4978:
discussion has focused on identifying "fastest" as "best," and the augmented canonical form meets that criterion flawlessly, but sometimes other factors predominate. The designer may have a primary goal of minimizing the number of gates, and/or of minimizing the fanouts of signals to other gates since big fanouts reduce resilience to a degraded power supply or other environmental factors. In such a case, a designer may develop the canonical-form design as a baseline, then try a bottom-up development, and finally compare the results.
6050:
143:
5025:′ in 2 gate delays. If the circuit inventory actually includes 4-input NOR gates, the top-down canonical design looks like a winner in both gate count and speed. But if (contrary to our convenient supposition) the circuits are actually 3-input NOR gates, of which two are required for each 4-input NOR function, then the canonical design takes 14 gates compared to 12 for the bottom-up approach, but still produces the sum digit
47:
88:
5201:′ out of the leftmost bit position will probably have to be complemented as part of the logic determining whether the addition overflowed. But using 3-input NOR gates, the bottom-up design is very nearly as fast for doing parallel addition on a non-trivial word length, cuts down on the gate count, and uses lower fanouts ... so it wins if gate count and/or fanout are paramount!
2382:
collector voltage (the output) very near to ground. That result is independent of the other inputs. Only when all 3 input signals are 0 (low voltage) do the emitter-collector impedances of all 3 transistors remain very high. Then very little current flows, and the voltage-divider effect with the load impedance imposes on the collector point a high voltage very near to V
4969:
through a 1-input NOR gate. However, this approach not only increases the number of gates used, but also doubles the number of gate delays processing the signals, cutting the processing speed in half. Consequently, whenever performance is vital, going beyond canonical forms and doing the
Boolean algebra to make the unenhanced NOR gates do the job is well worthwhile.
5197:′ by a 1-input NOR, is slower but for any word length the design only pays that penalty once (when the leftmost sum digit is developed). That's because those calculations overlap, each in what amounts to its own little pipeline without affecting when the next bit position's sum bit can be calculated. And, to be sure, the
2410:, always produce minterms, never maxterms—that is, of the 8 gates required to process all combinations of 3 input variables, only one has the output value 1. That's because a NOR gate, despite its name, could better be viewed (using De Morgan's law) as the AND of the complements of its input signals.
4968:
Now we could have implemented those functions exactly according to their SoP and PoS canonical forms, by turning NOR gates into the functions specified. A NOR gate is made into an OR gate by passing its output through a 1-input NOR gate; and it is made into an AND gate by passing each of its inputs
4143:
in this respect, because they are static throughout the addition and thus are normally held in latch circuits that routinely have both direct and complement outputs. (The simplest latch circuit made of NOR gates is a pair of gates cross-coupled to make a flip-flop: the output of each is wired as one
2369:
The sample truth tables for minterms and maxterms above are sufficient to establish the canonical form for a single bit position in the addition of binary numbers, but are not sufficient to design the digital logic unless your inventory of gates includes AND and OR. Where performance is an issue (as
4977:
We have now seen how the minterm/maxterm tools can be used to design an adder stage in canonical form with the addition of some
Boolean algebra, costing just 2 gate delays for each of the outputs. That's the "top-down" way to design the digital circuit for this function, but is it the best way? The
2381:
through a load impedance. Each base is connected to an input signal, and the common collector point presents the output signal. Any input that is a 1 (high voltage) to its base shorts its transistor's emitter to its collector, causing current to flow through the load impedance, which brings the
520:
variables, since a variable in the minterm expression can be in either its direct or its complemented form—two choices per variable. Minterms are often numbered by a binary encoding of the complementation pattern of the variables, where the variables are written in a standard order, usually
1338:
variables, since a variable in the maxterm expression can also be in either its direct or its complemented form—two choices per variable. The numbering is chosen so that the complement of a minterm is the respective maxterm. That is, each maxterm is assigned an index based on the opposite
2389:
The complementing property of these gate circuits may seem like a drawback when trying to implement a function in canonical form, but there is a compensating bonus: such a gate with only one input implements the complementing function, which is required frequently in digital logic.
2025:
It is often the case that the canonical minterm form is equivalent to a smaller SoP form. This smaller form would still consist of a sum of product terms, but have fewer product terms and/or product terms that contain fewer variables. For example, the following 3-variable function:
1204:
4358:
3553:
2698:
2370:
in the Apollo
Guidance Computer), the available parts are more likely to be NAND and NOR because of the complementing action inherent in transistor logic. The values are defined as voltage states, one near ground and one near the DC supply voltage V
4134:
One might suppose that the work of designing an adder stage is now complete, but we haven't addressed the fact that all 3 of the input variables have to appear in both their direct and complement forms. There's no difficulty about the addends
2012:
2393:
This example assumes the Apollo parts inventory: 3-input NOR gates only, but the discussion is simplified by supposing that 4-input NOR gates are also available (in Apollo, those were compounded out of pairs of 3-input NORs).
652:
5427:
The authors demonstrate a proof that any
Boolean (logic) function can be expressed in either disjunctive or conjunctive normal form (cf pages 5–6); the proof simply proceeds by creating all 2 rows of
330:) which is a conjunction (AND) of maxterms. These forms can be useful for the simplification of Boolean functions, which is of great importance in the optimization of Boolean formulas in general and
2510:
2297:. In general, there may be multiple minimal SoP forms, none clearly smaller or larger than another. In a similar manner, a canonical maxterm form can be reduced to various minimal PoS forms.
3362:
5971:
4156:′, but that would add a gate delay in the worst possible place, slowing down the rippling of carries from right to left. An additional 4-input NOR gate building the canonical form of
4148:. However, the carry out of one bit position must be passed as the carry into the next bit position in both direct and complement forms. The most straightforward way to do this is to pass
1508:
5270:, which pioneered the application of integrated circuits in the 1960s, was built with only one type of gate, a 3-input NOR, whose output is true only when all 3 inputs are false.
1268:
427:
4170:
5204:
We'll leave the exact circuitry of the bottom-up design of which all these statements are true as an exercise for the interested reader, assisted by one more algebraic formula:
3370:
943:
2518:
1778:
1448:
2287:
2347:
2206:
1552:
4934:
The trade-off to maintain full speed in this way includes an unexpected cost (in addition to having to use a bigger gate). If we'd just used that 1-input gate to complement
1406:
576:
5394:
To see how NOR gate logic was used in the Apollo
Guidance Computer's ALU, select any of the 4-BIT MODULE entries in the Index to Drawings, and expand images as desired.
683:
1370:
4963:
1805:
970:
718:
546:
2235:
975:
455:
379:
2377:
Specifically, a 3-input NOR gate may consist of 3 bipolar junction transistors with their emitters all grounded, their collectors tied together and linked to V
2413:
The reason this is not a problem is the duality of minterms and maxterms, i.e. each maxterm is the complement of the like-indexed minterm, and vice versa.
1813:
5248:
One application of
Boolean algebra is digital circuit design, with one goal to minimize the number of gates and another to minimize the settling time.
5327:
2512:
but to perform this with a 4-input NOR gate we need to restate it as a product of sums (PoS), where the sums are the opposite maxterms. That is,
5978:
581:
5181:′, which means that the carry propagation ripples along the bit positions just as fast as in the canonical design without ever developing
5251:
There are sixteen possible functions of two variables, but in digital logic hardware, the simplest gate circuits implement only four of them:
5531:
5300:
1302:
value for just one combination of the input variables, i.e. it is true at the maximal number of possibilities. For example, the maxterm
5763:
5431:
Boolean variables and demonstrates that each row ("minterm" or "maxterm") has a unique
Boolean expression. Any Boolean function of the
1570:
of a logical function, it is possible to write the function as a "product of sums" or "product of maxterms". This is a special form of
207:
5801:
2374:, e.g. +5 VDC. If the higher voltage is defined as the 1 "true" value, a NOR gate is the simplest possible useful logical element.
