Elsa is a lambda calculus evaluator
elsa
is a tiny language designed to build
intuition about how the Lambda Calculus, or
more generally, computation-by-substitution works.
Rather than the usual interpreter that grinds
lambda terms down to values, elsa
aims to be
a light-weight proof checker that determines
whether, under a given sequence of definitions,
a particular term reduces to to another.
You can try elsa
online at this link
You can locally build and run elsa
by
elsa
with stack
.That is, to say
$ curl -sSL https://get.haskellstack.org/ | sh
$ git clone https://github.com/ucsd-progsys/elsa.git
$ cd elsa
$ stack install
elsa
programs look like:
-- id_0.lc
let id = \x -> x
let zero = \f x -> x
eval id_zero :
id zero
=d> (\x -> x) (\f x -> x) -- expand definitions
=a> (\z -> z) (\f x -> x) -- alpha rename
=b> (\f x -> x) -- beta reduce
=d> zero -- expand definitions
eval id_zero_tr :
id zero
=*> zero -- transitive reductions
When you run elsa
on the above, you should get the following output:
$ elsa ex1.lc
OK id_zero, id_zero_tr.
If instead you write a partial sequence of reductions, i.e. where the last term can still be further reduced:
-- succ_1_bad.lc
let one = \f x -> f x
let two = \f x -> f (f x)
let incr = \n f x -> f (n f x)
eval succ_one :
incr one
=d> (\n f x -> f (n f x)) (\f x -> f x)
=b> \f x -> f ((\f x -> f x) f x)
=b> \f x -> f ((\x -> f x) x)
Then elsa
will complain that
$ elsa ex2.lc
ex2.lc:11:7-30: succ_one can be further reduced
11 | =b> \f x -> f ((\x -> f x) x)
^^^^^^^^^^^^^^^^^^^^^^^^^
You can fix the error by completing the reduction
-- succ_1.lc
let one = \f x -> f x
let two = \f x -> f (f x)
let incr = \n f x -> f (n f x)
eval succ_one :
incr one
=d> (\n f x -> f (n f x)) (\f x -> f x)
=b> \f x -> f ((\f x -> f x) f x)
=b> \f x -> f ((\x -> f x) x)
=b> \f x -> f (f x) -- beta-reduce the above
=d> two -- optional
Similarly, elsa
rejects the following program,
-- id_0_bad.lc
let id = \x -> x
let zero = \f x -> x
eval id_zero :
id zero
=b> (\f x -> x)
=d> zero
with the error
$ elsa ex4.lc
ex4.lc:7:5-20: id_zero has an invalid beta-reduction
7 | =b> (\f x -> x)
^^^^^^^^^^^^^^^
You can fix the error by inserting the appropriate
intermediate term as shown in id_0.lc
above.
elsa
ProgramsAn elsa
program has the form
-- definitions
[let <id> = <term>]+
-- reductions
[<reduction>]*
where the basic elements are lambda-calulus term
s
<term> ::= <id>
\ <id>+ -> <term>
(<term> <term>)
and id
are lower-case identifiers
<id> ::= x, y, z, ...
A <reduction>
is a sequence of term
s chained together
with a <step>
<reduction> ::= eval <id> : <term> (<step> <term>)*
<step> ::= =a> -- alpha equivalence
=b> -- beta equivalence
=d> -- def equivalence
=*> -- trans equivalence
=~> -- normalizes to
elsa
programsA reduction
of the form t_1 s_1 t_2 s_2 ... t_n
is valid if
t_i s_i t_i+1
is valid, andt_n
is in normal form (i.e. cannot be further beta-reduced.)Furthermore, a step
of the form
t =a> t'
is valid if t
and t'
are equivalent up to alpha-renaming,t =b> t'
is valid if t
beta-reduces to t'
in a single step,t =d> t'
is valid if t
and t'
are identical after let-expansion.t =*> t'
is valid if t
and t'
are in the reflexive, transitive closure
of the union of the above three relations.t =~> t'
is valid if t
normalizes to t'
.(Due to Michael Borkowski)
The difference between =*>
and =~>
is as follows.
t =*> t'
is any sequence of zero or more steps from t
to t'
.
So if you are working forwards from the start, backwards from the end,
or a combination of both, you could use =*>
as a quick check to see
if you're on the right track.
t =~> t'
says that t
reduces to t'
in zero or more steps and
that t'
is in normal form (i.e. t'
cannot be reduced further).
This means you can only place it as the final step.
So elsa
would accept these three
eval ex1:
(\x y -> x y) (\x -> x) b
=*> b
eval ex2:
(\x y -> x y) (\x -> x) b
=~> b
eval ex3:
(\x y -> x y) (\x -> x) (\z -> z)
=*> (\x -> x) (\z -> z)
=b> (\z -> z)
but elsa
would not accept
eval ex3:
(\x y -> x y) (\x -> x) (\z -> z)
=~> (\x -> x) (\z -> z)
=b> (\z -> z)
because the right hand side of =~>
can still be reduced further.