Exanoke

Exanoke is a pure functional language which is syntactically restricted to expressing the primitive recursive functions.

I'll assume you know what a primitive recursive function is. If not, go look it up, as it's quite interesting, if only for the fact that it demonstrates even a genius like Kurt Gödel can sometimes be mistaken. (He initially thought that all functions could be expressed primitive recursively, until Ackermann came up with a counterexample.)

So, you have a program. There are two ways that you can ensure that it implements a primtive recursive function:

• You can statically analyze the bastard, and prove that all of its loops eventually terminate, and so forth; or
• You can write it in a language which is inherently restricted to expressing only primitive recursive functions.

The second option is the route PL-{GOTO} takes. But that's an imperative language, and it's fairly easy to restrict an imperative language in this way. In PL-{GOTO}'s case, they just took PL and removed the GOTO command. The rest of the language essentially contains only for loops, so what you get is something in which you can only express primitive recursive functions. (That imperative programs consisting of only for loops can express only and exactly the primitive recursive functions was established by Meyer and Ritchie in "The complexity of loop programs".)

But what about functional languages?

The approach I've taken in TPiS, and that I wanted to take in Pixley and Robin, is to provide an unrestricted functional language to the programmer, and statically analyze it to see if you're going and writing primitive recursive functions in it or not.

Thing is, that's kind of difficult. Is it possible to take the same approach PL-{GOTO} takes, and syntactically restrict a functional language to the primitive recursive functions?

I mean, in a trivial sense, it must be; in the original definition, primitive recursive functions were functions. (Duh.) But these have a highly arithmetical flavour, with bounded sums and products and whatnot. What would primitive recursion look like in the setting of general (and symbolic) functional programming?

Functional languages don't do the for loop thing, they do the recursion thing, and there are no natural bounds on that recursion, so some restriction on recursion would have to be captured by the grammar, and... well, it sounds somewhat interesting, and doable, so let's try it.

Ground Rules

Here are some ground rules about how to tell if a functional program is primitive recursive:

• It doesn't perform mutual recursion.
• When recursion happens, it's always with arguments that are strictly "smaller" values than the arguments the function received.
• There is a "smallest" value that an argument can take on, so that there is always a base case to the recursion, so that it always eventually terminates.
• Higher-order functions are not used.

The first point can be enforced simply by providing a token that refers to the function currently being defined (self is a reasonable choice) to permit recursion, but to disallow calling any function that has not yet occurred, lexically, in the program source.

The second point can be enforced by stating syntactic rules for "smallerness". (Gee, typing that made me feel a bit like George W. Bush!)

The third point can be enforced by providing some default behaviour when functions are called with the "smallest" kinds of values. This could be as simple as terminating the program if you try to find a value "smaller" than the "smallest" value.

The fourth point can be enforced by simply disallowing functions to be passed to, or returned from, functions.

Note on these Criteria

In fact, these four criteria taken together do not strictly speaking define primitive recursion. They don't exclude functional programs which always terminate but which aren't primitive recursive (for example, the Ackermann function.) However, determining that such functions terminate requires a more sophisticated notion of "smallerness" — a reduction ordering on their arguments. Our notion of "smallerness" will be simple enough that it will be easy to express syntactically, and will only capture primitive recursion.

Note on Critical Arguments

I should note, though, that the second point is an oversimplification. Not all arguments need to be strictly "smaller" upon recursion — only those arguments which are used to determine if the function recurses. I'll call those the critical arguments. Other arguments can take on any value (which is useful for having "accumulator" arguments and such.)

When statically analyzing a function for primitive recursive-ness, you need to check how it decides to recurse, to find out which arguments are the critical arguments, so you can check that those ones always get "smaller".

But we can proceed in a simpler fashion here — we can simply say that the first argument to every function is the critical argument, and all the rest aren't. This is without loss of generality, as we can always split some functionality which would require more than one critical argument across multiple functions, each of which only has one critical argument. (Much like every for loop has only one loop variable.)

