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

Self Organising Maps for Anatomical Joint Constraint

The accurate simulation of anatomical joint models is becoming increasingly important for both realistic animation and diagnostic medical applications. Recent models have exploited unit quaternions to eliminate ingularities when modelling orientations between limbs at a joint. This has led to the development of quaternion based joint constraint validation and correction methods. In this paper a novel method for implicitly modelling unit quaternion joint constraints using Self Organizing Maps (SOMs) is proposed which attempts to address the limitations of current constraint validation and correction approaches. Initial results show that the resulting SOMs are capable of modelling regular spherical constraints on the orientation of the limb

Vector Reachability Problem in SL(2,Z)\mathrm{SL}(2,\mathbb{Z})

The decision problems on matrices were intensively studied for many decades as matrix products play an essential role in the representation of various computational processes. However, many computational problems for matrix semigroups are inherently difficult to solve even for problems in low dimensions and most matrix semigroup problems become undecidable in general starting from dimension three or four. This paper solves two open problems about the decidability of the vector reachability problem over a finitely generated semigroup of matrices from $\mathrm{SL}(2,\mathbb{Z})$ and the point to point reachability (over rational numbers) for fractional linear transformations, where associated matrices are from $\mathrm{SL}(2,\mathbb{Z})$. The approach to solving reachability problems is based on the characterization of reachability paths between points which is followed by the translation of numerical problems on matrices into computational and combinatorial problems on words and formal languages. We also give a geometric interpretation of reachability paths and extend the decidability results to matrix products represented by arbitrary labelled directed graphs. Finally, we will use this technique to prove that a special case of the scalar reachability problem is decidable

On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a nite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: \Is it decidable for a nitely generated semigroup S of integral square matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a nite set of matrices generates a group is also undecidable. We also answer several questions for matrices over di erent number elds. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words

Decidability of the Membership Problem for 2×22\times 2 integer matrices

The main result of this paper is the decidability of the membership problem for $2\times 2$ nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular $2\times 2$ integer matrices $M_1,\dots,M_n$ and $M$ decides whether $M$ belongs to the semigroup generated by $\{M_1,\dots,M_n\}$. Our algorithm relies on a translation of the numerical problem on matrices into combinatorial problems on words. It also makes use of some algebraical properties of well-known subgroups of $\mathrm{GL}(2,\mathbb{Z})$ and various new techniques and constructions that help to limit an infinite number of possibilities by reducing them to the membership problem for regular languages

Scalar Ambiguity and Freeness in Matrix Semigroups over Bounded Languages

There has been much research into freeness properties of finitely generated matrix semigroups under various constraints, mainly related to the dimensions of the generator matrices and the semiring over which the matrices are defined. A recent paper has also investigated freeness properties of matrices within a bounded language of matrices, which are of the form M1M2 · · · Mk ⊆ F n×n for some semiring F [9]. Most freeness problems have been shown to be undecidable starting from dimension three, even for upper-triangular matrices over the natural numbers. There are many open problems still remaining in dimension two. We introduce a notion of freeness and ambiguity for scalar reachability problems in matrix semigroups and bounded languages of matrices. Scalar reachability concerns the set {ρ TMτ |M ∈ S}, where ρ, τ ∈ F n are vectors and S is a finitely generated matrix semigroup. Ambiguity and freeness problems are defined in terms of uniqueness of factorizations leading to each scalar. We show various undecidability results

Towards Uniform Online Spherical Tessellations

The problem of uniformly placing points onto a sphere finds applications in many areas. For example, points on the sphere correspond to unit quaternions as well as to the group of rotations SO(3) and the online version of generating uniform rotations (known as “incremental generation”) plays a crucial role in a large number of engineering applications ranging from robotics and aeronautics to computer graphics. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The first online algorithm of Chen et al. was upper-bounded by 5.99 and later improved to 3.69, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. In this paper we provide a more efficient tessellation technique based on the regular icosahedron, which improves the upper-bound for the online version of this problem, decreasing it to approximately 2.84. Moreover, we show that the lower bound for the gap ratio of placing at least three points is and for at least four points is no less than 1.726.</jats:p

Towards Uniform Online Spherical Tessellations

The problem of uniformly placing N points onto a sphere finds applications in many areas. For example, points on the sphere correspond to unit quaternions as well as to the group of rotations SO(3) and the online version of generating uniform rotations (known as “incremental generation”) plays a crucial role in a large number of engineering applications ranging from robotics and aeronautics to computer graphics. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The first online algorithm of Chen et al. was upper-bounded by 5.99 and later improved to 3.69, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. In this paper we provide a more efficient tessellation technique based on the regular icosahedron, which improves the upper-bound for the online version of this problem, decreasing it to approximately 2.84. Moreover, we show that the lower bound for the gap ratio of placing at least three points is 1.618 and for at least four points is no less than 1.726

Weighted Automata on Infinite Words in the Context of Attacker-Defender Games

The paper is devoted to several infinite-state Attacker–Defender games with reachability objectives. We prove the undecidability of checking for the existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games. To prove these results, we consider a model of weighted automata operating on infinite words and prove that the universality problem is undecidable for this new class of weighted automata. We show that the universality problem is undecidable by using a non-standard encoding of the infinite Post correspondence problem

On the decidability of semigroup freeness

This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X of S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoid. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over three-by-three integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is three-fold: (i) to present general results concerning freeness problems, (ii) to study the decidability of freeness problems over various particular semigroups (special attention is devoted to multiplicative matrix semigroups), and (iii) to propose precise, challenging open questions in order to promote the study of the topic.Comment: 46 pages. 1 table. To appear in RAIR