An Algebraic View of Topological ∈-Machines
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Classifying Regular Languages Via Cascade Products of Automata
Classifying Regular Languages via Cascade Products of Automata Marcus Gelderie RWTH Aachen, Lehrstuhl f¨urInformatik 7, D-52056 Aachen [email protected] Abstract. Building on the celebrated Krohn-Rhodes Theorem we char- acterize classes of regular languages in terms of the cascade decomposi- tions of minimal DFA of languages in those classes. More precisely we provide characterizations for the classes of piecewise testable languages and commutative languages. To this end we use biased resets, which are resets in the classical sense, that can change their state at most once. Next, we introduce the concept of the scope of a cascade product of reset automata in order to capture a notion of locality inside a cascade prod- uct and show that there exist constant bounds on the scope for certain classes of languages. Finally we investigate the impact of biased resets in a product of resets on the dot-depth of languages recognized by this product. This investigation allows us to refine an upper bound on the dot-depth of a language, given by Cohen and Brzozowski. 1 Introduction A significant result in the structure theory of regular languages is the Krohn- Rhodes Theorem [7], which states that any finite automaton can be decomposed into simple \prime factors" (a detailed exposition is given in [4, 6, 9, 10]). We use the Krohn-Rhodes Theorem to characterize classes of regular lan- guages in terms of the decompositions of the corresponding minimal automata. In [8] this has been done for star-free languages by giving an alternative proof for the famous Sch¨utzenberger Theorem [11]. -
Computing Finite Semigroups
Computing finite semigroups J. East, A. Egri-Nagy, J. D. Mitchell, and Y. P´eresse August 24, 2018 Abstract Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and 0-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate the partial order of the D-classes, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup. Contents 1 Introduction 2 2 Mathematical prerequisites4 3 From transformation semigroups to arbitrary regular semigroups6 3.1 Actions on Green's classes . .6 3.2 Faithful representations of stabilisers . .7 3.3 A decomposition for Green's classes . .8 3.4 Membership testing . 11 3.5 Classes within classes . 13 4 Specific classes of semigroups 14 4.1 Transformation semigroups . 15 4.2 Partial permutation semigroups and inverse semigroups . 16 4.3 Matrix semigroups . -
CAI 2017 Book of Abstracts.Pdf
Table of Contents Track 1: Automata Theory and Logic ........................... 1 Invited speaker: Heiko Vogler . 1 Languages and formations generated by D4 and D8: Jean-Éric Pin, Xaro Soler-Escrivà . 2 Syntactic structures of regular languages: O. Klíma, L. Polák . 26 Improving witnesses for state complexity of catenation combined with boolean operations: P. Caron, J.-G. Luque, B. Patrou . 44 Track 2: Cryptography and Coding Theory ..................... 63 Invited speaker: Claude Carlet . 63 A topological approach to network coding: Cristina Martínez and Alberto Besana . 64 Pairing-friendly elliptic curves resistant to TNFS attacks: G. Fotiadis, E. Konstantinou . 65 Collaborative multi-authority key-policy attribute-based encryption for shorter keys and parameters: R. Longo, C. Marcolla, M. Sala 67 Conditional blind signatures: A. Zacharakis, P. Grontas, A. Pagourtzis . 68 Hash function design for cloud storage data auditing: Nikolaos Doukas, Oleksandr P. Markovskyi, Nikolaos G. Bardis . 69 Method for accelerated zero-knowledge identification of remote users based on standard block ciphers: Nikolaos G. Bardis, Oleksandr P. Markovskyi, Nikolaos Doukas . 81 Determining whether a given block cipher is a permutation of another given block cipher— a problem in intellectual property (Extended Abstract): G. V. Bard . 91 Track 3: Computer Algebra ..................................... 95 Invited speaker: Michael Wibmer . 95 Interpolation of syzygies for implicit matrix representations: Ioannis Z. Emiris, Konstantinos Gavriil, and Christos Konaxis . 97 Reduction in free modules: C. Fürst, G. Landsmann . 115 Instructing small cellular free resolutions for monomial ideals: J. Àlvarez Montaner, O. Fernández-Ramos, P. Gimenez . 117 Low autocorrelation binary sequences (LABS): lias S. Kotsireas . 123 A signature based border basis algorithm: J. Horáček, M. -
Free Idempotent Generated Semigroups and Endomorphism Monoids of Independence Algebras
