Symmetry operator of -<

[chansey97]

The operator -< currently is not symmetrical, i.e. lacks its mirror operator >-.

Two examples are provided below to demonstrate why it is useful.

The 1st one is the matrix addition.

Here is a possible solution:

;; 1 8 5
;; 3 4 3
;; 8 2 1
(define-flow D
  (~> (-< (~> (== (* 1) (* 8) (* 5)) +)
          (~> (== (* 3) (* 4) (* 3)) +)
          (~> (== (* 8) (* 2) (* 1)) +))))
          
;; 9 4 2
;; 5 4 7
;; 8 4 7
(define-flow E
  (~> (-< (~> (== (* 9) (* 4) (* 2)) +)
          (~> (== (* 5) (* 4) (* 7)) +)
          (~> (== (* 8) (* 4) (* 7)) +))))

(define-flow D+E
  (~>> (-< (~> E ▽) (~> D ▽)) (map +)))

The D+E seems ugly, because we have to collect them into a list, but I would like to write it like the following, if possible:

(define-flow D+E
  (~> (-< D E) (>- +)))

This example is inspired from Graphical Linear Algebra. In GLA, Pawel introduced a syntactic sugar called "m-wire". so the the string diagram of matrix addition can be simplified to:

Image from https://www.cs.ox.ac.uk/qpl2015/slides/sobocinski-tutorial.pdf p.25

The 2nd comes from meru.rkt

(define-flow meru-step
  (~>> (-< (~> (-< 0 _) ▽) 
           (~> (-< _ 0) ▽))
       (map +) △))

Using the expected operator >-, the solution would be more concise:

(define-flow meru-step
  (~>> (-< (~> (-< 0 _))
           (~> (-< _ 0)))
       (>- +) △))

P.S. This issue has been discussed on Racket Discord and @countvajhula proposed a solution. Currently we could write the following macro to workaround it.

(require (for-syntax racket/base syntax/parse))

(define-qi-syntax-rule (-<> f ... ((~datum >-) comb))
  (~>> (-< (~> f ▽) ...) (map comb)))

(define-flow D+E
  (~> (-<> D E (>- +))))

More details see https://discord.com/channels/571040468092321801/979642553471221790/1182246331922780251

[NoahStoryM]

Hello @chansey97

I agree that the -< operator should have a symmetric counterpart. In Qi, the values we work with can be viewed categorically as product objects, which naturally have a dual concept: sum objects. Therefore, not only -<, but other operators such as ><, ==*, fanout, n>, ⏚, etc., that handle values (product objects), should also have symmetric operators for handling covalues (sum objects), like <>, ==+, fanin, n<, ≂, etc.. Here is our previous discussion: https://github.com/drym-org/qi/issues/62#issuecomment-1204724194.

I’ve previously developed qi-cat, which implements these dual operators. Unfortunately, due to personal reasons, I haven’t been able to produce comprehensive documentation for it (I plan to address this when time permits).

To briefly introduce covalues, it's essentially values tagged with a natural number. For instance, (values 1 "a") represents a value in Number × String, whereas (covalues '(1 2) 0) and (covalues '(a b) 1) represent values in (Number × Number) + (Symbol × Symbol).

The -< operator corresponds to the categorical concept of pairing. For example, given a : G -> X and b : G -> Y, then (-< a b) : G -> X × Y.

The dual of -< is >-, known as copairing, which operates as follows: given a : X -> T and b : Y -> T, then (>- a b) : X + Y -> T.

[countvajhula]

Btw @NoahStoryM , unrelated to the present issue but, we are gearing up to release Qi 4 which includes the optimizing compiler. One of the big changes with backwards compatibility implications is the change from matching datum literals (~datum) to matching literals (~literal). See Literally Causing Problems for more details on how this can affect applications. If you have time, you may want to test qi-cat with the lets-write-a-qi-compiler branch to see if it is affected (but you may want to wait until later today as we are hoping to merge a few PRs including First Optimizations). It would also be helpful in general to see if we've broken anything else!

[NoahStoryM]

Hi @countvajhula ,

I’ve tried lets-write-a-qi-compiler and all the qi-cat tests passed. I didn’t notice any issues.

Also, I read @chansey97’s code and I found an interesting way. It seems that we can perform a matrix-like transpose operation on the values processed by qi. For example, (~> (1 1 1) (-< D E)) results in (values 14 10 11 15 16 19), which we can view as a 3×2 matrix. If we transpose it, we get (values 14 15 10 16 11 19), and then we can use ><, which can distribute input values (https://github.com/drym-org/qi/pull/64) , to get (values 29 26 30).

Here is the code:

#lang racket

(require qi/cat (for-syntax syntax/parse))

;; 1 8 5
;; 3 4 3
;; 8 2 1
(define-flow D
  (~> (-< (~> (== (* 1) (* 8) (* 5)) +)
          (~> (== (* 3) (* 4) (* 3)) +)
          (~> (== (* 8) (* 2) (* 1)) +))))

;; 9 4 2
;; 5 4 7
;; 8 4 7
(define-flow E
  (~> (-< (~> (== (* 9) (* 4) (* 2)) +)
          (~> (== (* 5) (* 4) (* 7)) +)
          (~> (== (* 8) (* 4) (* 7)) +))))

(define add (procedure-reduce-arity + 2))
(define-syntax for/values
  (syntax-parser
    [(_ (clause ...) body ...+)
     #'(apply values (for/list (clause ...) body ...))]))
(define (transpose* m n)
  (λ arg*
    (define v* (list->vector arg*))
    (for/values ([id (in-range (* m n))])
      (define-values (j i) (quotient/remainder id n))
      (vector-ref v* (+ (* i m) j)))))

(define-flow D+E (~> (-< D E) (transpose* 3 2) (>< add)))

And it's not hard to implement the dual operator transpose+, which deals with covalues.

By the way, for/values seems to be very useful for qi. Do you think it’s worth considering implementing it in qi?