Structures of M-Invariant Dual Subspaces with Respect to a Boolean Network
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911983187525632 |
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| author | Bi, Dongyao Zhang, Lijun Zhang, Kuize Zhang, Shenggui |
| author_facet | Bi, Dongyao Zhang, Lijun Zhang, Kuize Zhang, Shenggui |
| contents | This paper presents the following research findings on Boolean networks (BNs) and their dual subspaces.First, we establish a bijection between the dual subspaces of a BN and the partitions of its state set. Furthermore, we demonstrate that a dual subspace is $M$-invariant if and only if the associated partition is equitable (i.e., for every two cells of the partition, every two states in the former have the same number of out-neighbors in the latter) for the BN's state-transition graph (STG). Here $M$ represents the structure matrix of the BN.Based on the equitable graphic representation, we provide, for the first time, a complete structural characterization of the smallest $M$-invariant dual subspaces generated by a set of Boolean functions. Given a set of output functions, we prove that a BN is observable if and only if the partition corresponding to the smallest $M$-invariant dual subspace generated by this set of functions is trivial (i.e., all partition cells are singletons). Building upon our structural characterization, we also present a method for constructing output functions that render the BN observable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10961 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Structures of M-Invariant Dual Subspaces with Respect to a Boolean Network Bi, Dongyao Zhang, Lijun Zhang, Kuize Zhang, Shenggui Systems and Control This paper presents the following research findings on Boolean networks (BNs) and their dual subspaces.First, we establish a bijection between the dual subspaces of a BN and the partitions of its state set. Furthermore, we demonstrate that a dual subspace is $M$-invariant if and only if the associated partition is equitable (i.e., for every two cells of the partition, every two states in the former have the same number of out-neighbors in the latter) for the BN's state-transition graph (STG). Here $M$ represents the structure matrix of the BN.Based on the equitable graphic representation, we provide, for the first time, a complete structural characterization of the smallest $M$-invariant dual subspaces generated by a set of Boolean functions. Given a set of output functions, we prove that a BN is observable if and only if the partition corresponding to the smallest $M$-invariant dual subspace generated by this set of functions is trivial (i.e., all partition cells are singletons). Building upon our structural characterization, we also present a method for constructing output functions that render the BN observable. |
| title | Structures of M-Invariant Dual Subspaces with Respect to a Boolean Network |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2301.10961 |