4967:
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.
6039:
132:
5014:′ 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
36:
77:
5190:′ 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!
2371:
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
4958:
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.
5186:′ 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
2399:, 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.
4957:
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
4132:
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
2358:
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
4966:
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
2370:
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
509:
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
1327:
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
2378:
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.
2014:
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:
1193:
4347:
3542:
2687:
2359:
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
4123:
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
2001:
2382:
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).
641:
5416:
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
319:) 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
2499:
2286:. 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.
3351:
5960:
4145:′, 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
4137:. 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
1497:
5259:, 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.
1257:
416:
4159:
5193:
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:
3359:
932:
2507:
1767:
1437:
2276:
2336:
2195:
1541:
4923:
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
1395:
565:
5383:
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.
672:
1359:
4952:
1794:
959:
707:
535:
2224:
964:
444:
368:
2366:
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
2402:
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.
1802:
5237:
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.
5316:
2501:
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,
5967:
570:
5170:′, which means that the carry propagation ripples along the bit positions just as fast as in the canonical design without ever developing
5240:
There are sixteen possible functions of two variables, but in digital logic hardware, the simplest gate circuits implement only four of them:
5520:
5289:
1291:
value for just one combination of the input variables, i.e. it is true at the maximal number of possibilities. For example, the maxterm
5752:
5420:
Boolean variables and demonstrates that each row ("minterm" or "maxterm") has a unique
Boolean expression. Any Boolean function of the
1559:
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
196:
5790:
2363:, 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.
721:
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
2338:], in less obvious cases a convenient method for finding minimal PoS/SoP forms of a function with up to four variables is using a
168:
5692:
1287:. Instead of using ANDs and complements, we use ORs and complements and proceed similarly. It is apparent that a maxterm gives a
5795:
149:
49:
5326:
5299:
462:). A minterm gives a true value for just one combination of the input variables, the minimum nontrivial amount. For example,
175:
5221:)′]′. Decoupling the carry propagation from the sum formation in this way is what elevates the performance of a
5154:. Does that design simply never need the direct form of the carry out? Well, yes and no. At each stage, the calculation of
5627:
95:
87:
5740:
5639:
2408:
2391:
A set of 8 NOR gates, if their inputs are all combinations of the direct and complement forms of the 3 input variables
2350:
developed from the problem of finding optimal implementations of
Boolean functions, such as minimal PoS and SoP forms.
182:
5006:′ in 2 gate delays. The canonical baseline took eight 3-input NOR gates plus three 4-input NOR gates to produce
3353:
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,
5674:
5513:
5459:
5408:
5351:
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242:
215:
113:
63:
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164:
6093:
5686:
5622:
5953:
5268:
153:
5483:
1328:
conventional binary encoding used for minterms. The maxterm convention assigns the value 0 to the direct form
5812:
5680:
2343:
4342:{\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}).}
5506:
3537:{\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}).}
24:
2682:{\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}).}
1214:
373:
5929:
5824:
4998:. One such development takes twelve NOR gates in all: six 2-input gates and two 1-input gates to produce
881:
5844:
5802:
5424:
variables can be derived from a composite of the rows whose minterm or maxterm are logical 1s ("trues")
1719:
55:
5745:
5730:
5656:
5616:
4954:, and the gate that generated it could have been eliminated. Nevertheless, it is still a good trade.
6124:
6114:
5757:
5662:
5650:
5256:
189:
6003:
5998:
5807:
5585:
5580:
2229:
1560:
722:
299:
269:
256:
142:
2292:
2151:
1502:
1449:
5985:
5901:
5698:
19:
This article is about canonical forms particularly in
Boolean algebra. Not to be confused with
2278:, and the smaller form has both fewer product terms and fewer variables within each term. The
1712:
Observing that the rows that have an output of 0 are the 1st, 2nd, 3rd, and 5th, we can write
1364:
874:
Observing that the rows that have an output of 1 are the 2nd, 3rd, 5th, and 8th, we can write
6008:
5913:
5867:
5735:
5633:
5570:
1400:
1331:
335:
4133:
of the inputs to the other.) There is also no need to create the complement form of the sum
2282:
SoP representations of a function according to this notion of "smallest" are referred to as
540:
6018:
6012:
5993:
5839:
5834:
5817:
5537:
4930:
1772:
1188:{\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)}
937:
685:
513:
331:
2200:
646:
8:
5886:
5829:
5668:
5590:
5565:
5529:
5248:
5242:
1284:
5367:
6088:
6065:
6055:
5906:
5891:
5772:
5610:
5575:
2347:
1544:
429:
353:
6047:
5862:
5455:
5435:
5404:
5347:
5322:
5295:
1283:). Maxterms are a dual of the minterm idea, following the complementary symmetry of
6083:
6075:
5595:
5255:
Most gate circuits accept more than 2 input variables; for example, the spaceborne
1996:{\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)}
1204:
347:
260:
1279:
variables that employs only the complement operator and the disjunction operator (
458:
variables that employs only the complement operator and the conjunction operator (
6060:
5785:
5780:
5762:
5725:
5720:
5645:
5605:
2279:
320:
6119:
5715:
327:
290:
20:
4118:
2006:
evaluated for all 8 combinations of the three variables will match the table.
1195:
evaluated for all 8 combinations of the three variables will match the table.
6108:
5896:
5879:
5874:
6038:
5479:
5002:
in 5 gate delays, plus three 2-input gates and one 3-input gate to produce
2339:
636:{\displaystyle \sum \limits _{i=1}^{n}2^{i-1}\operatorname {value} (x_{i})}
423:
6028:
1556:
718:
459:
5945:
5252:(inclusive OR), and the respective complements of those (NAND and NOR).
2386:
2289:
While this example was simplified by applying normal algebraic methods [
5934:
5600:
5545:
1315:
is false—the input arrangement where a = 1, b = 0, c = 1 results in 0.
1280:
5498:
510:
alphabetical. This convention assigns the value 1 to the direct form (
5560:
131:
5555:
5550:
725:. For example, if given the truth table for the arithmetic sum bit
1567:
of one bit position's logic of an adder circuit, as a function of
729:
of one bit position's logic of an adder circuit, as a function of
5291:
Mathematical
Foundations of Computational Engineering: A Handbook
5439:
5344:
Journey to the Moon: The History of the Apollo Guidance Computer
1563:. For example, if given the truth table for the carry-out bit
5018:
considerably faster. The fanout comparison is tabulated as:
1271:(either in its complemented or uncomplemented form). Thus, a
450:(either in its complemented or uncomplemented form). Thus, a
5287:
4119:
Design trade-offs considered in addition to canonical forms
5454:(2nd ed.). NY: John Wiley & Sons. p. 101.
