# Module Determinism

Characterization and properties of deterministic semantics

Require Import Classical.
Require Import Coqlib.
Require Import AST.
Require Import Integers.
Require Import Values.
Require Import Events.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Behaviors.

This file uses classical logic (the axiom of excluded middle), as well as the following extensionality property over infinite sequences of events. All these axioms are sound in a set-theoretic model of Coq's logic.

Axiom traceinf_extensionality:
forall T T', traceinf_sim T T' -> T = T'.

# Deterministic worlds

One source of possible nondeterminism is that our semantics leave unspecified the results of system calls. Different results to e.g. a "read" operation can of course lead to different behaviors for the program. We address this issue by modeling a notion of deterministic external world that uniquely determines the results of external calls.

An external world is a function that, given the name of an external call and its arguments, returns either None, meaning that this external call gets stuck, or Some(r,w), meaning that this external call succeeds, has result r, and changes the world to w.

Inductive world: Type :=
World (io: ident -> list eventval -> option (eventval * world))
(vload: memory_chunk -> ident -> int -> option (eventval * world))
(vstore: memory_chunk -> ident -> int -> eventval -> option world).

Definition nextworld_io (w: world) (evname: ident) (evargs: list eventval) :
option (eventval * world) :=
match w with World io vl vs => io evname evargs end.

Definition nextworld_vload (w: world) (chunk: memory_chunk) (id: ident) (ofs: int) :
option (eventval * world) :=
match w with World io vl vs => vl chunk id ofs end.

Definition nextworld_vstore (w: world) (chunk: memory_chunk) (id: ident) (ofs: int) (v: eventval):
option world :=
match w with World io vl vs => vs chunk id ofs v end.

A trace is possible in a given world if all events correspond to non-stuck external calls according to the given world. Two predicates are defined, for finite and infinite traces respectively:
• possible_trace w t w', where w is the initial state of the world, t the finite trace of interest, and w' the state of the world after performing trace t.
• possible_traceinf w T, where w is the initial state of the world and T the infinite trace of interest.

Inductive possible_event: world -> event -> world -> Prop :=
| possible_event_syscall: forall w1 evname evargs evres w2,
nextworld_io w1 evname evargs = Some (evres, w2) ->
possible_event w1 (Event_syscall evname evargs evres) w2
| possible_event_vload: forall w1 chunk id ofs evres w2,
nextworld_vload w1 chunk id ofs = Some (evres, w2) ->
possible_event w1 (Event_vload chunk id ofs evres) w2
| possible_event_vstore: forall w1 chunk id ofs evarg w2,
nextworld_vstore w1 chunk id ofs evarg = Some w2 ->
possible_event w1 (Event_vstore chunk id ofs evarg) w2
| possible_event_annot: forall w1 id args,
possible_event w1 (Event_annot id args) w1.

Inductive possible_trace: world -> trace -> world -> Prop :=
| possible_trace_nil: forall w,
possible_trace w E0 w
| possible_trace_cons: forall w1 ev w2 t w3,
possible_event w1 ev w2 -> possible_trace w2 t w3 ->
possible_trace w1 (ev :: t) w3.

Lemma possible_trace_app:
forall t2 w2 w0 t1 w1,
possible_trace w0 t1 w1 -> possible_trace w1 t2 w2 ->
possible_trace w0 (t1 ** t2) w2.
Proof.
induction 1; simpl; intros.
auto.
econstructor; eauto.
Qed.

Lemma possible_trace_app_inv:
forall t2 w2 t1 w0,
possible_trace w0 (t1 ** t2) w2 ->
exists w1, possible_trace w0 t1 w1 /\ possible_trace w1 t2 w2.
Proof.
induction t1; simpl; intros.
exists w0; split. constructor. auto.
inv H. exploit IHt1; eauto. intros [w1 [A B]].
exists w1; split. econstructor; eauto. auto.
Qed.

Lemma match_possible_traces:
forall (F V: Type) (ge: Genv.t F V) t1 t2 w0 w1 w2,
match_traces ge t1 t2 -> possible_trace w0 t1 w1 -> possible_trace w0 t2 w2 ->
t1 = t2 /\ w1 = w2.
Proof.
intros. inv H; inv H1; inv H0.
auto.
inv H7; inv H6. inv H9; inv H10. split; congruence.
inv H7; inv H6. inv H9; inv H10. split; congruence.
inv H4; inv H3. inv H6; inv H7. split; congruence.
inv H4; inv H3. inv H7; inv H6. auto.
Qed.

CoInductive possible_traceinf: world -> traceinf -> Prop :=
| possible_traceinf_cons: forall w1 ev w2 T,
possible_event w1 ev w2 ->
possible_traceinf w2 T ->
possible_traceinf w1 (Econsinf ev T).