732:
of a logical function, it is possible to write the function as a "sum of products" or "sum of minterms". This is a special form of
2349:], in less obvious cases a convenient method for finding minimal PoS/SoP forms of a function with up to four variables is using a
179:
5703:
1298:. Instead of using ANDs and complements, we use ORs and complements and proceed similarly. It is apparent that a maxterm gives a
5806:
160:
60:
5337:
5310:
473:). A minterm gives a true value for just one combination of the input variables, the minimum nontrivial amount. For example,
186:
5232:)′]′. Decoupling the carry propagation from the sum formation in this way is what elevates the performance of a
5165:. Does that design simply never need the direct form of the carry out? Well, yes and no. At each stage, the calculation of
5638:
106:
98:
5751:
5650:
2419:
2402:
A set of 8 NOR gates, if their inputs are all combinations of the direct and complement forms of the 3 input variables
2361:
developed from the problem of finding optimal implementations of
Boolean functions, such as minimal PoS and SoP forms.
193:
5017:′ in 2 gate delays. The canonical baseline took eight 3-input NOR gates plus three 4-input NOR gates to produce
3364:
but to perform this with a 4-input NOR gate we need to notice the equality to the NOR of the same minterms. That is,
5685:
5524:
5470:
5419:
5362:
3277:
253:
226:
124:
74:
5987:
175:
6104:
5697:
5633:
5964:
5279:
164:
5494:
1339:
conventional binary encoding used for minterms. The maxterm convention assigns the value 0 to the direct form
5823:
5691:
2354:
4353:{\displaystyle co'(ci,x,y)=\mathrm {AND} (M_{3},M_{5},M_{6},M_{7})=\mathrm {NOR} (m_{3},m_{5},m_{6},m_{7}).}
5517:
3548:{\displaystyle co(ci,x,y)=\mathrm {AND} (M_{0},M_{1},M_{2},M_{4})=\mathrm {NOR} (m_{0},m_{1},m_{2},m_{4}).}
35:
2693:{\displaystyle u(ci,x,y)=\mathrm {AND} (M_{0},M_{3},M_{5},M_{6})=\mathrm {NOR} (m_{0},m_{3},m_{5},m_{6}).}
1225:
384:
5940:
5835:
5009:. One such development takes twelve NOR gates in all: six 2-input gates and two 1-input gates to produce
892:
5855:
5813:
5435:
variables can be derived from a composite of the rows whose minterm or maxterm are logical 1s ("trues")
1730:
66:
5756:
5741:
5667:
5627:
4965:, and the gate that generated it could have been eliminated. Nevertheless, it is still a good trade.
6135:
6125:
5768:
5673:
5661:
5267:
200:
6014:
6009:
5818:
5596:
5591:
2240:
1571:
733:
310:
280:
267:
153:
17:
2303:
2162:
1513:
1460:
5996:
5912:
5709:
30:
This article is about canonical forms particularly in
Boolean algebra. Not to be confused with
2289:, and the smaller form has both fewer product terms and fewer variables within each term. The
1723:
Observing that the rows that have an output of 0 are the 1st, 2nd, 3rd, and 5th, we can write
1375:
885:
Observing that the rows that have an output of 1 are the 2nd, 3rd, 5th, and 8th, we can write
6019:
5924:
5878:
5746:
5644:
5581:
1411:
1342:
346:
4144:
of the inputs to the other.) There is also no need to create the complement form of the sum
2293:
SoP representations of a function according to this notion of "smallest" are referred to as
551:
6029:
6023:
6004:
5850:
5845:
5828:
5548:
4941:
1783:
1199:{\displaystyle u(ci,x,y)=m_{1}+m_{2}+m_{4}+m_{7}=(ci',x',y)+(ci',x,y')+(ci,x',y')+(ci,x,y)}
948:
696:
524:
342:
2211:
657:
8:
5897:
5840:
5679:
5601:
5576:
5540:
5259:
5253:
1295:
5378:
6099:
6076:
6066:
5917:
5902:
5783:
5621:
5586:
2358:
1555:
440:
364:
6058:
5873:
5466:
5446:
5415:
5358:
5333:
5306:
1294:). Maxterms are a dual of the minterm idea, following the complementary symmetry of
6094:
6086:
5606:
5266:
Most gate circuits accept more than 2 input variables; for example, the spaceborne
2007:{\displaystyle co(ci,x,y)=M_{0}M_{1}M_{2}M_{4}=(ci+x+y)(ci+x+y')(ci+x'+y)(ci'+x+y)}
1215:
358:
271:
1290:
variables that employs only the complement operator and the disjunction operator (
469:
variables that employs only the complement operator and the conjunction operator (
6071:
5796:
5791:
5773:
5736:
5731:
5656:
5616:
2290:
331:
6130:
5726:
338:
301:
31:
4129:
2017:
evaluated for all 8 combinations of the three variables will match the table.
1206:
evaluated for all 8 combinations of the three variables will match the table.
6119:
5907:
5890:
5885:
6049:
5490:
5013:
in 5 gate delays, plus three 2-input gates and one 3-input gate to produce
2350:
647:{\displaystyle \sum \limits _{i=1}^{n}2^{i-1}\operatorname {value} (x_{i})}
434:
6039:
1567:
729:
470:
5956:
5263:(inclusive OR), and the respective complements of those (NAND and NOR).
2397:
2300:
While this example was simplified by applying normal algebraic methods [
5945:
5611:
5556:
1326:
is false—the input arrangement where a = 1, b = 0, c = 1 results in 0.
1291:
5509:
521:
alphabetical. This convention assigns the value 1 to the direct form (
5571:
142:
5566:
5561:
736:. For example, if given the truth table for the arithmetic sum bit
1578:
of one bit position's logic of an adder circuit, as a function of
740:
of one bit position's logic of an adder circuit, as a function of
5302:
Mathematical
Foundations of Computational Engineering: A Handbook
5450:
5355:
Journey to the Moon: The History of the Apollo Guidance Computer
1574:. For example, if given the truth table for the carry-out bit
5029:
considerably faster. The fanout comparison is tabulated as:
1282:(either in its complemented or uncomplemented form). Thus, a
461:(either in its complemented or uncomplemented form). Thus, a
5298:
4130:
Design trade-offs considered in addition to canonical forms
5465:(2nd ed.). NY: John Wiley & Sons. p. 101.
5243:
5305:. Springer Science & Business Media. pp. 15–.
5157:
The description of the bottom-up development mentions
4944:
4173:
3373:
3280:
2521:
2422:
2398:
Canonical and non-canonical consequences of NOR gates
2306:
2243:
2214:
2165:
1816:
1786:
1733:
1516:
1463:
1414:
1378:
1345:
1228:
978:
951:
895:
699:
660:
584:
554:
527:
443:
387:
367:
1408:. For example, we assign the index 6 to the maxterm
5463:
Introduction to Switching Theory and Logical Design
167:. Unsourced material may be challenged and removed.
4957:
4352:
3547:
3356:
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2504:
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2281:
2229:
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2006:
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1262:
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937:
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677:
646:
570:
540:
449:
421:
373:
4981:The bottom-up development involves noticing that
2505:{\displaystyle u(ci,x,y)=m_{1}+m_{2}+m_{4}+m_{7}}
2357:can solve slightly larger problems. The field of
6117:
5443:Introduction to the Theory of Switching Circuits
5409:
4972:
4152:through a 1-input NOR gate and label the output
341:include the complete sum of prime implicants or
5461:Hill, Fredrick J.; Peterson, Gerald R. (1974).
