505:
1433:
288:
1023:
248:
The operations on the log semiring can be defined extrinsically by mapping them to the non-negative real numbers, doing the operations there, and mapping them back. The non-negative real numbers with the usual operations of addition and multiplication form a
875:
1199:
1184:
1735:. Encyclopedia of Mathematics and Its Applications. Vol. 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot,
1689:
1631:
67:, etc. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base
905:
773:
555:, since logarithms take multiplication to addition; however, log addition depends on base. The units for usual addition and multiplication are 0 and 1; accordingly, the unit for log addition is
500:{\displaystyle {\begin{aligned}x\oplus _{b}y&=\log _{b}\left(b^{x}+b^{y}\right)\\x\otimes _{b}y&=\log _{b}\left(b^{x}\times b^{y}\right)=\log _{b}\left(b^{x+y}\right)=x+y.\end{aligned}}}
293:
693:
900:
768:
595:
553:
1475:
154:
220:, since it replaces the non-smooth maximum and minimum by a smooth operation. The log semiring also arises when working with numbers that are logarithms (measured on a
751:
719:
621:
188:
1076:
1428:{\displaystyle M_{\mathrm {lm} }(x,y):=(x\oplus y)\oslash 2=\log _{b}{\bigl (}(b^{x}+b^{y})/2{\bigr )}=\log _{b}(b^{x}+b^{y})-\log _{b}2=(x\oplus y)-\log _{b}2.}
1081:
1644:
1548:
1018:{\displaystyle {\begin{aligned}x\oplus y&=-\log \left(e^{-x}+e^{-y}\right)\\x\otimes _{b}y&=x+y.\end{aligned}}}
626:
1760:
870:{\displaystyle {\begin{aligned}x\oplus y&=\log \left(e^{x}+e^{y}\right)\\x\otimes y&=x+y.\end{aligned}}}
83:), which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a base
558:
55:
the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary
513:
1793:
1752:
1499:
with respect to log multiplication (usual addition, geometrically translation) with corresponds to the
48:
217:
1747:, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and
1441:
133:
724:
1485:
1190:
40:
698:
600:
167:
1744:
1504:
254:
1770:
8:
1788:
1748:
1500:
1055:
56:
1729:
197:
1756:
1496:
1481:
1186:
Thus log division ⊘ is well-defined, though log subtraction ⊖ is not always defined.
221:
193:
159:
121:
36:
24:
1766:
1526:
1516:
1492:
80:
233:
125:
63:
to reverse the initial exponentiation. Such operations are also known as, e.g.,
1740:
1736:
237:
1782:
200:("quantization") of the tropical semiring. Notably, the addition operation,
52:
229:
1724:
20:
1480:
A log semiring has the usual
Euclidean metric, which corresponds to the
1179:{\displaystyle x\otimes -x=\log _{b}(b^{x}\cdot b^{-x})=\log _{b}(1)=0.}
47:. That is, the operations of addition and multiplication are defined by
262:
510:
Regardless of base, log multiplication is the same as usual addition,
1521:
1041:
205:
60:
44:
1691:
is ambiguous, and is best left undefined, as is 0/0 in real numbers.
887:
The opposite convention is also common, and corresponds to the base
884:
and multiplicative unit 0; this corresponds to the max convention.
258:
250:
32:
225:
213:
209:
1189:
A mean can be defined by log addition and log division (as the
1477:
since logarithmic division corresponds to linear subtraction.
