Mika Hirvensalo - Academia.edu (original) (raw)

Papers by Mika Hirvensalo

Research paper thumbnail of Computational limitations of affine automata and generalized affine automata

Natural Computing, 2021

We present new results on the computational limitations of affine automata (AfAs). First, we show... more We present new results on the computational limitations of affine automata (AfAs). First, we show that using the endmarker does not increase the computational power of AfAs. Second, we show that the computation of bounded-error rational-valued AfAs can be simulated in logarithmic space. Third, we identify some logspace unary languages that are not recognized by algebraic-valued AfAs. Fourth, we show that using arbitrary real-valued transition matrices and state vectors does not increase the computational power of AfAs in the unbounded-error model. When focusing only the rational values, we obtain the same result also for bounded error. As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers only.

Research paper thumbnail of Workshop on Quantum and Classical Complexity Workshop on Quantum and Classical Complexity Organizers Program Commitee External Reviewers Tutorial Algebraic Algorithms: Ranks, Determinants and Derivatives Quantum Complexity of Matrix Multiplication Parameterized Quantum Query Complexity of Graph Coll

Probably the first algebraic algorithm is the famous result of Lovasz, who proposed a Monte Carlo... more Probably the first algebraic algorithm is the famous result of Lovasz, who proposed a Monte Carlo algorithm for finding the size of a maximum matching based on computing the rank of the Tutte matrix. During the talk we will cover some classical algebraic algorithms, the heart of which is finding rank or determinant of some matrix. Next we will investigate another algebraic tool, i.e. the Baur-Strassen’s theorem, where partial derivatives provide us with much more information within asymptotically same running time. We will show simple algebraic algorithms for problems such as computing the diameter, finding the shortest cycle or maximum weight perfect matching, where instances have integral weights from the range [−W,W ]. The talk is based on a joint work with Harold N. Gabow and Piotr Sankowski. Quantum Complexity of Matrix Multiplication

Research paper thumbnail of On the Moments of Squared Binomial Coe cients

Explicit recurrent formulas for ordinary and alternated power moments of the squared binomial coe... more Explicit recurrent formulas for ordinary and alternated power moments of the squared binomial coe cients are derived in this article. Every such moment proves to be a linear combination of the previous ones via a coe cient list of the relevant Krawtchouk polynomial. Introduction In this article, we study the sums of form

Research paper thumbnail of Preface, Fundamenta Informaticae

Research paper thumbnail of Acceptance Ambiguity for Quantum Automata

We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield... more 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's correspondence problem.

Research paper thumbnail of Alternating, private alternating, and quantum alternating realtime automata

Log. Methods Comput. Sci., 2019

We present new results on realtime alternating, private alternating, and quantum alternating auto... more We present new results on realtime alternating, private alternating, and quantum alternating automaton models. Firstly, we show that the emptiness problem for alternating one-counter automata on unary alphabets is undecidable. Then, we present two equivalent definitions of realtime private alternating finite automata (PAFAs). We show that the emptiness problem is undecidable for PAFAs. Furthermore, PAFAs can recognize some nonregular unary languages, including the unary squares language, which seems to be difficult even for some classical counter automata with two-way input. Regarding quantum finite automata (QFAs), we show that the emptiness problem is undecidable both for universal QFAs on general alphabets, and for alternating QFAs with two alternations on unary alphabets. On the other hand, the same problem is decidable for nondeterministic QFAs on general alphabets. We also show that the unary squares language is recognized by alternating QFAs with two alternations.

Research paper thumbnail of The Identity Problem for Matrix Semigroups in SL2(ℤ) is NP-complete

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 2017

In this paper, we show that the problem of determining if the identity matrix belongs to a finite... more In this paper, we show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of 2 × 2 matrices from the modular group PSL 2 (Z) and thus the Special Linear group SL 2 (Z) is solvable in NP. From this fact, we can immediately derive that the fundamental problem of whether a given finite set of matrices from SL 2 (Z) or PSL 2 (Z) generates a group or free semigroup is also decidable in NP. The previous algorithm for these problems, shown in 2005 by Choffrut and Karhumäki, was in EXPSPACE mainly due to the translation of matrices into exponentially long words over a binary alphabet {s, r} and further constructions with a large nondeterministic finite state automaton that is built on these words. Our algorithm is based on various new techniques that allow us to operate with compressed word representations of matrices without explicit expansions. When combined with the known NP-hard lower bound, this proves that the membership problem for the identity problem, the group problem and the freeness problem in SL 2 (Z) are NP-complete.