5455:Canonical expressions are defined and described
5410:Bender, Edward A.; Williamson, S. Gill (2005).
4938:, there would have been no use for the minterm
3357:{\displaystyle co(ci,x,y)=M_{0}M_{1}M_{2}M_{4}}
5292:
2237:. In this trivial example, it is obvious that
5972:
5525:
5477:Minterm and maxterm designation of functions
5460:
5445:. NY: McGraw–Hill Book Company. p. 78.
5299:Peter J. Pahl; Rudolf Damrath (2012-12-06).
2020:
5505:. Translated by Wilkins, David R.: 183–198.
5379:"APOLLO GUIDANCE COMPUTER (AGC) Schematics"
4993:), where XOR means eXclusive OR , and that
75:Learn how and when to remove these messages
5979:
5965:
5532:
5518:
242:presents an incomplete view of the subject
5986:
5499:Cambridge and Dublin Mathematical Journal
5440:
4160:′ (out of the opposite minterms as
2159:has the canonical minterm representation
1561:
723:
300:) as a disjunction (OR) of minterms. The
254:Learn how and when to remove this message
227:Learn how and when to remove this message
125:Learn how and when to remove this message
5414:. Mineola, NY: Dover Publications, Inc.
349:(also called Zhegalkin or Reed–Muller).
5704:Application-specific integrated circuit
5539:
3274:In the maxterm example above, we wrote
2416:In the minterm example above, we wrote
14:
6118:
5412:A Short Course in Discrete Mathematics
2364:
5960:
5513:
5489:
5332:. John Wiley & Sons. p. 78.
5244:Application in digital circuit design
496:is false—the input arrangement where
5639:Three-dimensional integrated circuit
5352:
5325:
2208:, but it has an equivalent SoP form
1329:
1263:{\displaystyle {x_{1},\dots ,x_{n}}}
511:
422:{\displaystyle {x_{1},\dots ,x_{n}}}
165:adding citations to reliable sources
136:
81:
40:
5329:Principles of Modern Digital Design
1586:from the addends and the carry in,
1274:is a sum term in which each of the
748:from the addends and the carry in,
586:
27:Standard forms of Boolean functions
24:
5651:Erasable programmable logic device
5403:
4288:
4285:
4282:
4219:
4216:
4213:
3483:
3480:
3477:
3414:
3411:
3408:
2628:
2625:
2622:
2559:
2556:
2553:
938:{\displaystyle m_{1},m_{2},m_{4},}
548:) and 0 to the complemented form (
97:tone or style may not reflect the
25:
6147:
5686:Complex programmable logic device
5483:
1773:{\displaystyle M_{0},M_{1},M_{2}}
1450:(110) and denote that maxterm as
306:canonical conjunctive normal form
276:canonical disjunctive normal form
56:This article has multiple issues.
6048:
141:
107:guide to writing better articles
86:
45:
6105:Normal form (natural deduction)
5698:Field-programmable object array
5634:Mixed-signal integrated circuit
1372:and 1 to the complemented form
152:needs additional citations for
64:or discuss these issues on the
5371:
5346:
5319:
5280:List of Boolean algebra topics
4344:
4292:
4275:
4223:
4206:
4185:
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3470:
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3401:
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2001:
1975:
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1917:
1914:
1893:
1844:
1823:
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1464:
1395:
1379:
1359:
1346:
1334:There are again 2 maxterms of
1193:
1172:
1166:
1135:
1129:
1098:
1092:
1061:
1003:
982:
641:
628:
13:
1:
5824:Hardware description language
5692:Field-programmable gate array
5326:Lala, Parag K. (2007-07-16).
5285:
5161:′ as an output but not
4973:Top-down vs. bottom-up design
1807:. If we wish to verify this:
972:. If we wish to verify this:
7:
5836:Formal equivalence checking
5273:
2282:{\displaystyle bc=a'bc+abc}
1314:′ is false only when
1286:is a logical expression of
1209:
465:is a logical expression of
352:
10:
6152:
5856:Hierarchical state machine
5814:Transaction-level modeling
2342:{\displaystyle f=(a'+a)bc}
2201:{\displaystyle f=a'bc+abc}
1547:{\displaystyle abc'=m_{6}}
1503:{\displaystyle (a'+b'+c)'}
29:
6085:
6057:
6046:
5995:
5933:
5866:
5782:
5757:Digital signal processing
5742:Logic in computer science
5719:
5668:Programmable logic device
5628:Hybrid integrated circuit
5547:
5441:McCluskey, E. J. (1965).
2355:Quine–McCluskey algorithm
2021:Minimal PoS and SoP forms
1727:as a product of maxterms
5769:Switching circuit theory
5674:Programmable Array Logic
5662:Programmable logic array
5268:Apollo Guidance Computer
5169:′ depends only on
5040:
5037:
5034:
4691:
4688:
4682:
4676:
4670:
4664:
4661:
4658:
4655:
4410:
4407:
4401:
4395:
4389:
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4380:
4377:
4374:
3886:
3883:
3877:
3871:
3865:
3859:
3856:
3853:
3850:
3605:
3602:
3596:
3590:
3584:
3578:
3575:
3572:
3569:
3031:
3028:
3022:
3016:
3010:
3004:
3001:
2998:
2995:
2750:
2747:
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2723:
2720:
2717:
2714:
2040:
2037:
2034:
2031:
1604:
1601:
1598:
1595:
1401:{\displaystyle (x'_{i})}
766:
763:
760:
757:
516:There are 2 minterms of
345:(and its dual), and the
274:can be expressed in the
6015:Disjunctive normal form
6010:Conjunctive normal form
5819:Register-transfer level
5495:"The Calculus of Logic"
5353:Hall, Eldon C. (1996).
4164:) solves this problem.
1572:conjunctive normal form
1443:{\displaystyle a'+b'+c}
1365:{\displaystyle (x_{i})}
734:disjunctive normal form
654:. For example, minterm
578:); the minterm is then
176:"Canonical normal form"
5710:Tensor Processing Unit
4959:
4354:
3549:
3358:
2694:
2506:
2343:
2283:
2231:
2202:
2008:
1801:
1774:
1562:Maxterm canonical form
1548:
1504:
1444:
1402:
1366:
1264:
1200:
966:
939:
724:Minterm canonical form
714:
679:
648:
605:
572:
571:{\displaystyle x'_{i}}
542:
451:
423:
375:
316:maxterm canonical form
286:minterm canonical form
6035:Canonical normal form
6020:Algebraic normal form
5925:Electronic literature
5879:Hardware acceleration
5747:Computer architecture
5645:Emitter-coupled logic
5582:Printed circuit board
5234:carry-lookahead adder
5189:, which does require
5185:. The calculation of
4960:
4958:{\displaystyle m_{7}}
4355:
3550:
3359:
2695:
2507:
2344:
2284:
2232:
2203:
2009:
1802:
1800:{\displaystyle M_{4}}
1775:
1549:
1505:
1445:
1403:
1367:
1265:
1201:
967:
965:{\displaystyle m_{7}}
940:
889:as a sum of minterms
715:
713:{\displaystyle m_{6}}
680:
649:
585:
573:
543:
541:{\displaystyle x_{i}}
452:
437:in which each of the
424:
376:
347:algebraic normal form
6030:Blake canonical form
6024:Zhegalkin polynomial
6005:Negation normal form
5851:Finite-state machine
5829:High-level synthesis
5764:Circuit minimization
4942:
4171:
3371:
3278:
2519:
2420:
2304:
2241:
2230:{\displaystyle f=bc}
2212:
2163:
1814:
1784:
1731:
1514:
1461:
1412:
1376:
1343:
1226:
976:
949:
893:
697:
678:{\displaystyle abc'}
658:
582:
552:
525:
484:, is true only when
441:
385:
365:
343:Blake canonical form
161:improve this article
5997:Propositional logic
5898:Digital photography
5680:Generic Array Logic
5602:Combinational logic
5577:Printed electronics
5541:Digital electronics
4367:
3562:
2707:
2365:Application example
1394:
567:
6100:Modal clausal form
6077:Prenex normal form
6067:Skolem normal form
5846:Asynchronous logic
5622:Integrated circuit
5587:Electronic circuit
5238:ripple carry adder
4955:
4363:
4350:
3558:
3545:
3354:
2703:
2690:
2502:
2359:logic optimization
2339:
2279:
2227:
2198:
2004:
1797:
1770:
1566:If one is given a
1544:
1500:
1440:
1398:
1382:
1362:
1322:both are true and
1278:variables appears
1260:
1196:
962:
935:
710:
675:
644:
568:
555:
538:
508:= 1 results in 1.
