Post's problem

theories/PostsProblem/limit_computability.v

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+From SyntheticComputability Require Import ArithmeticalHierarchySemantic PostsTheorem reductions SemiDec TuringJump OracleComputability Definitions partial Pigeonhole.
+
+Require Import stdpp.list Vectors.VectorDef Arith.Compare_dec Lia.
+Import Vector.VectorNotations.
+
+Local Notation vec := Vector.t.
+
+
+(* ########################################################################## *)
+(** * Limit Computability *)
+(* ########################################################################## *)
+
+(** This file contains the definition of limit computable and proves the
+Limit Lemma, i.e., limit computable is equivalence to reduciable to halting
+problem.
+
+Convention:
+
+   limit: limit computable
+   semi_dec(_K): semi decidable (on K)
+   turing_red_K: turing reduciable to halting problem
+   char[_X]: extensionality of X
+
+ **)
+
+  (* Definition of limit ciomputable *)
+
+  Definition limit_computable {X} (P: X → Prop) :=
+    ∃ f: X → nat → bool, ∀ x,
+      (P x ↔ ∃ N, ∀ n, n ≥ N → f x n = true) ∧
+        (¬ P x ↔ ∃ N, ∀ n, n ≥ N → f x n = false).
+
+  Definition char_rel_limit_computable {X} (P: X → bool → Prop) :=
+    ∃ f: X → nat → bool, ∀ x y, P x y ↔ ∃ N, ∀ n, n ≥ N → f x n = y.
+
+  Definition char_rel_limit_computable' {X} (P: X → bool → Prop) :=
+    ∃ f: X → nat → bool, ∀ x y, P x y → ∃ N, ∀ n, n ≥ N → f x n = y.    
+
+  Lemma char_rel_limit_equiv {X} (P: X → Prop):
+    char_rel_limit_computable (char_rel P) ↔ limit_computable P.
+  Proof.
+    split; intros [f Hf]; exists f; intros x.
+    - split; firstorder.
+    - intros []; destruct (Hf x) as [h1 h2]; eauto.
+  Qed.
+
+  Lemma char_rel_limit_equiv' {X} (P: X → Prop):
+    definite P → char_rel_limit_computable (char_rel P) ↔ char_rel_limit_computable' (char_rel P) .
+  Proof.
+    intros HP; split.
+    - intros [f Hf]. exists f; intros.
+      destruct (Hf x y) as [Hf' _].
+      now apply Hf'.
+    - intros [f Hf]. exists f. intros x y.
+      split. intro H. now apply Hf.
+      intros [N HN]. destruct (HP x).
+      destruct y; [easy|].
+      destruct (Hf x true H) as [N' HfN].
+      intros _. enough (true = false) by congruence.
+      specialize (HN (max N N')).
+      specialize (HfN (max N N')).
+      rewrite <- HN, <- HfN; eauto; lia.
+      destruct y; [|easy].
+      destruct (Hf x false H) as [N' HfN]. 
+      enough (true = false) by congruence.
+      specialize (HN (max N N')).
+      specialize (HfN (max N N')).
+      rewrite <- HN, <- HfN; eauto; lia.
+  Qed.
+
+
+  (* Naming the halting problem as K *)
+  Notation K := (­{0}^(1)).
+
+
+Section LimitLemma1.
+  (* Limit computable predicate P is reduciable to K *)
+
+  Variable vec_to_nat : ∀ k, vec nat k → nat.
+  Variable nat_to_vec : ∀ k, nat → vec nat k.
+  Variable vec_nat_inv : ∀ k v, nat_to_vec k (vec_to_nat v) = v.
+  Variable nat_vec_inv : ∀ k n, vec_to_nat (nat_to_vec k n) = n.
+
+  Variable list_vec_to_nat : ∀ k, list (vec nat k) → nat.
+  Variable nat_to_list_vec : ∀ k, nat → list (vec nat k).
+  Variable list_vec_nat_inv : ∀ k v, nat_to_list_vec k (list_vec_to_nat v) = v.
+  Variable nat_list_vec_inv : ∀ k n, list_vec_to_nat (nat_to_list_vec k n) = n.
+
+  Variable nat_to_list_bool : nat → list bool.
+  Variable list_bool_to_nat : list bool → nat.
+  Variable list_bool_nat_inv : ∀ l, nat_to_list_bool (list_bool_to_nat l) = l.
+  Variable nat_list_bool_inv : ∀ n, list_bool_to_nat (nat_to_list_bool n) = n.
+
+
+  Section def_K.
+
+    Hypothesis LEM_Σ_1: LEM_Σ 1.