Lemma possible_traceinf_app:
forall t2 w0 t1 w1,
possible_trace w0 t1 w1 -> possible_traceinf w1 t2 ->
possible_traceinf w0 (t1 *** t2).
Proof.
induction 1; simpl; intros.
auto.
econstructor; eauto.
Qed.

Lemma possible_traceinf_app_inv:
forall t2 t1 w0,
possible_traceinf w0 (t1 *** t2) ->
exists w1, possible_trace w0 t1 w1 /\ possible_traceinf w1 t2.
Proof.
induction t1; simpl; intros.
exists w0; split. constructor. auto.
inv H. exploit IHt1; eauto. intros [w1 [A B]].
exists w1; split. econstructor; eauto. auto.
Qed.

Ltac possibleTraceInv :=
match goal with
| [H: possible_trace _ E0 _ |- _] =>
inversion H; clear H; subst; possibleTraceInv
| [H: possible_trace _ (_ ** _) _ |- _] =>
let P1 := fresh "P" in
let w := fresh "w" in
let P2 := fresh "P" in
elim (possible_trace_app_inv _ _ _ _ H); clear H;
intros w [P1 P2];
possibleTraceInv
| [H: possible_traceinf _ (_ *** _) |- _] =>
let P1 := fresh "P" in
let w := fresh "w" in
let P2 := fresh "P" in
elim (possible_traceinf_app_inv _ _ _ H); clear H;
intros w [P1 P2];
possibleTraceInv
| [H: exists w, possible_trace _ _ w |- _] =>
let P := fresh "P" in let w := fresh "w" in
destruct H as [w P]; possibleTraceInv
| _ => idtac
end.

Definition possible_behavior (w: world) (b: program_behavior) : Prop :=
match b with
| Terminates t r => exists w', possible_trace w t w'
| Diverges t => exists w', possible_trace w t w'
| Reacts T => possible_traceinf w T
| Goes_wrong t => exists w', possible_trace w t w'
end.

CoInductive possible_traceinf': world -> traceinf -> Prop :=
| possible_traceinf'_app: forall w1 t w2 T,
possible_trace w1 t w2 -> t <> E0 ->
possible_traceinf' w2 T ->
possible_traceinf' w1 (t *** T).

Lemma possible_traceinf'_traceinf:
forall w T, possible_traceinf' w T -> possible_traceinf w T.
Proof.
cofix COINDHYP; intros. inv H. inv H0. congruence.
simpl. econstructor. eauto. apply COINDHYP.
inv H3. simpl. auto. econstructor; eauto. econstructor; eauto. unfold E0; congruence.
Qed.

# Definition and properties of deterministic semantics

Record sem_deterministic (L: semantics) := mk_deterministic {
det_step: forall s0 t1 s1 t2 s2,
Step L s0 t1 s1 -> Step L s0 t2 s2 -> s1 = s2 /\ t1 = t2;
det_initial_state: forall s1 s2,
initial_state L s1 -> initial_state L s2 -> s1 = s2;
det_final_state: forall s r1 r2,
final_state L s r1 -> final_state L s r2 -> r1 = r2;
det_final_nostep: forall s r,
final_state L s r -> Nostep L s
}.

Section DETERM_SEM.

Variable L: semantics.
Hypothesis DET: sem_deterministic L.

Ltac use_step_deterministic :=
match goal with
| [ S1: Step L _ ?t1 _, S2: Step L _ ?t2 _ |- _ ] =>
destruct (det_step L DET _ _ _ _ _ S1 S2) as [EQ1 EQ2]; subst
end.

Determinism for finite transition sequences.

Lemma star_step_diamond:
forall s0 t1 s1, Star L s0 t1 s1 ->
forall t2 s2, Star L s0 t2 s2 ->
exists t,
(Star L s1 t s2 /\ t2 = t1 ** t)
\/ (Star L s2 t s1 /\ t1 = t2 ** t).
Proof.
induction 1; intros.
exists t2; auto.
inv H2. exists (t1 ** t2); right.
split. econstructor; eauto. auto.
use_step_deterministic.
exploit IHstar. eexact H4. intros [t A]. exists t.
destruct A. left; intuition. traceEq. right; intuition. traceEq.
Qed.

Ltac use_star_step_diamond :=
match goal with
| [ S1: Star L _ ?t1 _, S2: Star L _ ?t2 _ |- _ ] =>
let t := fresh "t" in let P := fresh "P" in let EQ := fresh "EQ" in
destruct (star_step_diamond _ _ _ S1 _ _ S2)
as [t [ [P EQ] | [P EQ] ]]; subst
end.

Ltac use_nostep :=
match goal with
| [ S: Step L ?s _ _, NO: Nostep L ?s |- _ ] => elim (NO _ _ S)
end.

