The nonprofit Allen Institute for AI, led by a respected computer scientist who sold his company to Apple, is trying to democratize cutting-edge research. Ali Farhadi is no tech rebel. The 42-year-old computer scientist is a highly respected researcher, a professor at the University of Washington, and the founder of a start-up that was acquired by Apple, where he worked until four months ago. But Mr. Farhadi, who in July became chief executive of the Allen Institute for AI, is calling for “radical openness” to democratize research and development in a new wave of artificial intelligence that many believe is the most important technological advance in decades. The Allen Institute has begun an ambitious initiative to build a freely available AI alternative to tech giants like Google and start-ups like OpenAI. In an industry process called open source, other researchers will be allowed to scrutinize and use this new system and the data fed into it.