1765:
6810:
1882:
3418:
1183:
27:
2985:
4346:
3413:{\displaystyle {\begin{array}{lcl}f^{*}(00)&=&(f^{*})(00)&=&f(00)\\f^{*}(01)&=&(\partial _{1}f^{*})(00)&=&-f(00)+f(01)\\f^{*}(10)&=&(\partial _{2}f^{*})(00)&=&-f(00)+f(10)\\f^{*}(11)&=&(\partial _{1}\partial _{2}f^{*})(00)&=&f(00)-f(01)-f(10)+f(11)\\\end{array}}}
2341:
of a
Boolean function is a k-ary integer-valued function giving the correlation between a certain set of changes in the inputs and the function output. For a given bit vector it is related to the Hamming weight of the derivative in that direction. The maximal autocorrelation coefficient (in absolute
4163:
2707:
2224:
of the function to one of the arguments is a (k-1)-ary function that is true when the output of the function is sensitive to the chosen input variable; it is the XOR of the two corresponding cofactors. A derivative and a cofactor are used in a
2211:), which are the (k-1)-ary functions resulting from fixing one of the arguments (to zero or one). The general (k-ary) functions obtained by imposing a linear constraint on a set of inputs (a linear subspace) are known as
3966:
2967:). When all operands are independent (share no variables) a function's polynomial form can be found by repeatedly applying the polynomials of the operators in a Boolean formula. When the coefficients are calculated
3538:
3668:
1203:
2316:. The Walsh coefficient of a single bit vector is a measure for the correlation of that bit with the output of the Boolean function. The maximum (in absolute value) Walsh coefficient is known as the
4916:
3961:
2488:
2082:: for every combination of argument values, changing an argument from false to true can only cause the output to switch from false to true and not from true to false. A function is said to be
2534:
2843:
1521:
4185:) of the function (see above). The polynomial also has the same statistical interpretation as the one in the standard Boolean domain, except that it now deals with the expected values
3808:
1333:
4278:
546:
973:
488:
401:
613:
274:
932:
572:
248:
674:
2525:
1750:
755:
520:
360:
308:
188:
1085:
1033:
639:
890:
427:
1059:
334:
2733:
1600:
999:
222:
3879:
159:
110:
84:
2881:
1451:
1365:
864:
812:
453:
2965:
1547:
778:
705:
1651:
3729:
3568:
2907:
1419:
838:
2930:
728:
133:
3588:
2393:; it describes the correlation between different linear combinations of input and output bits. The set of autocorrelation coefficients of the components is the
1698:
1674:
1620:
1389:
2270:
Boolean functions are equal to their Möbius transform, i.e. their truth table (minterm) values equal their algebraic (monomial) coefficients. There are 2^2^(
1210:
4627:
Tarannikov, Yuriy; Korolev, Peter; Botev, Anton (2001). "Autocorrelation
Coefficients and Correlation Immunity of Boolean Functions". In Boyd, Colin (ed.).
4905:
5189:
2346:. If all autocorrelation coefficients are 0 (i.e. the derivatives are balanced) for a certain number of bits then the function is said to satisfy the
4572:
3431:
3604:
5864:
2229:. The concept can be generalized as a k-ary derivative in the direction dx, obtained as the difference (XOR) of the function at x and x + dx.
2320:
of the function. The highest number of bits (order) for which all Walsh coefficients are 0 (i.e. the subfunctions are balanced) is known as
5947:
5088:
2092:: for each variable, flipping the value of the variable either always makes a difference in the truth value or never makes a difference (a
4158:{\displaystyle g^{*}(x)=\sum _{a\in {\{-1,1\}}^{n}}g(a)\prod _{i:a_{i}=-1}{\frac {1-x_{i}}{2}}\prod _{i:a_{i}=1}{\frac {1+x_{i}}{2}}}
2738:
4833:
6261:
6419:
5048:
4806:
4678:
4646:
4536:
5207:
4935:"The Extended Autocorrelation and Boomerang Tables and Links Between Nonlinearity Properties of Vectorial Boolean Functions"
4592:
6274:
5597:
3888:
2011:
1196:
2421:
6279:
6269:
6006:
5859:
5212:
2361:
The Walsh coefficients of a
Boolean function and its autocorrelation coefficients are related by the equivalent of the
5203:
6415:
4983:
2039:
5757:
6512:
6256:
5081:
4941:
1456:
6849:
5817:
5510:
4293:
2234:
5251:
4855:
1239:
and result assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}). Alternative names are
6773:
6475:
6238:
6233:
6058:
5479:
5163:
4370:
3734:
2381:) individually, or more thoroughly, by looking at the set of all linear functions of output bits, known as its
5038:
1275:
6768:
6551:
6468:
6181:
6112:
5989:
5231:
5002:
4673:, Encyclopedia of Mathematics and its Applications, Cambridge: Cambridge University Press, pp. 257â397,
4188:
4166:
3882:
2982:
Direct expressions for the coefficients of the polynomial can be derived by taking an appropriate derivative:
2848:
2059:
2043:
525:
2188:
attempts to classify
Boolean functions with respect to the size or depth of circuits that can compute them.
2079:
945:
458:
373:
6693:
6519:
6205:
5839:
5438:
2197:
1186:
585:
253:
4881:
Fast
Software Encryption: Second International Workshop. Leuven, Belgium, 14-16 December 1994, Proceedings
6571:
6566:
6176:
5915:
5844:
5173:
5074:
4997:
2702:{\displaystyle f^{*}(x)=\sum _{a\in {\{0,1\}}^{n}}f(a)\prod _{i:a_{i}=1}x_{i}\prod _{i:a_{i}=0}(1-x_{i})}
2362:
2292:
of a
Boolean function is a k-ary integer-valued function giving the coefficients of a decomposition into
903:
551:
4561:
227:
6839:
6500:
6090:
5484:
5452:
5143:
4879:
Daemen, Joan; Govaerts, René; Vandewalle, Joos (1994). "Correlation matrices". In
Preneel, Bart (ed.).
2355:
2099:
2016:
644:
4992:
4799:
Proceedings of the 19th
International Conference on Theory and Application of Cryptographic Techniques
2501:
6790:
6739:
6636:
6134:
6095:
5572:
5217:
4631:. Lecture Notes in Computer Science. Vol. 2248. Berlin, Heidelberg: Springer. pp. 460â479.
2226:
2123:
1703:
733:
5246:
499:
339:
287:
164:
6834:
6631:
6561:
6100:
5952:
5935:
5658:
5138:
4701:"Mobius transforms, coincident Boolean functions and non-coincidence property of Boolean functions"
4663:
2251:
2157:
1064:
1012:
618:
869:
6463:
6440:
6401:
6287:
6228:
5874:
5794:
5638:
5582:
5195:
4460:
4365:
4360:
4323:
4312:
2168:
1984:
1970:
1944:
1938:
1909:
1133:
406:
31:
1038:
313:
6753:
6480:
6458:
6425:
6318:
6164:
6149:
6122:
6073:
5957:
5892:
5717:
5683:
5678:
5552:
5383:
5360:
3819:
2712:
2495:
2263:
1571:
1236:
1232:
1103:
978:
201:
3849:
138:
89:
63:
6683:
6536:
6328:
6046:
5782:
5688:
5547:
5532:
5413:
5388:
3598:
3425:
2972:
2247:
2179:
1956:
1950:
1925:
1794:
1677:
1424:
1338:
843:
791:
432:
194:
4795:"Propagation characteristics and correlation-immunity of highly nonlinear boolean functions"
2935:
1526:
760:
687:
6844:
6656:
6618:
6495:
6299:
6139:
6063:
6041:
5869:
5827:
5726:
5693:
5557:
5345:
5256:
4331:
3591:
2976:
2401:(DDT) which lists the correlations between differences in input and output bits (see also:
2365:, which states that the autocorrelation and the power spectrum are a Walsh transform pair.
2329:
2325:
2243:
2143:
1994:
1960:
1932:
1629:
1161:
280:
3695:
8:
6785:
6676:
6661:
6641:
6598:
6485:
6435:
6361:
6306:
6243:
6036:
6031:
5979:
5747:
5736:
5408:
5308:
5236:
5227:
5223:
5158:
5153:
4380:
3834:. The polynomial form of a Boolean function can also be used as its natural extension to
3543:
2886:
2528:
2113:
2109:
1822:
1812:
1753:
1398:
1171:
817:
784:
56:
3421:
2912:
710:
115:
6814:
6583:
6546:
6531:
6524:
6507:
6311:
6293:
6159:
6085:
6068:
6021:
5834:
5743:
5577:
5562:
5522:
5474:
5459:
5447:
5403:
5378:
5148:
5097:
4856:"S-Boxes and Their Algebraic Representations â Sage 9.2 Reference Manual: Cryptography"
4775:
4728:
4385:
3573:
2968:
2220:
2185:
2105:
2089:
2029:
1802:
1783:
1683:
1659:
1605:
1374:
1166:
45:
5767:
6809:
6749:
6556:
6366:
6356:
6248:
6129:
5964:
5940:
5721:
5705:
5610:
5587:
5464:
5433:
5398:
5293:
5128:
5044:
4979:
4826:
4802:
4767:
4720:
4674:
4642:
4532:
4505:
4436:
4351:
2990:
2305:
2255:
2208:
2073:
1870:
1108:
4779:
1764:
6763:
6758:
6651:
6608:
6430:
6391:
6386:
6371:
6197:
6154:
6051:
5849:
5799:
5335:
5023:
4971:
4884:
4793:
Canteaut, Anne; Carlet, Claude; Charpin, Pascale; Fontaine, Caroline (2000-05-14).