5232:
5294:. Springer Science & Business Media. pp. 15–.
5146:
The description of the bottom-up development mentions
4933:
4162:
3362:
3269:
2510:
2411:
2387:
Canonical and non-canonical consequences of NOR gates
2295:
2232:
2203:
2154:
1805:
1775:
1722:
1505:
1452:
1403:
1367:
1334:
1217:
967:
940:
884:
688:
649:
573:
543:
516:
432:
376:
356:
1397:. For example, we assign the index 6 to the maxterm
5452:
Introduction to Switching Theory and Logical Design
156:. Unsourced material may be challenged and removed.
4946:
4341:
3536:
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2330:
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1995:
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666:
635:
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410:
362:
4970:The bottom-up development involves noticing that
2494:{\displaystyle u(ci,x,y)=m_{1}+m_{2}+m_{4}+m_{7}}
2346:can solve slightly larger problems. The field of
6106:
5432:Introduction to the Theory of Switching Circuits
5398:
4961:
4141:through a 1-input NOR gate and label the output
330:include the complete sum of prime implicants or
5450:Hill, Fredrick J.; Peterson, Gerald R. (1974).
5444:Canonical expressions are defined and described
5399:Bender, Edward A.; Williamson, S. Gill (2005).
4927:, there would have been no use for the minterm
3346:{\displaystyle co(ci,x,y)=M_{0}M_{1}M_{2}M_{4}}
5281:
2226:. In this trivial example, it is obvious that
5961:
5514:
5466:Minterm and maxterm designation of functions
5449:
5434:. NY: McGraw–Hill Book Company. p. 78.
5288:Peter J. Pahl; Rudolf Damrath (2012-12-06).
2009:
5494:. Translated by Wilkins, David R.: 183–198.
5368:"APOLLO GUIDANCE COMPUTER (AGC) Schematics"
4982:), where XOR means eXclusive OR , and that
64:Learn how and when to remove these messages
5968:
5954:
5521:
5507:
231:presents an incomplete view of the subject
5975:
5488:Cambridge and Dublin Mathematical Journal
5429:
4149:′ (out of the opposite minterms as
2148:has the canonical minterm representation
1550:
712:
289:) as a disjunction (OR) of minterms. The
243:Learn how and when to remove this message
216:Learn how and when to remove this message
114:Learn how and when to remove this message
5403:. Mineola, NY: Dover Publications, Inc.
338:(also called Zhegalkin or Reed–Muller).
5693:Application-specific integrated circuit
5528:
3263:In the maxterm example above, we wrote
2405:In the minterm example above, we wrote
6107:
5401:A Short Course in Discrete Mathematics
2353:
5949:
5502:
5478:
5321:. John Wiley & Sons. p. 78.
5233:Application in digital circuit design
485:is false—the input arrangement where
5628:Three-dimensional integrated circuit
5341:
5314:
2197:, but it has an equivalent SoP form
1318:
1252:{\displaystyle {x_{1},\dots ,x_{n}}}
500:
411:{\displaystyle {x_{1},\dots ,x_{n}}}
154:adding citations to reliable sources
125:
70:
29:
5318:Principles of Modern Digital Design
1575:from the addends and the carry in,
1263:is a sum term in which each of the
737:from the addends and the carry in,
575:
16:Standard forms of Boolean functions
13:
5640:Erasable programmable logic device
5392:
4277:
4274:
4271:
4208:
4205:
4202:
3472:
3469:
3466:
3403:
3400:
3397:
2617:
2614:
2611:
2548:
2545:
2542:
927:{\displaystyle m_{1},m_{2},m_{4},}
537:) and 0 to the complemented form (
86:tone or style may not reflect the
14:
6136:
5675:Complex programmable logic device
5472:
1762:{\displaystyle M_{0},M_{1},M_{2}}
1439:(110) and denote that maxterm as
295:canonical conjunctive normal form
265:canonical disjunctive normal form
45:This article has multiple issues.
6037:
130:
96:guide to writing better articles
75:
34:
6094:Normal form (natural deduction)
5687:Field-programmable object array
5623:Mixed-signal integrated circuit
1361:and 1 to the complemented form
141:needs additional citations for
53:or discuss these issues on the
5360:
5335:
5308:
5269:List of Boolean algebra topics
4333:
4281:
4264:
4212:
4195:
4174:
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1990:
1964:
1961:
1935:
1932:
1906:
1903:
1882:
1833:
1812:
1482:
1453:
1384:
1368:
1348:
1335:
1323:There are again 2 maxterms of
1182:
1161:
1155:
1124:
1118:
1087:
1081:
1050:
992:
971:
630:
617:
1:
5813:Hardware description language
5681:Field-programmable gate array
5315:Lala, Parag K. (2007-07-16).
5274:
5150:′ as an output but not
4962:Top-down vs. bottom-up design
1796:. If we wish to verify this:
961:. If we wish to verify this:
7:
5825:Formal equivalence checking
5262:
2271:{\displaystyle bc=a'bc+abc}
1303:′ is false only when
1275:is a logical expression of
1198:
454:is a logical expression of
341:
10:
6141:
5845:Hierarchical state machine
5803:Transaction-level modeling
2331:{\displaystyle f=(a'+a)bc}
2190:{\displaystyle f=a'bc+abc}
1536:{\displaystyle abc'=m_{6}}
1492:{\displaystyle (a'+b'+c)'}
18:
6074:
6046:
6035:
5984:
5922:
5855:
5771:
5746:Digital signal processing
5731:Logic in computer science
5708:
5657:Programmable logic device
5617:Hybrid integrated circuit
5536:
5430:McCluskey, E. J. (1965).
2344:Quine–McCluskey algorithm
2010:Minimal PoS and SoP forms
1716:as a product of maxterms
5758:Switching circuit theory
5663:Programmable Array Logic
5651:Programmable logic array
5257:Apollo Guidance Computer
5158:′ depends only on
5029:
5026:
5023:
4680:
4677:
4671:
4665:
4659:
4653:
4650:
4647:
4644:
4399:
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4390:
4384:
4378:
4372:
4369:
4366:
4363:
3875:
3872:
3866:
3860:
3854:
3848:
3845:
3842:
3839:
3594:
3591:
3585:
3579:
3573:
3567:
3564:
3561:
3558:
3020:
3017:
3011:
3005:
2999:
2993:
2990:
2987:
2984:
2739:
2736:
2730:
2724:
2718:
2712:
2709:
2706:
2703:
2029:
2026:
2023:
2020:
1593:
1590:
1587:
1584:
1390:{\displaystyle (x'_{i})}
755:
752:
749:
746:
505:There are 2 minterms of
334:(and its dual), and the
263:can be expressed in the
6004:Disjunctive normal form
5999:Conjunctive normal form
5808:Register-transfer level
5484:"The Calculus of Logic"
5342:Hall, Eldon C. (1996).