756:
More concisely, the unit log semiring can be defined for base
261:
of the operations on the probability semiring, and these are
90:
is equivalent to using a negative sign and using the inverse
98:. If not qualified, the base is conventionally taken to be
1684:{\displaystyle \infty \otimes -\infty =\infty +(-\infty )}
1626:{\displaystyle b^{-x}=\left(b^{-1}\right)^{x}=(1/b)^{x}}
1641:
Usually only one infinity is included, not both, since
1705:
1647:
1551:
1444:
1202:
1084:
1058:
903:
771:
727:
701:
629:
603:
561:
516:
291:
170:
136:
1728:
1683:
1625:
1469:
1427:
1178:
1070:
1017:
869:
745:
713:
687:
615:
589:
547:
499:
257:, so the log semiring operations can be viewed as
182:
148:
128:", "dequantization") as the base goes to infinity
1044:, since all numbers other than the additive unit
688:{\displaystyle \log _{b}0=-\log _{1/b}0=+\infty }
1780:
1319:
1275:
268:Formally, given the extended real numbers
1052:) has a multiplicative inverse, given by
721:, and the unit for log multiplication is
1723:
1711:
1491:Similarly, a log semiring has the usual
253:(there are no negatives), known as the
216:. The log semiring has applications in
1781:
1743:, Michael Waterman, Philippe Jacquet,
16:Semiring arising in tropical analysis
208:) can be viewed as a deformation of
1193:corresponding to the exponent), as
590:{\displaystyle \log _{b}0=-\infty }
59:on real numbers, and then take the
13:
1675:
1663:
1657:
1648:
1212:
1209:
682:
584:
548:{\displaystyle x\otimes _{b}y=x+y}
143:
14:
1805:
1438:This is just addition shifted by
73:for the exponent and logarithm (
196:), and thus can be viewed as a
1731:Applied combinatorics on words
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39:, obtained by considering the
1:
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1035:
243:
1470:{\displaystyle -\log _{b}2,}
1040:A log semiring is in fact a
149:{\displaystyle b\to \infty }
7:
1510:
1032:and multiplicative unit 0.
10:
1810:
1753:Cambridge University Press
894:, the minimum convention:
218:mathematical optimization
120:The log semiring has the
1532:
746:{\displaystyle \log 1=0}
230:Decibel § Addition
111:, which corresponds to
1685:
1627:
1471:
1429:
1180:
1072:
1019:
871:
753:, regardless of base.
747:
715:
714:{\displaystyle b<1}
689:
617:
616:{\displaystyle b>1}
591:
549:
501:
184:
183:{\displaystyle b\to 0}
150:
1686:
1628:
1486:positive real numbers
1472:
1430:
1191:quasi-arithmetic mean
1181:
1073:
1020:
872:
748:
716:
690:
618:
592:
550:
502:
204:(for multiple terms,
185:
151:
41:extended real numbers
1745:Wojciech Szpankowski
1645:
1549:
1505:probability semiring
1442:
1200:
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1056:
901:
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601:
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289:
255:probability semiring
168:
134:
65:logarithmic addition
57:algebraic operations
1501:logarithmic measure
1071:{\displaystyle -x,}
1028:with additive unit
880:with additive unit
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495:
180:
146:
23:, in the field of
1794:Tropical analysis
1497:invariant measure
1482:logarithmic scale
222:logarithmic scale
194:min-plus semiring
160:max-plus semiring
122:tropical semiring
117:with a negative.
37:logarithmic scale
35:structure on the
25:tropical analysis
1801:
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1517:Logarithmic mean
1493:Lebesgue measure
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157:
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147:
116:
110:
103:
97:
89:
81:logarithmic unit
78:
72:
1809:
1808:
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1803:
1802:
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1799:
1798:
1779:
1778:
1777:
1763:
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1590:
1577:
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1547:
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1540:
1535:
1513:
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1201:
1198:
1197:
1152:
1148:
1133:
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1120:
1116:
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1083:
1080:
1079:
1057:
1054:
1053:
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1029:
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292:
290:
287:
286:
282:, one defines:
276:
269:
246:
238:log-likelihoods
234:log probability
169:
166:
165:
163:
135:
132:
131:
129:
126:tropicalization
112:
105:
99:
91:
84:
79:is a choice of
74:
68:
17:
12:
11:
5:
1807:
1797:
1796:
1791:
1776:
1775:
1761:
1749:Valérie Berthé
1741:Sophie Schbath
1737:Gesine Reinert
1720:
1717:
1716:
1714:, p. 211.
1703:
1702:
1700:
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1694:
1693:
1680:
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1668:
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1555:
1537:
1536:
1534:
1531:
1530:
1529:
1524:
1519:
1512:
1509:
1495:, which is an
1466:
1463:
1460:
1455:
1451:
1447:
1436:
1435:
1424:
1421:
1416:
1412:
1408:
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1399:
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1387:
1384:
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1375:
1371:
1368:
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1359:
1355:
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1123:
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1090:
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1064:
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1037:
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1026:
1025:
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986:
982:
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1790:
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1786:
1784:
1772:
1768:
1764:
1762:0-521-84802-4
1758:
1754:
1751:. Cambridge:
1750:
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1712:Lothaire 2005
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835:
827:
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162:) or to zero
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1730:
1725:Lothaire, M.