492:both are true and
457:variables appears
447:
419:
371:
6113:
6112:
5954:
5953:
5903:Digital telephone
5874:Computer hardware
5841:Synchronous logic
5339:978-0-470-07296-7
5312:978-3-642-56893-0
5155:
5154:
4932:
4931:
4928:
4927:
4647:
4646:
4127:
4126:
4123:
4122:
3842:
3841:
3272:
3271:
3268:
3267:
2987:
2986:
2295:minimal SoP forms
2157:
2156:
1721:
1720:
1457:. The complement
1330:Indexing maxterms
883:
882:
512:Indexing minterms
450:{\displaystyle n}
374:{\displaystyle n}
264:
263:
256:
237:
236:
229:
211:
135:
134:
127:
101:used on Knowledge
99:encyclopedic tone
79:
16:(Redirected from
6143:
6095:Beta normal form
6052:
5981:
5974:
5967:
5958:
5957:
5607:Sequential logic
5534:
5527:
5520:
5511:
5510:
5506:
5479:
5457:
5437:
5397:
5396:
5391:
5390:
5375:
5369:
5368:
5350:
5344:
5343:
5323:
5317:
5316:
5296:
5193:to be made from
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4953:
4653:
4652:
4372:
4371:
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4329:
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4184:
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3486:
3469:
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3456:
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3355:
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2712:
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2699:
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2631:
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2588:
2587:
2575:
2574:
2562:
2511:
2509:
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2503:
2501:
2500:
2488:
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2474:
2462:
2461:
2348:
2346:
2345:
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2323:
2288:
2286:
2285:
2280:
2260:
2236:
2234:
2233:
2228:
2207:
2205:
2204:
2199:
2179:
2029:
2028:
2013:
2011:
2010:
2005:
1988:
1965:
1942:
1889:
1888:
1879:
1878:
1869:
1868:
1859:
1858:
1806:
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1803:
1798:
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1545:
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1542:
1530:
1509:
1507:
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1501:
1499:
1485:
1474:
1449:
1447:
1446:
1441:
1433:
1422:
1407:
1405:
1404:
1399:
1390:
1371:
1369:
1368:
1363:
1358:
1357:
1337:
1296:De Morgan's laws
1289:
1277:
1269:
1267:
1266:
1261:
1259:
1258:
1257:
1239:
1238:
1221:
1216:boolean function
1205:
1203:
1202:
1197:
1165:
1154:
1128:
1111:
1085:
1074:
1057:
1056:
1044:
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1031:
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1018:
1017:
971:
969:
968:
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931:
930:
918:
917:
905:
904:
755:
754:
719:
717:
716:
711:
709:
708:
684:
682:
681:
676:
674:
653:
651:
650:
645:
640:
639:
621:
620:
604:
599:
577:
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574:
569:
563:
547:
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456:
454:
453:
448:
428:
426:
425:
420:
418:
417:
416:
398:
397:
380:
378:
377:
372:
359:boolean function
332:digital circuits
272:Boolean function
259:
252:
232:
225:
221:
218:
212:
210:
169:
145:
137:
130:
123:
119:
116:
110:
109:for suggestions.
105:See Knowledge's
90:
89:
82:
71:
49:
48:
41:
21:
6151:
6150:
6146:
6145:
6144:
6142:
6141:
6140:
6136:Algebraic logic
6126:Boolean algebra
6116:
6115:
6114:
6109:
6081:
6072:Herbrandization
6059:Predicate logic
6053:
6044:
5991:
5985:
5955:
5950:
5929:
5862:
5797:Place and route
5792:Logic synthesis
5778:
5774:Gate equivalent
5737:Logic synthesis
5732:Boolean algebra
5715:
5657:Macrocell array
5617:Boolean circuit
5543:
5538:
5486:
5473:
5426:
5422:
5406:
5404:Further reading
5401:
5400:
5388:
5386:
5377:
5376:
5372:
5365:
5351:
5347:
5340:
5324:
5320:
5313:
5297:
5293:
5288:
5276:
5246:
5236:over that of a
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4405:
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2583:
2579:
2570:
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2483:
2479:
2470:
2466:
2457:
2453:
2421:
2418:
2417:
2400:
2385:
2380:
2373:
2367:
2316:
2305:
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2301:
2253:
2242:
2239:
2238:
2213:
2210:
2209:
2172:
2164:
2161:
2160:
2023:
1981:
1958:
1935:
1884:
1880:
1874:
1870:
1864:
1860:
1854:
1850:
1815:
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1811:
1791:
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1781:
1764:
1760:
1751:
1747:
1738:
1734:
1732:
1729:
1728:
1564:
1556:de Morgan's law
1538:
1534:
1523:
1515:
1512:
1511:
1510:is the minterm
1492:
1478:
1467:
1462:
1459:
1458:
1456:
1426:
1415:
1413:
1410:
1409:
1386:
1377:
1374:
1373:
1353:
1349:
1344:
1341:
1340:
1335:
1332:
1287:
1275:
1253:
1249:
1234:
1230:
1229:
1227:
1224:
1223:
1219:
1212:
1158:
1147:
1121:
1104:
1078:
1067:
1052:
1048:
1039:
1035:
1026:
1022:
1013:
1009:
977:
974:
973:
956:
952:
950:
947:
946:
926:
922:
913:
909:
900:
896:
894:
891:
890:
726:
704:
700:
698:
695:
694:
692:
688:
685:is numbered 110
667:
659:
656:
655:
635:
631:
610:
606:
600:
589:
583:
580:
579:
559:
553:
550:
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532:
528:
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523:
522:
514:
442:
439:
438:
412:
408:
393:
389:
388:
386:
383:
382:
366:
363:
362:
355:
339:canonical forms
334:in particular.
