Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains

A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of the decomposition. Secondly, the computation uses regular chains theory to first build a cylindrical decomposition of complex space (CCD) incrementally by polynomial. Significant modification of the regular chains technology was used to achieve the more sophisticated invariance criteria. Experimental results on an implementation in the RegularChains Library for Maple verify that combining these advances gives an algorithm superior to its individual components and competitive with the state of the art

Algorithmic Thomas Decomposition of Algebraic and Differential Systems

In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these systems into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems, simplicity means triangularity, square-freeness and non-vanishing initials. Differential simplicity extends algebraic simplicity with involutivity. We build upon the constructive ideas of J. M. Thomas and develop them into a new algorithm for disjoint decomposition. The given paper is a revised version of a previous paper and includes the proofs of correctness and termination of our decomposition algorithm. In addition, we illustrate the algorithm with further instructive examples and describe its Maple implementation together with an experimental comparison to some other triangular decomposition algorithms.Comment: arXiv admin note: substantial text overlap with arXiv:1008.376

The complexity of the normal surface solution space

Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number of such surfaces grows in relation to the size of the underlying triangulation. Here we address this problem in both theory and practice. In theory, we tighten the exponential upper bound substantially; furthermore, we construct pathological triangulations that prove an exponential bound to be unavoidable. In practice, we undertake a comprehensive analysis of millions of triangulations and find that in general the number of vertex normal surfaces is remarkably small, with strong evidence that our pathological triangulations may in fact be the worst case scenarios. This analysis is the first of its kind, and the striking behaviour that we observe has important implications for the feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2 tables; v2: added minor clarification