4153:) solves this problem.
1561:conjunctive normal form
1432:{\displaystyle a'+b'+c}
1354:{\displaystyle (x_{i})}
723:disjunctive normal form
643:. For example, minterm
567:); the minterm is then
165:"Canonical normal form"
5699:Tensor Processing Unit
4948:
4343:
3538:
3347:
2683:
2495:
2332:
2272:
2220:
2191:
1997:
1790:
1763:
1551:Maxterm canonical form
1537:
1493:
1433:
1391:
1355:
1253:
1189:
955:
928:
713:Minterm canonical form
703:
668:
637:
594:
561:
560:{\displaystyle x'_{i}}
531:
440:
412:
364:
305:maxterm canonical form
275:minterm canonical form
6024:Canonical normal form
6009:Algebraic normal form
5914:Electronic literature
5868:Hardware acceleration
5736:Computer architecture
5634:Emitter-coupled logic
5571:Printed circuit board
5223:carry-lookahead adder
5178:, which does require
5174:. The calculation of
4949:
4947:{\displaystyle m_{7}}
4344:
3539:
3348:
2684:
2496:
2333:
2273:
2221:
2192:
1998:
1791:
1789:{\displaystyle M_{4}}
1764:
1538:
1494:
1434:
1392:
1356:
1254:
1190:
956:
954:{\displaystyle m_{7}}
929:
878:as a sum of minterms
704:
702:{\displaystyle m_{6}}
669:
638:
574:
562:
532:
530:{\displaystyle x_{i}}
441:
426:in which each of the
413:
365:
336:algebraic normal form
6019:Blake canonical form
6013:Zhegalkin polynomial
5994:Negation normal form
5840:Finite-state machine
5818:High-level synthesis
5753:Circuit minimization
4931:
4160:
3360:
3267:
2508:
2409:
2293:
2230:
2219:{\displaystyle f=bc}
2201:
2152:
1803:
1773:
1720:
1503:
1450:
1401:
1365:
1332:
1215:
965:
938:
882:
686:
667:{\displaystyle abc'}
647:
571:
541:
514:
473:, is true only when
430:
374:
354:
332:Blake canonical form
150:improve this article
5986:Propositional logic
5887:Digital photography
5669:Generic Array Logic
5591:Combinational logic
5566:Printed electronics
5530:Digital electronics
4356:
3551:
2696:
2354:Application example
1383:
556:
6089:Modal clausal form
6066:Prenex normal form
6056:Skolem normal form
5835:Asynchronous logic
5611:Integrated circuit
5576:Electronic circuit
5227:ripple carry adder
4944:
4352:
4339:
3547:
3534:
3343:
2692:
2679:
2491:
2348:logic optimization
2328:
2268:
2216:
2187:
1993:
1786:
1759:
1555:If one is given a
1533:
1489:
1429:
1387:
1371:
1351:
1311:both are true and
1267:variables appears
1249:
1185:
951:
924:
699:
664:
633:
557:
544:
527:
497:= 1 results in 1.
481:both are true and
446:variables appears
436:
408:
360:
6102:
6101:
5943:
5942:
5892:Digital telephone
5863:Computer hardware
5830:Synchronous logic
5328:978-0-470-07296-7
5301:978-3-642-56893-0
5144:
5143:
4921:
4920:
4917:
4916:
4636:
4635:
4116:
4115:
4112:
4111:
3831:
3830:
3261:
3260:
3257:
3256:
2976:
2975:
2284:minimal SoP forms
2146:
2145:
1710:
1709:
1446:. The complement
1319:Indexing maxterms
872:
871:
501:Indexing minterms
439:{\displaystyle n}
363:{\displaystyle n}
253:
252:
245:
226:
225:
218:
200:
124:
123:
116:
90:used on Knowledge
88:encyclopedic tone
68:
6132:
6084:Beta normal form
6041:
5970:
5963:
5956:
5947:
5946:
5596:Sequential logic
5523:
5516:
5509:
5500:
5499:
5495:
5468:
5446:
5426:
5386:
5385:
5380:
5379:
5364:
5358:
5357:
5339:
5333:
5332:
5312:
5306:
5305:
5285:
5182:to be made from
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5020:
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4945:
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4942:
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4641:
4361:
4360:
4357:
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4331:
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4280:
4263:
4262:
4250:
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4223:
4211:
4173:
3837:
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3488:
3487:
3475:
3458:
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3445:
3444:
3432:
3431:
3419:
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3406:
3352:
3350:
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3331:
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3321:
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3311:
2982:
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2701:
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2620:
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2590:
2589:
2577:
2576:
2564:
2563:
2551:
2500:
2498:
2497:
2492:
2490:
2489:
2477:
2476:
2464:
2463:
2451:
2450:
2337:
2335:
2334:
2329:
2312:
2277:
2275:
2274:
2269:
2249:
2225:
2223:
2222:
2217:
2196:
2194:
2193:
2188:
2168:
2018:
2017:
2002:
2000:
1999:
1994:
1977:
1954:
1931:
1878:
1877:
1868:
1867:
1858:
1857:
1848:
1847:
1795:
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1792:
1787:
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1784:
1768:
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1757:
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1744:
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1582:
1581:
1542:
1540:
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1534:
1532:
1531:
1519:
1498:
1496:
1495:
1490:
1488:
1474:
1463:
1438:
1436:
1435:
1430:
1422:
1411:
1396:
1394:
1393:
1388:
1379:
1360:
1358:
1357:
1352:
1347:
1346:
1326:
1285:De Morgan's laws
1278:
1266:
1258:
1256:
1255:
1250:
1248:
1247:
1246:
1228:
1227:
1210:
1205:boolean function
1194:
1192:
1191:
1186:
1154:
1143:
1117:
1100:
1074:
1063:
1046:
1045:
1033:
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958:
957:
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949:
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920:
919:
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906:
894:
893:
744:
743:
708:
706:
705:
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698:
697:
673:
671:
670:
665:
663:
642:
640:
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634:
629:
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609:
593:
588:
566:
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558:
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528:
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525:
445:
443:
442:
437:
417:
415:
414:
409:
407:
406:
405:
387:
386:
369:
367:
366:
361:
348:boolean function
321:digital circuits
261:Boolean function
248:
241:
221:
214:
210:
207:
201:
199:
158:
134:
126:
119:
112:
108:
105:
99:
98:for suggestions.
