Thus We Frustrate Charlemagne by R. A. Lafferty

Thus We Frustrate Charlemagne by R. A. Lafferty (Galaxy, September-October 1967) is one of his ‘Institute for Impure Science’ series. This one sees Epiktistes the Ktistec machine (an AI or computer) and a group of eight people attempt to alter history at the time of Charlemagne (778CE) in the hope of eradicating the four hundred years of darkness that occurred after a brief period of enlightenment. To achieve this they send an avatar (“partly of mechanical and partly of ghostly construction”) to intercept a man called Gano, whose ambush of Charlemagne’s rear-guard led him to close the borders to the East and initiate a period of cultural isolation.
After their intervention the timeline changes, but the group don’t realise it (and there are also three computers now, and ten people). So they have another go, this time by preventing John Lutterell’s denunciation of Ockham’s Commentary on the Sentences.
The next iteration leaves them once more oblivious to the changes they have wrought, and their world is now much more backward (they are down to three people and a computer made out of sticks and weed). When they make another change, things go back to the way they are (I think—the last short section isn’t that clear).
This is all told in Lafferty’s quirky and digressive style, and with the odd touch of humour, such as when they initially discuss the use of the avatar:

“I hope the Avatar isn’t expensive,” Willy McGilly said. “When I was a boy we got by with a dart whittled out of slippery elm wood.”
“This is no place for humor,” Glasser protested. “Who did you, as a boy, ever kill in time, Willy?”
“Lots of them. King Wu of the Manchu, Pope Adrian VII, President Hardy of our own country, King Marcel of Auvergne, the philosopher Gabriel Toeplitz. It’s a good thing we got them. They were a bad lot.”
“But I never heard of any of them, Willy,” Glasser insisted.
“Of course not. We killed them when they were kids.”
“Enough of your fooling, Willy,” Gregory cut it off.
“Willy’s not fooling,” the machine Epikt said. “Where do you think I got the idea?”  p. 259

This is an entertaining read for the most part, but the ending is weak.
** (Average). 4,200 words.