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1905fc7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.Computability.Logspace.Amplification
import OAI.Computability.Logspace.UniformEstimator
namespace OAI
/-!
A fixed-precision deterministic estimator and the equality L = RL = BPL.
From OpenAI, *Exact Derandomization of Logarithmic Space: L = RL = BPL*.
-/
set_option autoImplicit true
section
namespace ExactDerandomization.Umbra
open Analysis Relay Counter Rill Fret At Hie
open Arena Amber Song Fret.Lyric Reed Lu Est Net
open Grain Terrain Bank Helix Helix.Age Helix.Mold Helix.Foil Helix.Moon Helix.Night Helix.Orbit Chime Dome Scheme Finset Stage
open SiteX Line Fret.Seats
open Vessel Gamut
noncomputable section
open scoped Classical
variable {q w h:ℕ}
open Elect
namespace Scope.Sane
lemma shard (t:ℝ)(z:ℤ)(hp:in01 t)(hz:|(z:ℝ) - (2:ℝ)^6*t|≤2):
|((min 64 z.toNat:ℕ):ℝ)/64-t|≤1/32:=by
have h:=abs_le.1 hz; obtain ⟨h1,h2⟩:=hp
have he:(2:ℝ)^6=64:=by norm_num
rw [he] at h
apply abs_le.2
by_cases hc : z<0
· have hh:(z:ℝ)<0:=by exact_mod_cast hc
have hm:z.toNat=0:=by omega
rw [hm,min_eq_right (by omega),Nat.cast_zero,zero_div];constructor <;> linarith
have hm:(z:ℝ)=(z.toNat:ℝ):=by exact_mod_cast (show z= (z.toNat:ℤ) by omega)
by_cases hs:z.toNat<64
· rw [min_eq_right hs.le]; constructor<;>linarith
have hh:64 ≤ z.toNat:= le_of_not_gt hs
have hh':(64:ℝ) ≤ z.toNat:=by exact_mod_cast hh
rw [min_eq_left hh]
push_cast; constructor<;>linarith
lemma plump (r:Rock q w h) (x:Word) (s: Pool (r.n x)) :
let m:=r.matter0 x
in01 (m.oracleV s m.i):=by
intro m
let u:= m.view
unfold Matter.oracleV forecast current;dsimp only
split
· unfold Peek.activeV; unfold Axis.color
split;exact unit_mem _; exact ⟨le_rfl,by norm_num⟩
exact ⟨le_rfl,by norm_num⟩
theorem delivered (r:Rock q w h)(p:MintV)(v:Tags){k:ℕ} (hs:Sane r p v k)(hS:r.P.Space):
∃ f:PT ℕ ℕ, Kn f ∧ ∀ (x:Word) (s:Pool (r.n x)),
let m:=r.matter0 x
|r.survey f x s-m.oracleV s m.i| ≤ 1/32:=by
obtain ⟨f,hf,h⟩:=realization (prog0 p r v)
exact ⟨solve r v f,shake r v hf,
fun x s=> shard _ _ (plump r x s) (hs.bells0 x s hS (show Agent f r p v from h))⟩
end Scope.Sane
/-- One deterministic estimator works for every input and every finite sampling environment,
with additive error at most `1 / 32` for the normalized oracle value. -/
theorem fixed_precision_estimator (r:Rock q w h)(e:r.w.Proof)(hp:6≤r.pad)(he:e.T.bank≤Est.box r.pad)(hS:r.P.Space):
∃ f:PT ℕ ℕ, Kn f ∧ ∀ (x:Word) (s:Pool (r.n x)),
let m:=r.matter0 x
|r.survey f x s-m.oracleV s m.i| ≤ 1/32:=by
obtain ⟨p,v,k,h1,h2⟩:=Codex.enacted r 6 e
have h: Scope.Sane r p v k:=⟨h1,h2.ge,fun x=>⟨⟨e,r.prompts e hp he x⟩,h1.coast0⟩⟩
exact Scope.Sane.delivered r p v h hS
end
end ExactDerandomization.Umbra
namespace ExactDerandomization
open Analysis Relay Grain Hie Stage Net Umbra
/-- Every bounded-error probabilistic logspace language has a deterministic logspace decider. -/
theorem BPL_subset_L : BPL ⊆ L:=by
classical
intro A hA
obtain ⟨q,w,h,M,h1,c,k,hc,hx⟩:= hA
obtain ⟨cs,hz,hl⟩:=h1
let P:Base.Source q w h:=⟨M,cs,c,k⟩
obtain ⟨v,⟨p⟩⟩:= Wares.choose
obtain ⟨m,hm⟩:= exists_nat_gt p.T.bank
let r:Rock q w h:= ⟨P,v,max m 6⟩
have he:6≤r.pad:=le_max_right ..
have hr:p.T.bank ≤ Est.box r.pad:=by
have hh:m ≤ r.pad:=le_max_left ..
have hp : (r.pad:ℝ) ≤ Est.box r.pad :=by have := Est.box_big r.pad;linarith
refine (hm.trans_le (le_trans ?_ hp)).le; exact_mod_cast hh
have ht:r.P.Space:=hl
obtain ⟨f,hf,hQ⟩:= fixed_precision_estimator r p he hr ht
apply r.convince ht p he hr f hf (fun x s _=> hQ x s)
· intro x; exact (hx x).2.1
intro x; exact (hx x).2.2
/-- Deterministic, one-sided randomized, and two-sided randomized logarithmic space coincide. -/
theorem exact_logarithmic_space_derandomization :
L=RL ∧ RL=BPL:=by
have ht:= RL_subset_BPL.trans BPL_subset_L
exact ⟨Set.Subset.antisymm L_subset_RL ht,
Set.Subset.antisymm RL_subset_BPL (BPL_subset_L.trans L_subset_RL)⟩
end ExactDerandomization
end
end OAI
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