Research paper thumbnail of On the Generating Function of Discrete Chebyshev Polynomials

Journal of Mathematical Sciences, 2017

We give a closed form for the generating function of the discrete Chebyshev polynomials. It is th... more We give a closed form for the generating function of the discrete Chebyshev polynomials. It is the MacWilliams transform of Jacobi polynomials together with a binomial multiplicative factor. It turns out that the desired closed form is a solution to a special case of the Heun differential equation, and that it implies combinatorial identities that appear quite challenging to prove directly. Bibliography: 9 titles.

Research paper thumbnail of Decision Problems on Unary Probabilistic and Quantum Automata

Baltic Journal of Modern Computing, 2016

It is well known that the emptiness problem for binary probabilistic automata and so for quantum ... more It is well known that the emptiness problem for binary probabilistic automata and so for quantum automata is undecidable. We present the current status of the emptiness problems for unary probabilistic and quantum automata with connections with Skolem's and positivity problems. We also introduce the concept of linear recurrence automata in order to show the connection naturally. Then, we also give possible generalizations of linear recurrence relations and automata on vectors.

Research paper thumbnail of Improved Undecidability Results on the Emptiness Problem of Probabilistic and Quantum Cut-Point Languages

Lecture Notes in Computer Science, 2007

Research paper thumbnail of Mortality for 2 ×2 Matrices Is NP-Hard

Mathematical Foundations of Computer Science 2012, 2012

We study the computational complexity of determining whether the zero matrix belongs to a finitel... more 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.

Research paper thumbnail of On the Joint Spectral Radius for Bounded Matrix Languages

Lecture Notes in Computer Science, 2010

ABSTRACT We show several problems concerning probabilistic finite automata with fixed numbers of ... more ABSTRACT We show several problems concerning probabilistic finite automata with fixed numbers of letters and of fixed dimensions for bounded cut-point and strict cut-point languages are algorithmically undecidable by a reduction of Hilbert’s tenth problem using formal power series. For a finite set of matrices {M1, M2, ¼, Mk} Í \mathbbQt ×t\{M_1, M_2, \ldots, M_k\} \subseteq \mathbb{Q}^{t \times t}, we then consider the decidability of computing the joint spectral radius (which characterises the maximal asymptotic growth rate of a set of matrices) of the set X = {M1j1 M2j2 ¼Mkjk| j1, j2, ¼, jk ³ 0}X = \{M_1^{j_1} M_2^{j_2} \cdots M_k^{j_k}| j_1, j_2, \ldots, j_k \geq 0\}, which we term a bounded matrix language. Using an encoding of a probabilistic finite automaton shown in the paper, we prove the surprising result that determining whether the joint spectral radius of a bounded matrix language is less than or equal to one is undecidable, but determining if it is strictly less than one is in fact decidable (which is similar to a result recently shown for quantum automata). This has an interpretation in terms of a control problem for a switched linear system with a fixed and finite number of switching operations; if we fix the maximum number of switching operations in advance, then determining convergence to the origin for all initial points is decidable whereas determining boundedness of all initial points is undecidable.

Research paper thumbnail of Generalized PCP is decidable for marked morphisms

Lecture Notes in Computer Science, 1999

We prove that the generalized Post Correspondence Problem (GPCP) is decidable for marked morphism... more We prove that the generalized Post Correspondence Problem (GPCP) is decidable for marked morphisms. This result gives as a corollary a shorter proof for the decidability of the binary PCP, proved in 1982 by Ehrenfeucht, Karhumäki and Rozenberg.

Research paper thumbnail of Quantum Automata with Open Time Evolution

International Journal of Natural Computing Research, 2010

In this paper, a model for finite automaton with an open quantum evolution is introduced, and its... more In this paper, a model for finite automaton with an open quantum evolution is introduced, and its basic properties are studied. It is shown that the (fuzzy) languages accepted by open evolution quantum automata obey various closure properties. More importantly, it is shown that major other models of finite automata, including probabilistic, measure once quantum, measure many quantum, and Latvian quantum automata can be simulated by the open quantum evolution automata without increasing the number of the states.