Free idempotent generated semigroups and endomorphism monoids of independence algebras Yang Dandan & Victoria Gould Semigroup Forum ISSN 0037-1912 Semigroup Forum DOI 10.1007/s00233-016-9802-0 1 23 Your article is published under the Creative Commons Attribution license which allows users to read, copy, distribute and make derivative works, as long as the author of the original work is cited. You may self- archive this article on your own website, an institutional repository or funder’s repository and make it publicly available immediately. 1 23 Semigroup Forum DOI 10.1007/s00233-016-9802-0 RESEARCH ARTICLE Free idempotent generated semigroups and endomorphism monoids of independence algebras Yang Dandan1 · Victoria Gould2 Received: 18 January 2016 / Accepted: 3 May 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We study maximal subgroups of the free idempotent generated semigroup IG(E), where E is the biordered set of idempotents of the endomorphism monoid End A of an independence algebra A, in the case where A has no constants and has finite rank n. It is shown that when n ≥ 3 the maximal subgroup of IG(E) containing a rank 1 idempotent ε is isomorphic to the corresponding maximal subgroup of End A containing ε. The latter is the group of all unary term operations of A. Note that the class of independence algebras with no constants includes sets, free group acts and affine algebras. Keywords Independence algebra · Idempotent · Biordered set 1 Introduction Let S be a semigroup with set E = E(S) of idempotents, and let E denote the subsemigroup of S generated by E. -
Arxiv:1905.10901V1 [Cs.LG] 26 May 2019
Seeing Convolution Through the Eyes of Finite Transformation Semigroup Theory: An Abstract Algebraic Interpretation of Convolutional Neural Networks Andrew Hryniowski1;2;3, Alexander Wong1;2;3 1 Video and Image Processing Research Group, Systems Design Engineering, University of Waterloo 2 Waterloo Artificial Intelligence Institute, Waterloo, ON 3 DarwinAI Corp., Waterloo, ON fapphryni, [email protected] Abstract of applications, particularly for prediction using structured data. Despite such successes, a major challenge with lever- Researchers are actively trying to gain better insights aging convolutional neural networks is the sheer number of into the representational properties of convolutional neural learnable parameters within such networks, making under- networks for guiding better network designs and for inter- standing and gaining insights about them a daunting task. As preting a network’s computational nature. Gaining such such, researchers are actively trying to gain better insights insights can be an arduous task due to the number of pa- and understanding into the representational properties of rameters in a network and the complexity of a network’s convolutional neural networks, especially since it can lead architecture. Current approaches of neural network inter- to better design and interpretability of such networks. pretation include Bayesian probabilistic interpretations and One direction that holds a lot of promise in improving information theoretic interpretations. In this study, we take a understanding of convolutional neural networks, but is much different approach to studying convolutional neural networks less explored than other approaches, is the construction of by proposing an abstract algebraic interpretation using finite theoretical models and interpretations of such networks. Cur- transformation semigroup theory. -
Transformation Semigroups and Their Automorphisms
Transformation Semigroups and their Automorphisms Wolfram Bentz University of Hull Joint work with Jo~aoAra´ujo(CEMAT, Universidade Nova de Lisboa) Peter J. Cameron (University of St Andrews) York Semigroup York, October 9, 2019 X Let X be a set and TX = X be full transformation monoid on X , under composition. Note that TX contains the symmetric group over X (as its group of units) We will only consider finite X , and more specifically Tn = TXn , Sn = SXn , where Xn = f1;:::; ng for n 2 N For S ≤ Tn, we are looking at the interaction of S \ Sn and S Transformation Monoids and Permutation Groups Wolfram Bentz (Hull) Transformation Semigroups October 9, 2019 2 / 23 Note that TX contains the symmetric group over X (as its group of units) We will only consider finite X , and more specifically Tn = TXn , Sn = SXn , where Xn = f1;:::; ng for n 2 N For S ≤ Tn, we are looking at the interaction of S \ Sn and S Transformation Monoids and Permutation Groups X Let X be a set and TX = X be full transformation monoid on X , under composition. Wolfram Bentz (Hull) Transformation Semigroups October 9, 2019 2 / 23 We will only consider finite X , and more specifically Tn = TXn , Sn = SXn , where Xn = f1;:::; ng for n 2 N For S ≤ Tn, we are looking at the interaction of S \ Sn and S Transformation Monoids and Permutation Groups X Let X be a set and TX = X be full transformation monoid on X , under composition. Note that TX contains the symmetric group over X (as its group of units) Wolfram Bentz (Hull) Transformation Semigroups October 9, 2019 2 / 23 Transformation Monoids and Permutation Groups X Let X be a set and TX = X be full transformation monoid on X , under composition. -
Quantum Conditional Strategies for Prisoners' Dilemmata Under The
