Applying Theory to Practice

By making use of three IBM case studies involving the author and colleagues, this paper is about applying theory to practice. In the first case study, the system builders (or practitioners) initiated the interaction. This interaction led to the following problem. Assume that there is a set of objects, each with multiple attributes, and there is a numerical score assigned to each attribute of each object. In the spirit of real-valued logics, there is a scoring function (such as the min or the average), and a ranking of the objects is obtained by applying the scoring function to the scores of each object’s attributes The problem is to find the top $k$ objects, while minimizing the number of database accesses. An algorithm is given that is optimal in an extremely strong sense: not just in the worst case or the average case, but (up to a constant factor) in every case! Even though the algorithm is only 8 lines long (!), the paper containing the algorithm won the 2014 Gödel Prize, the top prize for a paper in theoretical computer science. The interaction in the second case study was initiated by theoreticians, who wanted to lay the foundations for ‘data exchange’, in which data is converted from one format to another. Although this problem may sound mundane, the issues that arise are fascinating, and this work made data exchange a new subfield, with special sessions in every major database conference. This work won the 2020 Alonzo Church Award, the highest prize for research in logic and computation. The third case study, specifically on real-valued (or ‘fuzzy’) logic, arose as part of a large ‘Logical Neural Nets’ (LNN) project at IBM. The inputs to, say, an ‘and’ gate could each be any numbers in the interval [0,1]. The system builders of LNN wanted a sound and complete axiomatization for real-valued logic, so that they could arrive at truth values given other truth values whenever possible. This recent work provides a sound and complete axiomatization for a large class of real-valued logics, including the most common ones. It also allows weights, where the importance of some subformulas can be greater than that of other subformulas. This paper is aimed at both theoreticians and system builders, to show them the mutual benefits of working together. This is via the three case studies mentioned above: two initiated by the system builders, and one by the theoreticians. The moral for the theoreticians is to show by example how to apply theory to practice, and why applying theory to practice can lead to better theory. The moral for the system builders is the value of theory, and the value of involving theoreticians. This paper is written in a very informal style. In fact, it is based closely on a talk on ‘Applying theory to practice’ that the author has presented a number of times.