X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=exercises%2Fassignment7.mdwn;h=f0213aa28ba3f9708a725a2202d87c125b536954;hp=f26f5d834912a1d846855645033f2514a82f3538;hb=c51047cd5851147777c81822d00b1dd6e35988d5;hpb=7c6b9557ee34b8106123687679fe43092fc48103 diff --git a/exercises/assignment7.mdwn b/exercises/assignment7.mdwn index f26f5d83..f0213aa2 100644 --- a/exercises/assignment7.mdwn +++ b/exercises/assignment7.mdwn @@ -1,8 +1,8 @@ -# 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.) +[[!toc]] + ## Evaluation order in Combinatory Logic 1. Give a term that the lazy evaluators (either [[the Haskell @@ -44,15 +44,26 @@ 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. +The first place you need to add code is in the `free_in` function. You already +wrote such a function [[in a previous homework|assignment5#occurs_free]], so this +part should be easy. The intended behavior is for the function to return results +like this: + + # free_in "x" (App (Lambda("x, Var "x"), Var "x"));; + - : bool = true + +The second place you need to add code is in the `reduce_head_once` function. As we +explain in the code comments, this is similar to the `reduce_if_redex` function in the +combinatory logic interpreters. Three of the clauses near the end of this function are incomplete. + +As we explain in the code comments, this interpreter uses an eager/call-by-value reduction +strategy. What are its other evaluation order properties? Does it perform beta-reduction +underneath lambdas? Does it perform eta-reduction? ## Evaluation in the untyped lambda calculus: environments -The previous strategy is nice because it corresponds so closely to the +The previous interpreter 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 @@ -67,8 +78,31 @@ 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. +This part of the exercise is the "VB" part of that code. + +You'll see that in the `eval` function, a new kind of value `Closure (ident) (term) (env)` +is used. What's that about? + +[[Read about Closures here.|/topics/week7_environments_and_closures]] + + +So that's what's going on with those `Closure`s. In the simple code we gave you to work with, we just made these another clause in the `term` datatype. That's really not correct. `Closure`s aren't terms. The syntax for our language doesn't have any constituents that get parsed into `Closure`s. `Closure`s are only created *during the course of evaluating terms*: specifically, when a variable gets bound to an abstract, which may itself contain variables that are locally free (not bound by the abstract itself). So really we should have two datatypes, one for terms, and another for the *results* (sometimes called "values") that terms can evaluate to. `Closure`s are results, but they aren't terms. `App`s are terms, but not results. If we had primitive numbers or other constants in our language, they might be both terms and results. In the fuller code from which your homework is simplified, this is how the types are in fact defined. But it makes things more complicated. So to keep things simple for the homework, we just pretended like `Closure`s were a new, exotic kind of `term`. + +In any case, now you know what's going on with the `Closure`s, and you should be able to complete the missing pieces of the `eval` function in the code skeleton linked above. + +If you've completed all the missing parts correctly (there are six gaps for the previous stage of the homework, and two gaps for this stage), then you should be able to compile the code skeleton, and use it as described in the comments at the start of the code. + + +## Fuller interpreter + +We've also prepared a fuller version of the interpreter, 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. +There's nothing you need to do with this; it's just for you to play with. If you're interested, +you can compare the code you completed for the previous two segments of the homework +to the (only moderately more complex) code in the `engine.ml` file of this fuller program. ## Monads @@ -78,7 +112,7 @@ boxed types. Here, a "boxed type" is a type function with one unsaturated hole (which may have several occurrences, as in `(Î±,Î±) 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 `â§ (* mid *) : P-> P`, and a +Recall that a monad requires a singleton function `â§ (\* 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