State-efficient realization of fault-tolerant FSSP algorithms
The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly so...
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| Published in | Natural computing Vol. 18; no. 4; pp. 827 - 844 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.12.2019
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1567-7818 1572-9796 |
| DOI | 10.1007/s11047-019-09765-3 |
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| Abstract | The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly some defective cells and present the first state-efficient implementations of fault-tolerant FSSP algorithms for one-dimensional (1D) and two-dimensional (2D) cellular arrays. It is shown that, under some constraints on the length and distribution of defective cells, any 1D cellular array of length
n
with
p
defective cell segments can be synchronized in
2
n
-
2
+
p
steps and the algorithm is realized on a 1D cellular automaton of length
n
,
2
≤
n
≤
50
, having 164 states and 4792 transition rules. In addition, we give by far a smaller-state implementation of a 2D FSSP algorithm that can synchronize any 2D rectangular array of size
m
×
n
, possibly including at most O(
mn
) isolated defective zones, exactly in
2
(
m
+
n
)
-
4
steps on a cellular automaton with only 6 states and 935 transition rules. |
|---|---|
| AbstractList | The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly some defective cells and present the first state-efficient implementations of fault-tolerant FSSP algorithms for one-dimensional (1D) and two-dimensional (2D) cellular arrays. It is shown that, under some constraints on the length and distribution of defective cells, any 1D cellular array of length
n
with
p
defective cell segments can be synchronized in
2
n
-
2
+
p
steps and the algorithm is realized on a 1D cellular automaton of length
n
,
2
≤
n
≤
50
, having 164 states and 4792 transition rules. In addition, we give by far a smaller-state implementation of a 2D FSSP algorithm that can synchronize any 2D rectangular array of size
m
×
n
, possibly including at most O(
mn
) isolated defective zones, exactly in
2
(
m
+
n
)
-
4
steps on a cellular automaton with only 6 states and 935 transition rules. The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly some defective cells and present the first state-efficient implementations of fault-tolerant FSSP algorithms for one-dimensional (1D) and two-dimensional (2D) cellular arrays. It is shown that, under some constraints on the length and distribution of defective cells, any 1D cellular array of length n with p defective cell segments can be synchronized in \[2n-2+p\] steps and the algorithm is realized on a 1D cellular automaton of length \[n, 2 \le n \le 50\], having 164 states and 4792 transition rules. In addition, we give by far a smaller-state implementation of a 2D FSSP algorithm that can synchronize any 2D rectangular array of size \[m \times n\], possibly including at most O(mn) isolated defective zones, exactly in \[2(m+n)-4\] steps on a cellular automaton with only 6 states and 935 transition rules. |
| Author | Umeo, Hiroshi Maeda, Masashi Kamikawa, Naoki Fujita, Gen |
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| Keywords | Cellular automaton FSSP Firing squad synchronization problem Fault-tolerant cellular automaton |
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| References | Balzer (CR1) 1967; 10 CR2 CR4 CR6 CR19 CR7 CR18 CR9 Kutrib, Vollmar (CR11) 1995; E78–D CR14 CR12 Mazoyer (CR13) 1987; 50 Waksman (CR20) 1966; 9 CR22 Gács (CR5) 1986; 32 Nishio, Kobuchi (CR15) 1975; 11 Harao, Noguchi (CR8) 1975; 11 Umeo (CR16) 2004; E87–D Yunès (CR21) 2006; 1 Coan, Dolev, Dwork, Stockmeyer (CR3) 1989; 18 Umeo, Maeda, Hisaoka, Teraoka (CR17) 2006; 74 Kutrib, Vollmar (CR10) 1991; ELK 27 9765_CR19 9765_CR18 R Balzer (9765_CR1) 1967; 10 H Umeo (9765_CR16) 2004; E87–D H Umeo (9765_CR17) 2006; 74 H Nishio (9765_CR15) 1975; 11 M Kutrib (9765_CR11) 1995; E78–D 9765_CR22 M Kutrib (9765_CR10) 1991; ELK 27 9765_CR12 BA Coan (9765_CR3) 1989; 18 M Harao (9765_CR8) 1975; 11 9765_CR14 P Gács (9765_CR5) 1986; 32 9765_CR7 J Mazoyer (9765_CR13) 1987; 50 J-B Yunès (9765_CR21) 2006; 1 9765_CR6 9765_CR4 9765_CR2 A Waksman (9765_CR20) 1966; 9 9765_CR9 |