320:Product of Sums
290:Sum of Products
268:Boolean algebra
260:
249:
248:
247:
233:
222:
216:
213:
170:
168:
158:
146:
131:
120:
114:
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104:
95:This article's
91:
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28:
23:
22:
15:
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11:
5:
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5992:
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5943:
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5927:
5922:
5921:
5920:
5915:
5913:cinematography
5905:
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5893:
5883:
5882:
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5863:
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5809:
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5794:
5788:
5786:
5780:
5779:
5777:
5776:
5771:
5766:
5761:
5760:
5759:
5752:Digital signal
5749:
5744:
5739:
5734:
5729:
5727:Digital signal
5723:
5721:
5717:
5716:
5714:
5713:
5707:
5701:
5695:
5689:
5683:
5677:
5671:
5665:
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5648:
5642:
5636:
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5619:
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5609:
5604:
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5536:
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5522:
5514:
5508:
5507:
5485:
5484:External links
5482:
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3150:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3121:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3092:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3063:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3034:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2989:
2988:
2985:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2956:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2927:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2898:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2869:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2840:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2811:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2782:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2753:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2701:
2700:
2689:
2686:
2681:
2677:
2673:
2668:
2664:
2660:
2655:
2651:
2647:
2642:
2638:
2634:
2630:
2627:
2624:
2620:
2617:
2612:
2608:
2604:
2599:
2595:
2591:
2586:
2582:
2578:
2573:
2569:
2565:
2561:
2558:
2555:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2499:
2495:
2491:
2486:
2482:
2478:
2473:
2469:
2465:
2460:
2456:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2399:
2396:
2383:
2378:
2371:
2366:
2363:
2338:
2335:
2332:
2329:
2326:
2322:
2319:
2315:
2312:
2309:
2278:
2275:
2272:
2269:
2266:
2263:
2259:
2256:
2252:
2249:
2246:
2226:
2223:
2220:
2217:
2197:
2194:
2191:
2188:
2185:
2182:
2178:
2175:
2171:
2168:
2155:
2154:
2151:
2148:
2145:
2141:
2140:
2137:
2134:
2131:
2127:
2126:
2123:
2120:
2117:
2113:
2112:
2109:
2106:
2103:
2099:
2098:
2095:
2092:
2089:
2085:
2084:
2081:
2078:
2075:
2071:
2070:
2067:
2064:
2061:
2057:
2056:
2053:
2050:
2047:
2043:
2042:
2039:
2036:
2033:
2022:
2019:
2015:
2014:
2003:
2000:
1997:
1994:
1991:
1987:
1984:
1980:
1977:
1974:
1971:
1968:
1964:
1961:
1957:
1954:
1951:
1948:
1945:
1941:
1938:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1887:
1883:
1877:
1873:
1867:
1863:
1857:
1853:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1794:
1790:
1767:
1763:
1759:
1754:
1750:
1746:
1741:
1737:
1719:
1718:
1715:
1712:
1709:
1705:
1704:
1701:
1698:
1695:
1691:
1690:
1687:
1684:
1681:
1677:
1676:
1673:
1670:
1667:
1663:
1662:
1659:
1656:
1653:
1649:
1648:
1645:
1642:
1639:
1635:
1634:
1631:
1628:
1625:
1621:
1620:
1617:
1614:
1611:
1607:
1606:
1603:
1600:
1597:
1563:
1560:
1541:
1537:
1533:
1529:
1526:
1522:
1519:
1498:
1495:
1491:
1488:
1484:
1481:
1477:
1473:
1470:
1466:
1454:
1439:
1436:
1432:
1429:
1425:
1421:
1418:
1397:
1393:
1389:
1385:
1381:
1361:
1356:
1352:
1348:
1331:
1328:
1256:
1252:
1248:
1245:
1242:
1237:
1233:
1211:
1208:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1164:
1161:
1157:
1153:
1150:
1146:
1143:
1140:
1137:
1134:
1131:
1127:
1124:
1120:
1117:
1114:
1110:
1107:
1103:
1100:
1097:
1094:
1091:
1088:
1084:
1081:
1077:
1073:
1070:
1066:
1063:
1060:
1055:
1051:
1047:
1042:
1038:
1034:
1029:
1025:
1021:
1016:
1012:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
959:
955:
934:
929:
925:
921:
916:
912:
908:
903:
899:
881:
880:
877:
874:
871:
867:
866:
863:
860:
857:
853:
852:
849:
846:
843:
839:
838:
835:
832:
829:
825:
824:
821:
818:
815:
811:
810:
807:
804:
801:
797:
796:
793:
790:
787:
783:
782:
779:
776:
773:
769:
768:
765:
762:
759:
725:
722:
707:
703:
690:
689: = 6
686:
673:
670:
666:
663:
643:
638:
634:
630:
627:
624:
619:
616:
613:
609:
603:
598:
595:
592:
588:
566:
562:
558:
535:
531:
513:
510:
446:
415:
411:
407:
404:
401:
396:
392:
370:
354:
351:
302:De Morgan dual
262:
261:
246:
245:
235:
234:
149:
147:
140:
133:
132:
94:
92:
85:
80:
54:
53:
51:
44:
32:Canonical form
26:
9:
6:
4:
3:
2:
6148:
6137:
6134:
6132:
6129:
6127:
6124:
6123:
6121:
6106:
6103:
6101:
6098:
6096:
6093:
6092:
6090:
6088:
6084:
6078:
6075:
6073:
6070:
6068:
6065:
6064:
6062:
6060:
6056:
6051:
6041:
6038:
6036:
6033:
6031:
6028:
6025:
6021:
6018:
6016:
6013:
6011:
6008:
6006:
6003:
6002:
6000:
5998:
5994:
5989:
5982:
5977:
5975:
5970:
5968:
5963:
5962:
5959:
5947:
5944:
5942:
5941:Metastability
5939:
5938:
5936:
5934:Design issues
5932:
5926:
5923:
5919:
5916:
5914:
5911:
5910:
5909:
5908:Digital video
5906:
5904:
5901:
5899:
5896:
5892:
5889:
5888:
5887:
5886:Digital audio
5884:
5880:
5877:
5876:
5875:
5872:
5871:
5869:
5865:
5857:
5854:
5853:
5852:
5849:
5847:
5844:
5842:
5839:
5837:
5834:
5830:
5827:
5825:
5822:
5821:
5820:
5817:
5815:
5812:
5808:
5805:
5803:
5800:
5799:
5798:
5795:
5793:
5790:
5789:
5787:
5785:
5781:
5775:
5772:
5770:
5767:
5765:
5762:
5758:
5755:
5754:
5753:
5750:
5748:
5745:
5743:
5740:
5738:
5735:
5733:
5730:
5728:
5725:
5724:
5722:
5718:
5711:
5708:
5705:
5702:
5699:
5696:
5693:
5690:
5687:
5684:
5681:
5678:
5675:
5672:
5669:
5666:
5663:
5660:
5658:
5655:
5652:
5649:
5646:
5643:
5640:
5637:
5635:
5632:
5629:
5626:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5605:
5603:
5600:
5598:
5595:
5593:
5590:
5588:
5585:
5583:
5580:
5578:
5575:
5573:
5570:
5568:
5565:
5563:
5560:
5558:
5555:
5554:
5552:
5550:
5546:
5542:
5535:
5530:
5528:
5523:
5521:
5516:
5515:
5512:
5504:
5500:
5496:
5492:
5491:Boole, George
5488:
5487:
5478:
5474:
5472:0-471-39882-9
5468:
5464:
5459:
5456:
5452:
5448:
5444:
5439:
5436:
5434:
5430:
5423:
5421:0-486-43946-1
5417:
5413:
5408:
5407:
5395:
5384:
5380:
5374:
5366:
5364:1-56347-185-X
5360:
5356:
5349:
5341:
5335:
5331:
5330:
5322:
5314:
5308:
5304:
5303:
5295:
5291:
5281:
5278:
5277:
5271:
5269:
5264:
5262:
5261:
5256:
5255:
5249:
5241:
5239:
5235:
5231:
5227:
5223:
5219:
5215:
5211:
5207:
5202:
5200:
5196:
5192:
5188:
5184:
5180:
5176:
5172:
5168:
5164:
5160:
5150:
5147:
5144:
5143:
5139:
5136:
5133:
5132:
5128:
5125:
5122:
5121:
5117:
5114:
5111:
5110:
5106:
5103:
5100:
5099:
5095:
5092:
5089:
5088:
5084:
5081:
5078:
5077:
5073:
5070:
5067:
5066:
5062:
5059:
5056:
5055:
5051:
5048:
5045:
5044:
5033:
5030:
5028:
5024:
5020:
5016:
5012:
5008:
5004:
5000:
4996:
4992:
4988:
4984:
4979:
4970:
4966:
4950:
4946:
4937:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4898:
4894:
4891:
4888:
4885:
4882:
4879:
4876:
4873:
4870:
4869:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4840:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4811:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4786:
4783:
4782:
4778:
4775:
4772:
4769:
4766:
4763:
4760:
4757:
4754:
4753:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4724:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4699:
4696:
4695:
4654:
4651:
4650:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4617:
4613:
4610:
4607:
4604:
4601:
4598:
4595:
4592:
4589:
4588:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4559:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4530:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4501:
4497:
4494:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4472:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4443:
4439:
4436:
4433:
4430:
4427:
4424:
4421:
4418:
4415:
4414:
4373:
4370:
4369:
4366:
4347:
4339:
4335:
4331:
4326:
4322:
4318:
4313:
4309:
4305:
4300:
4296:
4278:
4270:
4266:
4262:
4257:
4253:
4249:
4244:
4240:
4236:
4231:
4227:
4209:
4203:
4200:
4197:
4194:
4191:
4188:
4181:
4178:
4174:
4167:
4166:
4165:
4163:
4159:
4155:
4151:
4147:
4142:
4138:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4093:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4064:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4035:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4006:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3977:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3948:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3919:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3890:
3849:
3846:
3845:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3812:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3783:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3754:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3725:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3696:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3667:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3638:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3609:
3568:
3565:
3564:
3561:
3542:
3534:
3530:
3526:
3521:
3517:
3513:
3508:
3504:
3500:
3495:
3491:
3473:
3465:
3461:
3457:
3452:
3448:
3444:
3439:
3435:
3431:
3426:
3422:
3404:
3398:
3395:
3392:
3389:
3386:
3383:
3377:
3374:
3367:
3366:
3365:
3349:
3345:
3339:
3335:
3329:
3325:
3319:
3315:
3311:
3305:
3302:
3299:
3296:
3293:
3290:
3284:
3281:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3238:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3209:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3180:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3151:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3122:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3093:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3064:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3035:
2994:
2991:
2990:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2957:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2928:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2899:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2870:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2841:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2812:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2783:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2754:
2713:
2710:
2709:
2706:
2687:
2679:
2675:
2671:
2666:
2662:
2658:
2653:
2649:
2645:
2640:
2636:
2618:
2610:
2606:
2602:
2597:
2593:
2589:
2584:
2580:
2576:
2571:
2567:
2549:
2543:
2540:
2537:
2534:
2531:
2528:
2522:
2515:
2514:
2513:
2497:
2493:
2489:
2484:
2480:
2476:
2471:
2467:
2463:
2458:
2454:
2450:
2444:
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2432:
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2414:
2411:
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2250:
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2221:
2218:
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2195:
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2173:
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2110:
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2100:
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2018:
1998:
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240:This article
239:
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231:
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220:
209:
206:
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199:
195:
192:
188:
185:
181:
178: –
177:
173:
172:Find sources:
166:
162:
156:
155:
150:This article
148:
144:
139:
138:
129:
126:
118:
115:February 2009
108:
102:
100:
93:
84:
83:
78:
76:
69:
68:
63:
62:
57:
52:
43:
42:
37:
33:
19:
6034:
5988:Normal forms
5867:Applications
5502:
5498:
5476:
5462:
5454:
5442:
5432:
5428:
5425:
5411:
5393:
5387:. Retrieved
5382:
5373:
5354:
5348:
5328:
5321:
5301:
5294:
5265:
5258:
5252:
5250:
5247:
5237:
5233:
5229:
5225:
5221:
5217:
5213:
5209:
5205:
5203:
5198:
5194:
5190:
5186:
5182:
5178:
5177:′ and
5174:
5170:
5166:
5162:
5158:
5156:
5026:
5022:
5018:
5014:
5010:
5006:
5002:
4998:
4994:
4990:
4986:
4982:
4980:
4976:
4967:
4935:
4933:
4692:co'(ci,x,y)
4411:co'(ci,x,y)
4365:Truth tables
4364:
4161:
4157:
4153:
4149:
4145:
4140:
4136:
4133:
3560:Truth tables
3559:
3273:
2705:Truth tables
2704:
2415:
2412:
2407:
2403:
2401:
2392:
2388:
2376:
2368:
2351:Karnaugh map
2299:
2294:
2158:
2024:
2016:
1724:
1722:
1587:
1583:
1579:
1575:
1565:
1451:
1333:
1323:
1319:
1315:
1311:
1307:
1303:
1299:
1283:
1280:exactly once
1279:
1271:
1213:
886:
884:
749:
745:
741:
737:
727:
693:and denoted
517:
515:
505:
501:
497:
493:
489:
485:
481:
477:
474:
466:
462:
459:exactly once
458:
435:product term
430:
356:
336:
327:
323:
319:
315:
309:
305:
297:
293:
289:
285:
279:
275:
265:
250:
241:
223:
217:October 2010
214:
204:
197:
190:
183:
171:
159:Please help
154:verification
151:
121:
112:
96:
72:
65:
59:
58:Please help
55:
6040:Horn clause
5597:Memory cell
5385:. Rich Katz
5260:disjunction
5254:conjunction
3887:co(ci,x,y)
3606:co(ci,x,y)
1605:co(ci,x,y)
1568:truth table
730:truth table
471:logical AND
36:Normal form
6120:Categories
5946:Runt pulse
5918:television
5612:Logic gate
5557:Transistor
5549:Components
5389:2021-06-19
5286:References
5041:Bottom-up
5035:Variables
3032:u(ci,x,y)
2751:u(ci,x,y)
1306:′ +
1292:logical OR
1222:variables
767:u(ci,x,y)
728:Given the
381:variables
187:newspapers
61:improve it
5802:Placement
5592:Flip-flop
5572:Capacitor
5383:klabs.org
5173:′,
5038:Top-down
2041:f(a,b,c)
1244:…
1210:Maxterms
626:
615:−
587:∑
403:…
353:Minterms
67:talk page
5990:in logic
5567:Inductor
5562:Resistor
5493:(1848).
5451:65-17394
5357:. AIAA.
