How to understand these codes?

Questions
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1

Hello,

I see these codes in my book:

FixedPointList[# /. {s[x_][y_][z_] -> x[z][y[z]], k[x_][y_] -> x} &, 
s[s][k][s[s[s]][s]][s]]

In my thought:

s[s][k][s[s[s]][s]][s]] only has one part, how can it generates s[x_][y_][z_] and k[x_][y_].

2

Hi Zhenyu

Not sure what you mean by

s[s][k][s[s[s]][s]][s]] only has one part

To help understand the pattern matching take a look at

s[s][k][s[s[s]][s]][s] /. s[x_][y_][z_] :> {x, y, z}
(* {s, k, s[s[s]][s]}[s] *)

to see what matches x, y and z.

There are several combinator related functions in the Wolfram Function Repository to help visualize the evolution. e.g.

Magnify[ResourceFunction[
   "CombinatorEvolutionPlot"][(ResourceFunction["CombinatorPlot"][#, 
      "FramedMatches", "SKGlyphs" -> {s, k}] & /@ 
    ResourceFunction["CombinatorFixedPointList"][s[s][k][s[s[s]][s]][s], "SKGlyphs" -> {s, k}]), 
  "StatesDisplay"], .9]

Have you seen this.

3

Hi Rohit,

May you tell me why should the red part be separated from the right part of it?

Screenshot2022-02-22075723.jpg
Screenshot2022-02-22075723.jpg693×203 36.8 KB

4

Zhenyu,

The red s means the S combinator was applied. Similarly, the blue k means the K combinator was applied. There may be many parts in the expression that match S or K, the order in which they are applied results in a different evolution, but the same final result. This is discussed in detail in the “Updating Schemes and Multiway Systems” section in Stephen’s book.

5

Hi Rohit,

It’s too difficult to me. I decided to learn it after.