### Abstract

Coding plays an important role in the design of parallel algorithms. When each digit of a result of a desired operation depends only on a part of digits of operands under a coding scheme, we say the operation is locally computable under the coding scheme. There is a closed relation between local computability and redundancy of codes. Several excellent algorithms utilizing local computability by redundant coding schemes are developed and used practically. The problem to examine the relation among coding schemes, local computability and algebraic structures of target operations is a basic problem on the design of parallel algorithms. In this paper, we discuss a relation between redundancy of coding schemes and local computability of unary operations defined on finite sets. We show that it is generally impossible to realize local computability by nonredundant coding schemes. If we introduce redundancy into coding, we can construct a coding under which every digit of a result depends only on 2 digits of an operand for any unary operations. These results are closely related with a state assignment problem of finite state machines. The above 2-locally computable coding derives a state assignment of a pipelined sequential circuit for an arbitrary sequential machine.

Original language | English |
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Title of host publication | Concurrency |

Subtitle of host publication | Theory, Language, and Architecture - UK/Japan Workshop, Proceedings |

Editors | Takayasu Ito, Akinori Yonezawa |

Publisher | Springer Verlag |

Pages | 312-323 |

Number of pages | 12 |

ISBN (Print) | 9783540464525 |

DOIs | |

Publication status | Published - Jan 1 1991 |

Externally published | Yes |

Event | UK/Japan Workshop on Concurrency Theory, 1989 - Oxford, United Kingdom Duration: Sep 25 1989 → Sep 27 1989 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 491 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | UK/Japan Workshop on Concurrency Theory, 1989 |
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Country | United Kingdom |

City | Oxford |

Period | 9/25/89 → 9/27/89 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Concurrency: Theory, Language, and Architecture - UK/Japan Workshop, Proceedings*(pp. 312-323). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 491 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-53932-8_58

**Locally computable coding for unary operations.** / Yasuura, Hiroto.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Concurrency: Theory, Language, and Architecture - UK/Japan Workshop, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 491 LNCS, Springer Verlag, pp. 312-323, UK/Japan Workshop on Concurrency Theory, 1989, Oxford, United Kingdom, 9/25/89. https://doi.org/10.1007/3-540-53932-8_58

}

TY - GEN

T1 - Locally computable coding for unary operations

AU - Yasuura, Hiroto

PY - 1991/1/1

Y1 - 1991/1/1

N2 - Coding plays an important role in the design of parallel algorithms. When each digit of a result of a desired operation depends only on a part of digits of operands under a coding scheme, we say the operation is locally computable under the coding scheme. There is a closed relation between local computability and redundancy of codes. Several excellent algorithms utilizing local computability by redundant coding schemes are developed and used practically. The problem to examine the relation among coding schemes, local computability and algebraic structures of target operations is a basic problem on the design of parallel algorithms. In this paper, we discuss a relation between redundancy of coding schemes and local computability of unary operations defined on finite sets. We show that it is generally impossible to realize local computability by nonredundant coding schemes. If we introduce redundancy into coding, we can construct a coding under which every digit of a result depends only on 2 digits of an operand for any unary operations. These results are closely related with a state assignment problem of finite state machines. The above 2-locally computable coding derives a state assignment of a pipelined sequential circuit for an arbitrary sequential machine.

AB - Coding plays an important role in the design of parallel algorithms. When each digit of a result of a desired operation depends only on a part of digits of operands under a coding scheme, we say the operation is locally computable under the coding scheme. There is a closed relation between local computability and redundancy of codes. Several excellent algorithms utilizing local computability by redundant coding schemes are developed and used practically. The problem to examine the relation among coding schemes, local computability and algebraic structures of target operations is a basic problem on the design of parallel algorithms. In this paper, we discuss a relation between redundancy of coding schemes and local computability of unary operations defined on finite sets. We show that it is generally impossible to realize local computability by nonredundant coding schemes. If we introduce redundancy into coding, we can construct a coding under which every digit of a result depends only on 2 digits of an operand for any unary operations. These results are closely related with a state assignment problem of finite state machines. The above 2-locally computable coding derives a state assignment of a pipelined sequential circuit for an arbitrary sequential machine.

UR - http://www.scopus.com/inward/record.url?scp=85032878580&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85032878580&partnerID=8YFLogxK

U2 - 10.1007/3-540-53932-8_58

DO - 10.1007/3-540-53932-8_58

M3 - Conference contribution

SN - 9783540464525

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 312

EP - 323

BT - Concurrency

A2 - Ito, Takayasu

A2 - Yonezawa , Akinori

PB - Springer Verlag

ER -