The nature and limits of computation: a celebration of Alan Turing

The Nature and Limits of Computation: A Celebration of Alan Turing As a figure of speech we sometimes describe advances in math and science as “celebrated results,” but rarely does that mean that an actual celebration took place. Well, this time we mean it. Alan Turing (19121954) is widely regarded as the “Father of Computer Science.” To commemorate the 100th anniversary of his birth, 2012 has been named the “Alan Turing Year.” In honor of his landmark contributions to science and humanity, events are being held throughout 2012 and across the globe. But you won’t have to travel far to be a part of the celebrations: within these pages of XRDS is our tribute to Alan Turing. Turing was fascinated, among many things, by two questions: “What is the nature of computation?” and “What are the limits of computation?” Turing, along with his advisor Alonzo Church, answered the former with his development of the Universal Turing Machine and answered the latter with his proof of the undecidability of the halting problem. Both of these breakthroughs have stood the test of time. (It is astonishing that the Turing machine is still one of the basic models of abstract computation since it predates the modern digital computer.) As is usually the case with great answers to deep questions, Turing’s work was not the end of the story but the beginning of a rich field of inquiry. This issue of XRDS surveys our understanding of the nature and limits of computation— our current answers to those two questions. Ian Horswill of Northwestern University starts us off with “What is Computation?” As he explains, the idea of computation is both mathematical and cultural, and gets renegotiated as we understand more about the world (including the world between our ears). In a special interview Robert Soare of the University of Chicago (who studied with Church while an undergraduate at Princeton) takes a look at the past and future of logic and computability, and explains why Turing is like Michelangelo. Oded Goldreich of the Weizmann Institute of Science guides us through a whirlwind tour of computational complexity in “Invitation to Complexity Theory.” As an homage to Alan Turing’s work in breaking the Enigma during the Second World War, we include a sidebar by Goldreich, “On the Use of Complexity in Cryptography.” Dana Moshkovitz of MIT continues our mini-tour of complexity with a truly surprising result: Every mathematical proof can be written so that one can check its validity probabilistically by looking at only two statements in the proof. She explains more in “The Tale of the PCP Theorem.” Luca Trevisan of Stanford rounds out our section on complexity with “Pseudorandomness and Derandomization,” which is based on the question: Does having access to randomness help to compute things efficiently? No overview of computation would be complete without a discussion of quantum computing, and for this Aram Harrow of the University of Washington is our guide. He explains how quantum computing is not just a promising technology but a new way of looking at the world. In our last article we look forward. Dennis Shasha of New York University speculates about what’s coming next in “Future of Computing: Inspiration from Nature.” We’re truly excited to bring you such a diverse lineup in recognition of the Turing Centenary. Our contributors cover such topics as mathematical logic, neuroscience, cryptography, complexity, quantum physics, and even evolutionary algorithms. But perhaps it’s only fitting. As Andrew Miller MP has said, “There isn’t a discipline in science that Turing has not had an impact upon.” One hundred years after his birth, the transformative power of Alan Turing’s work continues to be felt around the world.