An MIU Decision Procedure in Lean #
The MIU formal system was introduced by Douglas
Hofstadter in the first chapter of his 1979 book,
Gödel, Escher, Bach.
The system is defined by four rules of inference, one axiom, and an alphabet of three symbols:
M
, I
, and U
.
Hofstadter's central question is: can the string "MU"
be derived?
It transpires that there is a simple decision procedure for this system. A string is derivable if
and only if it starts with M
, contains no other M
s, and the number of I
s in the string is
congruent to 1 or 2 modulo 3.
The principal aim of this project is to give a Lean proof that the derivability of a string is a decidable predicate.
The MIU System #
In Hofstadter's description, an atom is any one of M
, I
or U
. A string is a finite
sequence of zero or more symbols. To simplify notation, we write a sequence [I,U,U,M]
,
for example, as IUUM
.
The four rules of inference are:
- xI → xIU,
- Mx → Mxx,
- xIIIy → xUy,
- xUUy → xy,
where the notation α → β is to be interpreted as 'if α is derivable, then β is derivable'.
Additionally, he has an axiom:
MI
is derivable.
In Lean, it is natural to treat the rules of inference and the axiom on an equal footing via an
inductive data type Derivable
designed so that Derivable x
represents the notion that the string
x
can be derived from the axiom by the rules of inference. The axiom is represented as a
nonrecursive constructor for Derivable
. This mirrors the translation of Peano's axiom '0 is a
natural number' into the nonrecursive constructor zero
of the inductive type Nat
.
References #
- [Jeremy Avigad, Leonardo de Moura and Soonho Kong, Theorem Proving in Lean] [AdMK17]
- Douglas R Hofstadter, Gödel, Escher, Bach
Tags #
miu, derivable strings
The atoms of MIU can be represented as an enumerated type in Lean.
- M: Miu.MiuAtom
- I: Miu.MiuAtom
- U: Miu.MiuAtom
Instances For
Equations
- Miu.instDecidableEqMiuAtom x y = if h : x.toCtorIdx = y.toCtorIdx then isTrue ⋯ else isFalse ⋯
The annotation deriving DecidableEq
above indicates that Lean will automatically derive that
MiuAtom
is an instance of DecidableEq
. The use of deriving
is crucial in this project and will
lead to the automatic derivation of decidability.
We show that the type MiuAtom
is inhabited, giving M
(for no particular reason) as the default
element.
Equations
- Miu.miuAtomInhabited = { default := Miu.MiuAtom.M }
MiuAtom.repr
is the 'natural' function from MiuAtom
to String
.
Equations
- Miu.MiuAtom.M.repr = "M"
- Miu.MiuAtom.I.repr = "I"
- Miu.MiuAtom.U.repr = "U"
Instances For
Using MiuAtom.repr
, we prove that MiuAtom
is an instance of Repr
.
Equations
- Miu.instReprMiuAtom = { reprPrec := fun (u : Miu.MiuAtom) (x : ℕ) => Std.Format.text u.repr }
Equations
- Miu.instMembershipMiuAtomMiustr = id inferInstance
For display purposes, an Miustr
can be represented as a String
.
Equations
- Miu.Miustr.mrepr [] = ""
- Miu.Miustr.mrepr (c :: cs) = c.repr ++ Miu.Miustr.mrepr cs
Instances For
Equations
- Miu.miurepr = { reprPrec := fun (u : Miu.Miustr) (x : ℕ) => Std.Format.text u.mrepr }
In the other direction, we set up a coercion from String
to Miustr
.
Equations
- Miu.lcharToMiustr [] = []
- Miu.lcharToMiustr ('M' :: cs) = Miu.MiuAtom.M :: Miu.lcharToMiustr cs
- Miu.lcharToMiustr ('I' :: cs) = Miu.MiuAtom.I :: Miu.lcharToMiustr cs
- Miu.lcharToMiustr ('U' :: cs) = Miu.MiuAtom.U :: Miu.lcharToMiustr cs
- Miu.lcharToMiustr (c :: cs) = []
Instances For
Equations
- Miu.stringCoeMiustr = { coe := fun (st : String) => Miu.lcharToMiustr st.data }
Derivability #
The inductive type Derivable
has five constructors. The nonrecursive constructor mk
corresponds
to Hofstadter's axiom that "MI"
is derivable. Each of the constructors r1
, r2
, r3
, r4
corresponds to the one of Hofstadter's rules of inference.
- mk: Miu.Derivable (Miu.lcharToMiustr "MI".data)
- r1: ∀ {x : Miu.Miustr}, Miu.Derivable (x ++ [Miu.MiuAtom.I]) → Miu.Derivable (x ++ [Miu.MiuAtom.I, Miu.MiuAtom.U])
- r2: ∀ {x : List Miu.MiuAtom}, Miu.Derivable (Miu.MiuAtom.M :: x) → Miu.Derivable (Miu.MiuAtom.M :: x ++ x)
- r3: ∀ {x y : Miu.Miustr}, Miu.Derivable (x ++ [Miu.MiuAtom.I, Miu.MiuAtom.I, Miu.MiuAtom.I] ++ y) → Miu.Derivable (x ++ Miu.MiuAtom.U :: y)
- r4: ∀ {x y : Miu.Miustr}, Miu.Derivable (x ++ [Miu.MiuAtom.U, Miu.MiuAtom.U] ++ y) → Miu.Derivable (x ++ y)