Data types

Let's just go with pairs and atoms for now, although natural numbers would be easy to add too. Following Ruby, atoms are preceded by a colon; while I find this syntax somewhat obnoxious, it is less obnoxious than requiring that atoms are in ALL CAPS, which is what Exanoke originally had. In truth, there would be no real problem with allowing atoms, arguments, and function names (and even self) to all be arbitrarily alphanumeric, but it would require more static context checking to sort them all out, and we're trying to be as syntactic as reasonably possible here.

:true is the only truthy atom. Lists are by convention only, and, by convention, lists compose via the second element of each pair, and :nil is the agreed-upon list-terminating atom, much love to it.

Grammar

Exanoke     ::= {FunDef} Expr.
FunDef      ::= "def" Ident "(" "#" {"," Ident} ")" Expr.
Expr        ::= "cons" "(" Expr "," Expr ")"
              | "head" "(" Expr ")"
              | "tail" "(" Expr ")"
              | "if" Expr "then" Expr "else" Expr
              | "self" "(" Smaller {"," Expr} ")"
              | "eq?" "(" Expr "," Expr")"
              | "cons?" "(" Expr ")"
              | "not" "(" Expr ")"
              | "#"
              | ":" Ident
              | Ident ["(" Expr {"," Expr} ")"]
              | Smaller.
Smaller     ::= "<head" SmallerTerm
              | "<tail" SmallerTerm
              | "<if" Expr "then" Smaller "else" Smaller.
SmallerTerm ::= "#"
              | Smaller.
Ident       ::= name.

The first argument to a function does not have a user-defined name; it is simply referred to as #. Again, there would be no real problem if we were to allow the programmer to give it a better name, but more static context checking would be involved.

Note that <if is not strictly necessary. Its only use is to embed a conditional into the first argument being passed to a recursive call. You could also use a regular if and make the recursive call in both branches, one with :true as the first argument and the other with :false.

Examples

-> Tests for functionality "Evaluate Exanoke program"

cons can be used to make lists and trees and things.

| cons(:hi, :there)
= (:hi :there)

| cons(:hi, cons(:there, :nil))
= (:hi (:there :nil))

head extracts the first element of a cons cell.

| head(cons(:hi, :there))
= :hi

| head(:bar)
? head: Not a cons cell

tail extracts the second element of a cons cell.

| tail(cons(:hi, :there))
= :there

| tail(tail(cons(:hi, cons(:there, :nil))))
= :nil

| tail(:foo)
? tail: Not a cons cell

<head and <tail and syntactic variants of head and tail which expect their argument to be "smaller than or equal in size to" a critical argument.

| <head cons(:hi, :there)
? Expected <smaller>, found "cons"

| <tail :hi
? Expected <smaller>, found ":hi"

if is used for descision-making.

| if :true then :hi else :there
= :hi

| if :hi then :here else :there
= :there

eq? is used to compare atoms.

| eq?(:hi, :there)
= :false

| eq?(:hi, :hi)
= :true

eq? only compares atoms; it can't deal with cons cells.

| eq?(cons(:one, :nil), cons(:one, :nil))
= :false

cons? is used to detect cons cells.

| cons?(:hi)
= :false

| cons?(cons(:wagga, :nil))
= :true

not does the expected thing when regarding atoms as booleans.

| not(:true)
= :false

| not(:false)
= :true

Cons cells are falsey.

| not(cons(:wanga, :nil))
= :true

self and # can only be used inside function definitions.

| #
? Use of "#" outside of a function body

| self(:foo)
? Use of "self" outside of a function body

We can define functions. Here's the identity function.

| def id(#)
|     #
| id(:woo)
= :woo

Functions must be called with the appropriate arity.

| def id(#)
|     #
| id(:foo, :bar)
? Arity mismatch (expected 1, got 2)

| def snd(#, another)
|     another
| snd(:foo)
? Arity mismatch (expected 2, got 1)

