Symbolic Pattern Solving for Equational Reasoning

Symbolic pattern solving is a new approach for nding symbolic solutions of equa-tional constraints over arbitrary algebraic structures. Despite the deep results and methods already produced in equational constraint solving, real-world applications still cannot be tackled. To reduce the original complexity, the approach advocated here uses the solution pattern, the skeleton of the solution often suggested by the problem instance. Pattern solving is successful if some substitution, of the unknown coeecients within the pattern, turns the pattern into a solution. The core of the method is the possibility of straightforward elimination of universally quantiied variables from certain formulas related to the input problem and solution pattern. It is shown that fast elimination of universal quantiiers is complete in innnite integral domains, torsion-free modules over an innnite domain, nite dimensional division algebras over innnite elds, and in graded exterior algebras, if the underlying scalar eld is innnite. 1 Motivation This paper introduces a new approach for nding symbolic solutions of equational constraints over arbitrary algebraic structures 4]. Solving constraints symbolically in a spe-ciic algebraic structure means either deciding their truth, or nding terms of the language that, substituted for the free variables, make the constraints true. Intensive research, for instance in uniication theory 5, 1] and in computer algebra 6], has produced a wealth of deep results and there are now several computational methods available. However, all this is not yet suucient. The state of the art symbolic methods usually cannot tackle real-world applications. As soon as the problems increase in complexity, whether for the number of constraints, or their degree or for the number of symbols involved, the symbolic methods fail to perform as eeciently as the numeric methods do. Depending on the algebraic structure, the complexity of symbolic solving of equa-tional constraints ranges from polynomial-time decidable to undecidable. As example, take Hilbert's Tenth Problem: given a polynomial equation with integer coeecients, nd whether there are solutions over some algebraic structure. The famous results of 7] and of 9] show that it is unsolvable over the integers and decidable over the real numbers. It is also believed to be undecidable over the rational numbers, where is still an open problem 8]. The basic observation leading to the method introduced here is that often the nature of the constraint solving problem at hand suggests a \skeleton" of the solution, a solution pattern. For instance, in the following problem from quantum chemistry, …