Research paper thumbnail of Decidability and Undecidability of Marked PCP

Lecture Notes in Computer Science, 1999

Research paper thumbnail of Quantum Computing

Quantum computing / M. Hirvensalo. p. cm.-(Natural computing series) Includes bibliographical ref... more Quantum computing / M. Hirvensalo. p. cm.-(Natural computing series) Includes bibliographical references and index.

Research paper thumbnail of Special Issue: Non-Classical Models of Automata and Applications III (NCMA-2011)

RAIRO - Theoretical Informatics and Applications, 2012

Many non-classical models of automata are natural objects of theoretical computer science. They a... more Many non-classical models of automata are natural objects of theoretical computer science. They are studied from different points of view in various areas, both as theoretical concepts and as formal models for applications. A deeper and interdisciplinary coverage of this particular area may lead to new insights and substantial progress. The Third Workshop on Non-Classical Models of Automata and Applications (NCMA 2011) was organized in order to bring together researchers working on different aspects of various variants of non-classical models of automata to exchange and develop novel ideas.

Research paper thumbnail of Undecidability Bounds for Integer Matrices Using Claus Instances

International Journal of Foundations of Computer Science, 2007

There are several known undecidable problems for 3 × 3 integer matrices the proof of which use a ... more There are several known undecidable problems for 3 × 3 integer matrices the proof of which use a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the number of matrices for the mortality, zero in the left upper corner, vector reachability, matrix reachability, scalar reachability and freeness problems. Also, we give a short proof for a strengthened result due to Bell and Potapov stating that the membership problem is undecidable for finitely generated matrix semigroups R ⊆ ℤ 4×4 whether or not kI4 ∈ R for any given |k| > 1. These bounds are obtained by using the Claus instances of the PCP.

Research paper thumbnail of On injectivity of quantum finite automata

Journal of Computer and System Sciences, 2021

We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield... more 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.

Research paper thumbnail of Computational Limitations of Affine Automata

Unconventional Computation and Natural Computation, 2019

We present two new results on the computational limitations of affine automata. First, we show th... more We present two new results on the computational limitations of affine automata. First, we show that the computation of bounded-error rational-values affine automata is simulated in logarithmic space. Second, we give an impossibility result for algebraic-valued affine automata. As a result, we identify some unary languages (in logarithmic space) that are not recognized by algebraic-valued affine automata with cutpoints.

Research paper thumbnail of Computational limitations of affine automata and generalized affine automata

Natural Computing, 2021

We present new results on the computational limitations of affine automata (AfAs). First, we show... more We present new results on the computational limitations of affine automata (AfAs). First, we show that using the endmarker does not increase the computational power of AfAs. Second, we show that the computation of bounded-error rational-valued AfAs can be simulated in logarithmic space. Third, we identify some logspace unary languages that are not recognized by algebraic-valued AfAs. Fourth, we show that using arbitrary real-valued transition matrices and state vectors does not increase the computational power of AfAs in the unbounded-error model. When focusing only the rational values, we obtain the same result also for bounded error. As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers only.

Research paper thumbnail of Workshop on Quantum and Classical Complexity Workshop on Quantum and Classical Complexity Organizers Program Commitee External Reviewers Tutorial Algebraic Algorithms: Ranks, Determinants and Derivatives Quantum Complexity of Matrix Multiplication Parameterized Quantum Query Complexity of Graph Coll

Probably the first algebraic algorithm is the famous result of Lovasz, who proposed a Monte Carlo... more Probably the first algebraic algorithm is the famous result of Lovasz, who proposed a Monte Carlo algorithm for finding the size of a maximum matching based on computing the rank of the Tutte matrix. During the talk we will cover some classical algebraic algorithms, the heart of which is finding rank or determinant of some matrix. Next we will investigate another algebraic tool, i.e. the Baur-Strassen’s theorem, where partial derivatives provide us with much more information within asymptotically same running time. We will show simple algebraic algorithms for problems such as computing the diameter, finding the shortest cycle or maximum weight perfect matching, where instances have integral weights from the range [−W,W ]. The talk is based on a joint work with Harold N. Gabow and Piotr Sankowski. Quantum Complexity of Matrix Multiplication

Research paper thumbnail of On the Moments of Squared Binomial Coe cients

Explicit recurrent formulas for ordinary and alternated power moments of the squared binomial coe... more Explicit recurrent formulas for ordinary and alternated power moments of the squared binomial coe cients are derived in this article. Every such moment proves to be a linear combination of the previous ones via a coe cient list of the relevant Krawtchouk polynomial. Introduction In this article, we study the sums of form

Research paper thumbnail of Preface, Fundamenta Informaticae

Research paper thumbnail of Acceptance Ambiguity for Quantum Automata

We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield... more 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's correspondence problem.

