Constraint Propagation as Information Maximization

This paper draws on diverse areas of computer science to develop a unified view of computation: • Optimization in operations research, where a numerical objective function is maximized under constraints, is generalized from the numerical total order to a non-numerical partial order that can be interpreted in terms of information. • Relations are generalized so that there are relations of which the constituent tuples have numerical indexes, whereas in other relations these indexes are variables. The distinction is essential in our definition of constraint satisfaction problems. • Constraint satisfaction problems are formulated in terms of semantics of conjunctions of atomic formulas of predicate logic. • Approximation structures, which are available for several important domains, are applied to solutions of constraint satisfaction problems. As application we treat constraint satisfaction problems over reals. These cover a large part of numerical analysis, most significantly nonlinear equations and inequalities. The chaotic algorithm analyzed in the paper combines the efficiency of floating-point computation with the correctness guarantees of arising from our logico-mathematical model of constraint-satisfaction problems. 1 Computation as maximization in information space The early history of constraint processing is written in three MIT theses: Sutherland’s, Waltz’s, and Steele’s [16, 20, 14]. Already in this small selection one can discern two radically different approaches. Sutherland and Steele use relaxation: starting form a guessed assignment of values to variables, constraints are successively used to adjust variables in such a way as to satisfy better the constraint under consideration. These authors followed an old idea brought into prominence under the name of relaxation by Southwell [15]. Research Report 746, Dept. of Computer Science, University of Western Ontario, Canada. Department of Computer Science, University of Western Ontario, Canada and INRIA Rocquencourt, France. Department of Computer Science, University of Victoria, Canada.