5274:See also
5224:′(
5123:x XOR y
4182:′
2321:′
2258:′
2177:′
1986:′
1963:′
1940:′
1554:, using
1528:′
1497:′
1483:′
1472:′
1431:′
1420:′
1392:′
1163:′
1152:′
1126:′
1109:′
1083:′
1072:′
672:′
565:′
5807:Routing
5641:(3D IC)
5257:(AND),
5115:4@1,4@2
5112:M or m
2291:minimal
1284:maxterm
1272:maxterm
463:minterm
431:minterm
304:is the
201:scholar
18:Minterm
5784:Design
5720:Theory
5706:(ASIC)
5700:(FPOA)
5694:(FPGA)
5688:(CPLD)
5653:(EPLD)
5469:
5449:
5418:
5361:
5336:
5309:
4983:u = ci
2404:ci, x,
2353:. The
1214:For a
357:For a
337:Other
270:, any
203:
196:
189:
182:
174:
6131:Logic
6087:Other
5891:radio
5712:(TPU)
5682:(GAL)
5676:(PAL)
5670:(PLD)
5664:(PLA)
5647:(ECL)
5630:(HIC)
5134:Misc
5019:u, co
4985:XOR (
1300:false
623:value
504:= 0,
500:= 1,
433:is a
318:, or
288:, or
208:JSTOR
194:books
5624:(IC)
5467:ISBN
5447:LCCN
5416:ISBN
5359:ISBN
5334:ISBN
5307:ISBN
5228:XOR
5220:) +
5216:XOR
5145:Max
5140:5@1
5118:N/A
5101:ci'
5021:and
5007:y ci
4999:ci x
4989:XOR
4689:NOR
4408:AND
4139:and
3884:NOR
3603:AND
3029:NOR
2748:AND
2406:and
1780:and
1582:and
1318:and
1270:, a
945:and
744:and
488:and
429:, a
311:CCNF
281:CDNF
180:news
5503:III
5137:N/A
5126:N/A
5090:ci
5079:y'
5057:x'
5003:x y
4656:ci
4375:ci
3851:ci
3570:ci
2996:ci
2715:ci
1596:ci
1218:of
758:ci
361:of
328:POS
326:or
324:PoS
314:),
298:SOP
296:or
294:SoP
284:),
266:In
163:by
34:or
6122::
5501:.
5497:.
5475:.
5453:.
5424:.
5392:.
5381:.
5240:.
5222:ci
5210:ci
5208:=
5199:co
5195:ci
5191:ci
5183:co
5171:ci
5167:co
5163:co
5159:co
5151:3
5129:2
5107:3
5096:1
5085:3
5074:1
5068:y
5063:3
5052:1
5046:x
5023:co
5015:co
5005:+
5001:+
4997:=
4995:co
4936:co
4924:0
4895:0
4866:0
4837:1
4808:0
4779:1
4750:1
4721:1
4662:y
4659:x
4643:0
4614:0
4585:0
4556:1
4527:0
4498:1
4469:1
4440:1
4381:y
4378:x
4162:co
4158:co
4154:co
4150:co
4119:1
4090:1
4061:1
4032:0
4003:1
3974:0
3945:0
3916:0
3857:y
3854:x
3838:1
3809:1
3780:1
3751:0
3722:1
3693:0
3664:0
3635:0
3576:y
3573:x
3264:1
3235:0
3206:0
3177:1
3148:0
3119:1
3090:1
3061:0
3002:y
2999:x
2983:1
2954:0
2925:0
2896:1
2867:0
2838:1
2809:1
2780:0
2721:y
2718:x
2386:.
2384:cc
2379:cc
2372:cc
2153:1
2139:0
2125:0
2111:0
2097:1
2083:0
2069:0
2055:0
2038:c
2035:b
2032:a
1725:co
1717:1
1703:1
1689:1
1675:0
1661:1
1647:0
1633:0
1619:0
1602:y
1599:x
1590::
1588:ci
1576:co
1558:.
1310:+
879:1
865:0
851:0
837:1
823:0
809:1
795:1
781:0
764:y
761:x
752::
750:ci
720:.
691:10
480:'
70:.
6026:)
6022:(
5980:e
5973:t
5966:v
5533:e
5526:t
5519:v
5433:N
5429:N
5367:.
5342:.
5315:.
5230:y
5226:x
5218:y
5214:x
5212:(
5206:u
5187:u
5179:y
5175:x
5148:4
5104:4
5093:4
5082:4
5071:4
5060:4
5049:4
5027:u
5011:u
4991:y
4987:x
4951:7
4947:m
4921:0
4918:1
4915:0
4912:0
4909:0
4906:1
4903:1
4900:1
4892:0
4889:0
4886:1
4883:0
4880:0
4877:0
4874:1
4871:1
4863:0
4860:0
4857:0
4854:1
4851:0
4848:1
4845:0
4842:1
4834:1
4831:0
4828:0
4825:0
4822:0
4819:0
4816:0
4813:1
4805:0
4802:0
4799:0
4796:0
4793:1
4790:1
4787:1
4784:0
4776:1
4773:0
4770:0
4767:0
4764:0
4761:0
4758:1
4755:0
4747:1
4744:0
4741:0
4738:0
4735:0
4732:1
4729:0
4726:0
4718:1
4715:0
4712:0
4709:0
4706:0
4703:0
4700:0
4697:0
4685:7
4683:m
4679:6
4677:m
4673:5
4671:m
4667:3
4665:m
4640:0
4637:0
4634:1
4631:1
4628:1
4625:1
4622:1
4619:1
4611:0
4608:1
4605:0
4602:1
4599:1
4596:0
4593:1
4590:1
4582:0
4579:1
4576:1
4573:0
4570:1
4567:1
4564:0
4561:1
4553:1
4550:1
4547:1
4544:1
4541:1
4538:0
4535:0
4532:1
4524:0
4521:1
4518:1
4515:1
4512:0
4509:1
4506:1
4503:0
4495:1
4492:1
4489:1
4486:1
4483:1
4480:0
4477:1
4474:0
4466:1
4463:1
4460:1
4457:1
4454:1
4451:1
4448:0
4445:0
4437:1
4434:1
4431:1
4428:1
4425:1
4422:0
4419:0
4416:0
4404:7
4402:M
4398:6
4396:M
4392:5
4390:M
4386:3
4384:M
4348:.
4345:)
4340:7
4336:m
4332:,
4327:6
4323:m
4319:,
4314:5
4310:m
4306:,
4301:3
4297:m
4293:(
4289:R
4286:O
4283:N
4279:=
4276:)
4271:7
4267:M
4263:,
4258:6
4254:M
4250:,
4245:5
4241:M
4237:,
4232:3
4228:M
4224:(
4220:D
4217:N
4214:A
4210:=
4207:)
4204:y
4201:,
4198:x
4195:,
4192:i
4189:c
4186:(
4179:o
4175:c
4146:u
4141:y
4137:x
4116:1
4113:0
4110:0
4107:0
4104:0
4101:1
4098:1
4095:1
4087:1
4084:0
4081:0
4078:0
4075:0
4072:0
4069:1
4066:1
4058:1
4055:0
4052:0
4049:0
4046:0
4043:1
4040:0
4037:1
4029:0
4026:1
4023:0
4020:0
4017:0
4014:0
4011:0
4008:1
4000:1
3997:0
3994:0
3991:0
3988:0
3985:1
3982:1
3979:0
3971:0
3968:0
3965:1
3962:0
3959:0
3956:0
3953:1
3950:0
3942:0
3939:0
3936:0
3933:1
3930:0
3927:1
3924:0
3921:0
3913:0
3910:0
3907:0
3904:0
3901:1
3898:0
3895:0
3892:0
3880:4
3878:m
3874:2
3872:m
3868:1
3866:m
3862:0
3860:m
3835:1
3832:1
3829:1
3826:1
3823:1
3820:1
3817:1
3814:1
3806:1
3803:1
3800:1
3797:1
3794:1
3791:0
3788:1
3785:1
3777:1
3774:1
3771:1
3768:1
3765:1
3762:1
3759:0
3756:1
3748:0
3745:0
3742:1
3739:1
3736:1
3733:0
3730:0
3727:1
3719:1
3716:1
3713:1
3710:1
3707:1
3704:1
3701:1
3698:0
3690:0
3687:1
3684:0
3681:1
3678:1
3675:0
3672:1
3669:0
3661:0
3658:1
3655:1
3652:0
3649:1
3646:1
3643:0
3640:0
3632:0
3629:1
3626:1
3623:1
3620:0
3617:0
3614:0
3611:0
3599:4
3597:M
3593:2
3591:M
3587:1
3585:M
3581:0
3579:M
3543:.