+
+    Lemma semi_dec_def {X} (p: X → Prop):
+      semi_decidable p → definite p.
+    Proof.
+      intros [f Hf]. unfold semi_decider in Hf.
+      destruct level1 as (_&H2&_).
+      assert principles.LPO as H by now rewrite <- H2.
+      intro x. destruct (H (f x)).
+      left. now rewrite Hf.
+      right. intros [k Hk]%Hf.
+      apply H0. now exists k.
+    Qed.
+
+    Lemma def_K: definite K.
+    Proof.
+      apply semi_dec_def. 
+      assert (isΣsem 1 (@jumpNK _ 1 1)).
+      eapply jump_in_Σn; eauto.
+      assert (@jumpNK _ 1 1 ≡ₘ ­{0}^(1)).
+      apply jumpNKspec.
+      rewrite <- semi_dec_iff_Σ1 in H.
+      destruct H0 as [_ [f Hf]].
+      unfold reduces_m in Hf.
+      destruct H as [g Hg].
+      unfold semi_decider in Hg.
+      exists (fun x => g (f x)).
+      split. now intros H%Hf%Hg. now intros H%Hg%Hf.
+    Qed.
+
+  End def_K.
+
+  (* Extensionality of Σ2, i.e. P t iff ∃ x. ∀ y. f(x, y, t) = true *)
+
+  Lemma char_Σ2 {k: nat} (P: vec nat k → Prop) :
+    (∃ f: nat → nat → vec nat k → bool, ∀ x, P x ↔ (∃ n, ∀ m, f n m x = true)) →
+    isΣsem 2 P.
+  Proof.
+    intros [f H].
+    eapply isΣsemS_ with (p := fun v => ∀ y, f (hd v) y (tl v) = true).
+    eapply isΠsemS_ with (p := fun v => f (hd (tl v)) (hd v) (tl (tl v)) = true).
+    eapply isΣsem0. all: easy.
+  Qed.
+
+  Lemma limit_Σ2 {k: nat} (P: vec nat k → Prop) :
+    limit_computable P → isΣsem 2 P ∧ isΣsem 2 (compl P).
+  Proof.
+    intros [f H]; split; eapply char_Σ2.
+    - exists (fun N n x => if lt_dec n N then true else f x n).
+      intro w. destruct (H w) as [-> _]; split; intros [N Hn]; exists N.
+      + intro m. destruct (lt_dec m N); try apply Hn; lia.
+      + intros n He. specialize (Hn n); destruct (lt_dec n N); auto; lia.
+    - exists (fun N n x => if lt_dec n N then true else negb (f x n)).
+      intro w. destruct (H w) as [_ ->]; split; intros [N Hn]; exists N.
+      + intro m. destruct (lt_dec m N); [auto| rewrite (Hn m); lia].
+      + intros n He. specialize (Hn n).
+        destruct (lt_dec n N); auto; [lia|destruct (f w n); easy].
+  Qed.
+
+  Lemma limit_semi_dec_K {k: nat} (P: vec nat k → Prop) :
+    LEM_Σ 1 →
+    limit_computable P →
+    OracleSemiDecidable K P ∧
+    OracleSemiDecidable K (compl P).
+  Proof.
+    intros LEM H%limit_Σ2.
+    rewrite <- !(Σ_semi_decidable_jump).
+    all: eauto.
+  Qed.
+
+  Lemma limit_turing_red_K' {k: nat} (P: vec nat k → Prop) :
+    LEM_Σ 1 →
+    definite K →
+    limit_computable P →
+    P ⪯ᴛ K.
+  Proof.
+    intros LEM D H % (limit_semi_dec_K LEM); destruct H as [h1 h2].
+    apply PT; try assumption.
+    apply Dec.nat_eq_dec.
+  Qed.
+
+  Fact elim_vec (P: nat → Prop):
+    P ⪯ₘ (fun x: vec nat 1 => P (hd x)) .
+  Proof. exists (fun x => [x]). now intros x. Qed.
+
+    (** ** The Limit Lemma 1 *)
+
+  Lemma limit_turing_red_K {k: nat} (P: nat → Prop) :
+    LEM_Σ 1 →
+    limit_computable P →
+    P ⪯ᴛ K.
+  Proof.
+    intros Hc [h Hh].
+    specialize (def_K Hc) as Hk.
+    eapply Turing_transitive; last eapply (@limit_turing_red_K' 1); eauto.
+    eapply red_m_impl_red_T. apply elim_vec.
+    exists (fun v n => h (hd v) n).
+    intros x; split; 
+    destruct (Hh (hd x)) as [Hh1 Hh2]; eauto.