4759:
4732:
4712:
4632:
4497:
4316:
4297:
1902:
Marquand diagram: truth table values arranged in a two-dimensional grid (used in a
1862:
1266:
1244:
1005:
4411:
2150:
th order: if the output is uncorrelated with all (linear) combinations of at most
6744:
6734:
6688:
6671:
6626:
6588:
6490:
6410:
6217:
6144:
6117:
6105:
6011:
5925:
5899:
5854:
5822:
5623:
5425:
5368:
5318:
5283:
5241:
4963:
4883:. Lecture Notes in Computer Science. Vol. 1008. Springer. pp. 275â285.
4794:
4716:
4524:
4375:
4281:
4170:
3831:
3827:
2337:
2293:
2288:
2167:
if it can be used to create (by composition) any arbitrary
Boolean function (see
2093:
2086:
in a certain variable if it is monotone with respect to changes in that variable.
2024:
1998:
1886:
1262:
1113:
20:
6729:
6708:
6666:
6646:
6541:
6396:
5994:
5984:
5974:
5969:
5903:
5777:
5653:
5542:
5537:
5515:
5116:
4700:
4485:
2297:
2131:
2083:
1842:
1777:
1368:
1249:
1123:
366:
4763:
1881:
6828:
6703:
6381:
5888:
5673:
5663:
5633:
5618:
5288:
4975:
4889:
4771:
4724:
4637:
4509:
4501:
2351:
2137:
1138:
4747:
4671:
Boolean Models and
Methods in Mathematics, Computer Science, and Engineering
1805:- which receives one input and returns true when that input is false ("not")
6603:
6450:
6351:
6343:
6223:
6171:
6080:
6016:
5999:
5930:
5789:
5648:
5350:
5133:
4308:
2047:
1915:
1903:
1832:
1562:
938:
5010:
JankoviÄ, Dragan; StankoviÄ, Radomir S.; Moraga, Claudio (November 2003).
2160:: if evaluation of the function always requires the value of all arguments
1835:- true when one of its inputs is true and the other is false ("not equal")
6713:
6593:
5772:
5762:
5709:
5393:
5313:
5298:
5178:
5123:
5028:
5011:
4906:"Chapter 5: Propagation and Correlation - Annex to AES Proposal Rijndael"
3835:
2491:
2397:, related by a Walsh transform of the components to the more widely used
2140:: its derivatives are all balanced (the autocorrelation spectrum is zero)
2127:
1918:, depicting the truth table values as a colouring of regions of the plane
1896:
1852:
1773:
1623:
1224:
1118:
578:
35:
4934:
1935:, an arbitrary mix of AND and ORs of the arguments and their complements
1899:: explicitly listing its value for all possible values of the arguments
5643:
5498:
5469:
5275:
4390:
4301:
2020:
1787:
1156:
4177:(XOR is multiplication). This polynomial form thus corresponds to the
3810:
gives the probability of a positive outcome when the Boolean function
6795:
6698:
5751:
5668:
5628:
5592:
5528:
5340:
5330:
5303:
5066:
4165:
Using the symmetric Boolean domain simplifies certain aspects of the
3690:
1858:
1838:
896:
26:
1845:- true when it is not the case that all inputs are true ("not both")
6780:
6578:
6026:
5731:
5325:
4174:
2301:
2117:
1988:
1974:
1848:
1828:
1808:
1798:
680:
5012:"Arithmetic expressions optimisation using dual polarity property"
2038:
In order to optimize electronic circuits, Boolean formulas can be
6376:
5168:
4746:
Nitaj, Abderrahmane; Susilo, Willy; Tonien, Joseph (2017-10-01).
1953:, a standardized formula which uniquely identifies the function:
1818:
2300:), analogous to the decomposition of real-valued functions into
2178:
of a function is the order of the highest order monomial in its
4664:"Boolean Functions for Cryptography and Error-Correcting Codes"
3533:{\displaystyle f^{*}(m)=\sum _{a\subseteq m}(-1)^{|a|+|m|}f(a)}
1912:, listing the truth table values at the bottom of a binary tree
4699:
Pieprzyk, Josef; Wang, Huaxiong; Zhang, Xian-Mo (2011-05-01).
2527:, constructed by summing the truth table values multiplied by
2385:. The set of Walsh transforms of the components is known as a
1987:, an AND of ORs each containing every argument or complement (
1973:, an OR of ANDs each containing every argument or complement (
5920:
5266:
5111:
4792:
4593:"Boolean functions â Sage 9.2 Reference Manual: Cryptography"
2402:
2076:: Is always true or always false regardless of its arguments.
1558:
1392:
1258:
4330:(voting games); this notion is applied to solve problems in
1963:, as a XOR of ANDs of the arguments (no complements allowed)
4801:. EUROCRYPT'00. Bruges, Belgium: Springer-Verlag: 507â522.
3663:{\displaystyle {\hat {f}}(m)=\bigoplus _{a\subseteq m}f(a)}
2350:
to that order; if they are all zero then the function is a
2250:), as a function of the monomial exponent vectors. It is a
4300:, where they are implemented in electronic circuits using
3881:, with false ("0") mapping to 1 and true ("1") to -1 (see
2242:) of a Boolean function is the set of coefficients of its
2102:: the value does not depend on the order of its arguments.
1892:
A Boolean function may be specified in a variety of ways:
2328:
to that order. The Walsh coefficients play a key role in
2134:
of the function is the number of ones in the truth table.
1947:, as an AND of ORs of the arguments and their complements
1941:, as an OR of ANDs of the arguments and their complements
4878:
5009:
4968:
Boolean Functions: Theory, Algorithms, and Applications
1756:
if and only if they express the same Boolean function.
4626:
4169:, since negation corresponds to multiplying by -1 and
2709:
For example, the extension of the binary XOR function
2354:. The autocorrelation coefficients play a key role in
4827:"A Tutorial on Linear and Differential Cryptanalysis"
4531:, GBR: John Wiley and Sons Ltd., pp. 1727â1731,
4191:
3969:
3956:{\displaystyle g(x):\{-1,1\}^{n}\rightarrow \{-1,1\}}
3891:
3852:
3737:
3698:
3670:
In both cases, the sum is taken over all bit-vectors
3607:
3576:
3546:
3434:
2988:
2938:
2915:
2889:
2851:
2741:
2715:
2537:
2504:
2424:
2069:
A Boolean function can have a variety of properties:
1706:
1686:
1662:
1632:
1608:
1574:
1529:
1459:
1427:
1401:
1377:
1341:
1278:
1067:
1041:
1015:
981:
948:
906:
872:
846:
820:
794:
763:
736:
713:
690:
647:
621:
588:
554:
528:
502:
461:
435:
409:
376:
342:
316:
290:
256:
230:
204:
167:
141:
118:
92:
66:
4341:
4307:
The properties of Boolean functions are critical in
4292:
Boolean functions play a basic role in questions of
2377:
Boolean functions by considering their output bits (
2254:
transform. It can be calculated efficiently using a
3689:When the domain is restricted to the n-dimensional
2483:{\displaystyle f(x):\{0,1\}^{n}\rightarrow \{0,1\}}
2007:Boolean formulas can also be displayed as a graph:
1855:- true when none of the inputs are true ("neither")
4272:
4157:
3955:
3873:
3802:
3723:
3662:
3582:
3562:
3532:
3412:
2959:
2924:
2901:
2875:
2837:
2727:
2701:
2519:
2482:
1744:
1692:
1668:
1645:
1614:
1594:
1541:
1515:
1445:
1413:
1383:
1359:
1327:
1079:
1053:
1027:
993:
967:
926:
884:
858:
832:
806:
772:
749:
722:
699:
668:
633:
607:
566:
540:
514:
482:
447:
421:
395:
354:
328:
302:
268:
242:
216:
182:
153:
127:
104:
78:
4745:
4698:
1869:An example of a more complicated function is the
6826:
4412:"Boolean function - Encyclopedia of Mathematics"
2130:contains an equal number of zeros and ones. The
2838:{\displaystyle 0(1-x)(1-y)+1x(1-y)+1(1-x)y+0xy}
2490:can be uniquely extended (interpolated) to the
4562:"Vectorial Boolean Functions for Cryptography"
4326:theory, monotone Boolean functions are called
1865:- true when both inputs are the same ("equal")
5082:
4705:International Journal of Computer Mathematics
3841:
2368:
1782:The rudimentary symmetric Boolean functions (
1204:
4752:Journal of Applied Mathematics and Computing
4020:
4005:
3950:
3935:
3923:
3907:
3868:
3853:
2585:
2573:
2477:
2465:
2453:
2440:
2373:These concepts can be extended naturally to
1676:-ary Boolean function can be expressed as a
1622:arguments; equal to the number of different
1516:{\displaystyle f:\{0,1\}^{k}\to \{0,1\}^{m}}
1504:
1491:
1479:
1466:
1453:. A Boolean function with multiple outputs,
1440:
1428:
1354:
1342:
1322:
1310:
1298:
1285:
5057:
4961:
3822:) variables, with individual probabilities
2196:A Boolean function may be decomposed using
5274:
5089:
5075:
1211:
1197:
5037:Arnold, Bradford Henry (1 January 2011).
5027:
5016:Serbian Journal of Electrical Engineering
4888:
4748:"Dirichlet product for boolean functions"
4636:
4522:
3803:{\displaystyle f^{*}(x):^{n}\rightarrow }
2507:
2281:
16:Function returning one of only two values
4490:IRE Transactions on Electronic Computers
4486:"Switching Functions of Three Variables"
4296:as well as the design of processors for
2413:
2120:with a single instance of each variable.