Parameter names must be defined in the function definition.

| def id(#)
|     woo
| id(:woo)
? Undefined argument "woo"

You can't call a parameter as if it were a function.

| def wat(#, woo)
|     woo(#)
| wat(:woo)
? Undefined function "woo"

You can't define two functions with the same name.

| def wat(#)
|     :there
| def wat(#)
|     :hi
| wat(:woo)
? Function "wat" already defined

You can't name a function with an atom.

| def :wat(#)
|     #
| :wat(:woo)
? Expected identifier, but found atom (':wat')

Every function takes at least one argument.

| def wat()
|     :meow
| wat()
? Expected '#', but found ')'

The first argument of a function must be #.

| def wat(meow)
|     meow
| wat(:woo)
? Expected '#', but found 'meow'

The subsequent arguments don't have to be called #, and in fact, they shouldn't be.

| def snd(#, another)
|     another
| snd(:foo, :bar)
= :bar

| def snd(#, #)
|     #
| snd(:foo, :bar)
? Expected identifier, but found goose egg ('#')

A function can call a built-in.

| def snoc(#, another)
|     cons(another, #)
| snoc(:there, :hi)
= (:hi :there)

Functions can call other user-defined functions.

| def double(#)
|     cons(#, #)
| def quadruple(#)
|     double(double(#))
| quadruple(:meow)
= ((:meow :meow) (:meow :meow))

Functions must be defined before they are called.

| def quadruple(#)
|     double(double(#))
| def double(#)
|     cons(#, #)
| :meow
? Undefined function "double"

Argument names may shadow previously-defined functions, because we can syntactically tell them apart.

| def snoc(#, other)
|     cons(other, #)
| def snocsnoc(#, snoc)
|     snoc(snoc(snoc, #), #)
| snocsnoc(:blarch, :glamch)
= (:blarch (:blarch :glamch))

A function may recursively call itself, as long as it does so with values which are smaller than or equal in size to the critical argument as the first argument.

| def count(#)
|     self(<tail #)
| count(cons(:alpha, cons(:beta, :nil)))
? tail: Not a cons cell

| def count(#)
|     if eq?(#, :nil) then :nil else self(<tail #)
| count(cons(:alpha, cons(:beta, :nil)))
= :nil

| def last(#)
|     if not(cons?(#)) then # else self(<tail #)
| last(cons(:alpha, cons(:beta, :graaap)))
= :graaap

| def count(#, acc)
|     if eq?(#, :nil) then acc else self(<tail #, cons(:one, acc))
| count(cons(:A, cons(:B, :nil)), :nil)
= (:one (:one :nil))

Arity must match when a function calls itself recursively.

| def urff(#)
|     self(<tail #, <head #)
| urff(:woof)
? Arity mismatch on self (expected 1, got 2)

| def urff(#, other)
|     self(<tail #)
| urff(:woof, :moo)
? Arity mismatch on self (expected 2, got 1)

The remaining tests demonstrate that a function cannot call itself if it does not pass a values which is smaller than or equal in size to the critical argument as the first argument.

| def urff(#)
|     self(cons(#, #))
| urff(:woof)
? Expected <smaller>, found "cons"

| def urff(#)
|     self(#)
| urff(:graaap)
? Expected <smaller>, found "#"

| def urff(#, boof)
|     self(boof)
| urff(:graaap, :skooorp)
? Expected <smaller>, found "boof"

| def urff(#, boof)
|     self(<tail boof)
| urff(:graaap, :skooorp)
? Expected <smaller>, found "boof"

| def urff(#)
|     self(:wanga)
| urff(:graaap)
? Expected <smaller>, found ":wanga"

| def urff(#)
|     self(if eq?(:alpha, :alpha) then <head # else <tail #)
| urff(:graaap)
? Expected <smaller>, found "if"

| def urff(#)
|     self(<if eq?(:alpha, :alpha) then <head # else <tail #)
| urff(:graaap)
? head: Not a cons cell

| def urff(#)
|     self(<if eq?(self(<head #), :alpha) then <head # else <tail #)
| urff(:graaap)
? head: Not a cons cell

| def urff(#)
|     self(<if self(<tail #) then <head # else <tail #)
| urff(cons(:graaap, :skooorp))
? tail: Not a cons cell

