26 research outputs found

    The Identity Correspondence Problem and its Applications

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    In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite 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 finitely generated semigroup S of square integral 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 finite set of matrices generates a group is also undecidable. We also answer several question for matrices over different number fields. 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.Comment: We have made some proofs clearer and fixed an important typo from the published journal version of this article, see footnote 3 on page 1

    Acceptance Ambiguity for Quantum Automata

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    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

    Computational problems in matrix semigroups

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    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

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

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    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 SL(2,Z)\mathrm{SL}(2,\mathbb{Z}) and the point to point reachability (over rational numbers) for fractional linear transformations, where associated matrices are from SL(2,Z)\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

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    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

    On injectivity of quantum finite automata

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    We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the injectivity problem of determining if the acceptance probability function of a MO-QFA is injective over all input words, i.e., giving a distinct probability for each input word. We show that the injectivity problem is undecidable for 8 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial state vector is real algebraic. We also show undecidability of this problem when the initial vector is rational, although with a huge increase in the number of states. We utilize properties of quaternions, free rotation groups, representations of tuples of rationals as linear sums of radicals and a reduction of the mixed modification of Post's correspondence problem, as well as a new result on rational polynomial packing functions which may be of independent interest.</div

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

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    The main result of this paper is the decidability of the membership problem for 2×22\times 2 nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular 2×22\times 2 integer matrices M1,…,MnM_1,\dots,M_n and MM decides whether MM belongs to the semigroup generated by {M1,…,Mn}\{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 GL(2,Z)\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

    Self Organising Maps for Anatomical Joint Constraint

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    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
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