Suppose you want the sum of the squares of the elements of a list congruent to 1 mod 3 you can write

*reduce(lambda t,x t+x,map(square,filter(lambda x: x % 3 == 1,[1,2,3,4,5])))*

Clear? As mud … (there are plenty of tutorials online about *map*, *filter* and *reduce).*

You can always write a loop in an imperative language

*ssq=0
for i in [1,2,3,4,5]:*

*if i % 3 == 1 :*

ssq += i**2

ssq += i**2

Which is better? Hard to say. But the *reduce* version is declarative and nonsequential: there is no specified order in which the operations are to be performed, especially if we take into account the fact that addition is associative.

On the other hand, the *lambda*s and the functions obscure what is being computed. We need a notation that is clear but not imperative. Enter Variable Binding Operators (VBOS).

VBOs are inspired by conventional mathematical notation, which has been honed over centuries to be both concise and readable. A mathematician would write

∑ i^{2
i∈[1,5]
i mod 3 = 1
}

(that’s as close as I can get in wordpress but you know what it looks like).

The higher order functions *map, reduce,* and *filter* are nowhere to be seen, nor is *lambda. *This is a first order expression and as a result is much easier for mere mortals to comprehend. My suggestion is to make something like it available in, say, pyLucid.

What form should it take? Two dimensional syntax is impractical and for a lot of reasons it’s best to stick with ASCII characters. So we need a linear ASCII expression that incorporates the variable varying over a range, the range, the filtering condition, and the expression being summed. Here is what I propose:

*sumfor i in [1,2,3,4,5]: i % 3 = 1 all **i**2 end
*

If there is no condition we just omit it:

*sumfor i in [1,2,3,4,5] all i**2 end
*

If we want the product, we use *prodfor;*

*prodfor i in [1,2,3,4,5]: i % 3 = 1 all** i**2 end
*

If we want the average we use *avgfor;* if we want to accumulate concatenating strings we use *concatfor. *You can surely guess the semantics of *maxfor, minfor, appendfor, consfor, unionfor,* and *intersectfor.*

There are also VBOs that don’t follow this pattern (because they don’t have an implied *reduce*): the meanings of *first i in R: i>0, last i in R: i>0, exist i in R: i>0, forall i in R: i>0, those i in R: i>0 * should be clear. Notice that *those* and *consfor* are list comprehensions with a different syntax.

Here are is the primes less than 100

*those p in range(2,100):*

* forall d in range (2,round(sqrt(p))+1):*

* p mod d ne 0*

* end*

*end*

Here is a definition of a function *perms* that returns a list of all permutations of its argument, in hypothetical extended pyLucid:

*perms(L) =*

* appendfor i in L all*

* consfor J in perms(those j in L: j ne i end) all*

* i :: J*

* end*

*end*

And here is a definition of the list of sublists of a list L

*sublists(L) =*

* if L eq [] then [] else*

* sublists(tl(L)) <>*

* appendfor J in sublists(tl(L)) all *

* hd(L) :: J*

* end *

* fi *

With the VBOs we can simply transcribe many mathematical formulas. Here is e**x approximated by the first 10 terms of the power series:

*sumfor n in range(10) all*

* prodfor i in range(n) all*

* x/(i+1)*

Of course this straight-from-the-book expression is inefficient but at least we’re sure that it’s right.

One simple extension is to allow to bound variables running in parallel over different ranges; for example

*sumfor a,b in A,B all a*b end*

computes a dot product.

We could also have a VBO that extracts the ‘final’ value from a converging series:

*limit n in range(1,100) all (1+1/n)**n end*

(which assumes some otherwise specified convergence criterion).

Great … so pyLucid has VBOS. Sadly, not yet (and these examples haven’t been tested). There are problems.

The first problem is that every VBO is an implied loop. PyLucid already has (nested) loops so how do they interact? The obvious implementation technique is to translate VBOs into nested loops but I haven’t figured out the details.

A more difficult question is, what sort of objects are ranges? In Python they would be iterators, and that works out fine. In Haskell they would be lists.

I think the best choice is to make them vectors (arrays) but what if the other expressions also vary in space? Do we need extra space dimensions?

For once I’m stumped. I’ll let you know when I get on top of it all.