Higher-order Multidimensional Programming
Talk I gave to the Lambda Montreal Meetup group on 7 March 2018.
Slides: lambda-montreal-2018-03-07
Talk I gave to the Lambda Montreal Meetup group on 7 March 2018.
Slides: lambda-montreal-2018-03-07
Talk I gave to the C++ Montréal Meetup group on 25 July 2017.
Summary: c++-montreal-2017-07-25
This release fully supports higher-order functions for TransLucid.
It is available on sourceforge.
The blog’s address is now cartesianprogramming.com.
The word “TransLucid” comes from an essay by Ralph Waldo Emerson:
This insight, which expresses itself by what is called Imagination, is a very high sort of seeing, which does not come by study, but by the intellect being where and what it sees; by sharing the path or circuit of things through forms, and so making them translucid to others.
It’s serendipitous.
The second release of TransLucid, version 0.2.0, is out. It is available at the following link.
http://sourceforge.net/projects/translucid/files/TransLucid/0.2.0/tl-0.2.0.tar.bz2/download
It includes intensions as first-class values, and higher-order functions fully work.
Tournament computation can also take place in two dimensions. Here, tournamentOp₂ applies a quaternary function g to a 2-dimensional variable X, and keeps doing this to the results until there is a single result.
fun tournamentOp₂.d₁.d₂.n.g X = Y @ [d <- 0] where dim t <- ilog.n ;; var Y = fby.t X (g.(NWofQuad.d₁.d₂ Y).(NEofQuad.d₁.d₂ Y). (SWofQuad.d₁.d₂ Y).(SEofQuad.d₁.d₂ Y)) ;; end ;;
As for the single-dimensional case, it is useful to have a way of filling a two-dimensional grid with a neutral element if we do not have a grid whose extent in every dimension is the same power of 2.
fun default₂.d₁.m₁.n₁.d₂.m₂.n₂.val X = Y where var Y [d₁ : m₁..n₁, d₂ : m₂..n₂] = X ;; var Y [d₁ : nat, d₂ : nat] = val ;; end ;;
In the post on factorial, the following code appears:
var f = tournamentOp₁.d.n.times (default₁.d.1.n.1 (#!d)) ;;
What is going on? Let us look at the definitions from the TransLucid Standard Header:
fun default₁.d.m.n.val X = Y where var Y [d : m..n] = X ;; var Y [d : nat] = val ;; end ;; fun tournamentOp₁.d.n.g X = Y @ [d <- 0] where dim t <- ilog.n ;; var Y = fby.t X (g.(LofPair.d Y).(RofPair.d Y)) ;; end ;;
The default₁ function creates a stream Y varying in dimension d such that in the interval [m,n], the result will be the value of X. Everywhere else, the value of Y is the default val.
As for tournamentOp₁, when #!t ≡ 0, the value of Y is X. When #!t > 0, each element of Y is the result of applying the binary function g to a pair of elements from Y when #!t was one less. This process is completed until there is just one element left. Since the number n is not necessarily a power of 2, we use default₁ to fill in the slots of X with the neutral element of g.
This form of computation is called tournament computation, and writing programs this way encourages parallel implementations.
The origins of Cartesian Programming came from what was called Intensional Programming, in which the behavior of a program was context-dependent: a context is a set of (dimension,ordinate) pairs,
and the program can change behavior if some of the ordinates are changed. Formally, a variable in an intensional programming language is an intension, i.e., a mapping from contexts to values.
In TransLucid, after several failed attempts at defining the semantics of functions over these intensions, it finally dawned on us that the intension itself needs to be a first-class value. What this means is that
the context in which an intension is created is as important as the context in which it is evaluated. Consider:
var tempAtLocation = ↑{location} temperature ;; var tempInInuvik = tempAtLocation @ [location ← "Inuvik"] ;;
What this means is that whatever the value of the location-ordinate, variable tempInInuvik would always give the temperature in Inuvik, allowing any other dimensions to vary freely. Hence
↓tempInInuvik @ [location ← "Paris", date ← #!date - 1] ;;
would give the temperature in Inuvik yesterday, not in Paris yesterday.
Here we give an example of programming with infinite arrays. We take the well-known factorial function, and calculate using tournament computation. The TransLucid source code is found below.
We build an array f which varies with respect to dimensions t and d, effectively creating a computation tree. For example, to compute the factorial of 3, the variable f becomes
d t 1 1 2 3 1 1 ... 1 6 1 1 1 1 ... 6 1 1 1 1 1 ...
and the answer is 6, picked up when t=2 and d=0.
Similarly, for the factorial of 6, f becomes
d t 1 1 2 3 4 5 6 1 1 ... 1 6 20 6 1 1 1 1 1 ... 6 120 1 1 1 1 1 1 1 ... 720 1 1 1 1 1 1 1 1 ...
and the answer is 720, picked up when t=3 and d=0.
When t = 0, the value of f is a d-stream such that f is the current d-index if it is between 1 to n, and 1 otherwise. When t > 0, the value of f is a d-stream such that f is the product of pairs from the (t-1) d-stream.
fun fact.n = f where dim d <- 0 ;; var f = tournamentOp₁.d.n.times (default₁.d.1.n.1 (#!d)) ;; end ;;
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