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Special Issue "Symmetry in Theoretical Computer Science"

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A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (28 February 2011)

Special Issue Editor

Guest Editor
Prof. Dr. Takeshi Koshiba

Area of Informatics, Division of Mathematics, Electronics and Informatics, Graduate School of Science and Engineering, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama 338-8570, Japan
Website | E-Mail
Interests: theory of cryptography; randomness and computation; quantum computation; computational complexity

Special Issue Information

Dear Colleagues,

Symmetry is a basic concept even in Theoretical Computer Science. For the design and analysis of the algorithms, the symmetry plays an important role. The symmetry often makes the probabilistic analysis of randomized algorithms easier and simpler. Sometimes, the incorporation of elegant methods of symmetry- or tie- breaking into algorithms leads to the efficiency. Moreover, symmetric structures and patterns are omnipresent and the study of their algorithmic or computational aspects gives us an understanding of the nature of the symmetry.

Contributions are invited on all aspects of symmetry in theoretical computer science. Those that involve other fields are welcomed if they are discussed from the algorithmic or computational point of view.

Prof. Dr. Takeshi Koshiba
Guest Editor

Keywords

  • randomized algorithm
  • probabilistic analysis
  • symmetric break / tie break
  • distributed computing
  • algorithmic game theory
  • symmetric network
  • symmetric structure / pattern

Published Papers (3 papers)

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Research

Open AccessArticle Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content
Symmetry 2011, 3(3), 402-442; doi:10.3390/sym3030402
Received: 11 January 2011 / Revised: 15 June 2011 / Accepted: 30 June 2011 / Published: 7 July 2011
Cited by 3 | PDF Full-text (1531 KB)
Abstract
The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M
[...] Read more.
The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M2(F2) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M2(F2) in 90 geometric types. The classification over Klein’s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M2(F2), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version. Full article
(This article belongs to the Special Issue Symmetry in Theoretical Computer Science)
Open AccessArticle Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D2
Symmetry 2011, 3(2), 325-364; doi:10.3390/sym3020325
Received: 15 March 2011 / Revised: 24 May 2011 / Accepted: 27 May 2011 / Published: 9 June 2011
PDF Full-text (1457 KB)
Abstract
We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have pmm, pmg, pgg or cmm symmetry [1]. These symmetry groups are
[...] Read more.
We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have pmm, pmg, pgg or cmm symmetry [1]. These symmetry groups are members of the crystal class D2 among the 17 two-dimensional symmetry groups [2]. We display the algorithms’ output and give enumeration tables for small values of n. This work is a continuation of our earlier works for the symmetry groups p3, p31m, p3m1, p4, p4g, p4m, p6, and p6m [3–5]. Full article
(This article belongs to the Special Issue Symmetry in Theoretical Computer Science)
Open AccessArticle Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in between
Symmetry 2011, 3(2), 305-324; doi:10.3390/sym3020305
Received: 11 January 2011 / Revised: 24 May 2011 / Accepted: 27 May 2011 / Published: 7 June 2011
Cited by 5 | PDF Full-text (202 KB)
Abstract
Whereas quantum computing circuits follow the symmetries of the unitary Lie group, classical reversible computation circuits follow the symmetries of a finite group, i.e., the symmetric group. We confront the decomposition of an arbitrary classical reversible circuit with w bits and the
[...] Read more.
Whereas quantum computing circuits follow the symmetries of the unitary Lie group, classical reversible computation circuits follow the symmetries of a finite group, i.e., the symmetric group. We confront the decomposition of an arbitrary classical reversible circuit with w bits and the decomposition of an arbitrary quantum circuit with w qubits. Both decompositions use the control gate as building block, i.e., a circuit transforming only one (qu)bit, the transformation being controlled by the other w−1 (qu)bits. We explain why the former circuit can be decomposed into 2w − 1 control gates, whereas the latter circuit needs 2w − 1 control gates. We investigate whether computer circuits, not based on the full unitary group but instead on a subgroup of the unitary group, may be decomposable either into 2w − 1 or into 2w − 1 control gates. Full article
(This article belongs to the Special Issue Symmetry in Theoretical Computer Science)

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