Documentation

Mathlib.Computability.Encoding

Encodings #

This file contains the definition of a (finite) encoding, a map from a type to strings in an alphabet, used in defining computability by Turing machines. It also contains several examples:

Examples #

structure Computability.Encoding (α : Type u) :
Type (max u (v + 1))

An encoding of a type in a certain alphabet, together with a decoding.

  • Γ : Type v
  • encode : αList self
  • decode : List selfOption α
  • decode_encode : ∀ (x : α), self.decode (self.encode x) = some x
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    theorem Computability.Encoding.decode_encode {α : Type u} (self : Computability.Encoding α) (x : α) :
    self.decode (self.encode x) = some x
    structure Computability.FinEncoding (α : Type u) extends Computability.Encoding :
    Type (max 1 u)

    An encoding plus a guarantee of finiteness of the alphabet.

    • Γ : Type
    • encode : αList self
    • decode : List selfOption α
    • decode_encode : ∀ (x : α), self.decode (self.encode x) = some x
    • ΓFin : Fintype self
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      A standard Turing machine alphabet, consisting of blank,bit0,bit1,bra,ket,comma.

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        An arbitrary section of the natural inclusion of bool in Γ'.

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          An encoding function of the binary numbers in bool.

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            An encoding function of ℕ in bool.

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              A decoding function from List Bool to the positive binary numbers.

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                A decoding function from List Bool to the binary numbers.

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                  A decoding function from List Bool to ℕ.

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                    A binary encoding of ℕ in bool.

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                      A binary fin_encoding of ℕ in bool.

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                        A binary encoding of ℕ in Γ'.

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                          A unary encoding function of ℕ in bool.

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                            A unary decoding function from List Bool to ℕ.

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                              A unary fin_encoding of ℕ.

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                                An encoding function of bool in bool.

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                                  A decoding function from List Bool to bool.

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                                    A fin_encoding of bool in bool.

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