Abstract
The following results are proved:
any nonconstant Boolean function may be implemented by an irredundant circuit of gates in the basis {x& y, x, x ⊕ y ⊕ z} admitting a single fault detection test of length at most 2 with respect to arbitrary stuck-at faults at outputs of gates,
there exists a six-place Boolean function ψ such that any nonconstant Boolean function may be implemented by an irredundant circuit of gates in the basis {ψ} admitting a single diagnostic test of length at most 3 with respect to arbitrary stuck-at faults at outputs of gates.
Funding
This research was carried out with the financial support of the Russian Science Foundation (RSF) grant 14-21-00025 P.
Originally published in Diskretnaya Matematika (2018) 30, №3, 99–116 (in Russian).
References
[1] Chegis I. A., Yablonskiy S. V., “Logical methods of electric circuits operation control”, Trudy Mat. Inst. Steklov., 51 (1958), 270–360 (in Russian).Search in Google Scholar
[2] Yablonskiy S. V., “Reliability and control of control systems”, Mater. Vsesoyuzn. sem. po diskr. matem. i ee pril., Izd-vo MGU, Moscow, 1986, 7–12 (in Russian).Search in Google Scholar
[3] Yablonskiy S. V., “Some issues of reliability and control of control systems”, Matematicheskie voprosy kibernetiki, 1988, № 1, 5–25 (in Russian).Search in Google Scholar
[4] Red’kin N. P., Reliability and circuit diagnostics, Izd-vo MGU, Moscow, 1992 (in Russian), 192 pp.Search in Google Scholar
[5] Reddy, S. M., “Easily testable realizations for logic functions”, IEEE Trans. Comput., C-21:11 (1972), 1183–1188.10.1109/T-C.1972.223475Search in Google Scholar
[6] Kolyada S. S., “Single fault detection tests for circuits of functional elements”, Moscow Univ. Math. Bulletin, 68:4 (2013), 192–193.10.3103/S0027132213040049Search in Google Scholar
[7] Kolyada S. S., Upper bounds for the length of validation tests for functional element circuits, Diss. na soisk. uch. st. k.f.-m.n., Moscow, 2013 (in Russian), 77 pp.Search in Google Scholar
[8] Romanov D. S., “Method of synthesis of easily testable circuits admitting single fault detection tests of constant length”, Discrete Math. Appl., 24:4 (2014), 227–251.10.1515/dma-2014-0021Search in Google Scholar
[9] Romanov D. S., Romanov E. Yu., “The method of synthesis of redundant circuits that allow short single diagnostic tests with constant failures at the outputs of the elements”, Izv. VUZov. Povolzhskiy region. Fiz.-matem. nauki, 2016, № 2, 87–102 (in Russian).Search in Google Scholar
[10] Popkov K. A., “Lower bounds for lengths of single tests for Boolean circuits”, Discrete Math. Appl., 29:1 (2019), 23–33.10.1515/dma-2019-0004Search in Google Scholar
[11] Red’kin N. P., “About complete validation tests for functional element circuits”, Moscow University Mechanics Bulletin, 1986, № 1, 72–74 (in Russian).Search in Google Scholar
[12] Red’kin N. P., “About complete validation tests for functional element circuits”, Matematicheskie voprosy kibernetiki. Vyp. 2, Nauka, Moscow, 1989, 198–222 (in Russian).Search in Google Scholar
[13] Romanov D. S., “On the synthesis of circuits admitting complete fault detection test sets of constant length under arbitrary constant faults at the outputs of the gates”, Discrete Math. Appl., 23:3-4 (2013), 343–362.10.1515/dma-2013-024Search in Google Scholar
[14] Popkov K. A., “Full length two validation tests for circuits with arbitrary constant element failures”, Prepr. IPM im. M. V. Keldysha, 2017, № 104, 16 (in Russian).Search in Google Scholar
[15] Popkov K. A., “Lower bounds for the lengths of complete diagnostic tests for circuits and circuit inputs”, Prikl. diskr. matem., 2016, № 4(34), 65–73 (in Russian).10.17223/20710410/34/5Search in Google Scholar
[16] Red’kin N. P., “About short test schemes”, Moscow University Mechanics Bulletin, 1988, № 2, 17–21 (in Russian).Search in Google Scholar
[17] Red’kin N. P., “On single diagnostic tests for the same type of constant faults at the outputs of functional elements”, Moscow University Mechanics Bulletin, 1992, № 5, 43–46 (in Russian).Search in Google Scholar
[18] Borodina Yu. V., “Synthesis of easily-tested circuits in the case of single-type constant malfunctions at the element outputs”, Moscow Univ. Comput. Math. Cyber, 32:1 (2008), 42–46.10.3103/s11968-008-1006-5Search in Google Scholar
[19] Popkov K. A., “On the exact value of the length of the minimum unit diagnostic test for one class of circuits”, Diskr. analiz i issled. operatsiy, 24:3 (2017), 80–103 (in Russian).10.1134/S1990478917030140Search in Google Scholar
[20] Popkov K. A., “Single validation tests for circuits of functional elements in the basis « conjunction-negation »”, Prikl. diskr. matem., 2017, № 38, 66–88 (in Russian).Search in Google Scholar
[21] Borodina Yu. V., “Lower estimate of the length of the complete test in the basis {x | y}”, Moscow Univ. Math. Bulletin, 70:4 (2015), 185–186.10.3103/S0027132215040063Search in Google Scholar
[22] Borodina Yu. V., “Circuits admitting single-fault tests of length 1 under constant faults at outputs of elements”, Moscow Univ. Math. Bulletin, 63:5 (2008), 202–204.10.3103/S0027132208050069Search in Google Scholar
[23] Borodina Yu. V., Borodin P. A., “Synthesis of easily testable circuits over the Zhegalkin basis in the case of constant faults of type 0 at outputs of elements”, Discrete Math. Appl., 20:4 (2010), 441–449.10.1515/dma.2010.027Search in Google Scholar
[24] Popkov K. A., “On single diagnostic tests for circuits of functional elements in the Zhegalkin basis”, Izv. VUZov. Povolzhskiy region. Fiz.-matem. nauki, 2016, № 3, 3–18 (in Russian).10.21685/2072-3040-2016-3-1Search in Google Scholar
[25] Yablonskiy S.V., Introduction to Discrete Mathematics, Nauka, Moscow, 1986 (in Russian), 384 pp.Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On closed classes in partial k-valued logic that contain the class of monotone functions
- On relationship between the parameters characterizing nonlinearity and nonhomomorphy of vector spaces transformation
- Analogues of Gluskin–Hosszú and Malyshev theorems for strongly dependent n-ary operations
- Generation of the alternating group by modular additions
- Formulas for a characteristic of spheres and balls in binary high-dimensional spaces
- Short single tests for circuits with arbitrary stuck-at faults at outputs of gates
- Bounds on the frequencies of tuples on parts of the period of linear recurring sequences over Galois rings
- Compositions of a numerical semigroup
Articles in the same Issue
- Frontmatter
- On closed classes in partial k-valued logic that contain the class of monotone functions
- On relationship between the parameters characterizing nonlinearity and nonhomomorphy of vector spaces transformation
- Analogues of Gluskin–Hosszú and Malyshev theorems for strongly dependent n-ary operations
- Generation of the alternating group by modular additions
- Formulas for a characteristic of spheres and balls in binary high-dimensional spaces
- Short single tests for circuits with arbitrary stuck-at faults at outputs of gates
- Bounds on the frequencies of tuples on parts of the period of linear recurring sequences over Galois rings
- Compositions of a numerical semigroup