94:See Knowledge's
79:
78:
71:
60:
38:
37:
30:
6140:
6139:
6135:
6134:
6133:
6131:
6130:
6129:
6125:Algebraic logic
6115:Boolean algebra
6105:
6104:
6103:
6098:
6070:
6061:Herbrandization
6048:Predicate logic
6042:
6033:
5980:
5974:
5944:
5939:
5918:
5851:
5786:Place and route
5781:Logic synthesis
5767:
5763:Gate equivalent
5726:Logic synthesis
5721:Boolean algebra
5704:
5646:Macrocell array
5606:Boolean circuit
5532:
5527:
5475:
5462:
5415:
5411:
5395:
5393:Further reading
5390:
5389:
5377:
5375:
5366:
5365:
5361:
5354:
5340:
5336:
5329:
5313:
5309:
5302:
5286:
5282:
5277:
5265:
5235:
5225:over that of a
4964:
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4929:
4928:
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4669:
4663:
4657:
4394:
4388:
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4323:
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4297:
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4284:
4270:
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4241:
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4166:
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4121:
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3589:
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3571:
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3518:
3509:
3505:
3496:
3492:
3483:
3479:
3465:
3453:
3449:
3440:
3436:
3427:
3423:
3414:
3410:
3396:
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3357:
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3333:
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3307:
3303:
3268:
3265:
3264:
3015:
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3003:
2997:
2734:
2728:
2722:
2716:
2667:
2663:
2654:
2650:
2641:
2637:
2628:
2624:
2610:
2598:
2594:
2585:
2581:
2572:
2568:
2559:
2555:
2541:
2509:
2506:
2505:
2485:
2481:
2472:
2468:
2459:
2455:
2446:
2442:
2410:
2407:
2406:
2389:
2374:
2369:
2362:
2356:
2305:
2294:
2291:
2290:
2242:
2231:
2228:
2227:
2202:
2199:
2198:
2161:
2153:
2150:
2149:
2012:
1970:
1947:
1924:
1873:
1869:
1863:
1859:
1853:
1849:
1843:
1839:
1804:
1801:
1800:
1780:
1776:
1774:
1771:
1770:
1753:
1749:
1740:
1736:
1727:
1723:
1721:
1718:
1717:
1553:
1545:de Morgan's law
1527:
1523:
1512:
1504:
1501:
1500:
1499:is the minterm
1481:
1467:
1456:
1451:
1448:
1447:
1445:
1415:
1404:
1402:
1399:
1398:
1375:
1366:
1363:
1362:
1342:
1338:
1333:
1330:
1329:
1324:
1321:
1276:
1264:
1242:
1238:
1223:
1219:
1218:
1216:
1213:
1212:
1208:
1201:
1147:
1136:
1110:
1093:
1067:
1056:
1041:
1037:
1028:
1024:
1015:
1011:
1002:
998:
966:
963:
962:
945:
941:
939:
936:
935:
915:
911:
902:
898:
889:
885:
883:
880:
879:
715:
693:
689:
687:
684:
683:
681:
677:
674:is numbered 110
656:
648:
645:
644:
624:
620:
599:
595:
589:
578:
572:
569:
568:
548:
542:
539:
538:
521:
517:
515:
512:
511:
503:
431:
428:
427:
401:
397:
382:
378:
377:
375:
372:
371:
355:
352:
351:
344:
328:canonical forms
323:in particular.
309:Product of Sums
279:Sum of Products
257:Boolean algebra
249:
238:
237:
236:
222:
211:
205:
202:
159:
157:
147:
135:
120:
109:
103:
100:
93:
84:This article's
80:
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35:
28:
17:
12:
11:
5:
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5958:
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5932:
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5919:
5917:
5916:
5911:
5910:
5909:
5904:
5902:cinematography
5894:
5889:
5884:
5883:
5882:
5872:
5871:
5870:
5859:
5857:
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5852:
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5793:
5783:
5777:
5775:
5769:
5768:
5766:
5765:
5760:
5755:
5750:
5749:
5748:
5741:Digital signal
5738:
5733:
5728:
5723:
5718:
5716:Digital signal
5712:
5710:
5706:
5705:
5703:
5702:
5696:
5690:
5684:
5678:
5672:
5666:
5660:
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5568:
5563:
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5548:
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5526:
5525:
5518:
5511:
5503:
5497:
5496:
5474:
5473:External links
5471:
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3990:
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3129:
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3100:
3097:
3094:
3091:
3088:
3085:
3081:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3052:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3023:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2978:
2977:
2974:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2945:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2916:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2887:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2858:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2829:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2800:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2771:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2742:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2690:
2689:
2678:
2675:
2670:
2666:
2662:
2657:
2653:
2649:
2644:
2640:
2636:
2631:
2627:
2623:
2619:
2616:
2613:
2609:
2606:
2601:
2597:
2593:
2588:
2584:
2580:
2575:
2571:
2567:
2562:
2558:
2554:
2550:
2547:
2544:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2488:
2484:
2480:
2475:
2471:
2467:
2462:
2458:
2454:
2449:
2445:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2388:
2385:
2372:
2367:
2360:
2355:
2352:
2327:
2324:
2321:
2318:
2315:
2311:
2308:
2304:
2301:
2298:
2267:
2264:
2261:
2258:
2255:
2252:
2248:
2245:
2241:
2238:
2235:
2215:
2212:
2209:
2206:
2186:
2183:
2180:
2177:
2174:
2171:
2167:
2164:
2160:
2157:
2144:
2143:
2140:
2137:
2134:
2130:
2129:
2126:
2123:
2120:
2116:
2115:
2112:
2109:
2106:
2102:
2101:
2098:
2095:
2092:
2088:
2087:
2084:
2081:
2078:
2074:
2073:
2070:
2067:
2064:
2060:
2059:
2056:
2053:
2050:
2046:
2045:
2042:
2039:
2036:
2032:
2031:
2028:
2025:
2022:
2011:
2008:
2004:
2003:
1992:
1989:
1986:
1983:
1980:
1976:
1973:
1969:
1966:
1963:
1960:
1957:
1953:
1950:
1946:
1943:
1940:
1937:
1934:
1930:
1927:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1876:
1872:
1866:
1862:
1856:
1852:
1846:
1842:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1783:
1779:
1756:
1752:
1748:
1743:
1739:
1735:
1730:
1726:
1708:
1707:
1704:
1701:
1698:
1694:
1693:
1690:
1687:
1684:
1680:
1679:
1676:
1673:
1670:
1666:
1665:
1662:
1659:
1656:
1652:
1651:
1648:
1645:
1642:
1638:
1637:
1634:
1631:
1628:
1624:
1623:
1620:
1617:
1614:
1610:
1609:
1606:
1603:
1600:
1596:
1595:
1592:
1589:
1586:
1552:
1549:
1530:
1526:
1522:
1518:
1515:
1511:
1508:
1487:
1484:
1480:
1477:
1473:
1470:
1466:
1462:
1459:
1455:
1443:
1428:
1425:
1421:
1418:
1414:
1410:
1407:
1386:
1382:
1378:
1374:
1370:
1350:
1345:
1341:
1337:
1320:
1317:
1245:
1241:
1237:
1234:
1231:
1226:
1222:
1200:
1197:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1153:
1150:
1146:
1142:
1139:
1135:
1132:
1129:
1126:
1123:
1120:
1116:
1113:
1109:
1106:
1103:
1099:
1096:
1092:
1089:
1086:
1083:
1080:
1077:
1073:
1070:
1066:
1062:
1059:
1055:
1052:
1049:
1044:
1040:
1036:
1031:
1027:
1023:
1018:
1014:
1010:
1005:
1001:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
948:
944:
923:
918:
914:
910:
905:
901:
897:
892:
888:
870:
869:
866:
863:
860:
856:
855:
852:
849:
846:
842:
841:
838:
835:
832:
828:
827:
824:
821:
818:
814:
813:
810:
807:
804:
800:
799:
796:
793:
790:
786:
785:
782:
779:
776:
772:
771:
768:
765:
762:
758:
757:
754:
751:
748:
714:
711:
696:
692:
679:
678: = 6
675:
662:
659:
655:
652:
632:
627:
623:
619:
616:
613:
608:
605:
602:
598:
592:
587:
584:
581:
577:
555:
551:
547:
524:
520:
502:
499:
435:
404:
400:
396:
393:
390:
385:
381:
359:
343:
340:
291:De Morgan dual
251:
250:
235:
234:
224:
223:
138:
136:
129:
122:
121:
83:
81:
74:
69:
43:
42:
40:
33:
21:Canonical form
15:
9:
6:
4:
3:
2:
6137:
6126:
6123:
6121:
6118:
6116:
6113:
6112:
6110:
6095:
6092:
6090:
6087:
6085:
6082:
6081:
6079:
6077:
6073:
6067:
6064:
6062:
6059:
6057:
6054:
6053:
6051:
6049:
6045:
6040:
6030:
6027:
6025:
6022:
6020:
6017:
6014:
6010:
6007:
6005:
6002:
6000:
5997:
5995:
5992:
5991:
5989:
5987:
5983:
5978:
5971:
5966:
5964:
5959:
5957:
5952:
5951:
5948:
5936:
5933:
5931:
5930:Metastability
5928:
5927:
5925:
5923:Design issues
5921:
5915:
5912:
5908:
5905:
5903:
5900:
5899:
5898:
5897:Digital video
5895:
5893:
5890:
5888:
5885:
5881:
5878:
5877:
5876:
5875:Digital audio
5873:
5869:
5866:
5865:
5864:
5861:
5860:
5858:
5854:
5846:
5843:
5842:
5841:
5838:
5836:
5833:
5831:
5828:
5826:
5823:
5819:
5816:
5814:
5811:
5810:
5809:
5806:
5804:
5801:
5797:
5794:
5792:
5789:
5788:
5787:
5784:
5782:
5779:
5778:
5776:
5774:
5770:
5764:
5761:
5759:
5756:
5754:
5751:
5747:
5744:
5743:
5742:
5739:
5737:
5734:
5732:
5729:
5727:
5724:
5722:
5719:
5717:
5714:
5713:
5711:
5707:
5700:
5697:
5694:
5691:
5688:
5685:
5682:
5679:
5676:
5673:
5670:
5667:
5664:
5661:
5658:
5655:
5652:
5649:
5647:
5644:
5641:
5638:
5635:
5632:
5629:
5626:
5624:
5621:
5618:
5615:
5612:
5609:
5607:
5604:
5602:
5599:
5597:
5594:
5592:
5589:
5587:
5584:
5582:
5579:
5577:
5574:
5572:
5569:
5567:
5564:
5562:
5559:
5557:
5554:
5552:
5549:
5547:
5544:
5543:
5541:
5539:
5535:
5531:
5524:
5519:
5517:
5512:
5510:
5505:
5504:
5501:
5493:
5489:
5485:
5481:
5480:Boole, George
5477:
5476:
5467:
5463:
5461:0-471-39882-9
5457:
5453:
5448:
5445:
5441:
5437:
5433:
5428:
5425:
5423:
5419:
5412:
5410:0-486-43946-1
5406:
5402:
5397:
5396:
5384:
5373:
5369:
5363:
5355:
5353:1-56347-185-X
5349:
5345:
5338:
5330:
5324:
5320:
5319:
5311:
5303:
5297:
5293:
5292:
5284:
5280:
5270:
5267:
5266:
5260:
5258:
5253:
5251:
5250:
5245:
5244:
5238:
5230:
5228:
5224:
5220:
5216:
5212:
5208:
5204:
5200:
5196:
5191:
5189:
5185:
5181:
5177:
5173:
5169:
5165:
5161:
5157:
5153:
5149:
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:
5040:
5037:
5034:
5033:
5022:
5019:
5017:
5013:
5009:
5005:
5001:
4997:
4993:
4989:
4985:
4981:
4977:
4973:
4968:
4959:
4955:
4939:
4935:
4926:
4912:
4909:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4887:
4883:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4858:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4829:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4800:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4775:
4772:
4771:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4742:
4738:
4735:
4732:
4729:
4726:
4723:
4720:
4717:
4714:
4713:
4709:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4684:
4643:
4640:
4639:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4606:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4577:
4573:
4570:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4548:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4519:
4515:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4490:
4486:
4483:
4480:
4477:
4474:
4471:
4468:
4465:
4462:
4461:
4457:
4454:
4451:
4448:
4445:
4442:
4439:
4436:
4433:
4432:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4403:
4362:
4359:
4358:
4355:
4336:
4328:
4324:
4320:
4315:
4311:
4307:
4302:
4298:
4294:
4289:
4285:
4267:
4259:
4255:
4251:
4246:
4242:
4238:
4233:
4229:
4225:
4220:
4216:
4198:
4192:
4189:
4186:
4183:
4180:
4177:
4170:
4167:
4163:
4156:
4155:
4154:
4152:
4148:
4144:
4140:
4136:
4131:
4127:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4082:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4053:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4024:
4020:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3995:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3966:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3937:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3908:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3879:
3838:
3835:
3834:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3801:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3772:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3743:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3714:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3685:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3656:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3627:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3598:
3557:
3554:
3553:
3550:
3531:
3523:
3519:
3515:
3510:
3506:
3502:
3497:
3493:
3489:
3484:
3480:
3462:
3454:
3450:
3446:
3441:
3437:
3433:
3428:
3424:
3420:
3415:
3411:
3393:
3387:
3384:
3381:
3378:
3375:
3372:
3366:
3363:
3356:
3355:
3354:
3338:
3334:
3328:
3324:
3318:
3314:
3308:
3304:
3300:
3294:
3291:
3288:
3285:
3282:
3279:
3273:
3270:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3227:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3198:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3169:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3140:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3111:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3082:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3053:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3024:
2983:
2980:
2979:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2946:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2917:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2888:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2859:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2830:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2801:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2772:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2743:
2702:
2699:
2698:
2695:
2676:
2668:
2664:
2660:
2655:
2651:
2647:
2642:
2638:
2634:
2629:
2625:
2607:
2599:
2595:
2591:
2586:
2582:
2578:
2573:
2569:
2565:
2560:
2556:
2538:
2532:
2529:
2526:
2523:
2520:
2517:
2511:
2504:
2503:
2502:
2486:
2482:
2478:
2473:
2469:
2465:
2460:
2456:
2452:
2447:
2443:
2439:
2433:
2430:
2427:
2424:
2421:
2418:
2412:
2403:
2400:
2398:
2394:
2384:
2380:
2376:
2364:
2351:
2349:
2345:
2341:
2325:
2322:
2316:
2313:
2309:
2306:
2299:
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2287:
2285:
2281:
2265:
2262:
2259:
2256:
2253:
2250:
2246:
2243:
2239:
2236:
2233:
2213:
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2207:
2204:
2184:
2181:
2178:
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2172:
2169:
2165:
2162:
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2155:
2141:
2138:
2135:
2132:
2131:
2127:
2124:
2121:
2118:
2117:
2113:
2110:
2107:
2104:
2103:
2099:
2096:
2093:
2090:
2089:
2085:
2082:
2079:
2076:
2075:
2071:
2068:
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2061:
2057:
2054:
2051:
2048:
2047:
2043:
2040:
2037:
2034:
2033:
2019:
2016:
2007:
1987:
1984:
1981:
1978:
1974:
1971:
1967:
1958:
1955:
1951:
1948:
1944:
1941:
1938:
1928:
1925:
1921:
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1915:
1912:
1909:
1900:
1897:
1894:
1891:
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1879:
1874:
1870:
1864:
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1854:
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1844:
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1485:
1478:
1475:
1471:
1468:
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1460:
1457:
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1426:
1423:
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1416:
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1127:
1121:
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1104:
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1075:
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1021:
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1012:
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1003:
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850:
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780:
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694:
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590:
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579:
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549:
545:
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518:
508:
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496:
492:
488:
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480:
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465:
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421:
402:
398:
394:
391:
388:
383:
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349:
339:
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333:
329:
324:
322:
318:
314:
310:
306:
302:
301:
296:
292:
288:
284:
280:
276:
272:
271:
266:
262:
258:
247:
244:
232:
229:This article
228:
227:
220:
217:
209:
198:
195:
191:
188:
184:
181:
177:
174:
170:
167: –
166:
162:
161:Find sources:
155:
151:
145:
144:
139:This article
137:
133:
128:
127:
118:
115:
107:
104:February 2009
97:
91:
89:
82:
73:
72:
67:
65:
58:
57:
52:
51:
46:
41:
32:
31:
26:
22:
6023:
5977:Normal forms
5856:Applications
5491:
5487:
5465:
5451:
5443:
5431:
5421:
5417:
5414:
5400:
5382:
5376:. Retrieved
5371:
5362:
5343:
5337:
5317:
5310:
5290:
5283:
5254:
5247:
5241:
5239:
5236:
5226:
5222:
5218:
5214:
5210:
5206:
5202:
5198:
5194:
5192:
5187:
5183:
5179:
5175:
5171:
5167:
5166:′ and
5163:
5159:
5155:
5151:
5147:
5145:
5015:
5011:
5007:
5003:
4999:
4995:
4991:
4987:
4983:
4979:
4975:
4971:
4969:
4965:
4956:
4924:
4922:
4681:co'(ci,x,y)
4400:co'(ci,x,y)
4354:Truth tables
4353:
4150:
4146:
4142:
4138:
4134:
4129:
4125:
4122:
3549:Truth tables
3548:
3262:
2694:Truth tables
2693:
2404:
2401:
2396:
2392:
2390:
2381:
2377:
2365:
2357:
2340:Karnaugh map
2288:
2283:
2147:
2013:
2005:
1713:
1711:
1576:
1572:
1568:
1564:
1554:
1440:
1322:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1272:
1269:exactly once
1268:
1260:
1202:
875:
873:
738:
734:
730:
726:
716:
682:and denoted
506:
504:
494:
490:
486:
482:
478:
474:
470:
466:
463:
455:
451:
448:exactly once
447:
424:product term
419:
345:
325:
316:
312:
308:
304:
298:
294:
286:
282:
278:
274:
268:
264:
254:
239:
230:
212:
206:October 2010
203:
193:
186:
179:
172:
160:
148:Please help
143:verification
140:
110:
101:
85:
61:
54:
48:
47:Please help
44:
6029:Horn clause
5586:Memory cell
5374:. Rich Katz
5249:disjunction
5243:conjunction
3876:co(ci,x,y)
3595:co(ci,x,y)
1594:co(ci,x,y)
1557:truth table
719:truth table
460:logical AND
25:Normal form
6109:Categories
5935:Runt pulse
5907:television
5601:Logic gate
5546:Transistor
5538:Components
5378:2021-06-19
5275:References
5030:Bottom-up
5024:Variables
3021:u(ci,x,y)
2740:u(ci,x,y)
1295:′ +
1281:logical OR
1211:variables
756:u(ci,x,y)
717:Given the
370:variables
176:newspapers
50:improve it
5791:Placement
5581:Flip-flop
5561:Capacitor
5372:klabs.org
5162:′,
5027:Top-down
2030:f(a,b,c)
1233:…
1199:Maxterms
615:
604:−
576:∑
392:…
342:Minterms
56:talk page
5979:in logic
5556:Inductor
5551:Resistor
5482:(1848).
5440:65-17394
5346:. AIAA.