Now, some practical examples, on Peano naturals. Addition:

| def inc(#)
|   cons(:one, #)
| def add(#, other)
|   if eq?(#, :nil) then other else self(<tail #, inc(other))
| 
| add(cons(:one, cons(:one, :nil)), cons(:one, :nil))
= (:one (:one (:one :nil)))

Multiplication:

| def inc(#)
|   cons(:one, #)
| def add(#, other)
|   if eq?(#, :nil) then other else self(<tail #, inc(other))
| def mul(#, other)
|   if eq?(#, :nil) then :nil else
|     add(other, self(<tail #, other))
| def three(#)
|   cons(:one, cons(:one, cons(:one, #)))
| 
| mul(three(:nil), three(:nil))
= (:one (:one (:one (:one (:one (:one (:one (:one (:one :nil)))))))))

Factorial! There are 24 :one's in this test's expectation.

| def inc(#)
|   cons(:one, #)
| def add(#, other)
|   if eq?(#, :nil) then other else self(<tail #, inc(other))
| def mul(#, other)
|   if eq?(#, :nil) then :nil else
|     add(other, self(<tail #, other))
| def fact(#)
|   if eq?(#, :nil) then cons(:one, :nil) else
|     mul(#, self(<tail #))
| def four(#)
|   cons(:one, cons(:one, cons(:one, cons(:one, #))))
| 
| fact(four(:nil))
= (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one (:one :nil))))))))))))))))))))))))

Discussion

So, what of it?

It was not a particularly challenging design goal to meet; it's one of those things that seems rather obvious after the fact, that you can just dictate that one of the arguments is a critical argument, and only call yourself with some smaller version of your critical argument in that position. Recursive calls map quite straightforwardly to for loops, and you end up with what is essentially a functional version of of a for program.

I guess the question is, is it worth doing this primitive-recursion check as a syntactic, rather than a static semantic, thing?

I think it is. If you're concerned at all with writing functions which are guaranteed to terminate, you probably have a plan in mind (however vague) for how they will accomplish this, so it seems reasonable to require that you mark up your function to indicate how it does this. And it's certainly easier to implement than analyzing an arbirarily-written function.

Of course, the exact syntactic mechanisms would likely see some improvement in a practical application of this idea. As alluded to in several places in this document, any actually-distinct lexical items (name of the critical argument, and so forth) could be replaced by simple static semantic checks (against a symbol table or whatnot.) Which arguments are the critical arguments for a particular function could be indicated in the source.

One criticism (if I can call it that) of primitive recursive functions is that, even though they can express any algorithm which runs in non-deterministic exponential time (which, if you believe "polynomial time = feasible", means, basically, all algorithms you'd ever care about), for any primitive recursively expressed algorithm, theye may be a (much) more efficient algorithm expressed in a general recursive way.

However, in my experience, there are many functions, generally non-, or minimally, numerical, which operate on data structures, where the obvious implementation is primitive recursive. In day-to-day database and web programming, there will be operations which are series of replacements, updates, simple transformations, folds, and the like, all of which "obviously" terminate, and which can readily be written primitive recursively.

Limited support for higher-order functions could be added, possibly even to Exanoke (as long as the "no mutual recursion" rule is still observed.) After all (and if you'll forgive the anthropomorphizing self-insertion in this sentence), if you pass me a primitive recursive function, and I'm primitive recursive, I'll remain primitive recursive no matter how many times I call your function.

Lastly, the requisite etymological denoument: the name "Exanoke" started life as a typo for the word "example".

Happy primitive recursing!
Chris Pressey
Cornwall, UK, WTF
Jan 5, 2013