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Lambda calculus & proof theory

A proof-of-concept implementation of the most fundamental concepts of lambda calculus:

  • nameless lambda terms with DeBruijn indexing (compressed representation)
  • lambda term parser (into intermediate named lambda terms, converted to nameless terms)
  • beta-reduction
  • type inference (Algorithm W) (!)
  • binary encoding/decoding
  • Church numerals, booleans & general practical uses

Example usage:

> :load lcpt.hs

The LCPT module serves as to unify the exported functionality of the other modules. It is recommended to import LCPT, rather than any of the others.

Parsing lambda terms

The parsing function returns an optional compressed nameless lambda term.

> :t ps
ps :: String -> Maybe Lambda
> let (Just t) = ps "lambda[x,y,z]xz(yz)"
> t
lambda[u,v,w]uw(vw)

The parsed format resembles closely the format used by humans - variables can only be named one of x,y,z,u,v,w (perhaps followed by a number), abstraction can bind multiple variables ("lambda[x,y].." can be used instead of "lambda[x]lambda[y]") and application of terms is done by simply concatenating them, without any other syntactic constructs. Of course, brackets can be placed at will and will be taken into account.

Nameless lambda terms - implementation details

The nameless lambda terms are compressed, meaning:

  • repeated abstraction lambda[x]lambda[y].. is internally represented as one node in the tree, with the number of successively bound variables
  • function application on a term on multiple arguments is also represented as one node, containing a list of terms (constructed with the Ap constructor). The tail of this list are the arguments, applied to the head, f.e. Ap [s,k,k] is equivalent to ((s k) k)
  • standard DeBruijn indexing is used, meaning alpha-equivalence comes free as pure syntactic equivalence

Beta reduction

> :t beta
beta :: Lambda -> Lambda
> let t2 = Ap [s,k,k]
> beta t2
λ[u]u
> i == beta t2
True
> w == beta w
True

Some standard combinators are exposed for convenience:

  • the identity function I = λ[u]u (exported as i)
  • the constant functions K = λ[u,v]u and K* = λ[u,v]v (k and ks)
  • the substitution operator S = λ[u,v,w]uw(vw) (s)
  • the irreducible terms ω = λ[u]uu and Ω = ωω = (λ[u]uu)(λ[u]uu) (exported as w and om)
  • the fixed-point combinator Y = λ[u](λ[v]u(vv))(λ[v]u(vv)) (y)
  • Chris Barker's iota combinator J (j) = L 1 (Ap [Var 0, s, k]):
> i == beta (Ap [j,j])
True
> ks == beta (Ap [j, Ap[j,j]])
True
> k == beta (Ap [j, Ap [j, Ap [j,j]]])
True
> s == beta (Ap [j, Ap [j, Ap [j, Ap [j,j]]]])
True

Beta reduction in details

  • the function betaStep performs one step of beta-reduction, if possible
  • beta performs a full reduction, but for ease-of-use acts as the identity on irreducibles
  • betaSteps gives a list of all intermediate steps
  • betaSteps_ prints all intermediate steps

Type inference

An implementation of the classic Algorithm W for type inference, adapted for compressed nameless lambda terms from this repo.

As expected, beta reduction preserves the type. The function infer does the heavy lifting, whereas infer_ simply pretty-prints the result. Upon failure, a detailed error message is printed.

> :t infer
infer :: Lambda -> Either String Type
> :t infer_
infer_ :: Lambda -> IO ()
> infer_ i
λ[u]u :: a -> a

> infer_ s
λ[u,v,w]uw(vw) :: (a -> b -> c) -> (a -> b) -> a -> c

> let t = Ap [s,k,k]
> infer_ t
(λ[u,v,w]uw(vw))(λ[u,v]u)(λ[u,v]u) :: a -> a

> mapM_ infer_ (betaSteps t)
(λ[u,v,w]uw(vw))(λ[u,v]u)(λ[u,v]u) :: a -> a

(λ[u,v](λ[w,x]w)v(uv))(λ[u,v]u) :: a -> a

λ[u](λ[v,w]v)u((λ[v,w]v)u) :: a -> a

λ[u](λ[v]u)((λ[v,w]v)u) :: a -> a

λ[u]u :: a -> a

> infer_ w
λ[u]uu
  occurs check fails: a vs. a -> b
  in uu
  in λ[u]uu

Binary encoding/decoding

[WIP]

Church numerals, booleans & arithmetic

[WIP]