Homotopy Type Theory for Big Data

Homotopy type theory has been developed by us for over two decades and is applied to big data to eliminate the inevitable bottleneck of big data implementation originating from inherent combinatorial explosion. An incrementally modular abstraction hierarchy, IMAH in short, is used in 7 layers starting from homotopy, then type, and ending with presentation. Big data is in cyber worlds that are being formed in cyberspaces as computational spaces. Now cyberspaces are rapidly expanding on the Web either intentionally or spontaneously, with or without design. It is quite different in its emphasis of homotopy type theory as recently reported with emphasis on mathematical proof automation and computer verification. Widespread and intensive local activities are melting each other on the web as big data globally to create cyber worlds. The major key players of big data in cyber worlds include e-finance that trades a GDP-equivalent a day and e-manufacturing that is transforming industrial production into Web shopping of product components and assembly factories. Lacking proper theory and design, big data has continued to grow chaotic and are now out of human understanding and control. This research first presents a generic theoretical framework as an incrementally modular abstraction hierarchy, based on homotopy type theory, provides an axiomatic approach to theorize the potentials of big data in cyber worlds, and shows that the incrementally modular abstraction hierarchy automate big data application development and eliminates the need for design verification and validation. It also makes the systems developed secure from the all sort of attacks.