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

Integer Vector Addition Systems with States

This paper studies reachability, coverability and inclusion problems for Integer Vector Addition Systems with States (ZVASS) and extensions and restrictions thereof. A ZVASS comprises a finite-state controller with a finite number of counters ranging over the integers. Although it is folklore that reachability in ZVASS is NP-complete, it turns out that despite their naturalness, from a complexity point of view this class has received little attention in the literature. We fill this gap by providing an in-depth analysis of the computational complexity of the aforementioned decision problems. Most interestingly, it turns out that while the addition of reset operations to ordinary VASS leads to undecidability and Ackermann-hardness of reachability and coverability, respectively, they can be added to ZVASS while retaining NP-completness of both coverability and reachability.Comment: 17 pages, 2 figure

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

Computational problems in matrix semigroups

This thesis deals with computational problems that are defined on matrix semigroups, which playa pivotal role in Mathematics and Computer Science in such areas as control theory, dynamical systems, hybrid systems, computational geometry and both classical and quantum computing to name but a few. Properties that researchers wish to study in such fields often turn out to be questions regarding the structure of the underlying matrix semigroup and thus the study of computational problems on such algebraic structures in linear algebra is of intrinsic importance. Many natural problems concerning matrix semigroups can be proven to be intractable or indeed even unsolvable in a formal mathematical sense. Thus, related problems concerning physical, chemical and biological systems modelled by such structures have properties which are not amenable to algorithmic procedures to determine their values. With such recalcitrant problems we often find that there exists a tight border between decidability and undecidability dependent upon particular parameters of the system. Examining this border allows us to determine which properties we can hope to derive algorithmically and those problems which will forever be out of our reach, regardless of any future advances in computational speed. There are a plethora of open problems in the field related to dynamical systems, control theory and number theory which we detail throughout this thesis. We examine undecidability in matrix semigroups for a variety of different problems such as membership and vector reachability problems, semigroup intersection emptiness testing and freeness, all of which are well known from the literature. We also formulate and survey decidability questions for several new problems such as vector ambiguity, recurrent matrix problems, the presence of any diagonal matrix and quaternion matrix semigroups, all of which we feel give a broader perspective to the underlying structure of matrix semigroups

Reachability games and related matrix and word problems

In this thesis, we study different two-player zero-sum games, where one player, called Eve, has a reachability objective (i.e., aims to reach a particular configuration) and the other, called Adam, has a safety objective (i.e., aims to avoid the configuration). We study a general class of games, called Attacker-Defender games, where the computational environment can vary from as simple as the integer line to n-dimensional topological braids. Similarly, the moves themselves can be simple vector addition or linear transformations defined by matrices. The main computational problem is to decide whether Eve has a winning strategy to reach the target configuration from the initial configuration, or whether the dual holds, that is, whether Adam can ensure that the target is never reached. The notion of a winning strategy is widely used in game semantics and its existence means that the player can ensure that his or her winning conditions are met, regardless of the actions of the opponent. It general, games provide a powerful framework to model and analyse interactive processes with uncontrollable adversaries. We formulated several Attacker-Defender games played on different mathematical domains with different transformations (moves), and identified classes of games, where the checking for existence of a winning strategy is undecidable. In other classes, where the problem is decidable, we established their computational complexity. In the thesis, we investigate four classes of games where determining the winner is undecidable: word games, where the players' moves are words over a group alphabet together with integer weights or where the moves are pairs of words over group alphabets; matrix games on vectors, where players transform a three-dimensional vector by linear transformations defined by 3×3 integer matrices; braid games, where players braid and unbraid a given braid; and last, but not least, games played on two-dimensional Z-VAS, closing the gap between decidable and undecidable cases and answering an existing open problem of the field. We also identified decidable fragments, such as word games, where the moves are over a single group alphabet, games on one-dimensional Z-VASS. For word games, we provide an upper-bound of EXPTIME , while for games on Z-VASS, tight bounds of EXPTIME-complete or EXPSPACE-complete, depending on the state structure. We also investigate single-player systems such as polynomial iteration and identity problem in matrix semigroups. We show that the reachability problem for polynomial iteration is PSPACE-complete while the identity problem for the Heisenberg group is in PTIME for dimension three and in EXPTIME for higher dimensions