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 May 2019 doi:10.20944/preprints201905.0366.v1 Peer-reviewed version available at Appl. Sci. 2019, 9, 2635; doi:10.3390/app9132635 Article Quantum conditional strategies for prisoners’ dilemmata under the EWL scheme Konstantinos Giannakis* , Georgia Theocharopoulou, Christos Papalitsas, Sofia Fanarioti, and Theodore Andronikos* Department of Informatics, Ionian University, Tsirigoti Square 7, Corfu, 49100, Greece; {kgiann, zeta.theo, c14papa, sofiafanar, andronikos}@ionio.gr * Correspondence: [email protected] (K.G.); [email protected] (Th.A.) 1 Abstract: Classic game theory is an important field with a long tradition of useful results. Recently, 2 the quantum versions of classical games, such as the Prisoner’s Dilemma (PD), have attracted a lot of 3 attention. Similarly, state machines and specifically finite automata have also been under constant 4 and thorough study for plenty of reasons. The quantum analogues of these abstract machines, like the 5 quantum finite automata, have been studied extensively. In this work, we examine some well-known 6 game conditional strategies that have been studied within the framework of the repeated PD game. 7 Then, we try to associate these strategies to proper quantum finite automata that receive them as 8 inputs and recognize them with probability 1, achieving some interesting results. We also study the 9 quantum version of PD under the Eisert-Wilkens-Lewenstein scheme, proposing a novel conditional 10 strategy for the repeated version of this game. 11 Keywords: quantum game theory, quantum automata, prisoner’s dilemma, conditional strategies, 12 quantum strategies 13 1. Introduction 14 Quantum game theory has gained a lot of research interest since the first pioneering works of the 15 late ’90s [1–5]. -
Green's Relations in Finite Transformation Semigroups
Green’s Relations in Finite Transformation Semigroups Lukas Fleischer Manfred Kufleitner FMI, University of Stuttgart∗ Universitätsstraße 38, 70569 Stuttgart, Germany {fleischer,kufleitner}@fmi.uni-stuttgart.de Abstract. We consider the complexity of Green’s relations when the semigroup is given by transformations on a finite set. Green’s relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected com- ponents. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of el- ements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups. 1 Introduction Let Q be a finite set with n elements. There are nn mappings from Q to Q. Such mappings are called transformations and the elements of Q are called states. The composition of transformations defines an associative operation. If Σ is some arbi- trary subset of transformations, we can consider the transformation semigroup S generated by Σ; this is the closure of Σ under composition.1 The set of all trans- arXiv:1703.04941v1 [cs.FL] 15 Mar 2017 formations on Q is called the full transformation semigroup on Q. One can view (Q, Σ) as a description of S. Since every element s of a semigroup S defines a trans- formation x 7→ x · s on S1 = S ∪ {1}, every semigroup S admits such a description ∗ This work was supported by the DFG grants DI 435/5-2 and KU 2716/1-1. -
On the Decomposition of Generalized Semiautomata
On the Decomposition of Generalized Semiautomata Merve Nur Cakir and Karl-Heinz Zimmermann∗ Department of Computer Engineering Hamburg University of Technology 21071 Hamburg, Germany April 21, 2020 Abstract Semiautomata are abstractions of electronic devices that are de- terministic finite-state machines having inputs but no outputs. Gen- eralized semiautomata are obtained from stochastic semiautomata by dropping the restrictions imposed by probability. It is well-known that each stochastic semiautomaton can be decomposed into a sequential product of a dependent source and deterministic semiautomaton mak- ing partly use of the celebrated theorem of Birkhoff-von Neumann. It will be shown that each generalized semiautomaton can be partitioned into a sequential product of a generalized dependent source and a de- terministic semiautomaton. AMS Subject Classification: 68Q70, 20M35, 15A04 Keywords: Semiautomaton, stochastic automaton, monoid, Birkhoff- von Neumann. 1 Introduction arXiv:2004.08805v1 [cs.FL] 19 Apr 2020 The theory of discrete stochastic systems has been initiated by the work of Shannon [14] and von Neumann [10]. While Shannon has considered memory-less communication channels and their generalization by introduc- ing states, von Neumann has studied the synthesis of reliable systems from unreliable components. The fundamental work of Rabin and Scott [12] about deterministic finite-state automata has led to two generalizations. First, the generalization of transition functions to conditional distributions studied by Carlyle [3] and Starke [15]. This in turn yields a generalization of discrete- time Markov chains in which the chains are governed by more than one ∗Email: [email protected] 1 transition probability matrix. Second, the generalization of regular sets by introducing stochastic automata as described by Rabin [11]. -
