Only four states  Alabama, Alaska, Massachusetts and Oklahoma  do not have laws in place specifically dealing with postconviction DNA testing, and Alabama recently enacted one limited to death row inmates that will become effective soon.

Many states that do allow postconviction testing impose conditions on who may seek it. Prosecutors often fight hard to deny access to DNA evidence even in states that nominally allow it, saying the prisoner in question had not met the statutory conditions.

Some laws, for instance, do not allow prisoners who have confessed to seek DNA evidence, though false confessions have been common among exonerated inmates. Other states allow testing only if it was unavailable at the time of trial.

Mr. Neufeld said the logic of Thursday’s decision might allow constitutional challenges to some of those laws.

In the case from Alaska, District Attorney’s Office v. Osborne, No. 08-6, Mr. Osborne sought to test biological evidence on a condom found at the crime scene, a snowbank near Anchorage International Airport. The victim was raped, beaten with an ax handle, shot in the head and left for dead. But the bullet only grazed her head, and she survived.

Rudimentary DNA testing on the condom in preparation for trial excluded two other suspects and included Mr. Osborne among those who might have committed the crime. The kind of testing used at the time, Chief Justice Roberts wrote, “generally cannot narrow the perpetrator down to less than 5 percent of the population.”

Image William G. Osborne sought more discriminating DNA testing after his conviction. Credit... Innocence Project

Mr. Osborne’s trial lawyer decided not to pursue a second kind of DNA testing that was more discriminating. The lawyer said she feared that the results might further incriminate her client. After his conviction, Mr. Osborne sued state officials in federal court seeking access to the DNA evidence for a third kind of yet-more-discriminating testing.