Reachability problems in low-dimensional nondeterministic polynomial maps over integers

We study reachability problems for various nondeterministic polynomial maps in Zn. We prove that the reachability problem for very simple three-dimensional affine maps (with independent variables) is undecidable and is PSPACE-hard for both two-dimensional affine maps and one-dimensional quadratic maps. Then we show that the complexity of the reachability problem for maps without functions of the form ±x+a0 is lower. In this case the reachability problem is PSPACE for any dimension and if the dimension is not fixed, then the problem is PSPACE-complete. Finally we extend the model by considering maps as language acceptors and prove that the universality problem is undecidable for two-dimensional affine maps

Model checking infinite-state systems: generic and specific approaches

Model checking is a fully-automatic formal verification method that has been extremely successful in validating and verifying safety-critical systems in the past three decades. In the past fifteen years, there has been a lot of work in extending many model checking algorithms over finite-state systems to finitely representable infinitestate systems. Unlike in the case of finite systems, decidability can easily become a problem in the case of infinite-state model checking. In this thesis, we present generic and specific techniques that can be used to derive decidability with near-optimal computational complexity for various model checking problems over infinite-state systems. Generic techniques and specific techniques primarily differ in the way in which a decidability result is derived. Generic techniques is a “top-down” approach wherein we start with a Turing-powerful formalismfor infinitestate systems (in the sense of being able to generate the computation graphs of Turing machines up to isomorphisms), and then impose semantic restrictions whereby the desired model checking problem becomes decidable. In other words, to show that a subclass of the infinite-state systems that is generated by this formalism is decidable with respect to the model checking problem under consideration, we will simply have to prove that this subclass satisfies the semantic restriction. On the other hand, specific techniques is a “bottom-up” approach in the sense that we restrict to a non-Turing powerful formalism of infinite-state systems at the outset. The main benefit of generic techniques is that they can be used as algorithmic metatheorems, i.e., they can give unified proofs of decidability of various model checking problems over infinite-state systems. Specific techniques are more flexible in the sense they can be used to derive decidability or optimal complexity when generic techniques fail. In the first part of the thesis, we adopt word/tree automatic transition systems as a generic formalism of infinite-state systems. Such formalisms can be used to generate many interesting classes of infinite-state systems that have been considered in the literature, e.g., the computation graphs of counter systems, Turing machines, pushdown systems, prefix-recognizable systems, regular ground-tree rewrite systems, PAprocesses, order-2 collapsible pushdown systems. Although the generality of these formalisms make most interesting model checking problems (even safety) undecidable, they are known to have nice closure and algorithmic properties. We use these nice properties to obtain several algorithmic metatheorems over word/tree automatic systems, e.g., for deriving decidability of various model checking problems including recurrent reachability, and Linear Temporal Logic (LTL) with complex fairness constraints. These algorithmic metatheorems can be used to uniformly prove decidability with optimal (or near-optimal) complexity of various model checking problems over many classes of infinite-state systems that have been considered in the literature. In fact, many of these decidability/complexity results were not previously known in the literature. In the second part of the thesis, we study various model checking problems over subclasses of counter systems that were already known to be decidable. In particular, we consider reversal-bounded counter systems (and their extensions with discrete clocks), one-counter processes, and networks of one-counter processes. We shall derive optimal complexity of various model checking problems including: model checking LTL, EF-logic, and first-order logic with reachability relations (and restrictions thereof). In most cases, we obtain a single/double exponential reduction in the previously known upper bounds on the complexity of the problems