1880:
1825:- true when any input is true ("either")
1815:- true when all inputs are true ("both")
1763:
1421:, the function is a constant element of
1328:{\displaystyle f:\{0,1\}^{k}\to \{0,1\}}
25:
4629:Advances in Cryptology â ASIACRYPT 2001
4273:{\displaystyle E(X)=P(X=1)-P(X=-1)\in }
2408:
1261:. Boolean functions are the subject of
541:{\displaystyle A\not \Leftrightarrow B}
6827:
5096:
5036:
4932:
4903:
4661:
4483:
3846:Often, the Boolean domain is taken as
968:{\displaystyle A{\underline {\lor }}B}
483:{\displaystyle {\overline {A\cdot B}}}
396:{\displaystyle A{\overline {\land }}B}
5070:
4850:
4848:
4846:
4820:
4818:
4434:
3826:. A special case of this fact is the
3570:denotes the weight of the bit vector
1928:using rudimentary Boolean functions:
1752:, and two propositional formulas are
1391:is a non-negative integer called the
608:{\displaystyle A{\overline {\lor }}B}
269:{\displaystyle A\leftrightharpoons B}
5058:Mano, M. M.; Ciletti, M. D. (2013),
4622:
4620:
4618:
4616:
4614:
4612:
4587:
4585:
4555:
4553:
2191:
2012:Propositional directed acyclic graph
1885:A Boolean function represented as a
1768:The sixteen binary Boolean functions
4523:McCluskey, Edward J. (2003-01-01),
3885:). The polynomial corresponding to
1395:of the function. In the case where
927:{\displaystyle A\ {\text{XNOR}}\ B}
567:{\displaystyle A\nleftrightarrow B}
13:
4955:
4933:Nyberg, Kaisa (December 1, 2019).
4843:
4815:
4559:
3311:
3301:
3199:
3097:
2883:Some other examples are negation (
1272:A Boolean function takes the form
691:
243:{\displaystyle A\Leftrightarrow B}
174:
171:
145:
14:
6861:
4609:
4582:
4550:
3682:form a subset of the one bits of
2324:, and the function is said to be
1876:
1602:different Boolean functions with
669:{\displaystyle {\overline {A+B}}}
6808:
4947:from the original on 2020-11-02.
4922:from the original on 2018-07-23.
4839:from the original on 2017-05-17.
4824:
4578:from the original on 2016-01-17.
4529:Encyclopedia of Computer Science
4344:
4311:, particularly in the design of
2520:{\displaystyle \mathbb {R} ^{n}}
1182:
1181:
4926:
4897:
4872:
4786:
4739:
4692:
4484:Davies, D. W. (December 1957).
4287:
4181:(in this context also known as
1745:{\displaystyle x_{1},...,x_{k}}
750:{\displaystyle {\overline {A}}}
4970:, Cambridge University Press,
4655:
4516:
4477:
4453:
4428:
4404:
4267:
4252:
4246:
4231:
4222:
4210:
4201:
4195:
4041:
4035:
3986:
3980:
3932:
3901:
3895:
3797:
3785:
3782:
3773:
3760:
3754:
3748:
3712:
3699:
3657:
3651:
3626:
3620:
3614:
3590:. Taken modulo 2, this is the
3556:
3548:
3527:
3521:
3512:
3504:
3496:
3488:
3483:
3473:
3451:
3445:
3403:
3397:
3388:
3382:
3373:
3367:
3358:
3352:
3339:
3333:
3330:
3297:
3287:
3281:
3264:
3258:
3249:
3243:
3227:
3221:
3218:
3195:
3185:
3179:
3162:
3156:
3147:
3141:
3125:
3119:
3116:
3093:
3083:
3077:
3060:
3054:
3041:
3035:
3032:
3019:
3009:
3003:
2817:
2805:
2796:
2784:
2772:
2760:
2757:
2745:
2696:
2677:
2606:
2600:
2554:
2548:
2462:
2434:
2428:
1873:(of an odd number of inputs).
1488:
1307:
1071:
1019:
625:
515:{\displaystyle A\not \equiv B}
413:
355:{\displaystyle A\rightarrow B}
346:
303:{\displaystyle A\Rightarrow B}
294:
260:
234:
183:{\displaystyle A\&\&B}
1:
6769:History of mathematical logic
4904:Daemen, Joan (10 June 1998).
4397:
3883:Analysis of Boolean functions
2399:Difference Distribution Table
2064:
2060:Analysis of Boolean functions
1080:{\displaystyle A\leftarrow B}
1028:{\displaystyle A\Leftarrow B}
634:{\displaystyle A\downarrow B}
38:of a ternary Boolean function
6694:Primitive recursive function
4717:10.1080/00207160.2010.509428
2274:â1) coincident functions of
885:{\displaystyle A\parallel B}
742:
661:
597:
475:
385:
7:
4998:Encyclopedia of Mathematics
4337:
2053:
1759:
1243:, used especially in older
422:{\displaystyle A\uparrow B}
10:
6866:
5758:SchröderâBernstein theorem
5485:Monadic predicate calculus
5144:Foundations of mathematics
5022:(71â80, number 1): 71â80.
3842:On the symmetric hypercube
2387:Linear Approximation Table
2369:Linear approximation table
2356:differential cryptanalysis
2057:
1771:
1054:{\displaystyle A\subset B}
329:{\displaystyle A\supset B}
18:
6804:
6791:Philosophy of mathematics
6740:Automated theorem proving
6722:
6617:
6449:
6342:
6194:
5911:
5887:
5865:Von NeumannâBernaysâGödel
5810:
5704:
5608:
5506:
5497:
5424:
5359:
5265:
5187:
5104:
5040:Logic and Boolean Algebra
4764:10.1007/s12190-016-1037-4
3678:, i.e. the "one" bits of
2728:{\displaystyle x\oplus y}
2200:in positive and negative
2198:Boole's expansion theorem
2044:QuineâMcCluskey algorithm
1595:{\displaystyle 2^{2^{k}}}
994:{\displaystyle A\oplus B}
217:{\displaystyle A\equiv B}
4976:10.1017/CBO9780511852008
4890:10.1007/3-540-60590-8_21
4638:10.1007/3-540-45682-1_27
4502:10.1109/TEC.1957.5222038
4313:symmetric key algorithms
3874:{\displaystyle \{-1,1\}}
3420:this generalizes as the
2163:A Boolean function is a
2108:: Can be expressed with
2032:, using only AND and NOT
154:{\displaystyle A\&B}
105:{\displaystyle A\cdot B}
79:{\displaystyle A\land B}
19:Not to be confused with
6441:Self-verifying theories
6262:Tarski's axiomatization
5213:Tarski's undefinability
5208:incompleteness theorems
5043:. Courier Corporation.
4662:Carlet, Claude (2010),
4366:Boolean-valued function
4361:Pseudo-Boolean function
2876:{\displaystyle x+y-2xy}
2363:WienerâKhinchin theorem
2342:value) is known as the
2169:functional completeness
1985:conjunctive normal form
1971:disjunctive normal form
1945:Conjunctive normal form
1939:Disjunctive normal form
1910:Binary decision diagram
1446:{\displaystyle \{0,1\}}
1360:{\displaystyle \{0,1\}}
1134:Functional completeness
859:{\displaystyle A\mid B}
807:{\displaystyle A\lor B}
448:{\displaystyle A\mid B}
32:binary decision diagram
6850:Programming constructs
6815:Mathematics portal
6426:Proof of impossibility
6074:propositional variable
5384:Propositional calculus
4416:encyclopediaofmath.org
4371:Boolean algebra topics
4274:
4159:
3957:
3875:
3804:
3725:
3664:
3584:
3564:
3534:
3414:
2961:
2960:{\displaystyle x+y-xy}
2926:
2903:
2877:
2839:
2729:
2703:
2521:
2496:multilinear polynomial
2484:
2282:Cryptographic analysis
2264:Fast Fourier Transform
2240:Boole-Möbius transform
1889:
1769:
1746:
1694:
1670:
1647:
1616:
1596:
1543:
1542:{\displaystyle m>1}
1517:
1447:
1415:
1385:
1361:
1329:
1104:Propositional calculus
1081:
1055:
1029:
995:
969:
928:
886:
860:
834:
808:
774:
773:{\displaystyle \sim A}
751:
724:
701:
700:{\displaystyle \neg A}
670:
635:
609:
568:
542:
516:
484:
449:
423:
397:
356:
330:
304:
270:
244:
218:
184:
155:
129:
106:
80:
39:
6684:Kolmogorov complexity
6637:Computably enumerable
6537:Model complete theory
6329:Principia Mathematica
5389:Propositional formula
5218:BanachâTarski paradox
4465:TheFreeDictionary.com
4441:mathworld.wolfram.com
4275:
4160:
3958:
3876:
3805:
3726:
3665:
3599:algebraic normal form
3585:
3565:
3535:
3426:partially ordered set
3415:
2973:algebraic normal form
2962:
2927:
2904:
2878:
2840:
2730:
2704:
2529:indicator polynomials
2522:
2485:
2418:Any Boolean function
2414:On the unit hypercube
2395:autocorrelation table
2348:propagation criterion
2262:"), analogous to the
2260:Fast Möbius Transform
2248:algebraic normal form
2227:ReedâMuller expansion
2180:algebraic normal form
1957:Algebraic normal form
1951:Canonical normal form
1926:propositional formula
1884:
1833:exclusive disjunction
1767:
1747:
1695:
1678:propositional formula
1671:
1648:
1646:{\displaystyle 2^{k}}
1617:
1597:
1557:Boolean function (an
1544:
1518:
1448:
1416:
1386:
1362:
1330:
1162:Programming languages
1082:
1056:
1030:
996:
970:
929:
887:
861:
835:
809:
775:
752:
725:
702:
671:
636:
610:
569:
543:
517:
485:
450:
424:
398:
357:
331:
305:
271:
245:
219:
185:
156:
130:
107:
81:
29:
6632:ChurchâTuring thesis
6619:Computability theory
5828:continuum hypothesis
5346:Square of opposition
5204:Gödel's completeness
5029:10.2298/SJEE0301071J
4461:"switching function"
4332:social choice theory
4189:
3967:
3889:
3850:
3818:independent random (
3735:
3724:{\displaystyle ^{n}}
3696:
3605:
3574:
3544:
3432:
2986:
2977:Zhegalkin polynomial
2936:
2913:
2887:
2849:
2739:
2713:
2535:
2502:
2422:
2409:Real polynomial form
2330:linear cryptanalysis
2308:. Its square is the
1997:, the OR of all the
1995:Blake canonical form
1961:Zhegalkin polynomial
1933:Negation normal form
1924:Algebraically, as a
1754:logically equivalent
1704:
1684:
1660:
1630:
1606:
1572:
1527:
1457:
1425:
1399:
1375:
1339:
1276:
1065:
1039:
1013:
979:
946:
904:
870:
844:
818:
792:
761:
734:
711:
688:
645:
619:
586:
552:
526:
500:
459:
433:
407:
374:
340:
314:
288:
254:
228:
202:
165:
139:
116:
90:
64:
6786:Mathematical object
6677:P versus NP problem
6642:Computable function
6436:Reverse mathematics
6362:Logical consequence
6239:primitive recursive
6234:elementary function
6007:Free/bound variable
5860:TarskiâGrothendieck
5379:Logical connectives
5309:Logical equivalence
5159:Logical consequence
4569:University of Paris
4435:Weisstein, Eric W.
