The Carotid Occlusion Surgery Study (COSS) evaluates whether a surgical operation, EC-IC bypass surgery, can reduce the chance of a stroke in someone who has complete blockage in one main artery in the neck that supplies blood to the brain (the carotid artery). The operation bypasses the blockage so more blood can flow to the brain. Only people with decreased blood flow to the brain, as demonstrated on a PET (positron emission tomographic) scan, are randomized into the COSS study. Among patients randomized into the COSS study, RECON will evaluate whether restoring the blood flow to the brain (with EC-IC bypass surgery) will also improve mental functioning. Participants in both the surgical and medical groups of the COSS study will participate in the RECON study. By comparing the mental functioning of the participants in both treatment groups over the course of 2 years, the investigators hope to determine whether the EC-IC bypass operation also helps improve or maintain mental functioning.