| import Mathlib |
| import OAI.MathematicalPhysics.RecorderFlows.Observation |
|
|
| namespace OAI |
|
|
| /-! Finite oracle programs and effective mixed-derivative evaluation. -/ |
|
|
| namespace Solenoidal |
| namespace Effective |
| inductive OracleCode |
| | zero | succ | left | right | query |
| | pair (f g : OracleCode) |
| | comp (f g : OracleCode) |
| | prec (f g : OracleCode) |
| | rfind (f : OracleCode) |
|
|
| |
| noncomputable def OracleCode.eval (o : ℕ → ℕ) : OracleCode → ℕ → Part ℕ |
| | .zero => fun _ => .some 0 |
| | .succ => fun n => .some (n + 1) |
| | .left => fun n => .some n.unpair.1 |
| | .right => fun n => .some n.unpair.2 |
| | .query => fun n => .some (o n) |
| | .pair f g => fun n => Nat.pair <$> f.eval o n <*> g.eval o n |
| | .comp f g => fun n => g.eval o n >>= f.eval o |
| | .prec f g => fun n => |
| let (a, k) := n.unpair |
| k.rec (f.eval o a) fun j ih => do |
| let b ← ih |
| g.eval o (Nat.pair a (Nat.pair j b)) |
| | .rfind f => fun a => Nat.rfind fun n => (fun k => k = 0) <$> f.eval o (Nat.pair a n) |
|
|
| |
| def decodeRat (n : ℕ) : ℚ := (Encodable.decode (α := ℚ) n).getD 0 |
|
|
| noncomputable def tolerance (n : ℕ) : ℝ := ((2 : ℝ) ^ n)⁻¹ |
|
|
| def Names (x : ℝ) (o : ℕ → ℕ) : Prop := |
| ∀ n : ℕ, |(decodeRat (o n) : ℝ) - x| ≤ tolerance n |
|
|
| def ComputableReal (x : ℝ) : Prop := |
| ∃ c : Nat.Partrec.Code, ∀ n : ℕ, ∃ a : ℕ, |
| a ∈ c.eval n ∧ |(decodeRat a : ℝ) - x| ≤ tolerance n |
|
|
| |
|
|
| def NamesArguments (ν : ℝ) (p : Fluid.SpaceTime) (o : ℕ → ℕ) : Prop := |
| Names ν (fun n => o (Nat.pair 0 n)) ∧ |
| ∀ i : Fin 4, Names (p i) (fun n => o (Nat.pair (i.val + 1) n)) |
|
|
| |
| def machineCode (M : Machine) : ℕ := |
| let qs := List.finRange (M.states + 1) |
| let as := List.finRange (M.symbols + 1) |
| let moves : Move → ℕ := fun | .left => 0 | .stay => 1 | .right => 2 |
| let table := qs.flatMap fun q => as.map fun a => |
| let r := M.table q a |
| (r.1.val, r.2.1.val, moves r.2.2) |
| Encodable.encode (M.states, M.symbols, M.initial.val, M.blank.val, qs.map M.halt, table) |
|
|
| def wordCode {M : Machine} (w : List M.Symbol) : ℕ := |
| Encodable.encode (w.map Fin.val) |
|
|
| def request (M : Machine) (w : List M.Symbol) (a : List (Fin 4)) (j : Fin 3) (n : ℕ) : ℕ := |
| Encodable.encode (machineCode M, wordCode w, a.map Fin.val, j.val, n) |
|
|
| def boundRequest (M : Machine) (w : List M.Symbol) (a : List (Fin 4)) (R : ℕ) : ℕ := |
| Encodable.encode (machineCode M, wordCode w, a.map Fin.val, R) |
|
|
| abbrev ForceFamily := (ν : ℝ) → (M : Machine) → List M.Symbol → Fluid.Field |
|
|
| |
|
|
| def RelativeEffective (F : ForceFamily) (eval bounds : OracleCode) : Prop := |
| (∀ (ν : ℝ), 0 < ν → ∀ (M : Machine) (w : List M.Symbol) (p : Fluid.SpaceTime), |
| 0 ≤ p 0 → ∀ o : ℕ → ℕ, NamesArguments ν p o → |
| ∀ (a : List (Fin 4)) (j : Fin 3) (n : ℕ), ∃ r : ℕ, |
| r ∈ eval.eval o (request M w a j n) ∧ |
| |(decodeRat r : ℝ) - Fluid.mixedD a (F ν M w) p j| ≤ tolerance n) ∧ |
| (∀ (ν : ℝ), 0 < ν → ∀ o : ℕ → ℕ, Names ν o → |
| ∀ (M : Machine) (w : List M.Symbol) (a : List (Fin 4)) (R : ℕ), ∃ C : ℕ, |
| C ∈ bounds.eval o (boundRequest M w a R) ∧ |
| ∀ p : Fluid.SpaceTime, 0 ≤ p 0 → (∀ i, |p i| ≤ (R : ℝ)) → |
| ‖Fluid.mixedD a (F ν M w) p‖ ≤ (C : ℝ)) |
|
|
| |
|
|
| def SuppliesArgument (c : Nat.Partrec.Code) (p : Fluid.SpaceTime) : Prop := |
| ∀ (i : Fin 4) (n : ℕ), ∃ r : ℕ, r ∈ c.eval (Nat.pair i.val n) ∧ |
| |(decodeRat r : ℝ) - p i| ≤ tolerance n |
|
|
| |
|
|
| def EffectiveAt (F : ForceFamily) (ν : ℝ) : Prop := |
| ∃ eval bounds : Nat.Partrec.Code, |
| (∀ (M : Machine) (w : List M.Symbol) (p : Fluid.SpaceTime), 0 ≤ p 0 → |
| ∀ c : Nat.Partrec.Code, SuppliesArgument c p → |
| ∀ (a : List (Fin 4)) (j : Fin 3) (n : ℕ), ∃ r : ℕ, |
| r ∈ eval.eval (Nat.pair (request M w a j n) (Encodable.encode c)) ∧ |
| |(decodeRat r : ℝ) - Fluid.mixedD a (F ν M w) p j| ≤ tolerance n) ∧ |
| (∀ (M : Machine) (w : List M.Symbol) (a : List (Fin 4)) (R : ℕ), ∃ C : ℕ, |
| C ∈ bounds.eval (boundRequest M w a R) ∧ |
| ∀ p : Fluid.SpaceTime, 0 ≤ p 0 → (∀ i, |p i| ≤ (R : ℝ)) → |
| ‖Fluid.mixedD a (F ν M w) p‖ ≤ (C : ℝ)) |
| end Effective |
| end Solenoidal |
|
|
| end OAI |
|
|