Lemma star_step_triangle:
forall s0 t1 s1 t2 s2,
Star L s0 t1 s1 ->
Star L s0 t2 s2 ->
Nostep L s2 ->
exists t,
Star L s1 t s2 /\ t2 = t1 ** t.
Proof.
intros. use_star_step_diamond.
exists t; auto.
inv P. exists E0. split. constructor. traceEq.
use_nostep.
Qed.

Ltac use_star_step_triangle :=
match goal with
| [ S1: Star L _ ?t1 _, S2: Star L _ ?t2 ?s2, NO: Nostep L ?s2 |- _ ] =>
let t := fresh "t" in let P := fresh "P" in let EQ := fresh "EQ" in
destruct (star_step_triangle _ _ _ _ _ S1 S2 NO)
as [t [P EQ]]; subst
end.

Lemma steps_deterministic:
forall s0 t1 s1 t2 s2,
Star L s0 t1 s1 -> Star L s0 t2 s2 ->
Nostep L s1 -> Nostep L s2 ->
t1 = t2 /\ s1 = s2.
Proof.
intros. use_star_step_triangle. inv P.
split; auto; traceEq. use_nostep.
Qed.

Lemma terminates_not_goes_wrong:
forall s t1 s1 r t2 s2,
Star L s t1 s1 -> final_state L s1 r ->
Star L s t2 s2 -> Nostep L s2 ->
(forall r, ~final_state L s2 r) -> False.
Proof.
intros.
assert (t1 = t2 /\ s1 = s2).
eapply steps_deterministic; eauto. eapply det_final_nostep; eauto.
destruct H4; subst. elim (H3 _ H0).
Qed.

Determinism for infinite transition sequences.

Lemma star_final_not_forever_silent:
forall s t s', Star L s t s' ->
Nostep L s' ->
Forever_silent L s -> False.
Proof.
induction 1; intros.
inv H0. use_nostep.
inv H3. use_step_deterministic. eauto.
Qed.

Lemma star2_final_not_forever_silent:
forall s t1 s1 t2 s2,
Star L s t1 s1 -> Nostep L s1 ->
Star L s t2 s2 -> Forever_silent L s2 ->
False.
Proof.
intros. use_star_step_triangle.
eapply star_final_not_forever_silent. eexact P. eauto. auto.
Qed.

Lemma star_final_not_forever_reactive:
forall s t s', Star L s t s' ->
forall T, Nostep L s' -> Forever_reactive L s T -> False.
Proof.
induction 1; intros.
inv H0. inv H1. congruence. use_nostep.
inv H3. inv H4. congruence.
use_step_deterministic.
eapply IHstar with (T := t4 *** T0). eauto.
eapply star_forever_reactive; eauto.
Qed.

Lemma star_forever_silent_inv:
forall s t s', Star L s t s' ->
Forever_silent L s ->
t = E0 /\ Forever_silent L s'.
Proof.
induction 1; intros.
auto.
subst. inv H2. use_step_deterministic. eauto.
Qed.

Lemma forever_silent_reactive_exclusive:
forall s T,
Forever_silent L s -> Forever_reactive L s T -> False.
Proof.
intros. inv H0. exploit star_forever_silent_inv; eauto.
Qed.

Lemma forever_reactive_inv2:
forall s t1 s1, Star L s t1 s1 ->
forall t2 s2 T1 T2,
Star L s t2 s2 ->
t1 <> E0 -> t2 <> E0 ->
Forever_reactive L s1 T1 ->
Forever_reactive L s2 T2 ->
exists s', exists t, exists T1', exists T2',
t <> E0 /\
Forever_reactive L s' T1' /\
Forever_reactive L s' T2' /\
t1 *** T1 = t *** T1' /\
t2 *** T2 = t *** T2'.
Proof.
induction 1; intros.
congruence.
inv H2. congruence. use_step_deterministic.
destruct t3.
simpl in *. eapply IHstar; eauto.
exists s5; exists (e :: t3);
exists (t2 *** T1); exists (t4 *** T2).
split. unfold E0; congruence.
split. eapply star_forever_reactive; eauto.
split. eapply star_forever_reactive; eauto.
split; traceEq.
Qed.

Lemma forever_reactive_determ':
forall s T1 T2,
Forever_reactive L s T1 ->
Forever_reactive L s T2 ->
traceinf_sim' T1 T2.
Proof.
cofix COINDHYP; intros.
inv H. inv H0.
destruct (forever_reactive_inv2 _ _ _ H t s2 T0 T)
as [s' [t' [T1' [T2' [A [B [C [D E]]]]]]]]; auto.
rewrite D; rewrite E. constructor. auto.
eapply COINDHYP; eauto.
Qed.