5263:See also
5213:′(
5112:x XOR y
4171:′
2310:′
2247:′
2166:′
1975:′
1952:′
1929:′
1543:, using
1517:′
1486:′
1472:′
1461:′
1420:′
1409:′
1381:′
1152:′
1141:′
1115:′
1098:′
1072:′
1061:′
661:′
554:′
5796:Routing
5630:(3D IC)
5246:(AND),
5104:4@1,4@2
5101:M or m
2280:minimal
1273:maxterm
1261:maxterm
452:minterm
420:minterm
293:is the
190:scholar
5773:Design
5709:Theory
5695:(ASIC)
5689:(FPOA)
5683:(FPGA)
5677:(CPLD)
5642:(EPLD)
5458:
5438:
5407:
5350:
5325:
5298:
4972:u = ci
2393:ci, x,
2342:. The
1203:For a
346:For a
326:Other
259:, any
192:
185:
178:
171:
163:
6120:Logic
6076:Other
5880:radio
5701:(TPU)
5671:(GAL)
5665:(PAL)
5659:(PLD)
5653:(PLA)
5636:(ECL)
5619:(HIC)
5123:Misc
5008:u, co
4974:XOR (
1289:false
612:value
493:= 0,
489:= 1,
422:is a
307:, or
277:, or
197:JSTOR
183:books
5613:(IC)
5456:ISBN
5436:LCCN
5405:ISBN
5348:ISBN
5323:ISBN
5296:ISBN
5217:XOR
5209:) +
5205:XOR
5134:Max
5129:5@1
5107:N/A
5090:ci'
5010:and
4996:y ci
4988:ci x
4978:XOR
4678:NOR
4397:AND
4128:and
3873:NOR
3592:AND
3018:NOR
2737:AND
2395:and
1769:and
1571:and
1307:and
1259:, a
934:and
733:and
477:and
418:, a
300:CCNF
270:CDNF
169:news
5492:III
5126:N/A
5115:N/A
5079:ci
5068:y'
5046:x'
4992:x y
4645:ci
4364:ci
3840:ci
3559:ci
2985:ci
2704:ci
1585:ci
1207:of
747:ci
350:of
317:POS
315:or
313:PoS
303:),
287:SOP
285:or
283:SoP
273:),
255:In
152:by
23:or
6111::
5490:.
5486:.
5464:.
5442:.
5413:.
5381:.
5370:.
5229:.
5211:ci
5199:ci
5197:=
5188:co
5184:ci
5180:ci
5172:co
5160:ci
5156:co
5152:co
5148:co
5140:3
5118:2
5096:3
5085:1
5074:3
5063:1
5057:y
5052:3
5041:1
5035:x
5012:co
5004:co
4994:+
4990:+
4986:=
4984:co
4925:co
4913:0
4884:0
4855:0
4826:1
4797:0
4768:1
4739:1
4710:1
4651:y
4648:x
4632:0
4603:0
4574:0
4545:1
4516:0
4487:1
4458:1
4429:1
4370:y
4367:x
4151:co
4147:co
4143:co
4139:co
4108:1
4079:1
4050:1
4021:0
3992:1
3963:0
3934:0
3905:0
3846:y
3843:x
3827:1
3798:1
3769:1
3740:0
3711:1
3682:0
3653:0
3624:0
3565:y
3562:x
3253:1
3224:0
3195:0
3166:1
3137:0
3108:1
3079:1
3050:0
2991:y
2988:x
2972:1
2943:0
2914:0
2885:1
2856:0
2827:1
2798:1
2769:0
2710:y
2707:x
2375:.
2373:cc
2368:cc
2361:cc
2142:1
2128:0
2114:0
2100:0
2086:1
2072:0
2058:0
2044:0
2027:c
2024:b
2021:a
1714:co
1706:1
1692:1
1678:1
1664:0
1650:1
1636:0
1622:0
1608:0
1591:y
1588:x
1579::
1577:ci
1565:co
1547:.
1299:+
868:1
854:0
840:0
826:1
812:0
798:1
784:1
770:0
753:y
750:x
741::
739:ci
709:.
680:10
469:'
59:.
6015:)
6011:(
5969:e
5962:t
5955:v
5522:e
5515:t
5508:v
5422:N
5418:N
5356:.
5331:.
5304:.
5219:y
5215:x
5207:y
5203:x
5201:(
5195:u
5176:u
5168:y
5164:x
5137:4
5093:4
5082:4
5071:4
5060:4
5049:4
5038:4
5016:u
5000:u
4980:y
4976:x
4940:7
4936:m
4910:0
4907:1
4904:0
4901:0
4898:0
4895:1
4892:1
4889:1
4881:0
4878:0
4875:1
4872:0
4869:0
4866:0
4863:1
4860:1
4852:0
4849:0
4846:0
4843:1
4840:0
4837:1
4834:0
4831:1
4823:1
4820:0
4817:0
4814:0
4811:0
4808:0
4805:0
4802:1
4794:0
4791:0
4788:0
4785:0
4782:1
4779:1
4776:1
4773:0
4765:1
4762:0
4759:0
4756:0
4753:0
4750:0
4747:1
4744:0
4736:1
4733:0
4730:0
4727:0
4724:0
4721:1
4718:0
4715:0
4707:1
4704:0
4701:0
4698:0
4695:0
4692:0
4689:0
4686:0
4674:7
4672:m
4668:6
4666:m
4662:5
4660:m
4656:3
4654:m
4629:0
4626:0
4623:1
4620:1
4617:1
4614:1
4611:1
4608:1
4600:0
4597:1
4594:0
4591:1
4588:1
4585:0
4582:1
4579:1
4571:0
4568:1
4565:1
4562:0
4559:1
4556:1
4553:0
4550:1
4542:1
4539:1
4536:1
4533:1
4530:1
4527:0
4524:0
4521:1
4513:0
4510:1
4507:1
4504:1
4501:0
4498:1
4495:1
4492:0
4484:1
4481:1
4478:1
4475:1
4472:1
4469:0
4466:1
4463:0
4455:1
4452:1
4449:1
4446:1
4443:1
4440:1
4437:0
4434:0
4426:1
4423:1
4420:1
4417:1
4414:1
4411:0
4408:0
4405:0
4393:7
4391:M
4387:6
4385:M
4381:5
4379:M
4375:3
4373:M
4337:.
4334:)
4329:7
4325:m
4321:,
4316:6
4312:m
4308:,
4303:5
4299:m
4295:,
4290:3
4286:m
4282:(
4278:R
4275:O
4272:N
4268:=
4265:)
4260:7
4256:M
4252:,
4247:6
4243:M
4239:,
4234:5
4230:M
4226:,
4221:3
4217:M
4213:(
4209:D
4206:N
4203:A
4199:=
4196:)
4193:y
4190:,
4187:x
4184:,
4181:i
4178:c
4175:(
4168:o
4164:c
4135:u
4130:y
4126:x
4105:1
4102:0
4099:0
4096:0
4093:0
4090:1
4087:1
4084:1
4076:1
4073:0
4070:0
4067:0
4064:0
4061:0
4058:1
4055:1
4047:1
4044:0
4041:0
4038:0
4035:0
4032:1
4029:0
4026:1
4018:0
4015:1
4012:0
4009:0
4006:0
4003:0
4000:0
3997:1
3989:1
3986:0
3983:0
3980:0
3977:0
3974:1
3971:1
3968:0
3960:0
3957:0
3954:1
3951:0
3948:0
3945:0
3942:1
3939:0
3931:0
3928:0
3925:0
3922:1
3919:0
3916:1
3913:0
3910:0
3902:0
3899:0
3896:0
3893:0
3890:1
3887:0
3884:0
3881:0
3869:4
3867:m
3863:2
3861:m
3857:1
3855:m
3851:0
3849:m
3824:1
3821:1
3818:1
3815:1
3812:1
3809:1
3806:1
3803:1
3795:1
3792:1
3789:1
3786:1
3783:1
3780:0
3777:1
3774:1
3766:1
3763:1
3760:1
3757:1
3754:1
3751:1
3748:0
3745:1
3737:0
3734:0
3731:1
3728:1
3725:1
3722:0
3719:0
3716:1
3708:1
3705:1
3702:1
3699:1
3696:1
3693:1
3690:1
3687:0
3679:0
3676:1
3673:0
3670:1
3667:1
3664:0
3661:1
3658:0
3650:0
3647:1
3644:1
3641:0
3638:1
3635:1
3632:0
3629:0
3621:0
3618:1
3615:1
3612:1
3609:0
3606:0
3603:0
3600:0
3588:4
3586:M
3582:2
3580:M
3576:1
3574:M
3570:0
3568:M
3532:.
