Y again: compete!

In the post Y again: conflict!  the chemlambda computation based on a very crude model ended in conflict.

Conflict means that the same graphical element appears in two LEFT g-patterns (see, in the series of expository posts  the part II for the g-patterns and the part III for the moves) .

In the next figure we see this conflict, in the upper part (that’s where we were left in the previous post), followed by a fork: in the lower left part of the figure we see what we get if we apply the beta move and in the lower right part we see what happens if we apply the DIST move.




Recall that (or look again at the upper side of the picture) the conflict was between LEFT patterns of a beta move and of a DIST move.

I rearranged the drawing of the g-patterns a bit (mind that this is not affecting in any way the graphs, because the drawings on paper or screen are one thing and the graphs another thing, excuse me for being trivially obvious). In this pretty picture we see well that already the Y gun shot a second pair of nodes A and FO.

The differences now:

  • for the graph from the lower left part of the picture, obtained after the beta move, we see that A[i.i] produces a loop: A[i,i] –COMB–> loop. We can disregard the loop further (because we allow only moves from LEFT to RIGHT, therefore the loop will not interact with anything in the future of the computation). We are left with a very simple graph, made of the two pairs of nodes A and FO which the gun already shot and with a very interesting pair:  A[w,a1,o] FO[o,q,a1] .
  • for the graph from the lower right part of the picture, obtained after the DIST move, we see the two pairs of nodes A and FO whih the Y gun already shot and a graph made by 6 nodes.

Which way is the best? What to do?

Let’s make them compete!  Who shoots faster?


Imagine the scene: in the Lambda Saloon, somewhere in the Wide Wild West, enter two new cowboys.

“Who called these guys?” asks the sheriff of the Functional City, where the reputed saloon resides.

“They are strangers! They both look like Lucky Luke, though…”

The cowboys and cowgirls from the saloon nod in approval: they all remember what happened when Lucky Luke — the one, the single cowboy who shoots faster than his shadow — was put between the two parallel mirrors from the saloon stage. What a show! That day, the Functional City got a reputation, and everybody knows that reputation is something as good as gold. Let the merchants from Imperative County sell windows panes for the whole US, nobody   messes with a cowboy, or cowgirl from the Functional City. Small, but elegant. Yes sir, style is the right word…

Let’s go back to the two strangers.

“I’m faster than Master Y” says the stranger from the right.

“I’m MUCH faster than Master Y” says the one from the left, releasing from his cigar a loop of smoke.

“Who the … is Master Y?” asks the sheriff.

“Why, it’s Lucky Luke. He trained us both, then sent us to Functional City. He says hello to you and tells you that he got new tricks to show” says one of the strangers.

“… things not learned from church…” says the other.

“I need to see how fast are you, or else I call you both liars” shouts one of the bearded, long haired  cowboys.

The stranger from the right started first. What a show!



He first makes a clever DIST move (not that there was anything else to do). Then he is presented with 4 simultaneous moves to do (FAN-IN, 2 betas and a DIST). He shoots and freezes. Nothing moves, excepting two loops of smoke, rising from his gun.

“I could continue like this forever, but I stopped, to let my colleague show you what he’s good at.” said the stranger from the right.

“Anyway, I am a bit slow  with the first shot, but after that I am faster.” he continued.

“Wait, said the sheriff, you sure you really shot?”

“Yep, sure, look better how I stand”, said the stranger from the right, only slightly modifying his posture, so that everybody could clearly see the shot:



“Wow, true!” said the cowboys.

“I’m fast from the first shoot!” said the stranger from the left. “Look!”



“I only did a DIST move.” said the stranger from the left, freezing in his posture.

“Nice show, guys! Hey, look, I can’t tell now which is which, they look the same. I got it, the guy from the right is a bit slower at the first shoot (however he is dazzlingly fast) but then he is as  good as his fellow from the left.”

“Hm, said the sheriff, true. Only one thing: I have never seen in the Lambda Saloon  anything like this. It’s fast, but it does not seem to belong here.”



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