1707:
1637:
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53:exponentiate
29:log semiring
28:
18:
224:), such as
198:deformation
124:as limit ("
49:conjugation
21:mathematics
1789:Logarithms
1783:Categories
1771:1133.68067
1699:References
1036:Properties
265:as rings.
263:isomorphic
244:Definition
45:logarithms
1676:∞
1673:−
1664:∞
1658:∞
1655:−
1652:⊗
1649:∞
1579:−
1558:−
1522:LogSumExp
1459:
1446:−
1420:
1407:−
1398:⊕
1383:
1370:−
1338:
1271:
1252:⊘
1243:⊕
1159:
1135:−
1127:⋅
1111:
1092:−
1089:⊗
1060:−
1042:semifield
981:⊗
960:−
944:−
931:
925:−
912:⊕
839:⊗
796:
780:⊕
732:
683:∞
671:
650:−
641:
585:∞
582:−
573:
522:⊗
452:
421:×
403:
374:⊗
330:
301:⊕
273:∪ {–∞, +∞
259:pullbacks
206:LogSumExp
175:→
144:∞
141:→
61:logarithm
1727:(2005).
1511:See also
251:semiring
226:decibels
33:semiring
1527:Softmax
1503:on the
1484:on the
214:minimum
210:maximum
190:
164:
156:
130:
31:is the
1769:
1759:
1545:Since
1078:since
202:logadd
96:> 1
88:< 1
27:, the
1533:Notes
236:, or
228:(see
1757:ISBN
1048:(or
762:as:
706:<
695:for
623:and
608:>
597:for
1767:Zbl
1450:log
1411:log
1374:log
1329:log
1262:log
1150:log
1102:log
928:log
793:log
729:log
654:log
632:log
564:log
443:log
394:log
321:log
280:≠ 1
232:),
212:or
104:or
43:as
19:In
1785::
1765:.
1755:.
1739:,
1507:.
1488:.
1423:2.
1234::=
1174:0.
1050:+∞
1046:−∞
1030:+∞
889:1/
882:−∞
240:.
106:1/
92:1/
51::
1773:.
1679:)
1670:(
1667:+
1661:=
1619:x
1615:)
1611:b
1607:/
1603:1
1600:(
1597:=
1592:x
1587:)
1582:1
1575:b
1571:(
1566:=
1561:x
1554:b
1465:,
1462:2
1454:b
1415:b
1404:)
1401:y
1395:x
1392:(
1389:=
1386:2
1378:b
1367:)
1362:y
1358:b
1354:+
1349:x
1345:b
1341:(
1333:b
1325:=
1320:)
1315:2
1311:/
1307:)
1302:y
1298:b
1294:+
1289:x
1285:b
1281:(
1276:(
1266:b
1258:=
1255:2
1249:)
1246:y
1240:x
1237:(
1231:)
1228:y
1225:,
1222:x
1219:(
1213:m
1210:l
1205:M
1171:=
1168:)
1165:1
1162:(
1154:b
1146:=
1143:)
1138:x
1131:b
1122:x
1118:b
1114:(
1106:b
1098:=
1095:x
1086:x
1066:,
1063:x
1009:.
1006:y
1003:+
1000:x
997:=
990:y
985:b
977:x
969:)
963:y
956:e
952:+
947:x
940:e
935:(
922:=
915:y
909:x
891:e
861:.
858:y
855:+
852:x
849:=
842:y
836:x
828:)
822:y
818:e
814:+
809:x
805:e
800:(
790:=
783:y
777:x
759:e
741:0
738:=
735:1
709:1
703:b
680:+
677:=
674:0
666:b
662:/
658:1
647:=
644:0
636:b
611:1
605:b
579:=
576:0
568:b
543:y
540:+
537:x
534:=
531:y
526:b
518:x
491:.
488:y
485:+
482:x
479:=
475:)
470:y
467:+
464:x
460:b
456:(
447:b
439:=
435:)
429:y
425:b
416:x
412:b
407:(
398:b
390:=
383:y
378:b
370:x
362:)
356:y
352:b
348:+
343:x
339:b
334:(
325:b
317:=
310:y
305:b
297:x
278:b
271:R
192:(
178:0
172:b
158:(
138:b
114:e
108:e
101:e
94:b
86:b
76:b
70:b
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