+  Qed.
+
+End LimitLemma1.
+
+Section Σ1Approximation.
+
+  (* Turing jump of a trivial decidable problem is semi decidable *)
+
+  Lemma semi_dec_halting : semi_decidable K.
+  Proof.
+    eapply OracleSemiDecidable_semi_decidable with (q := ­{0}).
+    - exists (λ n, match n with | O => true | _ => false end); intros [|n]; easy.
+    - eapply semidecidable_J.
+  Qed.
+
+
+  (* Stabilizing the semi decider allows the semi decider
+     to be used as a Σ1 approximation *)
+
+  Definition stable (f: nat → bool) := ∀ n m, n ≤ m → f n = true → f m = true.
+
+  Fixpoint stabilize_step {X} (f: X → nat → bool) x n :=
+    match n with
+    | O => false
+    | S n => if f x n then true else stabilize_step f x n
+    end.
+
+  Lemma stabilize {X} (P: X → Prop) :
+    semi_decidable P → ∃ f, semi_decider f P ∧ ∀ x, stable (f x).
+  Proof.
+    intros [f Hf].
+    exists (fun x n => stabilize_step f x n); split.
+    - intro x; split; intro h.
+      rewrite (Hf x) in h.
+      destruct h as [c Hc].
+      now exists (S c); cbn; rewrite Hc.
+      rewrite (Hf x).
+      destruct h as [c Hc].
+      induction c; cbn in Hc; [congruence|].
+      destruct (f x c) eqn: E; [now exists c|now apply IHc].
+    - intros x n m Lenm Hn.
+      induction Lenm; [trivial|].
+      cbn; destruct (f x m) eqn: E; [trivial|assumption].
+  Qed.
+
+  (* The Σ1 approximation output correct answers for arbitray list of questions *)
+  Definition approximation_list {A} (P: A → Prop) (f: A → bool) L :=
+    ∀ i, List.In i L → P i ↔ f i = true.
+
+  Definition approximation_Σ1 {A} (P: A → Prop) :=
+    ∃ P_ : nat → A → bool,
+    ∀ L, ∃ c, ∀ c', c' ≥ c → approximation_list P (P_ c') L.
+
+  Definition approximation_Σ1_strong {A} (P: A → Prop) :=
+     ∃ P_ : nat → A → bool,
+       (∀ L, ∃ c, ∀ c', c' ≥ c → approximation_list P (P_ c') L) ∧
+       (∀ tau q a, @interrogation _ _ _ bool tau (char_rel P) q a → ∃ n, ∀ m, m ≥ n → interrogation tau (fun q a => P_ m q = a) q a).
+
+  Definition approximation_Σ1_weak {A} (P: A → Prop) :=
+    ∃ P_ : nat → A → bool,
+      (∀ tau q a, @interrogation _ _ _ bool tau (char_rel P) q a → ∃ n, ∀ m, m ≥ n → interrogation tau (λ q a, P_ m q = a) q a).
+
+  Lemma semi_dec_approximation_Σ1 {X} (P: X → Prop) :
+    definite P →
+    semi_decidable P → approximation_Σ1 P.
+  Proof.
+    intros defP semiP; unfold approximation_Σ1, approximation_list.
+    destruct (stabilize semiP)  as [h [Hh HS]].
+    exists (fun n x => h x n). intro l. induction l as [|a l [c Hc]].
+    - exists 42; eauto.
+    - destruct (defP a) as [h1| h2].
+      + destruct (Hh a) as [H _].
+        destruct (H h1) as [N HN].
+        exists (max c N); intros c' Hc' e [->| He].
+        split; [intros _|easy].
+        eapply HS; [|eapply HN]; lia.
+        rewrite <- (Hc c'); [trivial|lia | assumption].
+      + exists c; intros c' Hc' e [->| He].
+        split; [easy| intros h'].
+        unfold semi_decider in Hh.
+        now rewrite Hh; exists c'.
+        rewrite Hc; eauto.
+  Qed.
+
+  Lemma semi_dec_approximation_Σ1_strong {X} (P: X → Prop) :
+    definite P →
+    semi_decidable P → approximation_Σ1_strong P.
+  Proof.
+    intros defP semiP.
+    destruct (semi_dec_approximation_Σ1 defP semiP) as [P_ HP_].
+    exists P_; split; [apply HP_|].
+    intros tau q ans Htau.
+    destruct (HP_ q) as [w Hw].
+    exists w. intros m Hm. rewrite interrogation_ext.
+    exact Htau. eauto.
+    intros q_ a H1.