Applications of Semigroups by P
Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 3 Version 1.0 Year 2015 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Applications of Semigroups By P. Sreenivasulu Reddy & Mulugeta Dawud Samara University, Ethiopia Abstract- This section deals with the applications of semigroups in general and regular semigroups in particular. The theory of semigroups attracts many algebraists due to their applications to automata theory, formal languages, network analogy etc. In section 2 we have seen different areas of applications of semigroups. We identified some examples in biology, sociology etc. whose semigroup structures are nothing but regular, E-inversive and inverse semigroup etc. GJSFR -F Classification : FOR Code : MSC 2010: 16W22 ApplicationsofS emigroups Strictly as per the compliance and regulations of : © 2015. P. Sreenivasulu Reddy & Mulugeta Dawud. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Applications of Semigroups Notes P. Sreenivasulu Reddy α & Mulugeta Dawud σ 2015 Absract- This section deals with the applications of semigroups in general and regular semigroups in particular. The r ea theory of semigroups attracts many algebraists due to their applications to automata theory, formal languages, network Y analogy etc. In section 2 we have seen different areas of applications of semigroups. We identified some examples in biology, sociology etc. -
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CHAPTER 1 Introduction The concept of a fundamental semigroup, which is a semigroup that cannot be shrunk homomorphically without collapsing idempotents together, was introduced by Munn [25] in 1966, who developed an elegant theory within the class of inverse semigroups, which inspired many researchers in subsequent decades. There is a natural partial order on the set of idempotents of any semigroup, which is given by e ≤ f ⇐⇒ ef = fe = e. In any inverse semigroup idempotents commute, so the product of any two idempotents is idempotent. The set of idempotents becomes a semilattice under ≤, where the greatest lower bound of two idempotents e and f is the product ef. Munn constructed a semigroup TE from an arbitrary semilattice E, which he proved is the maximum fundamental inverse semigroup with semilattice of idempotents E. Any fundamental inverse semigroup with semilattice of idempotents E must be isomorphic to a full subsemigroup of TE. Also, for any inverse semigroup with semilattice of idempotents E, Munn constructed a fundamental representation φ : S → TE with kernel µ = {(a, b) ∈ S × S | (∀e ∈ E) a−1ea = b−1eb} which is the maximum idempotent separating congruence on S. These results demonstrate the importance of TE, since every inverse semigroup with semilattice of idempotents E is a coextension of a fundamental inverse semigroup S/µ with the same semilattice of idem- potents, which is a subsemigroup of TE. For any set X the symmetric inverse semigroup IX = {partial one-one mappings : X → X} is a fundamental inverse semigroup, which is an example of Munn’s construction TE. For more details about Munn’s theory of fundamental inverse semigroups see [22] Section 5.4. -
Primitive Permutation Groups and Strongly Factorizable
Primitive Permutation Groups and Strongly Factorizable Transformation Semigroups Jo˜ao Ara´ujo∗†, Wolfram Bentz‡, and Peter J. Cameron§ Abstract Let Ω be a finite set and T (Ω) be the full transformation monoid on Ω. The rank of a transformation t T (Ω) is the natural number ∈ Ωt . Given A T (Ω), denote by A the semigroup generated by A. | | ⊆ h i Let k be a fixed natural number such that 2 k Ω . In the first ≤ ≤ | | part of this paper we (almost) classify the permutation groups G on Ω such that for all rank k transformation t T (Ω), every element ∈ 2 in St := G, t can be written as a product eg, where e = e St h i ∈ and g G. In the second part we prove, among other results, that if ∈ S T (Ω) and G is the normalizer of S in the symmetric group on Ω, ≤ then the semigroup SG is regular if and only if S is regular. (Recall that a semigroup S is regular if for all s S there exists s′ S such ′ ∈ ∈ that s = ss s.) The paper ends with a list of problems. ∗ arXiv:1910.08335v1 [math.GR] 18 Oct 2019 Departamento de Matem´atica and CMA, Faculdade de Ciˆencias e Tecnologia (FCT) Universidade Nova de Lisboa (UNL), 2829-516 Caparica, Portugal; [email protected] †CEMAT-Ciˆencias, Faculdade de Ciˆencias, Universidade de Lisboa 1749–016, Lisboa, Portugal; [email protected] ‡School of Mathematics & Physical Sciences, University of Hull, Kingston upon Hull, HU6 7RX, UK; [email protected] §School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK; [email protected] 1 1 Introduction A semigroup S with set of idempotents E and group of units G is said to be strongly factorizable if S = EG.