# Assignment 7 There is no separate assignment 6. (There was a single big assignment for weeks 5 and 6, and we're going to keep the assignment numbers in synch with the weeks.) ## Evaluation order in Combinatory Logic 1. Give a term that the lazy evaluators (either [[the Haskell evaluator|code/ski_evaluator.hs]], or the lazy version of [[the OCaml evaluator|code/ski_evaluator.ml]]) do not evaluate all the way to a normal form, that is, that contains a redex somewhere inside of it after it has been reduced. 2. One of the [[criteria we established for classifying reduction strategies|topics/week3_evaluation_order]] strategies is whether they reduce subterms hidden under lambdas. That is, for a term like `(\x y. x z) (\x. x)`, do we reduce to `\y.(\x.x) z` and stop, or do we reduce further to `\y.z`? Explain what the corresponding question would be for CL. Using the eager version of the OCaml CL evaluator (`reduce_try2`), prove that the evaluator does reduce terms inside of at least some "functional" CL terms. Then provide a modified evaluator that does not perform reductions in those positions. (Just give the modified version of your recursive reduction function.) ## Evaluation in the untyped lambda calculus: substitute-and-repeat Having sketched the issues with a discussion of Combinatory Logic, we're going to begin to construct an evaluator for a simple language that includes lambda abstraction. In this problem set, we're going to work through the issues twice: once with a function that does substitution in the obvious way, and keeps reducing-and-repeating until there are no more eligible redexes. You'll see it's somewhat complicated. The complications come from the need to worry about variable capture. (Seeing these complications should give you an inkling of why we presented the evaluation order discussion using Combinatory Logic, since we don't need to worry about variables in CL.) We're not going to ask you to write the entire program yourself. Instead, we're going to give you almost the complete program, with a few gaps in it that you have to complete. You have to understand enough to add the last pieces to make the program function. You can find the skeleton code [[here|/code/untyped_evaluator.ml]]. We've also prepared a much fuller version, that has user-friendly input and printing of results. We'll provide a link to that shortly. It will be easiest for you to understand that code if you've completed the gaps in the simplified skeleton linked above. ## Evaluation in the untyped lambda calculus: environments The previous strategy is nice because it corresponds so closely to the reduction rules we give when specifying our lambda calculus. (Including specifying evaluation order, which redexes it's allowed to reduce, and so on.) But keeping track of free and bound variables, computing fresh variables when needed, that's all a pain. Here's a better strategy. Instead of keeping all of the information about which variables have been bound or are still free implicitly inside of the terms, we'll keep a separate scorecard, which we will call an "environment". This is a familiar strategy for philosophers of language and for linguists, since it amounts to evaluating terms relative to an assignment function. The difference between the substitute-and-repeat approach above, and this approach, is one huge step towards monads. The skeleton code for this is at the [[same link as before|/code/untyped_evaluator.ml]]. This part of the exercise is the "V2" part of that code. ## Monads Mappables (functors), MapNables (applicative functors), and Monads (composables) are ways of lifting computations from unboxed types into boxed types. Here, a "boxed type" is a type function with one unsaturated hole (which may have several occurrences, as in `(α,α) tree`). We can think of the box type as a function from a type to a type. Recall that a monad requires a singleton function `mid : P-> P`, and a composition operator like `>=> : (P->Q) -> (Q->R) -> (P->R)`. As we said in the notes, we'll move freely back and forth between using `>=>` and using `<=<` (aka `mcomp`), which is just `>=>` with its arguments flipped. `<=<` has the virtue that it corresponds more closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue that its types flow more naturally from left to right. Anyway, `mid` and (let's say) `<=<` have to obey the Monad Laws: mid <=< k = k k <=< mid = k j <=< (k <=< l) = (j <=< k) <=< l For example, the Identity monad has the identity function `I` for `mid` and ordinary function composition `○` for `<=<`. It is easy to prove that the laws hold for any terms `j`, `k`, and `l` whose types are suitable for `mid` and `<=<`: mid <=< k == I ○ k == \p. I (k p) ~~> \p. k p ~~> k k <=< mid == k ○ I == \p. k (I p) ~~> \p. k p ~~> k (j <=< k) <=< l == (\p. j (k p)) ○ l == \q. (\p. j (k p)) (l q) ~~> \q. j (k (l q)) j <=< (k <=< l) == j ○ (k ○ l) == j ○ (\p. k (l p)) == \q. j ((\p. k (l p)) q) ~~> \q. j (k (l q)) 1. On a number of occasions, we've used the Option/Maybe type to make our conceptual world neat and tidy (for instance, think of [[our discussion of Kaplan's Plexy|topics/week6_plexy]]). As we learned in class, there is a natural monad for the Option type. Using the vocabulary of OCaml, let's say that `'a option` is the type of a boxed `'a`, whatever type `'a` is. More specifically, type 'a option = None | Some 'a (If you have trouble keeping straight what is the OCaml terminology for this and what is the Haskell terminology, don't worry, we do too.) Now the obvious singleton for the Option monad is `\p. Some p`. Give (or reconstruct) either of the composition operators `>=>` or `<=<`. Show that your composition operator obeys the Monad Laws. 2. Do the same with lists. That is, given an arbitrary type `'a`, let the boxed type be `['a]` or `'a list`, that is, lists of values of type `'a`. The `mid` is the singleton function `\p. [p]`, and the composition operator is: let (>=>) (j : 'a -> 'b list) (k : 'b -> 'c list) : 'a -> 'c list = fun a -> List.flatten (List.map k (j a)) For example: let j a = [a; a+1];; let k b = [b*b; b+b];; (j >=> k) 7 (* which OCaml evaluates to: - : int list = [49; 14; 64; 16] *)