Research paper thumbnail of Alternating, private alternating, and quantum alternating realtime automata

Log. Methods Comput. Sci., 2019

We present new results on realtime alternating, private alternating, and quantum alternating auto... more We present new results on realtime alternating, private alternating, and quantum alternating automaton models. Firstly, we show that the emptiness problem for alternating one-counter automata on unary alphabets is undecidable. Then, we present two equivalent definitions of realtime private alternating finite automata (PAFAs). We show that the emptiness problem is undecidable for PAFAs. Furthermore, PAFAs can recognize some nonregular unary languages, including the unary squares language, which seems to be difficult even for some classical counter automata with two-way input. Regarding quantum finite automata (QFAs), we show that the emptiness problem is undecidable both for universal QFAs on general alphabets, and for alternating QFAs with two alternations on unary alphabets. On the other hand, the same problem is decidable for nondeterministic QFAs on general alphabets. We also show that the unary squares language is recognized by alternating QFAs with two alternations.

Research paper thumbnail of The Identity Problem for Matrix Semigroups in SL2(ℤ) is NP-complete

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 2017

In this paper, we show that the problem of determining if the identity matrix belongs to a finite... more In this paper, we show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of 2 × 2 matrices from the modular group PSL 2 (Z) and thus the Special Linear group SL 2 (Z) is solvable in NP. From this fact, we can immediately derive that the fundamental problem of whether a given finite set of matrices from SL 2 (Z) or PSL 2 (Z) generates a group or free semigroup is also decidable in NP. The previous algorithm for these problems, shown in 2005 by Choffrut and Karhumäki, was in EXPSPACE mainly due to the translation of matrices into exponentially long words over a binary alphabet {s, r} and further constructions with a large nondeterministic finite state automaton that is built on these words. Our algorithm is based on various new techniques that allow us to operate with compressed word representations of matrices without explicit expansions. When combined with the known NP-hard lower bound, this proves that the membership problem for the identity problem, the group problem and the freeness problem in SL 2 (Z) are NP-complete.

Research paper thumbnail of On the Generating Function of Discrete Chebyshev Polynomials

Journal of Mathematical Sciences, 2017

We give a closed form for the generating function of the discrete Chebyshev polynomials. It is th... more We give a closed form for the generating function of the discrete Chebyshev polynomials. It is the MacWilliams transform of Jacobi polynomials together with a binomial multiplicative factor. It turns out that the desired closed form is a solution to a special case of the Heun differential equation, and that it implies combinatorial identities that appear quite challenging to prove directly. Bibliography: 9 titles.

Research paper thumbnail of Decision Problems on Unary Probabilistic and Quantum Automata

Baltic Journal of Modern Computing, 2016

It is well known that the emptiness problem for binary probabilistic automata and so for quantum ... more It is well known that the emptiness problem for binary probabilistic automata and so for quantum automata is undecidable. We present the current status of the emptiness problems for unary probabilistic and quantum automata with connections with Skolem's and positivity problems. We also introduce the concept of linear recurrence automata in order to show the connection naturally. Then, we also give possible generalizations of linear recurrence relations and automata on vectors.

Research paper thumbnail of Improved Undecidability Results on the Emptiness Problem of Probabilistic and Quantum Cut-Point Languages

Lecture Notes in Computer Science, 2007

Research paper thumbnail of Mortality for 2 ×2 Matrices Is NP-Hard

Mathematical Foundations of Computer Science 2012, 2012

We study the computational complexity of determining whether the zero matrix belongs to a finitel... more 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.

