Registration is being held until Tuesday for the fall semester of George Washington University's After-School Program for Gifted Children. This semester's focus, the Middle Ages, will be approached through reading, mathematics, music, art and drama. At the end of the term, the students will produce a play they have created, accompanied by music they have written. The program is sponsored by GWU's Reading Center and supplements the formal education of selected gifted children between the ages of 5 and 14. Classes start Tuesday and will be held every Tuesday and Thursday from 3:45 to 5:45 p.m. until Dec. 10. The meeting place is 3500 Woodley Rd. NW. For cost and registration information, call Judy Findlay at 994-6286.

The 11th annual John Marshall Library Pet Show is scheduled for 1 p.m. Saturday. It is strictly dogs to Ring 1, cats to Ring 2; but pets that slither, fly or crawl are also welcome to compete for 14 awards. Dr. Richard Hawe, an Alexandria veterinarian, will give free answers to questions at his Ask the Vet Booth. A free professional demonstration of dog obedience will also be featured. Events include a dog parade, cat parade, small pet review and shelf pet review. Registration for pets will be held at 12:45 on the day of the show. The rain date is Oct. 3. The John Marshall Library is at 6209 Rose Hill Dr., Alexandria. For more information, call 971-0010 or 971-0011.

The annual Hill School Horse and Pony Show will be held from 9 a.m. to 5 p.m. at the Foxcroft School's showgrounds on Sunday. The show, part of the Short Circuit Organization, offers children and adults the chance to demonstrate their skills in competition. Participants may compete in any of 32 classes. Silver prizes and six ribbons will be awarded to the winners of each class. In addition, the News Reel Championship Trophy will be awarded to the champion of the Small/Medium Pony Hunter Division, and the Karbon Kopy Perpetual Trophy will be awarded to the winner in the Large Pony Hunter Division. Each class entry costs $6, but admission is free. Food will be available from a snack stand run by Hill School eighth graders. Proceeds from the stand will help fund their graduation present to Hill School. Foxcroft School is on Rte. 626, four miles north of Middleburg. For more information, call Margaret New at (703) 687-3077.