4381:Decision tree model
3563:{\displaystyle |a|}
2902:{\displaystyle 1-x}
2256:butterfly algorithm
1784:logical connectives
1414:{\displaystyle k=0}
1172:Philosophy of logic
833:{\displaystyle A+B}
46:Logical connectives
6584:Transfer principle
6547:Semantics of logic
6532:Categorical theory
6508:Non-standard model
6022:Logical connective
5149:Information theory
5098:Mathematical logic
4993:"Boolean function"
4525:"Switching theory"
4437:"Boolean Function"
4386:Indicator function
4270:
4155:
4129:
4075:
4031:
3953:
3871:
3800:
3721:
3660:
3647:
3580:
3560:
3530:
3472:
3410:
3408:
2957:
2925:{\displaystyle xy}
2922:
2899:
2873:
2835:
2725:
2699:
2676:
2637:
2596:
2517:
2480:
2391:correlation matrix
2344:absolute indicator
2326:correlation immune
2221:Boolean derivative
2186:Circuit complexity
2144:Correlation immune
2030:And-inverter graph
1890:
1770:
1742:
1690:
1666:
1643:
1612:
1592:
1539:
1513:
1443:
1411:
1381:
1357:
1325:
1241:switching function
1167:Mathematical logic
1077:
1051:
1025:
991:
965:
960:
924:
882:
856:
830:
804:
770:
747:
723:{\displaystyle -A}
720:
697:
666:
631:
605:
564:
538:
512:
480:
445:
419:
393:
352:
326:
300:
266:
240:
214:
180:
151:
128:{\displaystyle AB}
125:
102:
76:
40:
6840:Binary arithmetic
6822:
6821:
6754:Abstract category
6557:Theories of truth
6367:Rule of inference
6357:Natural deduction
6338:
6337:
5883:
5882:
5588:Cartesian product
5493:
5492:
5399:Many-valued logic
5374:Boolean functions
5257:Russell's paradox
5232:diagonal argument
5129:First-order logic
5050:978-0-486-48385-6
4808:978-3-540-67517-4
4680:978-0-521-84752-0
4648:978-3-540-45682-7
4538:978-0-470-86412-8
4352:Philosophy portal
4298:digital computers
4294:complexity theory
4284:for an example).
4183:Fourier transform
4153:
4101:
4099:
4044:
3992:
3963:is then given by:
3731:, the polynomial
3632:
3617:
3583:{\displaystyle a}
3457:
2648:
2609:
2560:
2306:Fourier transform
2209:Shannon expansion
2192:Derived functions
1871:majority function
1693:{\displaystyle k}
1669:{\displaystyle k}
1615:{\displaystyle k}
1384:{\displaystyle k}
1255:logical function)
1221:
1220:
1090:
1089:
953:
920:
916:
912:
745:
664:
600:
478:
388:
6857:
6813:
6812:
6764:History of logic
6759:Category of sets
6652:Decision problem
6431:Ordinal analysis
6372:Sequent calculus
6270:Boolean algebras
6210:
6209:
6184:
6155:logical/constant
5909:
5908:
5895:
5818:ZermeloâFraenkel
5569:Set operations:
5504:
5503:
5441:
5272:
5271:
5252:LöwenheimâSkolem
5139:Formal semantics
5091:
5084:
5077:
5068:
5067:
5063:
5054:
5033:
5031:
5006:
4988:
4964:Hammer, Peter L.
4949:
4948:
4946:
4939:
4930:
4924:
4923:
4921:
4910:
4901:
4895:
4894:
4892:
4876:
4870:
4869:
4867:
4866:
4860:doc.sagemath.org
4852:
4841:
4840:
4838:
4831:
4825:Heys, Howard M.
4822:
4813:
4812:
4790:
4784:
4783:
4743:
4737:
4736:
4711:(7): 1398â1416.
4696:
4690:
4689:
4688:
4687:
4668:
4659:
4653:
4652:
4640:
4624:
4607:
4606:
4604:
4603:
4597:doc.sagemath.org
4589:
4580:
4579:
4577:
4566:
4560:Carlet, Claude.
4557:
4548:
4547:
4546:
4545:
4520:
4514:
4513:
4481:
4475:
4474:
4472:
4471:
4457:
4451:
4450:
4448:
4447:
4432:
4426:
4425:
4423:
4422:
4408:
4354:
4349:
4348:
4347:
4324:cooperative game
4317:substitution box
4279:
4277:
4276:
4271:
4171:linear functions
4164:
4162:
4161:
4156:
4154:
4149:
4148:
4147:
4131:
4128:
4121:
4120:
4100:
4095:
4094:
4093:
4077:
4074:
4064:
4063:
4030:
4029:
4028:
4023:
3979:
3978:
3962:
3960:
3959:
3954:
3931:
3930:
3880:
3878:
3877:
3872:
3832:parity functions
3809:
3807:
3806:
3801:
3781:
3780:
3747:
3746:
3730:
3728:
3727:
3722:
3720:
3719:
3669:
3667:
3666:
3661:
3646:
3619:
3618:
3610:
3594:Möbius transform
3589:
3587:
3586:
3581:
3569:
3567:
3566:
3561:
3559:
3551:
3539:
3537:
3536:
3531:
3517:
3516:
3515:
3507:
3499:
3491:
3471:
3444:
3443:
3422:Möbius inversion
3419:
3417:
3416:
3411:
3409:
3329:
3328:
3319:
3318:
3309:
3308:
3280:
3279:
3217:
3216:
3207:
3206:
3178:
3177:
3115:
3114:
3105:
3104:
3076:
3075:
3031:
3030:
3002:
3001:
2971:one obtains the
2966:
2964:
2963:
2958:
2931:
2929:
2928:
2923:
2908:
2906:
2905:
2900:
2882:
2880:
2879:
2874:
2844:
2842:
2841:
2836:
2734:
2732:
2731:
2726:
2708:
2706:
2705:
2700:
2695:
2694:
2675:
2668:
2667:
2647:
2646:
2636:
2629:
2628:
2595:
2594:
2593:
2588:
2547:
2546:
2526:
2524:
2523:
2518:
2516:
2515:
2510:
2489:
2487:
2486:
2481:
2461:
2460:
2294:linear functions
2235:Möbius transform
2176:algebraic degree
2165:Sheffer function
1999:prime implicants
1863:logical equality
1751:
1749:
1748:
1743:
1741:
1740:
1716:
1715:
1699:
1697:
1696:
1691:
1675:
1673:
1672:
1667:
1652:
1650:
1649:
1644:
1642:
1641:
1621:
1619:
1618:
1613:
1601:
1599:
1598:
1593:
1591:
1590:
1589:
1588:
1548:
1546:
1545:
1540:
1522:
1520:
1519:
1514:
1512:
1511:
1487:
1486:
1452:
1450:
1449:
1444:
1420:
1418:
1417:
1412:
1390:
1388:
1387:
1382:
1367:is known as the
1366:
1364:
1363:
1358:
1334:
1332:
1331:
1326:
1306:
1305:
1267:switching theory
1247:literature, and
1245:computer science
1229:Boolean function
1213:
1206:
1199:
1185:
1184:
1129:Boolean function
1095:Related concepts
1086:
1084:
1083:
1078:
1060:
1058:
1057:
1052:
1034:
1032:
1031:
1026:
1000:
998:
997:
992:
974:
972:
971:
966:
961:
933:
931:
930:
925:
918:
917:
914:
910:
891:
889:
888:
883:
865:
863:
862:
857:
839:
837:
836:
831:
813:
811:
810:
805:
779:
777:
776:
771:
756:
754:
753:
748:
746:
738:
729:
727:
726:
721:
706:
704:
703:
698:
675:
673:
672:
667:
665:
660:
649:
640:
638:
637:
632:
614:
612:
611:
606:
601:
593:
573:
571:
570:
565:
547:
545:
544:
539:
521:
519:
518:
513:
489:
487:
486:
481:
479:
474:
463:
454:
452:
451:
446:
428:
426:
425:
420:
402:
400:
399:
394:
389:
381:
361:
359:
358:
353:
335:
333:
332:
327:
309:
307:
306:
301:
275:
273:
272:
267:
249:
247:
246:
241:
223:
221:
220:
215:
189:
187:
186:
181:
160:
158:
157:
152:
134:
132:
131:
126:
111:
109:
108:
103:
85:
83:
82:
77:
53:
52:
42:
41:
6865:
6864:
6860:
6859:
6858:
6856:
6855:
6854:
6835:Boolean algebra
6825:
6824:
6823:
6818:
6807:
6800:
6745:Category theory
6735:Algebraic logic
6718:
6689:Lambda calculus
6627:Church encoding
6613:
6589:Truth predicate
6445:
6411:Complete theory
6334:
6203:
6199:
6195:
6190:
6182:
5902: and
5898:
5893:
5879:
5855:New Foundations
5823:axiom of choice
5806:
5768:Gödel numbering
5708: and
5700:
5604:
5489:
5439:
5420:
5369:Boolean algebra
5355:
5319:Equiconsistency
5284:Classical logic
5261:
5242:Halting problem
5230: and
5206: and
5194: and
5193:
5188:Theorems (
5183:
5100:
5095:
5051:
4991:
4986:
4958:
4956:Further reading
4953:
4952:
4944:
4937:
4931:
4927:
4919:
4908:
4902:
4898:
4877:
4873:
4864:
4862:
4854:
4853:
4844:
4836:
4829:
4823:
4816:
4809:
4791:
4787:
4744:
4740:
4697:
4693:
4685:
4683:
4681:
4666:
4660:
4656:
4649:
4625:
4610:
4601:
4599:
4591:
4590:
4583:
4575:
4564:
4558:
4551:
4543:
4541:
4539:
4521:
4517:
4482:
4478:
4469:
4467:
4459:
4458:
4454:
4445:
4443:
4433:
4429:
4420:
4418:
4410:
4409:
4405:
4400:
4395:
4376:Algebra of sets
4350:
4345:
4343:
4340:
4290:
4282:piling-up lemma
4190:
4187:
4186:
4179:Walsh transform
4143:
4139:
4132:
4130:
4116:
4112:
4105:
4089:
4085:
4078:
4076:
4059:
4055:
4048:
4024:
4004:
4003:
3996:
3974:
3970:
3968:
3965:
3964:
3926:
3922:
3890:
3887:
3886:
3851:
3848:
3847:
3844:
3828:piling-up lemma
3776:
3772:
3742:
3738:
3736:
3733:
3732:
3715:
3711:
3697:
3694:
3693:
3636:
3609:
3608:
3606:
3603:
3602:
3575:
3572:
3571:
3555:
3547:
3545:
3542:
3541:
3511:
3503:
3495:
3487:
3486:
3482:
3461:
3439:
3435:
3433:
3430:
3429:
3428:of bit vectors:
3407:
3406:
3347:
3342:
3324:
3320:
3314:
3310:
3304:
3300:
3295:
3290:
3275:
3271:
3268:
3267:
3235:
3230:
3212:
3208:
3202:
3198:
3193:
3188:
3173:
3169:
3166:
3165:
3133:
3128:
3110:
3106:
3100:
3096:
3091:
3086:
3071:
3067:
3064:
3063:
3049:
3044:
3026:
3022:
3017:
3012:
2997:
2993:
2989:
2987:
2984:
2983:
2937:
2934:
2933:
2914:
2911:
2910:
2888:
2885:
2884:
2850:
2847:
2846:
2740:
2737:
2736:
2714:
2711:
2710:
2690:
2686:
2663:
2659:
2652:
2642:
2638:
2624:
2620:
2613:
2589:
2572:
2571:
2564:
2542:
2538:
2536:
2533:
2532:
2511:
2506:
2505:
2503:
2500:
2499:
2456:
2452:
2423:
2420:
2419:
2416:
2411:
2371:
2338:autocorrelation
2298:Walsh functions
2289:Walsh transform
2284:
2194:
2094:parity function
2067:
2062:
2056:
2025:Boolean circuit
2017:Digital circuit
2001:of the function
1887:Boolean circuit
1879:
1780:
1762:
1736:
1732:
1711:
1707:
1705:
1702:
1701:
1685:
1682:
1681:
1661:
1658:
1657:
1637:
1633:
1631:
1628:
1627:
1607:
1604:
1603:
1584:
1580:
1579:
1575:
1573:
1570:
1569:
1528:
1525:
1524:
1507:
1503:
1482:
1478:
1458:
1455:
1454:
1426:
1423:
1422:
1400:
1397:
1396:
1376:
1373:
1372:
1340:
1337:
1336:
1301:
1297:
1277:
1274:
1273:
1263:Boolean algebra
1217:
1176:
1143:
1114:Boolean algebra
1109:Predicate logic
1066:
1063:
1062:
1040:
1037:
1036:
1014:
1011:
1010:
980:
977:
976:
952:
947:
944:
943:
913:
905:
902:
901:
871:
868:
867:
845:
842:
841:
819:
816:
815:
793:
790:
789:
762:
759:
758:
737:
735:
732:
731:
712:
709:
708:
689:
686:
685:
650:
648:
646:
643:
642:
620:
617:
616:
592:
587:
584:
583:
553:
550:
549:
527:
524:
523:
501:
498:
497:
464:
462:
460:
457:
456:
434:
431:
430:
408:
405:
404:
380:
375:
372:
371:
341:
338:
337:
315:
312:
311:
289:
286:
285:
255:
252:
251:
229:
226:
225:
203:
200:
199:
166:
163:
162:
140:
137:
136:
117:
114:
113:
91:
88:
87:
65:
62:
61:
24:
21:Binary function
17:
12:
11:
5:
6863:
6853:
6852:
6847:
6842:
6837:
6820:
6819:
6805:
6802:
6801:
6799:
6798:
6793:
6788:
6783:
6778:
6777:
6776:
6766:
6761:
6756:
6747:
6742:
6737:
6732:
6730:Abstract logic
6726:
6724:
6720:
6719:
6717:
6716:
6711:
6709:Turing machine
6706:
6701:
6696:
6691:
6686:
6681:
6680:
6679:
6674:
6669:
6664:
6659:
6649:
6647:Computable set
6644:
6639:
6634:
6629:
6623:
6621:
6615:
6614:
6612:
6611:
6606:
6601:
6596:
6591:
6586:
6581:
6576:
6575:
6574:
6569:
6564:
6554:
6549:
6544:
6542:Satisfiability
6539:
6534:
6529:
6528:
6527:
6517:
6516:
6515:
6505:
6504:
6503:
6498:
6493:
6488:
6483:
6473:
6472:
6471:
6466:
6459:Interpretation
6455:
6453:
6447:
6446:
6444:
6443:
6438:
6433:
6428:
6423:
6413:
6408:
6407:
6406:
6405:
6404:
6394:
6389:
6379:
6374:
6369:
6364:
6359:
6354:
6348:
6346:
6340:
6339:
6336:
6335:
6333:
6332:
6324:
6323:
6322:
6321:
6316:
6315:
6314:
6309:
6304:
6284:
6283:
6282:
6280:minimal axioms
6277:
6266:
6265:
6264:
6253:
6252:
6251:
6246:
6241:
6236:
6231:
6226:
6213:
6211:
6192:
6191:
6189:
6188:
6187:
6186:
6174:
6169:
6168:
6167:
6162:
6157:
6152:
6142:
6137:
6132:
6127:
6126:
6125:
6120:
6110:
6109:
6108:
6103:
6098:
6093:
6083:
6078:
6077:
6076:
6071:
6066:
6056:
6055:
6054:
6049:
6044:
6039:
6034:
6029:
6019:
6014:
6009:
6004:
6003:
6002:
5997:
5992:
5987:
5977:
5972:
5970:Formation rule
5967:
5962:
5961:
5960:
5955:
5945:
5944:
5943:
5933:
5928:
5923:
5918:
5912:
5906:
5889:Formal systems
5885:
5884:
5881:
5880:
5878:
5877:
5872:
5867:
5862:
5857:
5852:
5847:
5842:
5837:
5832:
5831:
5830:
5825:
5814:
5812:
5808:
5807:
5805:
5804:
5803:
5802:
5792:
5787:
5786:
5785:
5778:Large cardinal
5775:
5770:
5765:
5760:
5755:
5741:
5740:
5739:
5734:
5729:
5714:
5712:
5702:
5701:
5699:
5698:
5697:
5696:
5691:
5686:
5676:
5671:
5666:
5661:
5656:
5651:
5646:
5641:
5636:
5631:
5626:
5621:
5615:
5613:
5606:
5605:
5603:
5602:
5601:
5600:
5595:
5590:
5585:
5580:
5575:
5567:
5566:
5565:
5560:
5550:
5545:
5543:Extensionality
5540:
5538:Ordinal number
5535:
5525:
5520:
5519:
5518:
5507:
5501:
5495:
5494:
5491:
5490:
5488:
5487:
5482:
5477:
5472:
5467:
5462:
5457:
5456:
5455:
5445:
5444:
5443:
5430:
5428:
5422:
5421:
5419:
5418:
5417:
5416:
5411:
5406:
5396:
5391:
5386:
5381:
5376:
5371:
5365:
5363:
5357:
5356:
5354:
5353:
5348:
5343:
5338:
5333:
5328:
5323:
5322:
5321:
5311:
5306:
5301:
5296:
5291:
5286:
5280:
5278:
5269:
5263:
5262:
5260:
5259:
5254:
5249:
5244:
5239:
5234:
5222:Cantor's
5220:
5215:
5210:
5200:
5198:
5185:
5184:
5182:
5181:
5176:
5171:
5166:
5161:
5156:
5151:
5146:
5141:
5136:
5131:
5126:
5121:
5120:
5119:
5108:
5106:
5102:
5101:
5094:
5093:
5086:
5079:
5071:
5065:
5064:
5060:Digital Design
5055:
5049:
5034:
5007:
4989:
4984:
4957:
4954:
4951:
4950:
4925:
4896:
4871:
4842:
4814:
4807:
4785:
4758:(1): 293â312.
4738:
4691:
4679:
4654:
4647:
4608:
4581:
4549:
4537:
4515:
4496:(4): 265â275.