Research paper thumbnail of On the Joint Spectral Radius for Bounded Matrix Languages

Lecture Notes in Computer Science, 2010

ABSTRACT We show several problems concerning probabilistic finite automata with fixed numbers of ... more ABSTRACT We show several problems concerning probabilistic finite automata with fixed numbers of letters and of fixed dimensions for bounded cut-point and strict cut-point languages are algorithmically undecidable by a reduction of Hilbert’s tenth problem using formal power series. For a finite set of matrices {M1, M2, ¼, Mk} Í \mathbbQt ×t\{M_1, M_2, \ldots, M_k\} \subseteq \mathbb{Q}^{t \times t}, we then consider the decidability of computing the joint spectral radius (which characterises the maximal asymptotic growth rate of a set of matrices) of the set X = {M1j1 M2j2 ¼Mkjk| j1, j2, ¼, jk ³ 0}X = \{M_1^{j_1} M_2^{j_2} \cdots M_k^{j_k}| j_1, j_2, \ldots, j_k \geq 0\}, which we term a bounded matrix language. Using an encoding of a probabilistic finite automaton shown in the paper, we prove the surprising result that determining whether the joint spectral radius of a bounded matrix language is less than or equal to one is undecidable, but determining if it is strictly less than one is in fact decidable (which is similar to a result recently shown for quantum automata). This has an interpretation in terms of a control problem for a switched linear system with a fixed and finite number of switching operations; if we fix the maximum number of switching operations in advance, then determining convergence to the origin for all initial points is decidable whereas determining boundedness of all initial points is undecidable.

Research paper thumbnail of Generalized PCP is decidable for marked morphisms

Lecture Notes in Computer Science, 1999

We prove that the generalized Post Correspondence Problem (GPCP) is decidable for marked morphism... more We prove that the generalized Post Correspondence Problem (GPCP) is decidable for marked morphisms. This result gives as a corollary a shorter proof for the decidability of the binary PCP, proved in 1982 by Ehrenfeucht, Karhumäki and Rozenberg.

Research paper thumbnail of Quantum Automata with Open Time Evolution

International Journal of Natural Computing Research, 2010

In this paper, a model for finite automaton with an open quantum evolution is introduced, and its... more In this paper, a model for finite automaton with an open quantum evolution is introduced, and its basic properties are studied. It is shown that the (fuzzy) languages accepted by open evolution quantum automata obey various closure properties. More importantly, it is shown that major other models of finite automata, including probabilistic, measure once quantum, measure many quantum, and Latvian quantum automata can be simulated by the open quantum evolution automata without increasing the number of the states.

Research paper thumbnail of Decidability and Undecidability of Marked PCP

Lecture Notes in Computer Science, 1999

Research paper thumbnail of Quantum Computing

Quantum computing / M. Hirvensalo. p. cm.-(Natural computing series) Includes bibliographical ref... more Quantum computing / M. Hirvensalo. p. cm.-(Natural computing series) Includes bibliographical references and index.

Research paper thumbnail of Special Issue: Non-Classical Models of Automata and Applications III (NCMA-2011)

RAIRO - Theoretical Informatics and Applications, 2012

Many non-classical models of automata are natural objects of theoretical computer science. They a... more Many non-classical models of automata are natural objects of theoretical computer science. They are studied from different points of view in various areas, both as theoretical concepts and as formal models for applications. A deeper and interdisciplinary coverage of this particular area may lead to new insights and substantial progress. The Third Workshop on Non-Classical Models of Automata and Applications (NCMA 2011) was organized in order to bring together researchers working on different aspects of various variants of non-classical models of automata to exchange and develop novel ideas.

Research paper thumbnail of Undecidability Bounds for Integer Matrices Using Claus Instances

International Journal of Foundations of Computer Science, 2007

There are several known undecidable problems for 3 × 3 integer matrices the proof of which use a ... more There are several known undecidable problems for 3 × 3 integer matrices the proof of which use a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the number of matrices for the mortality, zero in the left upper corner, vector reachability, matrix reachability, scalar reachability and freeness problems. Also, we give a short proof for a strengthened result due to Bell and Potapov stating that the membership problem is undecidable for finitely generated matrix semigroups R ⊆ ℤ 4×4 whether or not kI4 ∈ R for any given |k| > 1. These bounds are obtained by using the Claus instances of the PCP.

Research paper thumbnail of On injectivity of quantum finite automata

Journal of Computer and System Sciences, 2021

We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield... more 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.

Research paper thumbnail of Computational Limitations of Affine Automata

Unconventional Computation and Natural Computation, 2019

We present two new results on the computational limitations of affine automata. First, we show th... more We present two new results on the computational limitations of affine automata. First, we show that the computation of bounded-error rational-values affine automata is simulated in logarithmic space. Second, we give an impossibility result for algebraic-valued affine automata. As a result, we identify some unary languages (in logarithmic space) that are not recognized by algebraic-valued affine automata with cutpoints.