+    specialize (Hw m Hm q_ H1).
+    unfold char_rel; cbn.
+    destruct a; eauto; split; intro h2.
+    intro h. rewrite Hw in h. congruence.
+    firstorder.
+  Qed.
+
+  Lemma approximation_Σ1_halting : definite K → approximation_Σ1 K.
+  Proof. now intros H; apply semi_dec_approximation_Σ1, semi_dec_halting. Qed.
+
+  Lemma approximation_Σ1_halting_strong: definite K → approximation_Σ1_strong K.
+  Proof. now intros H; apply semi_dec_approximation_Σ1_strong, semi_dec_halting. Qed.
+
+
+End Σ1Approximation.
+
+
+Section LimitLemma2.
+
+  (* A predicate P is reduciable to K if P is limit computable *)
+
+  Section Construction.
+
+    Variable f : nat → nat → bool.
+    Variable tau : nat → tree nat bool bool.
+    Hypothesis Hf: ∀ L, ∃ c, ∀ c', c' ≥ c → approximation_list K (f c') L.
+
+    Definition K_ n := fun i o => f n i = o.
+    Definition char_K_ n := fun i => ret (f n i).
+
+    Lemma dec_K_ n : decidable (λ i, f n i = true).
+    Proof.
+      exists (f n). easy.
+    Qed.
+
+    Lemma pcomputes_K_ n: pcomputes (char_K_ n) (λ i o, f n i = o).
+    Proof.
+      intros i o; split; intro H.
+      now apply ret_hasvalue_inv.
+      now apply ret_hasvalue'.
+    Qed.
+
+  End Construction.
+
+    (** ** The Limit Lemma 2 *)
+
+  Theorem turing_red_K_lim (P: nat → Prop) :
+    P ⪯ᴛ K →
+    definite K →
+    definite P →
+    limit_computable P.
+  Proof.
+    intros [F [H HF]] defK defP.
+    rewrite <- char_rel_limit_equiv.
+    destruct (approximation_Σ1_halting_strong defK) as [k_ [_ Hk_2]].
+    destruct H as [tau Htau].
+    pose (char_K_ n := char_K_ k_ n).
+    pose (K_ n := K_ k_ n).
+    pose (Phi x n := evalt_comp (tau x) (k_ n) n n).
+    assert (∀ x y, char_rel P x y → ∃ N : nat, ∀ n : nat, n ≥ N → (evalt_comp (tau x) (k_ n)) n n = Some (inr y)) as HL.
+    {
+    intros x y H.
+    rewrite HF in H.
+    rewrite Htau in H.
+    destruct H as (qs & ans & Hint & Out).
+    specialize (Hk_2 (tau x) qs ans Hint).
+    destruct Hk_2 as [nth Hnth].
+    assert (interrogation (tau x)
+              (fun (q : nat) (a : bool) => (k_ nth) q = a) qs ans) as Hnthbase.
+    eapply Hnth. lia.
+    edestruct (interrogation_evalt_comp_limit (tau x) k_ qs ans y) as [L Hlimt].
+    exists nth. intros. eapply Hnth. easy.
+    eapply Out.
+    exists L. intros. now apply Hlimt.
+    }
+    assert (∃ f, ∀ x y, char_rel P x y → ∃ N : nat, ∀ n : nat, n ≥ N → f x n = y) as [f HL'].
+    {
+      exists (λ x n, match (Phi x n) with
+               | Some (inr y) => y | _ => false end).
+    intros x y Hxy%HL.
+    destruct (Hxy) as [N HN].
+    exists N; intros.
+    unfold Phi. rewrite HN; eauto.
+    }
+    exists f. intros x y; split.
+    - now intros; apply HL'.
+    - intro H0. destruct y; cbn.
+      destruct (defP x); [easy|].
+      assert (char_rel P x false); [easy|].
+      apply HL' in H1.
+      destruct H0 as [N1 HN1].
+      destruct H1 as [N2 HN2].
+      specialize (HN1 (S (max N1 N2))).
+      specialize (HN2 (S (max N1 N2))).
+      enough (true = false) by congruence.
+      rewrite <- HN1, HN2; lia.
+      destruct (defP x); [|easy].
+      assert (char_rel P x true); [easy|].
+      apply HL' in H1.
+      destruct H0 as [N1 HN1].
+      destruct H1 as [N2 HN2].
+      specialize (HN1 (S (max N1 N2))).
+      specialize (HN2 (S (max N1 N2))).
+      enough (true = false) by congruence.
+      rewrite <- HN2, HN1; lia.
+  Qed.
+
+End LimitLemma2.