4476:
4452:
4427:
4402:
4401:
4399:
4396:
4394:
4393:
4388:
4383:
4378:
4373:
4368:
4363:
4357:
4356:
4355:
4339:
4336:
4289:
4286:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4152:
4146:
4142:
4138:
4135:
4127:
4124:
4119:
4115:
4111:
4108:
4104:
4098:
4092:
4088:
4084:
4081:
4073:
4070:
4067:
4062:
4058:
4054:
4051:
4047:
4043:
4040:
4037:
4034:
4027:
4022:
4019:
4016:
4013:
4010:
4007:
4002:
3999:
3995:
3991:
3988:
3985:
3982:
3977:
3973:
3952:
3949:
3946:
3943:
3940:
3937:
3934:
3929:
3925:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3870:
3867:
3864:
3861:
3858:
3855:
3843:
3840:
3814:is applied to
3799:
3796:
3793:
3790:
3787:
3784:
3779:
3775:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3745:
3741:
3718:
3714:
3710:
3707:
3704:
3701:
3659:
3656:
3653:
3650:
3645:
3642:
3639:
3635:
3631:
3628:
3625:
3622:
3616:
3613:
3579:
3558:
3554:
3550:
3529:
3526:
3523:
3520:
3514:
3510:
3506:
3502:
3498:
3494:
3490:
3485:
3481:
3478:
3475:
3470:
3467:
3464:
3460:
3456:
3453:
3450:
3447:
3442:
3438:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3346:
3343:
3341:
3338:
3335:
3332:
3327:
3323:
3317:
3313:
3307:
3303:
3299:
3296:
3294:
3291:
3289:
3286:
3283:
3278:
3274:
3270:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3234:
3231:
3229:
3226:
3223:
3220:
3215:
3211:
3205:
3201:
3197:
3194:
3192:
3189:
3187:
3184:
3181:
3176:
3172:
3168:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3132:
3129:
3127:
3124:
3121:
3118:
3113:
3109:
3103:
3099:
3095:
3092:
3090:
3087:
3085:
3082:
3079:
3074:
3070:
3066:
3065:
3062:
3059:
3056:
3053:
3050:
3048:
3045:
3043:
3040:
3037:
3034:
3029:
3025:
3021:
3018:
3016:
3013:
3011:
3008:
3005:
3000:
2996:
2992:
2991:
2956:
2953:
2950:
2947:
2944:
2941:
2921:
2918:
2898:
2895:
2892:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2724:
2721:
2718:
2698:
2693:
2689:
2685:
2682:
2679:
2674:
2671:
2666:
2662:
2658:
2655:
2651:
2645:
2641:
2635:
2632:
2627:
2623:
2619:
2616:
2612:
2608:
2605:
2602:
2599:
2592:
2587:
2584:
2581:
2578:
2575:
2570:
2567:
2563:
2559:
2556:
2553:
2550:
2545:
2541:
2514:
2509:
2479:
2476:
2473:
2470:
2467:
2464:
2459:
2455:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2415:
2412:
2410:
2407:
2370:
2367:
2314:Walsh spectrum
2310:power spectrum
2283:
2280:
2193:
2190:
2183:
2182:
2172:
2161:
2155:
2141:
2135:
2132:Hamming weight
2121:
2103:
2097:
2087:
2077:
2066:
2063:
2055:
2052:
2036:
2035:
2034:
2033:
2027:
2005:
2004:
2003:
2002:
1992:
1978:
1964:
1948:
1942:
1936:
1922:
1921:
1920:
1919:
1913:
1907:
1878:
1877:Representation
1875:
1867:
1866:
1856:
1846:
1843:Sheffer stroke
1836:
1826:
1816:
1806:
1778:Truth function
1761:
1758:
1739:
1735:
1731:
1728:
1725:
1722:
1719:
1714:
1710:
1689:
1665:
1640:
1636:
1611:
1587:
1583:
1578:
1538:
1535:
1532:
1510:
1506:
1502:
1499:
1496:
1493:
1490:
1485:
1481:
1477:
1474:
1471:
1468:
1465:
1462:
1442:
1439:
1436:
1433:
1430:
1410:
1407:
1404:
1380:
1369:Boolean domain
1356:
1353:
1350:
1347:
1344:
1324:
1321:
1318:
1315:
1312:
1309:
1304:
1300:
1296:
1293:
1290:
1287:
1284:
1281:
1250:truth function
1219:
1218:
1216:
1215:
1208:
1201:
1193:
1190:
1189:
1178:
1177:
1175:
1174:
1169:
1164:
1159:
1153:
1150:
1149:
1145:
1144:
1142:
1141:
1136:
1131:
1126:
1124:Truth function
1121:
1116:
1111:
1106:
1100:
1097:
1096:
1092:
1091:
1088:
1087:
1076:
1073:
1070:
1050:
1047:
1044:
1024:
1021:
1018:
1008:
1002:
1001:
990:
987:
984:
964:
959:
956:
951:
941:
935:
934:
923:
909:
899:
893:
892:
881:
878:
875:
855:
852:
849:
829:
826:
823:
803:
800:
797:
787:
781:
780:
769:
766:
744:
741:
719:
716:
696:
693:
683:
677:
676:
663:
659:
656:
653:
630:
627:
624:
604:
599:
596:
591:
581:
575:
574:
563:
560:
557:
537:
534:
531:
511:
508:
505:
495:
491:
490:
477:
473:
470:
467:
444:
441:
438:
418:
415:
412:
392:
387:
384:
379:
369:
363:
362:
351:
348:
345:
325:
322:
319:
299:
296:
293:
283:
277:
276:
265:
262:
259:
239:
236:
233:
213:
210:
207:
197:
191:
190:
179:
176:
173:
170:
150:
147:
144:
124:
121:
101:
98:
95:
75:
72:
69:
59:
49:
48:
15:
9:
6:
4:
3:
2:
6862:
6851:
6848:
6846:
6843:
6841:
6838:
6836:
6833:
6832:
6830:
6817:
6816:
6811:
6803:
6797:
6794:
6792:
6789:
6787:
6784:
6782:
6779:
6775:
6772:
6771:
6770:
6767:
6765:
6762:
6760:
6757:
6755:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6727:
6725:
6721:
6715:
6712:
6710:
6707:
6705:
6704:Recursive set
6702:
6700:
6697:
6695:
6692:
6690:
6687:
6685:
6682:
6678:
6675:
6673:
6670:
6668:
6665:
6663:
6660:
6658:
6655:
6654:
6653:
6650:
6648:
6645:
6643:
6640:
6638:
6635:
6633:
6630:
6628:
6625:
6624:
6622:
6620:
6616:
6610:
6607:
6605:
6602:
6600:
6597:
6595:
6592:
6590:
6587:
6585:
6582:
6580:
6577:
6573:
6570:
6568:
6565:
6563:
6560:
6559:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6526:
6523:
6522:
6521:
6518:
6514:
6513:of arithmetic
6511:
6510:
6509:
6506:
6502:
6499:
6497:
6494:
6492:
6489:
6487:
6484:
6482:
6479:
6478:
6477:
6474:
6470:
6467:
6465:
6462:
6461:
6460:
6457:
6456:
6454:
6452:
6448:
6442:
6439:
6437:
6434:
6432:
6429:
6427:
6424:
6421:
6420:from ZFC
6417:
6414:
6412:
6409:
6403:
6400:
6399:
6398:
6395:
6393:
6390:
6388:
6385:
6384:
6383:
6380:
6378:
6375:
6373:
6370:
6368:
6365:
6363:
6360:
6358:
6355:
6353:
6350:
6349:
6347:
6345:
6341:
6331:
6330:
6326:
6325:
6320:
6319:non-Euclidean
6317:
6313:
6310:
6308:
6305:
6303:
6302:
6298:
6297:
6295:
6292:
6291:
6289:
6285:
6281:
6278:
6276:
6273:
6272:
6271:
6267:
6263:
6260:
6259:
6258:
6254:
6250:
6247:
6245:
6242:
6240:
6237:
6235:
6232:
6230:
6227:
6225:
6222:
6221:
6219:
6215:
6214:
6212:
6207:
6201:
6196:Example
6193:
6185:
6180:
6179:
6178:
6175:
6173:
6170:
6166:
6163:
6161:
6158:
6156:
6153:
6151:
6148:
6147:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6124:
6121:
6119:
6116:
6115:
6114:
6111:
6107:
6104:
6102:
6099:
6097:
6094:
6092:
6089:
6088:
6087:
6084:
6082:
6079:
6075:
6072:
6070:
6067:
6065:
6062:
6061:
6060:
6057:
6053:
6050:
6048:
6045:
6043:
6040:
6038:
6035:
6033:
6030:
6028:
6025:
6024:
6023:
6020:
6018:
6015:
6013:
6010:
6008:
6005:
6001:
5998:
5996:
5993:
5991:
5988:
5986:
5983:
5982:
5981:
5978:
5976:
5973:
5971:
5968:
5966:
5963:
5959:
5956:
5954:
5953:by definition
5951:
5950:
5949:
5946:
5942:
5939:
5938:
5937:
5934:
5932:
5929:
5927:
5924:
5922:
5919:
5917:
5914:
5913:
5910:
5907:
5905:
5901:
5896:
5890:
5886:
5876:
5873:
5871:
5868:
5866:
5863:
5861:
5858:
5856:
5853:
5851:
5848:
5846:
5843:
5841:
5840:KripkeâPlatek
5838:
5836:
5833:
5829:
5826:
5824:
5821:
5820:
5819:
5816:
5815:
5813:
5809:
5801:
5798:
5797:
5796:
5793:
5791:
5788:
5784:
5781:
5780:
5779:
5776:
5774:
5771:
5769:
5766:
5764:
5761:
5759:
5756:
5753:
5749:
5745:
5742:
5738:
5735:
5733:
5730:
5728:
5725:
5724:
5723:
5719:
5716:
5715:
5713:
5711:
5707:
5703:
5695:
5692:
5690:
5687:
5685:
5684:constructible
5682:
5681:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5650:
5647:
5645:
5642:
5640:
5637:
5635:
5632:
5630:
5627:
5625:
5622:
5620:
5617:
5616:
5614:
5612:
5607:
5599:
5596:
5594:
5591:
5589:
5586:
5584:
5581:
5579:
5576:
5574:
5571:
5570:
5568:
5564:
5561:
5559:
5556:
5555:
5554:
5551:
5549:
5546:
5544:
5541:
5539:
5536:
5534:
5530:
5526:
5524:
5521:
5517:
5514:
5513:
5512:
5509:
5508:
5505:
5502:
5500:
5496:
5486:
5483:
5481:
5478:
5476:
5473:
5471:
5468:
5466:
5463:
5461:
5458:
5454:
5451:
5450:
5449:
5446:
5442:
5437:
5436:
5435:
5432:
5431:
5429:
5427:
5423:
5415:
5412:
5410:
5407:
5405:
5402:
5401:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5370:
5367:
5366:
5364:
5362:
5361:Propositional
5358:
5352:
5349:
5347:
5344:
5342:
5339:
5337:
5334:
5332:
5329:
5327:
5324:
5320:
5317:
5316:
5315:
5312:
5310:
5307:
5305:
5302:
5300:
5297:
5295:
5292:
5290:
5289:Logical truth
5287:
5285:
5282:
5281:
5279:
5277:
5273:
5270:
5268:
5264:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5238:
5235:
5233:
5229:
5225:
5221:
5219:
5216:
5214:
5211:
5209:
5205:
5202:
5201:
5199:
5197:
5191:
5186:
5180:
5177:
5175:
5172:
5170:
5167:
5165:
5162:
5160:
5157:
5155:
5152:
5150:
5147:
5145:
5142:
5140:
5137:
5135:
5132:
5130:
5127:
5125:
5122:
5118:
5115:
5114:
5113:
5110:
5109:
5107:
5103:
5099:
5092:
5087:
5085:
5080:
5078:
5073:
5072:
5069:
5061:
5056:
5052:
5046:
5042:
5041:
5035:
5030:
5025:
5021:
5017:
5013:
5008:
5004:
5000:
4999:
4994:
4990:
4987:
4985:9780511852008
4981:
4977:
4973:
4969:
4965:
4962:Crama, Yves;
4960:
4959:
4943:
4936:
4929:
4918:
4914:
4907:
4900:
4891:
4886:
4882:
4875:
4861:
4857:
4851:
4849:
4847:
4835:
4828:
4821:
4819:
4810:
4804:
4800:
4796:
4789:
4781:
4777:
4773:
4769:
4765:
4761:
4757:
4753:
4749:
4742:
4734:
4730:
4726:
4722:
4718:
4714:
4710:
4706:
4702:
4695:
4682:
4676:
4672:
4665:
4658:
4650:
4644:
4639:
4634:
4630:
4623:
4621:
4619:
4617:
4615:
4613:
4598:
4594:
4588:
4586:
4574:
4570:
4563:
4556:
4554:
4540:
4534:
4530:
4526:
4519:
4511:
4507:
4503:
4499:
4495:
4491:
4487:
4480:
4466:
4462:
4456:
4442:
4438:
4431:
4417:
4413:
4407:
4403:
4392:
4389:
4387:
4384:
4382:
4379:
4377:
4374:
4372:
4369:
4367:
4364:
4362:
4359:
4358:
4353:
4342:
4335:
4333:
4329:
4325:
4320:
4318:
4314:
4310:
4305:
4303:
4299:
4295:
4285:
4283:
4264:
4261:
4258:
4255:
4249:
4243:
4240:
4237:
4234:
4228:
4225:
4219:
4216:
4213:
4207:
4204:
4198:
4192:
4184:
4180:
4176:
4172:
4168:
4150:
4144:
4140:
4136:
4133:
4125:
4122:
4117:
4113:
4109:
4106:
4102:
4096:
4090:
4086:
4082:
4079:
4071:
4068:
4065:
4060:
4056:
4052:
4049:
4045:
4038:
4032:
4025:
4017:
4014:
4011:
4008:
4000:
3997:
3993:
3989:
3983:
3975:
3971:
3947:
3944:
3941:
3938:
3927:
3919:
3916:
3913:
3910:
3904:
3898:
3892:
3884:
3865:
3862:
3859:
3856:
3839:
3837:
3833:
3829:
3825:
3821:
3817:
3813:
3794:
3791:
3788:
3777:
3769:
3766:
3763:
3757:
3751:
3743:
3739:
3716:
3708:
3705:
3702:
3692:
3687:
3685:
3681:
3677:
3673:
3654:
3648:
3643:
3640:
3637:
3633:
3629:
3623:
3611:
3601:coefficients:
3600:
3597:, giving the
3596:
3595:
3577:
3552:
3524:
3518:
3508:
3500:
3492:
3479:
3476:
3468:
3465:
3462:
3458:
3454:
3448:
3440:
3436:
3427:
3423:
3400:
3394:
3391:
3385:
3379:
3376:
3370:
3364:
3361:
3355:
3349:
3344:
3336:
3325:
3321:
3315:
3305:
3292:
3284:
3276:
3272:
3261:
3255:
3252:
3246:
3240:
3237:
3232:
3224:
3213:
3209:
3203:
3190:
3182:
3174:
3170:
3159:
3153:
3150:
3144:
3138:
3135:
3130:
3122:
3111:
3107:
3101:
3088:
3080:
3072:
3068:
3057:
3051:
3046:
3038:
3027:
3023:
3014:
3006:
2998:
2994:
2980:
2978:
2974:
2970:
2954:
2951:
2948:
2945:
2942:
2939:
2919:
2916:
2896:
2893:
2890:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2832:
2829:
2826:
2823:
2820:
2814:
2811:
2808:
2802:
2799:
2793:
2790:
2787:
2781:
2778:
2775:
2769:
2766:
2763:
2754:
2751:
2748:
2742:
2722:
2719:
2716:
2691:
2687:
2683:
2680:
2672:
2669:
2664:
2660:
2656:
2653:
2649:
2643:
2639:
2633:
2630:
2625:
2621:
2617:
2614:
2610:
2603:
2597:
2590:
2582:
2579:
2576:
2568:
2565:
2561:
2557:
2551:
2543:
2539:
2530:
2512:
2497:
2493:
2474:
2471:
2468:
2457:
2449:
2446:
2443:
2437:
2431:
2425:
2406:
2404:
2400:
2396:
2392:
2388:
2384:
2380:
2376:
2366:
2364:
2359:
2357:
2353:
2352:bent function
2349:
2345:
2340:
2339:
2333:
2331:
2327:
2323:
2319:
2315:
2311:
2307:
2303:
2299:
2295:
2291:
2290:
2279:
2277:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2236:
2230:
2228:
2223:
2222:
2216:
2214:
2210:
2206:
2203:
2199:
2189:
2187:
2181:
2177:
2173:
2170:
2166:
2162:
2159:
2156:
2153:
2149:
2145:
2142:
2139:
2136:
2133:
2129:
2125:
2122:
2119:
2115:
2111:
2107:
2104:
2101:
2098:
2095:
2091:
2088:
2085:
2081:
2078:
2075:
2072:
2071:
2070:
2061:
2051:
2049:
2045:
2041:
2031:
2028:
2026:
2022:
2018:
2015:
2014:
2013:
2010:
2009:
2008:
2000:
1996:
1993:
1990:
1986:
1982:
1979:
1976:
1972:
1968:
1965:
1962:
1958:
1955:
1954:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1930:
1929:
1927:
1917:
1914:
1911:
1908:
1905:
1901:
1900:
1898:
1895:
1894:
1893:
1888:
1883:
1874:
1872:
1864:
1860:
1857:
1854:
1850:
1847:
1844:
1840:
1837:
1834:
1830:
1827:
1824:
1820:
1817:
1814:
1810:
1807:
1804:
1800:
1796:
1793:
1792:
1791:
1789:
1785:
1779:
1775:
1766:
1757:
1755:
1737:
1733:
1729:
1726:
1723:
1720:
1717:
1712:
1708:
1687:
1679:
1663:
1654:
1638:
1634:
1625:
1609:
1585:
1581:
1576:
1566:
1564:
1561:in symmetric
1560:
1556:
1555:vector-valued
1552:
1536:
1533:
1530:
1508:
1500:
1497:
1494:
1483:
1475:
1472:
1469:
1463:
1460:
1437:
1434:
1431:
1408:
1405:
1402:
1394:
1378:
1370:
1351:
1348:
1345:
1319:
1316:
1313:
1302:
1294:
1291:
1288:
1282:
1279:
1270:
1268:
1264:
1260:
1256:
1252:
1251:
1246:
1242:
1238:
1234:
1230:
1226:
1214:
1209:
1207:
1202:
1200:
1195:
1194:
1192:
1191:
1188:
1180:
1179:
1173:
1170:
1168:
1165:
1163:
1160:
1158:
1157:Digital logic
1155:
1154:
1152:
1151:
1147:
1146:
1140:
1139:Scope (logic)
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1101:
1099:
1098:
1094:
1093:
1074:
1068:
1048:
1045:
1042:
1022:
1016:
1009:
1007:
1004:
1003:
988:
985:
982:
962:
957:
954:
949:
942:
940:
937:
936:
921:
907:
900:
898:
895:
894:
879:
876:
873:
853:
850:
847:
827:
824:
821:
801:
798:
795:
788:
786:
783:
782:
767:
764:
739:
717:
714:
694:
684:
682:
679:
678:
657:
654:
651:
628:
622:
602:
594:
589:
582:
580:
577:
576:
561:
558:
555:
535:
532:
529:
509:
506:
503:
496:
494:nonequivalent
493:
492:
471:
468:
465:
442:
439:
436:
416:
410:
390:
382:
377:
370:
368:
365:
364:
349:
343:
323:
320:
317:
297:
291:
284:
282:
279:
278:
263:
257:
237:
231:
211:
208:
205:
198:
196:
193:
192:
177:
168:
148:
142:
122:
119:
99:
96:
93:
73:
70:
67:
60:
58:
55:
54:
51:
50:
47:
44:
43:
37:
33:
28:
22:
6806:
6604:Ultraproduct
6451:Model theory
6416:Independence
6352:Formal proof
6344:Proof theory
6327:
6300:
6257:real numbers
6229:second-order
6140:Substitution