Lemma forever_reactive_determ:
forall s T1 T2,
Forever_reactive L s T1 ->
Forever_reactive L s T2 ->
traceinf_sim T1 T2.
Proof.
intros. apply traceinf_sim'_sim. eapply forever_reactive_determ'; eauto.
Qed.

Lemma star_forever_reactive_inv:
forall s t s', Star L s t s' ->
forall T, Forever_reactive L s T ->
exists T', Forever_reactive L s' T' /\ T = t *** T'.
Proof.
induction 1; intros.
exists T; auto.
inv H2. inv H3. congruence.
use_step_deterministic.
exploit IHstar. eapply star_forever_reactive. 2: eauto. eauto.
intros [T' [A B]]. exists T'; intuition. traceEq. congruence.
Qed.

Lemma forever_silent_reactive_exclusive2:
forall s t s' T,
Star L s t s' -> Forever_silent L s' ->
Forever_reactive L s T ->
False.
Proof.
intros. exploit star_forever_reactive_inv; eauto.
intros [T' [A B]]. subst T.
eapply forever_silent_reactive_exclusive; eauto.
Qed.

Determinism for program executions

Lemma state_behaves_deterministic:
forall s beh1 beh2,
state_behaves L s beh1 -> state_behaves L s beh2 -> beh1 = beh2.
Proof.
generalize (det_final_nostep L DET); intro dfns.
intros until beh2; intros BEH1 BEH2.
inv BEH1; inv BEH2.
assert (t = t0 /\ s' = s'0). eapply steps_deterministic; eauto.
destruct H3. f_equal; auto. subst. eapply det_final_state; eauto.
byContradiction. eapply star2_final_not_forever_silent with (s1 := s') (s2 := s'0); eauto.
byContradiction. eapply terminates_not_goes_wrong with (s1 := s') (s2 := s'0); eauto.
byContradiction. eapply star2_final_not_forever_silent with (s2 := s') (s1 := s'0); eauto.
f_equal. use_star_step_diamond.
exploit star_forever_silent_inv. eexact P. eauto.
intros [A B]. subst; traceEq.
exploit star_forever_silent_inv. eexact P. eauto.
intros [A B]. subst; traceEq.
byContradiction. eapply star2_final_not_forever_silent with (s1 := s'0) (s2 := s'); eauto.
f_equal. apply traceinf_extensionality.
eapply forever_reactive_determ; eauto.
byContradiction. eapply terminates_not_goes_wrong with (s1 := s'0) (s2 := s'); eauto.
byContradiction. eapply star2_final_not_forever_silent with (s1 := s') (s2 := s'0); eauto.
assert (t = t0 /\ s' = s'0). eapply steps_deterministic; eauto.
destruct H5. congruence.
Qed.

Theorem program_behaves_deterministic:
forall beh1 beh2,
program_behaves L beh1 -> program_behaves L beh2 ->
beh1 = beh2.
Proof.
intros until beh2; intros BEH1 BEH2. inv BEH1; inv BEH2.
assert (s = s0) by (eapply det_initial_state; eauto). subst s0.
eapply state_behaves_deterministic; eauto.
elim (H1 _ H).
elim (H _ H0).
auto.
Qed.

End DETERM_SEM.

# Integrating an external world in a semantics.

Given a transition semantics, we can build another semantics that integrates an external world in its state and allows only world-possible transitions.

Section WORLD_SEM.

Variable L: semantics.
Variable initial_world: world.

Notation "s #1" := (fst s) (at level 9, format "s '#1'") : pair_scope.
Notation "s #2" := (snd s) (at level 9, format "s '#2'") : pair_scope.
Local Open Scope pair_scope.

Definition world_sem : semantics := @Semantics
(state L * world)%type
(funtype L)
(vartype L)
(fun ge s t s' => step L ge s#1 t s'#1 /\ possible_trace s#2 t s'#2)
(fun s => initial_state L s#1 /\ s#2 = initial_world)
(fun s r => final_state L s#1 r)
(globalenv L).

If the original semantics is determinate, the world-aware semantics is deterministic.

Hypothesis D: determinate L.

Theorem world_sem_deterministic: sem_deterministic world_sem.
Proof.
constructor; simpl; intros.
destruct H; destruct H0.
exploit (sd_determ D). eexact H. eexact H0. intros [A B].
exploit match_possible_traces; eauto. intros [EQ1 EQ2]. subst t2.
split; auto.
rewrite (surjective_pairing s1). rewrite (surjective_pairing s2). intuition congruence.
destruct H; destruct H0.
rewrite (surjective_pairing s1). rewrite (surjective_pairing s2). decEq.
eapply (sd_initial_determ D); eauto.
congruence.
eapply (sd_final_determ D); eauto.
red; simpl; intros. red; intros [A B]. exploit (sd_final_nostep D); eauto.
Qed.

End WORLD_SEM.