Reachability problems for systems with linear dynamics

This thesis deals with reachability and freeness problems for systems with linear dynamics, including hybrid systems and matrix semigroups. Hybrid systems are a type of dynamical system that exhibit both continuous and discrete dynamic behaviour. Thus they are particularly useful in modelling practical real world systems which can both flow (continuous behaviour) and jump (discrete behaviour). Decision questions for matrix semigroups have attracted a great deal of attention in both the Mathematics and Theoretical Computer Science communities. They can also be used to model applications with only discrete components. For a computational model, the reachability problem asks whether we can reach a target point starting from an initial point, which is a natural question both in theoretical study and for real-world applications. By studying this problem and its variations, we shall prove in a formal mathematical sense that many problems are intractable or even unsolvable. Thus we know when such a problem appears in other areas like Biology, Physics or Chemistry, either the problem itself needs to be simplified, or it should by studied by approximation. In this thesis we concentrate on a specific hybrid system model, called an HPCD, and its variations. The objective of studying this model is twofold: to obtain the most expressive system for which reachability is algorithmically solvable and to explore the simplest system for which it is impossible to solve. For the solvable sub-cases, we shall also study whether reachability is in some sense easy or hard by determining which complexity classes the problem belongs to, such as P, NP(-hard) and PSPACE(-hard). Some undecidable results for matrix semigroups are also shown, which both strengthen our knowledge of the structure of matrix semigroups, and lead to some undecidability results for other models

Matrix Semigroup Freeness Problems in SL(2, Z)

In this paper we study decidability and complexity of decision problems on matrices from the special linear group SL(2,Z). In particular, we study the freeness problem: given a finite set of matrices G generating a multiplicative semigroup S, decide whether each element of S has at most one factorization over G. In other words, is G a code? We show that the problem of deciding whether a matrix semigroup in SL(2,Z) is non-free is NP-hard. Then, we study questions about the number of factorizations of matrices in the matrix semigroup such as the finite freeness problem, the recurrent matrix problem, the unique factorizability problem, etc. Finally, we show that some factorization problems could be even harder in SL(2,Z), for example we show that to decide whether every prime matrix has at most k factorizations is PSPACE-hard

Vector Ambiguity and Freeness Problems in SL (2, ℤ).

We study the vector ambiguity problem and the vector freeness problem in SL(2,Z). Given a finitely generated n×n matrix semigroup S and an n-dimensional vector x, the vector ambiguity problem is to decide whether for every target vector y=Mx, where M∈S, M is unique. We also consider the vector freeness problem which is to show that every matrix M which is transforming x to Mx has a unique factorization with respect to the generator of S. We show that both problems are NP-complete in SL(2,Z), which is the set of 2×2 integer matrices with determinant 1. Moreover, we generalize the vector ambiguity problem and extend to the finite and k-vector ambiguity problems where we consider the degree of vector ambiguity of matrix semigroups

Acceptance Ambiguity for Quantum Automata

We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post\u27s correspondence problem

Matrix Semigroup Freeness Problems in SL(2,Z)\mathrm{SL}(2,\mathbb{Z})

In this paper we study decidability and complexity of decision problems on matrices from the special linear group $\mathrm{SL}(2,\mathbb{Z})$. In particular, we study the freeness problem: given a finite set of matrices $G$ generating a multiplicative semigroup $S$, decide whether each element of $S$ has at most one factorization over $G$. In other words, is $G$ a code? We show that the problem of deciding whether a matrix semigroup in $\mathrm{SL}(2,\mathbb{Z})$ is non-free is NP-hard. Then, we study questions about the number of factorizations of matrices in the matrix semigroup such as the finite freeness problem, the recurrent matrix problem, the unique factorizability problem, etc. Finally, we show that some factorization problems could be even harder in $\mathrm{SL}(2,\mathbb{Z})$, for example we show that to decide whether every prime matrix has at most $k$ factorizations is PSPACE-hard

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

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

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

The symmetric Post Correspondence Problem, and errata for the freeness problem for matrix semigroups

We define the symmetric Post Correspondence Problem (PCP) and prove that it is undecidable. As an application we show that the original proof of undecidability of the freeness problem for 3-by-3 integer matrix semigroups works for the symmetric PCP, but not for the PCP in general.Comment: 12 pages. Small corrections and clarifications were added in version