Paul C. Bell

The continuous Skolem-Pisot problem

We study decidability and complexity questions related to a continuous analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of a linear recurrent sequence. In particular, we show that the continuous version of the nonnegativity problem is NP-hard in general and we show that the presence of a zero is decidable for several subcases, including instances of depth two or less, although the decidability in general is left open. The problems may also be stated as reachability problems related to real zeros of exponential polynomials or solutions to initial value problems of linear differential equations, which are interesting problems in their own right.

Reachability problems in quaternion matrix and rotation semigroups

We examine computational problems on quaternion matrix and rotation semigroups. It is shown that in the ultimate case of quaternion matrices, in which multiplication is still associative, most of the decision problems for matrix semigroups are undecidable in dimension two. The geometric interpretation of matrix problems over quaternions is presented in terms of rotation problems for the 2- and 3-sphere. In particular, we show that the reachability of the rotation problem is undecidable on the 3-sphere and other rotation problems can be formulated as matrix problems over complex and hypercomplex numbers.

On the Computational Complexity of Matrix Semigroup Problems

Most computational problems for matrix semigroups and groups are inherently difficult to solve and even undecidable starting from dimension three. The questions about the decidability and complexity of problems for two-dimensional matrix semigroups remain open and are directly linked with other challenging problems in the field. In this paper we study the computational complexity of the problem of determining whether the identity matrix belongs to a matrix semigroup (the Identity Problem) generated by a finite set of 2 × 2 integral unimodular matrices. The Identity Problem for matrix semigroups is a well-known challenging problem, which has remained open in any dimension until recently. It is currently known that the problem is decidable in dimension two and undecidable starting from dimension four. In particular, we show that the Identity Problem for 2 × 2 integral unimodular matrices is NP-hard by a reduction of the Subset Sum Problem and several new encoding techniques. An upper bound for the nontrivial decidability result by C. Choffrut and J. Karhumaki is unknown. However, we derive a lower bound on the minimum length solution to the Identity Problem for a constructible set of instances, which is exponential in the number of matrices of the generator set and the maximal element of the matrices. This shows that the most obvious candidate for an NP algorithm, which is to guess the shortest sequence of matrices which multiply to give the identity matrix, does not work correctly since the certificate would have a length which is exponential in the size of the instance. Both results lead to a number of corollaries confirming the same bounds for vector reachability, scalar reachability and zero in the right upper corner problems.

Mortality for 2 ×2 matrices is NP-Hard

We study the computational complexity of determining whether the zero matrix belongs to a finitely generated semigroup of two dimensional integer matrices (the mortality problem). We show that this problem is NP-hard to decide in the two-dimensional case by using a new encoding and properties of the projective special linear group. The decidability of the mortality problem in two dimensions remains a long standing open problem although in dimension three is known to be undecidable as was shown by Paterson in 1970. We also show a lower bound on the minimum length solution to the Mortality Problem, which is exponential in the number of matrices of the generator set and the maximal element of the matrices.

Periodic and infinite traces in matrix semigroups

In this paper we provide several new results concerning word and matrix semigroup problems using counter automaton models. As a main result, we prove a new version of Posts correspondence problem to be undecidable and show its application to matrix semigroup problems, such as Any Diagonal Matrix Problem and Recurrent Matrix Problem. We also use infinite periodic traces in counter automaton models to show the undecidability of a new variation of the Infinite Post Correspondence Problem and Vector Ambiguity Problem for matrix semigroups.

On the membership of invertible diagonal and scalar matrices

In this paper, we consider decidability questions that are related to the membership problem in matrix semigroups. In particular, we consider the membership of a given invertible diagonal matrix in a matrix semigroup and then a scalar matrix, which has a separate geometric interpretation. Both problems have been open for any dimensions and are shown to be undecidable in dimension 4 with integral matrices by a reduction of the Post Correspondence Problem (PCP). Although the idea of PCP reduction is standard for such problems, we suggest a new coding technique to cover the case of diagonal matrices.

Reachability Problems for Hierarchical Piecewise Constant Derivative Systems

In this paper, we investigate the computability and complexity of reachability problems for two-dimensional hierarchical piecewise constant derivative (HPCD) systems. The main interest in HPCDs stems from the fact that their reachability problem is on the border between decidability and undecidability, since it is equivalent to that of reachability for one-dimensional piecewise affine maps (PAMs) which is a long standing open problem. Understanding the most expressive hybrid system models that retain decidability for reachability has generated a great deal of interest over the past few years. In this paper, we show a restriction of HPCDs (called RHPCDs) which leads to the reachability problem becoming decidable. We then study which additional powers we must add to the RHPCD model to render it 1D PAM-equivalent. Finally, we show NP-hardness of reachability for nondeterministic RHPCDs.

Lowering undecidability bounds for decision questions in matrices

In this paper we consider several reachability problems such as vector reachability, membership in matrix semigroups and reachability problems in piecewise linear maps. Since all of these questions are undecidable in general, we work on lowering the bounds for undecidability. In particular, we show an elementary proof of undecidability of the reachability problem for a set of 7 two-dimensional affine transformations. Then, using a modified version of a standard technique, we also prove the vector reachability problem is undecidable for two (rational) matrices in dimension 16. The above result can be used to show that the system of piecewise linear functions of dimension 17 with only two intervals has an undecidable set-to-point reachability problem. We also show that the “zero in the upper right corner” problem is undecidable for two integral matrices of dimension 18 lowering the bound from 23.

On the membership of invertible diagonal matrices

In this paper we consider decidability questions that are related to the membership problem in matrix semigroups. In particular we consider the membership of a particular invertible diagonal matrix in a matrix semigroup and then a scalar matrix, which has a separate geometric interpretation. Both problems have been open for any dimensions and are shown to be undecidable in dimenesion 4 with integral matrices and in dimension 3 with rational matrices by a reduction of the Post Correspondence Problem (PCP). Although the idea of PCP reduction is standard for such problems, we suggest a new coding technique to cover the case of diagonal matrices.

Reachability and Mortality Problems for Restricted Hierarchical Piecewise Constant Derivatives

We show the NP-hardness of the reachability and mortality problems for a three dimensional variant of Piecewise Constant Derivative (PCD) system called a bounded 3-dimensional Restricted Hierarchical PCD (3-RHPCD). Both problems are shown to be in PSPACE, even for n-dimensional RHPCD. This is a restricted model with similarities to other models in the literature such as stopwatch automata, rectangular automata and PCDs. We also show that for an unbounded 3-RHPCD, both problems become undecidable via a simulation of a Minsky machine.