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Symmetry
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Symmetry in Theoretical Computer Science
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Symmetry in Theoretical Computer Science

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

Deadline for manuscript submissions: closed (28 February 2011) | Viewed by 18553

Special Issue Editor

Prof. Dr. Takeshi Koshiba

E-Mail Website
Guest Editor
Faculty of Education and Integrated Arts and Sciences, Waseda University, Shinjuku, Tokyo 169-8050, Japan
Interests: theory of cryptography; randomness and computation; quantum computation; computational complexity
Special Issues, Collections and Topics in MDPI journals

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

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Published Papers (3 papers)

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1531 KiB  
Article
Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content
by Mihai Prunescu
Symmetry 2011, 3(3), 402-442; https://doi.org/10.3390/sym3030402 - 7 Jul 2011
Cited by 4 | Viewed by 5298
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)
1457 KiB  
Article
Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D2
by Hiroshi Fukuda, Chiaki Kanomata, Nobuaki Mutoh, Gisaku Nakamura and Doris Schattschneider
Symmetry 2011, 3(2), 325-364; https://doi.org/10.3390/sym3020325 - 9 Jun 2011
Cited by 1 | Viewed by 5732
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)
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202 KiB  
Article
Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in between
by Alexis De Vos and Stijn De Baerdemacker
Symmetry 2011, 3(2), 305-324; https://doi.org/10.3390/sym3020305 - 7 Jun 2011
Cited by 10 | Viewed by 5415
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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