3529:)
3524:4
3520:m
3516:,
3511:2
3507:m
3503:,
3498:1
3494:m
3490:,
3485:0
3481:m
3477:(
3473:R
3470:O
3467:N
3463:=
3460:)
3455:4
3451:M
3447:,
3442:2
3438:M
3434:,
3429:1
3425:M
3421:,
3416:0
3412:M
3408:(
3404:D
3401:N
3398:A
3394:=
3391:)
3388:y
3385:,
3382:x
3379:,
3376:i
3373:c
3370:(
3367:o
3364:c
3339:4
3335:M
3329:2
3325:M
3319:1
3315:M
3309:0
3305:M
3301:=
3298:)
3295:y
3292:,
3289:x
3286:,
3283:i
3280:c
3277:(
3274:o
3271:c
3250:1
3247:0
3244:0
3241:0
3238:0
3235:1
3232:1
3229:1
3221:0
3218:1
3215:0
3212:0
3209:0
3206:0
3203:1
3200:1
3192:0
3189:0
3186:1
3183:0
3180:0
3177:1
3174:0
3171:1
3163:1
3160:0
3157:0
3154:0
3151:0
3148:0
3145:0
3142:1
3134:0
3131:0
3128:0
3125:1
3122:0
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3035:1
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3000:m
2996:0
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2751:0
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2715:0
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2677:.
2674:)
2669:6
2665:m
2661:,
2656:5
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2648:,
2643:3
2639:m
2635:,
2630:0
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2622:(
2618:R
2615:O
2612:N
2608:=
2605:)
2600:6
2596:M
2592:,
2587:5
2583:M
2579:,
2574:3
2570:M
2566:,
2561:0
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2553:(
2549:D
2546:N
2543:A
2539:=
2536:)
2533:y
2530:,
2527:x
2524:,
2521:i
2518:c
2515:(
2512:u
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2483:m
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2448:1
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2440:=
2437:)
2434:y
2431:,
2428:x
2425:,
2422:i
2419:c
2416:(
2413:u
2397:y
2326:c
2323:b
2320:)
2317:a
2314:+
2307:a
2303:(
2300:=
2297:f
2266:c
2263:b
2260:a
2257:+
2254:c
2251:b
2244:a
2240:=
2237:c
2234:b
2214:c
2211:b
2208:=
2205:f
2185:c
2182:b
2179:a
2176:+
2173:c
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2163:a
2159:=
2156:f
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2133:1
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2119:1
2111:1
2108:0
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2097:0
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2055:1
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2049:0
2041:0
2038:0
2035:0
1991:)
1988:y
1985:+
1982:x
1979:+
1972:i
1968:c
1965:(
1962:)
1959:y
1956:+
1949:x
1945:+
1942:i
1939:c
1936:(
1933:)
1926:y
1922:+
1919:x
1916:+
1913:i
1910:c
1907:(
1904:)
1901:y
1898:+
1895:x
1892:+
1889:i
1886:c
1883:(
1880:=
1875:4
1871:M
1865:2
1861:M
1855:1
1851:M
1845:0
1841:M
1837:=
1834:)
1831:y
1828:,
1825:x
1822:,
1819:i
1816:c
1813:(
1810:o
1807:c
1782:4
1778:M
1755:2
1751:M
1747:,
1742:1
1738:M
1734:,
1729:0
1725:M
1703:1
1700:1
1697:1
1689:0
1686:1
1683:1
1675:1
1672:0
1669:1
1661:0
1658:0
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1647:1
1644:1
1641:0
1633:0
1630:1
1627:0
1619:1
1616:0
1613:0
1605:0
1602:0
1599:0
1573:y
1569:x
1529:6
1525:m
1521:=
1514:c
1510:b
1507:a
1483:)
1479:c
1476:+
1469:b
1465:+
1458:a
1454:(
1444:6
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1427:c
1424:+
1417:b
1413:+
1406:a
1385:)
1377:i
1373:x
1369:(
1349:)
1344:i
1340:x
1336:(
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1301:c
1297:b
1293:a
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1265:n
1244:n
1240:x
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1230:,
1225:1
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1209:n
1183:)
1180:y
1177:,
1174:x
1171:,
1168:i
1165:c
1162:(
1159:+
1156:)
1149:y
1145:,
1138:x
1134:,
1131:i
1128:c
1125:(
1122:+
1119:)
1112:y
1108:,
1105:x
1102:,
1095:i
1091:c
1088:(
1085:+
1082:)
1079:y
1076:,
1069:x
1065:,
1058:i
1054:c
1051:(
1048:=
1043:7
1039:m
1035:+
1030:4
1026:m
1022:+
1017:2
1013:m
1009:+
1004:1
1000:m
996:=
993:)
990:y
987:,
984:x
981:,
978:i
975:c
972:(
969:u
947:7
943:m
922:,
917:4
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909:,
904:2
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891:1
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862:1
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792:1
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781:1
778:0
775:0
767:0
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761:0
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691:m
676:2
658:c
654:b
651:a
631:)
626:i
622:x
618:(
607:1
601:i
597:2
591:n
586:1
583:=
580:i
550:i
546:x
523:i
519:x
507:n
495:c
491:b
487:a
483:b
479:c
475:a
471:c
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464:a
456:n
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403:n
399:x
395:,
389:,
384:1
380:x
358:n
311:(
297:(
281:(
267:(
246:)
240:(
233:.
219:)
213:(
208:)
204:(
194:·
187:·
180:·
173:·
146:.
117:)
111:(
106:)
102:(
92:.
66:)
62:(
27:.
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