3540:)
3535:4
3531:m
3527:,
3522:2
3518:m
3514:,
3509:1
3505:m
3501:,
3496:0
3492:m
3488:(
3484:R
3481:O
3478:N
3474:=
3471:)
3466:4
3462:M
3458:,
3453:2
3449:M
3445:,
3440:1
3436:M
3432:,
3427:0
3423:M
3419:(
3415:D
3412:N
3409:A
3405:=
3402:)
3399:y
3396:,
3393:x
3390:,
3387:i
3384:c
3381:(
3378:o
3375:c
3350:4
3346:M
3340:2
3336:M
3330:1
3326:M
3320:0
3316:M
3312:=
3309:)
3306:y
3303:,
3300:x
3297:,
3294:i
3291:c
3288:(
3285:o
3282:c
3261:1
3258:0
3255:0
3252:0
3249:0
3246:1
3243:1
3240:1
3232:0
3229:1
3226:0
3223:0
3220:0
3217:0
3214:1
3211:1
3203:0
3200:0
3197:1
3194:0
3191:0
3188:1
3185:0
3182:1
3174:1
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3168:0
3165:0
3162:0
3159:0
3156:0
3153:1
3145:0
3142:0
3139:0
3136:1
3133:0
3130:1
3127:1
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3116:1
3113:0
3110:0
3107:0
3104:0
3101:0
3098:1
3095:0
3087:1
3084:0
3081:0
3078:0
3075:0
3072:1
3069:0
3066:0
3058:0
3055:0
3052:0
3049:0
3046:1
3043:0
3040:0
3037:0
3025:6
3023:m
3019:5
3017:m
3013:3
3011:m
3007:0
3005:m
2980:1
2977:1
2974:1
2971:1
2968:1
2965:1
2962:1
2959:1
2951:0
2948:0
2945:1
2942:1
2939:1
2936:0
2933:1
2930:1
2922:0
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2916:0
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2910:1
2907:1
2904:0
2901:1
2893:1
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2849:1
2846:1
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2820:0
2817:1
2814:0
2806:1
2803:1
2800:1
2797:1
2794:1
2791:1
2788:0
2785:0
2777:0
2774:1
2771:1
2768:1
2765:0
2762:0
2759:0
2756:0
2744:6
2742:M
2738:5
2736:M
2732:3
2730:M
2726:0
2724:M
2688:.
2685:)
2680:6
2676:m
2672:,
2667:5
2663:m
2659:,
2654:3
2650:m
2646:,
2641:0
2637:m
2633:(
2629:R
2626:O
2623:N
2619:=
2616:)
2611:6
2607:M
2603:,
2598:5
2594:M
2590:,
2585:3
2581:M
2577:,
2572:0
2568:M
2564:(
2560:D
2557:N
2554:A
2550:=
2547:)
2544:y
2541:,
2538:x
2535:,
2532:i
2529:c
2526:(
2523:u
2498:7
2494:m
2490:+
2485:4
2481:m
2477:+
2472:2
2468:m
2464:+
2459:1
2455:m
2451:=
2448:)
2445:y
2442:,
2439:x
2436:,
2433:i
2430:c
2427:(
2424:u
2408:y
2337:c
2334:b
2331:)
2328:a
2325:+
2318:a
2314:(
2311:=
2308:f
2277:c
2274:b
2271:a
2268:+
2265:c
2262:b
2255:a
2251:=
2248:c
2245:b
2225:c
2222:b
2219:=
2216:f
2196:c
2193:b
2190:a
2187:+
2184:c
2181:b
2174:a
2170:=
2167:f
2150:1
2147:1
2144:1
2136:0
2133:1
2130:1
2122:1
2119:0
2116:1
2108:0
2105:0
2102:1
2094:1
2091:1
2088:0
2080:0
2077:1
2074:0
2066:1
2063:0
2060:0
2052:0
2049:0
2046:0
2002:)
1999:y
1996:+
1993:x
1990:+
1983:i
1979:c
1976:(
1973:)
1970:y
1967:+
1960:x
1956:+
1953:i
1950:c
1947:(
1944:)
1937:y
1933:+
1930:x
1927:+
1924:i
1921:c
1918:(
1915:)
1912:y
1909:+
1906:x
1903:+
1900:i
1897:c
1894:(
1891:=
1886:4
1882:M
1876:2
1872:M
1866:1
1862:M
1856:0
1852:M
1848:=
1845:)
1842:y
1839:,
1836:x
1833:,
1830:i
1827:c
1824:(
1821:o
1818:c
1793:4
1789:M
1766:2
1762:M
1758:,
1753:1
1749:M
1745:,
1740:0
1736:M
1714:1
1711:1
1708:1
1700:0
1697:1
1694:1
1686:1
1683:0
1680:1
1672:0
1669:0
1666:1
1658:1
1655:1
1652:0
1644:0
1641:1
1638:0
1630:1
1627:0
1624:0
1616:0
1613:0
1610:0
1584:y
1580:x
1540:6
1536:m
1532:=
1525:c
1521:b
1518:a
1494:)
1490:c
1487:+
1480:b
1476:+
1469:a
1465:(
1455:6
1452:M
1438:c
1435:+
1428:b
1424:+
1417:a
1396:)
1388:i
1384:x
1380:(
1360:)
1355:i
1351:x
1347:(
1336:n
1324:b
1320:c
1316:a
1312:c
1308:b
1304:a
1288:n
1276:n
1255:n
1251:x
1247:,
1241:,
1236:1
1232:x
1220:n
1194:)
1191:y
1188:,
1185:x
1182:,
1179:i
1176:c
1173:(
1170:+
1167:)
1160:y
1156:,
1149:x
1145:,
1142:i
1139:c
1136:(
1133:+
1130:)
1123:y
1119:,
1116:x
1113:,
1106:i
1102:c
1099:(
1096:+
1093:)
1090:y
1087:,
1080:x
1076:,
1069:i
1065:c
1062:(
1059:=
1054:7
1050:m
1046:+
1041:4
1037:m
1033:+
1028:2
1024:m
1020:+
1015:1
1011:m
1007:=
1004:)
1001:y
998:,
995:x
992:,
989:i
986:c
983:(
980:u
958:7
954:m
933:,
928:4
924:m
920:,
915:2
911:m
907:,
902:1
898:m
887:u
876:1
873:1
870:1
862:0
859:1
856:1
848:1
845:0
842:1
834:0
831:0
828:1
820:1
817:1
814:0
806:0
803:1
800:0
792:1
789:0
786:0
778:0
775:0
772:0
746:y
742:x
738:u
706:6
702:m
687:2
669:c
665:b
662:a
642:)
637:i
633:x
629:(
618:1
612:i
608:2
602:n
597:1
594:=
591:i
561:i
557:x
534:i
530:x
518:n
506:c
502:b
498:a
494:b
490:c
486:a
482:c
478:b
475:a
467:n
445:n
414:n
410:x
406:,
400:,
395:1
391:x
369:n
322:(
308:(
292:(
278:(
257:)
251:(
244:.
230:)
224:(
219:)
215:(
205:·
198:·
191:·
184:·
157:.
128:)
122:(
117:)
113:(
103:.
77:)
73:(
38:.
20:)
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