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Despite the steady progress, Giffords has a long road to recovery. Doctors are not sure what, if any, disability she will have.

Sometimes, areas of the brain that seem damaged can recover, said Mark Sherer, a neuropsychologist at the rehab center.

“Some of the tissue is temporarily dysfunctional, so the patient appears very impaired very early on after the injury,” but may not be permanently damaged, he said.

Mr. Kelly said Mrs. Giffords would be proud of the way Tucson has responded. Memorials continued to grow Friday outside the hospital, in front of her office and at the scene of the shooting.

“I know one of the first things Gabby is going to want to do as soon as she’s able to is start writing thank you notes,” he said.

Medical Writer Marilynn Marchione and Ramit Plushnick-Masti reported from Houston. AP aerospace writer Marcia Dunn contributed to this story from Cape Canaveral, Fla.