6017:Metalanguage
5958:conservative
5931:Axiom schema
5875:Constructive
5845:MorseâKelley
5811:Set theories
5790:Aleph number
5783:inaccessible
5689:Grothendieck
5573:intersection
5460:Higher-order
5448:Second-order
5394:Truth tables
5373:
5351:Venn diagram
5134:Formal proof
5059:
5039:
5019:
5015:
4996:
4967:
4928:
4912:
4899:
4880:
4874:
4863:. Retrieved
4859:
4798:
4788:
4755:
4751:
4741:
4708:
4704:
4694:
4684:, retrieved
4670:
4657:
4628:
4600:. Retrieved
4596:
4568:
4542:, retrieved
4528:
4518:
4493:
4489:
4479:
4468:. Retrieved
4464:
4455:
4444:. Retrieved
4440:
4430:
4419:. Retrieved
4415:
4406:
4328:simple games
4327:
4321:
4309:cryptography
4306:
4291:
4288:Applications
4182:
4178:
3845:
3823:
3815:
3811:
3688:
3683:
3679:
3675:
3671:
3593:
2981:
2845:which equals
2417:
2398:
2394:
2390:
2386:
2382:
2378:
2374:
2372:
2360:
2347:
2343:
2336:
2334:
2321:
2317:
2313:
2309:
2287:
2285:
2275:
2271:
2267:
2259:
2252:self-inverse
2239:
2233:
2231:
2219:
2217:
2213:subfunctions
2212:
2204:
2201:
2195:
2184:
2175:
2164:
2151:
2147:
2068:
2048:Karnaugh map
2037:
2006:
1983:(canonical)
1980:
1969:(canonical)
1966:
1923:
1916:Venn diagram
1904:Karnaugh map
1891:
1868:
1781:
1655:
1624:truth tables
1567:
1563:cryptography
1554:
1550:
1271:
1254:
1248:
1240:
1228:
1222:
1148:Applications
1128:
6845:Logic gates
6714:Type theory
6662:undecidable
6594:Truth value
6481:equivalence
6160:non-logical
5773:Enumeration
5763:Isomorphism
5710:cardinality
5694:Von Neumann
5659:Ultrafilter
5624:Uncountable
5558:equivalence
5475:Quantifiers
5465:Fixed-point
5434:First-order
5314:Consistency
5299:Proposition
5276:Traditional
5247:Lindström's
5237:Compactness
5179:Type theory
5124:Cardinality
4302:logic gates
3836:fuzzy logic
3674:covered by
2492:real domain
2379:coordinates
2278:arguments.
2128:truth table
2114:disjunction
2110:conjunction
2021:logic gates
2019:diagram of
1897:Truth table
1853:logical nor
1823:disjunction
1813:conjunction
1788:logic gates
1774:Truth table
1225:mathematics
1119:Truth table
36:truth table
6829:Categories
6525:elementary
6218:arithmetic
6086:Quantifier
6064:functional
5936:Expression
5654:Transitive
5598:identities
5583:complement
5516:hereditary
5499:Set theory
4865:2021-05-04
4686:2021-05-17
4602:2021-05-01
4544:2021-05-03
4470:2021-05-03
4446:2021-05-03
4421:2021-05-03
4398:References
4391:Signed set
2932:) and OR (
2383:components
2322:resiliency
2268:Coincident
2244:polynomial
2065:Properties
2058:See also:
2042:using the
1803:complement
1772:See also:
1700:variables
1568:There are
1257:, used in
195:equivalent
6796:Supertask
6699:Recursion
6657:decidable
6491:saturated
6469:of models
6392:deductive
6387:axiomatic
6307:Hilbert's
6294:Euclidean
6275:canonical
6198:axiomatic
6130:Signature
6059:Predicate
5948:Extension
5870:Ackermann
5795:Operation
5674:Universal
5664:Recursive
5639:Singleton
5634:Inhabited
5619:Countable
5609:Types of
5593:power set
5563:partition
5480:Predicate
5426:Predicate
5341:Syllogism
5331:Soundness
5304:Inference
5294:Tautology
5196:paradoxes
5062:, Pearson
5003:EMS Press
4772:1865-2085
4725:0020-7160
4510:0367-9950
4256:−
4250:∈
4241:−
4226:−
4175:monomials
4103:∏
4083:−
4069:−
4046:∏
4009:−
4001:∈
3994:∑
3976:∗
3939:−
3933:→
3911:−
3857:−
3820:Bernoulli
3783:→
3744:∗
3691:hypercube
3641:⊆
3634:⨁
3615:^
3477:−
3466:⊆
3459:∑
3441:∗
3377:−
3362:−
3326:∗
3312:∂
3302:∂
3277:∗
3238:−
3214:∗
3200:∂
3175:∗
3136:−
3112:∗
3098:∂
3073:∗
3028:∗
2999:∗
2949:−
2894:−
2862:−
2812:−
2791:−
2767:−
2752:−
2720:⊕
2684:−
2650:∏
2611:∏
2569:∈
2562:∑
2544:∗
2463:→
2389:(LAT) or
2375:vectorial
2318:linearity
2302:harmonics
2205:cofactors
2154:arguments
2126:: if its
2106:Read-once
2100:Symmetric
2040:minimized
1653:entries.
1551:vectorial
1489:→
1308:→
1237:arguments
1072:←
1046:⊂
1020:⇐
986:⊕
958:_
955:∨
877:∥
851:∣
799:∨
765:∼
743:¯
715:−
692:¬
662:¯
626:↓
598:¯
595:∨
559:↮
476:¯
469:⋅
440:∣
414:↑
386:¯
383:∧
347:→
321:⊃
295:⇒
261:⇋
235:⇔
209:≡
175:&
172:&
146:&
97:⋅
71:∧
6781:Logicism
6774:timeline
6750:Concrete
6609:Validity
6579:T-schema
6572:Kripke's
6567:Tarski's
6562:semantic
6552:Strength
6501:submodel
6496:spectrum
6464:function
6312:Tarski's
6301:Elements
6288:geometry
6244:Robinson
6165:variable
6150:function
6123:spectrum
6113:Sentence
6069:variable
6012:Language
5965:Relation
5926:Automata
5916:Alphabet
5900:language
5754:-jection
5732:codomain
5718:Function
5679:Universe
5649:Infinite
5553:Relation
5336:Validity
5326:Argument
5224:theorem,
4966:(2011),
4942:Archived
4917:Archived
4834:Archived
4780:16760125
4573:Archived
4338:See also
4167:analysis
3592:Boolean
2969:modulo 2
2909:), AND (
2124:Balanced
2118:negation
2080:Monotone
2074:Constant
2054:Analysis
1989:maxterms
1975:minterms
1799:negation
1760:Examples
1335:, where
1233:function
1187:Category
1006:converse
533:⇎
507:≢
6723:Related
6520:Diagram
6418: (
6397:Hilbert
6382:Systems
6377:Theorem
6255:of the
6200:systems
5980:Formula
5975:Grammar
5891: (
5835:General
5548:Forcing
5533:Element
5453:Monadic
5228:paradox
5169:Theorem
5105:General
5005:, 2001
4733:9580510
3424:of the
2304:by the
2202:Shannon
2158:Evasive
1790:) are:
281:implies
6486:finite
6249:Skolem
6202:
6177:Theory
6145:Symbol
6135:String
6118:atomic
5995:ground
5990:closed
5985:atomic
5941:ground
5904:syntax
5800:binary
5727:domain
5644:Finite
5409:finite
5267:Logics
5226:
5174:Theory
5047:
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4805:
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4770:
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4508:
3540:where
2116:, and
2090:Linear
1656:Every
1235:whose
919:
911:
6476:Model
6224:Peano
6081:Proof
5921:Arity
5850:Naive
5737:image
5669:Fuzzy
5629:Empty
5578:union
5523:Class
5164:Model
5154:Lemma
5112:Axiom
4945:(PDF)
4938:(PDF)
4920:(PDF)
4909:(PDF)
4837:(PDF)
4830:(PDF)
4776:S2CID
4729:S2CID
4667:(PDF)
4576:(PDF)
4565:(PDF)
4315:(see
4280:(see
2494:by a
2403:S-box
2084:unate
1626:with
1559:S-box
1549:is a
1523:with
1393:arity
1259:logic
1231:is a
6599:Type
6402:list
6206:list
6183:list
6172:Term
6106:rank
6000:open
5894:list
5706:Maps
5611:sets
5470:Free
5440:list
5190:list
5117:list
5045:ISBN
4980:ISBN
4913:NIST
4803:ISBN
4768:ISSN
4721:ISSN
4675:ISBN
4643:ISBN
4533:ISBN
4506:ISSN
4494:EC-6
4173:are
3830:for
2335:The
2286:The
2238:(or
2232:The
2218:The
2174:The
2138:Bent
2023:, a
1981:Full
1967:Full
1859:XNOR
1839:NAND
1776:and
1534:>
1371:and
1265:and
1253:(or
1227:, a
915:XNOR
897:XNOR
367:NAND
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6286:of
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6216:of
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5511:Set
5024:doi
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2979:).
2498:in
2405:).
2312:or
2146:to
2046:or
1959:or
1861:or
1851:or
1849:NOR
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