition cond rs## args m = Some false ->
    step (State s f sp pc rs m)
    E0 (State s f sp ifnot rs m)
| exec_Ijumptable:
  forall s f sp pc rs m arg tbl n pc',
    (fn_code f)!pc = Some(Ijumptable arg tbl) ->
    rs# arg = Vint n ->
    list_nth_z tbl (Int.unsigned n) = Some pc' ->
    step (State s f sp pc rs m)
    E0 (State s f sp pc' rs m)
| exec_Ireturn:
  forall s f stk pc rs m or m',
    (fn_code f)!pc = Some(Ireturn or) ->
    Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
    step (State s f (Vptr stk Ptrofs.zero) pc rs m)
    E0 (Returnstate s (regmap_optget or Vundef rs) m')
| exec_function_internal:
  forall s f args m m' stk,
    Mem.alloc m 0 f.(fn_stacksize) = (m', stk) ->
    step (Callstate s (Internal f) args m)
    E0 (State s
      f
      (Vptr stk Ptrofs.zero)
      f.(fn_entrypoint)
      (init_regs args f.(fn_params))
      m')
| exec_function_external:
  forall s ef args res t m m',
    external_call ef ge args m t res m' ->
    step (Callstate s (External ef) args m)
    t (Returnstate s res m')
| exec_return:
  forall res f sp pc rs s vres m,
    step (Returnstate (Stackframe res f sp pc rs :: s) vres m)
    E0 (State s f sp pc (rs# res <- vres) m).

Hint Constructors step: core.

End RELSEM.

Execution of whole programs are described as sequences of transitions from an initial state to a final state. An initial state is a Callstate corresponding to the invocation of the ``main'' function of the program without arguments and with an empty call stack.

Variant initial_state (p: program): state -> Prop :=
  | initial_state_intro: forall b f m0,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      funsig f = signature_main ->
      initial_state p (Callstate nil f nil m0).

A final state is a Returnstate with an empty call stack.

Variant final_state: state -> int -> Prop :=
  | final_state_intro: forall r m, final_state (Returnstate nil (Vint r) m) r.

The small-step semantics for a program.

Definition semantics (p: program) :=
  Semantics step (initial_state p) final_state (Genv.globalenv p).