### Abstract

A sequential machine is defined to be linear if its next-state function and output function are linear transformations from their domains (vector spaces) to their ranges (also vector spaces). It is shown that this definition is equivalent to those given by other authors. Based on this definition, a set of necessary and sufficient conditions for the flow table of a sequential machine to be linear is obtained. Many properties of the flow table of a linear sequential machine are found, and in many cases they form very simple tests for the linearity of a flow table. A general procedure for testing the linearity of a flow table is established. This procedure, including the coding of the states, inputs, and outputs, either ends with a linear realization of the flow table with the minimum possible numbers of state variables, input variables, and output variables, or detects that such a linear realization for the flow table is impossible. The type of sequential machine considered in this paper is deterministic and synchronous, and both Moore model and Mealy model are studied in detail. The linearity of incompletely specified sequential machines is also discussed. COPYRIGHT

Original language | English (US) |
---|---|

Pages (from-to) | 337-354 |

Number of pages | 18 |

Journal | IEEE Transactions on Electronic Computers |

Volume | EC-15 |

Issue number | 3 |

DOIs | |

State | Published - 1966 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Hardware and Architecture
- Software
- Theoretical Computer Science

### Cite this

*IEEE Transactions on Electronic Computers*,

*EC-15*(3), 337-354. https://doi.org/10.1109/PGEC.1966.264492

**Linearity of Sequential Machines.** / Yau, Sik-Sang; Wang, K. C.; Wang, K. C.

Research output: Contribution to journal › Article

*IEEE Transactions on Electronic Computers*, vol. EC-15, no. 3, pp. 337-354. https://doi.org/10.1109/PGEC.1966.264492

}

TY - JOUR

T1 - Linearity of Sequential Machines

AU - Yau, Sik-Sang

AU - Wang, K. C.

AU - Wang, K. C.

PY - 1966

Y1 - 1966

N2 - A sequential machine is defined to be linear if its next-state function and output function are linear transformations from their domains (vector spaces) to their ranges (also vector spaces). It is shown that this definition is equivalent to those given by other authors. Based on this definition, a set of necessary and sufficient conditions for the flow table of a sequential machine to be linear is obtained. Many properties of the flow table of a linear sequential machine are found, and in many cases they form very simple tests for the linearity of a flow table. A general procedure for testing the linearity of a flow table is established. This procedure, including the coding of the states, inputs, and outputs, either ends with a linear realization of the flow table with the minimum possible numbers of state variables, input variables, and output variables, or detects that such a linear realization for the flow table is impossible. The type of sequential machine considered in this paper is deterministic and synchronous, and both Moore model and Mealy model are studied in detail. The linearity of incompletely specified sequential machines is also discussed. COPYRIGHT

AB - A sequential machine is defined to be linear if its next-state function and output function are linear transformations from their domains (vector spaces) to their ranges (also vector spaces). It is shown that this definition is equivalent to those given by other authors. Based on this definition, a set of necessary and sufficient conditions for the flow table of a sequential machine to be linear is obtained. Many properties of the flow table of a linear sequential machine are found, and in many cases they form very simple tests for the linearity of a flow table. A general procedure for testing the linearity of a flow table is established. This procedure, including the coding of the states, inputs, and outputs, either ends with a linear realization of the flow table with the minimum possible numbers of state variables, input variables, and output variables, or detects that such a linear realization for the flow table is impossible. The type of sequential machine considered in this paper is deterministic and synchronous, and both Moore model and Mealy model are studied in detail. The linearity of incompletely specified sequential machines is also discussed. COPYRIGHT

UR - http://www.scopus.com/inward/record.url?scp=84912244158&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84912244158&partnerID=8YFLogxK

U2 - 10.1109/PGEC.1966.264492

DO - 10.1109/PGEC.1966.264492

M3 - Article

VL - EC-15

SP - 337

EP - 354

JO - IEEE Transactions on Computers

JF - IEEE Transactions on Computers

SN - 0018-9340

IS - 3

ER -