Computation on Structures - The Challenge of Seamless Integration of Theory and Rigorous Scientific Practice
Frontiers in Computer Science
In 1985 Egon Börger published his influential monograph on computation theory, logic, and complexity (Börger, 1989) , which focused on the concept of formal language as a carrier of the precise expression of meaning, facts, and problems, and the concept of algorithm or calculus, i.e., a formally operating procedure for the solution of precisely described questions and problems. At that time, the text was at the forefront of a modern theory of these concepts, paving the way in which they
... d first in mathematical logic and computability theory and later in automata theory, the theory of formal languages and complexity theory. Nonetheless, the state of the theory left many open problems. Computing started to stretch out into many new application areas. Distributed computing over networks became possible, database systems facilitated concurrent computation, artificial intelligence ventured from a niche area to a useful technology, enabling inferential problem solving in diagnosis, controlling machines through software became possible, etc. Now, only 35 years later, the rapid progress in computing has led to a fascinating variety of interconnected systems that are used to support, manage, and control almost all aspects of our life. There is hardly an area that has not yet been penetrated by computing, and still there are many open challenges for the continuation of this success story. We are now dealing with systems of systems that are characterized by the following properties: 1) they operate in parallel either synchronously or asynchronously exploiting multiple processor cores and computing resources distributed over networks; 2) they are hybrid interacting with analogue systems with continuous behaviour; 3) they are adaptive changing their own behaviour; 4) they are intelligent reasoning about themselves and their environment and learning from data; 5) they are interactive communicating with their environment; and 6) they integrate random behaviour depending on probability distributions. For instance, if we consider advanced manufacturing, all equipment will integrate electronic and mechanical parts with continuous behaviour, software with discrete behaviours, and sensors and actuators for their integration. Each piece of equipment constitutes an autonomous system of its own, and all the equipment of a shopfloor form a system of systems. This can be taken further to systems integrating all systems within a city. Any collection of systems that collaborate to achieve a common task can be considered a system of systems. The key challenge for theoretical computer science is to provide the necessary scientific foundations centered around computation theory, complexity, and logic. It has to be ensured that modern computing, in practice with all its facets, is grounded in rigorous scientific methods. There is a huge gap between the abstraction level of an algorithmic system and the level of Turing machines. It is not sufficient to know that, deep inside the core of systems, we deal with computations that given a proper string encoding can be represented by Turing machines. Instead, we need a computation theory over arbitrary structures rather than strings. Structures in the sense of Tarski are omnipresent in all mathematical theories, and any extension should be conservative in the sense that the classical theory is preserved as a representation on the lowest level of abstraction.