In this, Mr. MacKenzie had a number of supporters, including the bishop of the Roman Catholic Diocese of Albany and a state judge, who this year began to fine the parole board for what she believed was its unwillingness to evaluate Mr. MacKenzie’s rehabilitation. “This case begs the question: If parole isn’t granted to this petitioner, when and under what circumstances would it be granted?” Justice Maria G. Rosa of State Supreme Court wrote in May.

The possibility of rehabilitation is a central premise underpinning New York State’s penal system, where open-ended sentences like the one Mr. MacKenzie received — 25 years to life — are typical. Such sentences suggest the possibility of release but leave the decision to the parole board. One recent state-by-state survey found that the parole board in New York has the power to leave more inmates — 9,262 as of January of this year — in prison until their dying breath than any state but California.

Mr. MacKenzie became eligible for release in 2000, after 25 years in prison. When he appeared before the parole board, he said that he wished he had been the one to die, and not Officer Giglio. “I wish that night it was me,” he said the first of many times that he would appear before the board.

But when pressed to explain his crime, a question he would be asked at every parole hearing, Mr. MacKenzie struggled. He said that he could not actually remember shooting Officer Giglio. He was in the middle of a blackout — he used the term “automatism” — which he attributed to the pills he was then taking. There were a lot — Darvon, Norgesic Forte, Valium and others.

It had taken years, in fact, for him to even accept that he had shot Officer Giglio. While reading thousands of pages of testimony from his trial, as he prepared for his appeals, “little bits and pieces started coming and I realize I did this,” he later told a parole board.

Image Officer Giglio died on Dec. 16, 1975. Credit... Nassau County Police Benevolent Association

John MacKenzie had been a small-time thief and fence until 2:30 a.m. on Oct. 7, 1975. At that moment, he was stealing 240 blouses from a women’s clothing store in West Hempstead, tossing them out a window into the trunk of a car, when the police drove by. They saw an idling car, strewed clothing, an open window — and called for backup.