| References_xml | – volume: 50 start-page: 183 year: 1987 end-page: 238 ident: CR13 article-title: A six-state minimal time solution to the firing squad synchronization problem publication-title: Theor Comput Sci doi: 10.1016/0304-3975(87)90124-1 – ident: CR22 – ident: CR19 – volume: ELK 27 start-page: 179 year: 1991 end-page: 196 ident: CR10 article-title: Minimal time synchronization in restricted defective cellular automata publication-title: J Inform Process Cybern – ident: CR18 – volume: 10 start-page: 22 year: 1967 end-page: 42 ident: CR1 article-title: An 8-state minimal time solution to the firing squad synchronization problem publication-title: Inf Control doi: 10.1016/S0019-9958(67)90032-0 – ident: CR4 – ident: CR14 – volume: 9 start-page: 66 year: 1966 end-page: 78 ident: CR20 article-title: An optimum solution to the firing squad synchronization problem publication-title: Inf Control doi: 10.1016/S0019-9958(66)90110-0 – ident: CR2 – ident: CR12 – volume: 74 start-page: 603 year: 2006 end-page: 623 ident: CR17 article-title: A state-efficient mapping scheme for designing two-dimensional firing squad synchronization algorithms publication-title: Fund Inf – volume: 18 start-page: 990 year: 1989 end-page: 1012 ident: CR3 article-title: The distributed firing squad problem publication-title: SIAM J Comput doi: 10.1137/0218068 – volume: E78–D start-page: 895 year: 1995 end-page: 900 ident: CR11 article-title: The firing squad synchronization problem in defective cellular automata publication-title: IEICE Trans Inf Syst – ident: CR9 – volume: E87–D start-page: 733 year: 2004 end-page: 739 ident: CR16 article-title: A simple design of time-efficient firing squad synchronization algorithms with fault-tolerance publication-title: IEICE Trans Inf Syst – ident: CR6 – volume: 11 start-page: 171 year: 1975 end-page: 185 ident: CR8 article-title: Fault tolerant cellular automata publication-title: J Comput Syst. Sci doi: 10.1016/S0022-0000(75)80066-3 – ident: CR7 – volume: 1 start-page: 253 year: 2006 end-page: 268 ident: CR21 article-title: Fault tolerant solutions to the firing squad synchronization problem in linear cellular automata publication-title: J Cell Autom – volume: 32 start-page: 15 year: 1986 end-page: 78 ident: CR5 article-title: Reliable computation with cellular automata publication-title: J Comput Syst Sci doi: 10.1016/0022-0000(86)90002-4 – volume: 11 start-page: 150 year: 1975 end-page: 170 ident: CR15 article-title: Fault tolerant cellular spaces publication-title: J Comput Syst. Sci doi: 10.1016/S0022-0000(75)80065-1 – volume: 74 start-page: 603 year: 2006 ident: 9765_CR17 publication-title: Fund Inf – volume: ELK 27 start-page: 179 year: 1991 ident: 9765_CR10 publication-title: J Inform Process Cybern – ident: 9765_CR6 – volume: E78–D start-page: 895 year: 1995 ident: 9765_CR11 publication-title: IEICE Trans Inf Syst – ident: 9765_CR12 – ident: 9765_CR2 – ident: 9765_CR14 – ident: 9765_CR9 – volume: 1 start-page: 253 year: 2006 ident: 9765_CR21 publication-title: J Cell Autom – ident: 9765_CR7 – ident: 9765_CR4 doi: 10.1007/978-3-319-44365-2_12 – volume: 11 start-page: 171 year: 1975 ident: 9765_CR8 publication-title: J Comput Syst. Sci doi: 10.1016/S0022-0000(75)80066-3 – ident: 9765_CR18 doi: 10.1007/978-3-319-99813-8-25 – volume: 18 start-page: 990 year: 1989 ident: 9765_CR3 publication-title: SIAM J Comput doi: 10.1137/0218068 – volume: 11 start-page: 150 year: 1975 ident: 9765_CR15 publication-title: J Comput Syst. Sci doi: 10.1016/S0022-0000(75)80065-1 – ident: 9765_CR19 doi: 10.1007/978-0-387-30440-3_211 – volume: 50 start-page: 183 year: 1987 ident: 9765_CR13 publication-title: Theor Comput Sci doi: 10.1016/0304-3975(87)90124-1 – volume: E87–D start-page: 733 year: 2004 ident: 9765_CR16 publication-title: IEICE Trans Inf Syst – ident: 9765_CR22 doi: 10.1007/3-540-35828-5_14 – volume: 10 start-page: 22 year: 1967 ident: 9765_CR1 publication-title: Inf Control doi: 10.1016/S0019-9958(67)90032-0 – volume: 9 start-page: 66 year: 1966 ident: 9765_CR20 publication-title: Inf Control doi: 10.1016/S0019-9958(66)90110-0 – volume: 32 start-page: 15 year: 1986 ident: 9765_CR5 publication-title: J Comput Syst Sci doi: 10.1016/0022-0000(86)90002-4 |
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| SubjectTerms | Algorithms Arrays Artificial Intelligence Cellular automata Complex Systems Computer Science Dimensional tolerances Evolutionary Biology Fault tolerance Processor Architectures Synchronism Theory of Computation Transition rules |
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| Title | State-efficient realization of fault-tolerant FSSP algorithms |
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