31:
427:
Properties of recursively defined functions and sets can often be proved by an induction principle that follows the recursive definition. For example, the definition of the natural numbers presented here directly implies the principle of mathematical induction for natural numbers: if a property holds
836:
1322:
1056:
1362:, for which it is essential that the second clause says "if and only if"; if it had just said "if", the primality of, for instance, the number 4 would not be clear, and the further application of the second clause would be impossible.
1153:
209:
954:
469:
In contrast, a circular definition may have no base case, and even may define the value of a function in terms of that value itself — rather than on other values of the function. Such a situation would lead to an
692:
1167:
1172:
1075:
973:
865:
697:
126:
684:
579:
592:. The formal criteria for what constitutes a valid recursive definition are more complex for the general case. An outline of the general proof and the criteria can be found in
385:
317:
412:
349:
280:
481:, the proof of which is non-trivial. Where the domain of the function is the natural numbers, sufficient conditions for the definition to be valid are that the value of
968:
1070:
121:
2147:
Warren, D.S. and
Denecker, M., 2023. A better logical semantics for prolog. In Prolog: The Next 50 Years (pp. 82-92). Cham: Springer Nature Switzerland.
860:
477:
That recursive definitions are valid – meaning that a recursive definition identifies a unique function – is a theorem of set theory known as the
104:
defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the
258:
An inductive definition of a set describes the elements in a set in terms of other elements in the set. For example, one definition of the set
462:
being defined in terms of the definition itself, and that all other instances in the inductive clauses must be "smaller" in some sense (i.e.,
831:{\displaystyle {\begin{aligned}h(1)&=a_{0}\\h(i)&=\rho \left(h|_{\{1,2,\ldots ,i-1\}}\right){\text{ for }}i>1.\end{aligned}}}
424:
satisfies the definition. However, condition (3) specifies the set of natural numbers by removing the sets with extraneous members.
1317:{\displaystyle {\begin{aligned}&{\binom {a}{0}}=1,\\&{\binom {1+a}{1+n}}={\frac {(1+a){\binom {a}{n}}}{1+n}}.\end{aligned}}}
2325:
2315:
2403:
2236:
1763:
The program can be used not only to check whether a query is true, but also to generate answers that are true. For example:
2300:
17:
2138:
Denecker, M., Ternovska, E.: A logic of nonmonotone inductive definitions. ACM Trans. Comput. Log. 9(2), 14:1–14:52 (2008)
1346:
any other positive integer is a prime number if and only if it is not divisible by any prime number smaller than itself.
2275:
1905:
Logic programs significantly extend recursive definitions by including the use of negative conditions, implemented by
2213:
2123:
224:
of 0. The definition may also be thought of as giving a procedure for computing the value of the function
2330:
2197:
648:
513:
2090:
2228:
1998:
466:
to those base cases that terminate the recursion) — a rule also known as "recur only with a simpler case".
2398:
2066:
1424:(wff) in propositional logic is defined recursively as the smallest set satisfying the three rules:
365:
297:
395:
332:
263:
2320:
2305:
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2342:
2310:
2268:
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1983:
252:
101:
1350:
The primality of the integer 2 is the base case; checking the primality of any larger integer
447:
Most recursive definitions have two foundations: a base case (basis) and an inductive clause.
2352:
2173:
1158:
1051:{\displaystyle {\begin{aligned}&0\cdot a=0,\\&(1+n)\cdot a=a+n\cdot a.\end{aligned}}}
589:
220:
63:
1578:. For example, the recursive definition of even number can be written as the logic program:
1463:
The definition can be used to determine whether any particular string of symbols is a wff:
2357:
2347:
2003:
478:
248:
1358:, which is well defined by this definition. That last point can be proved by induction on
604:
instead of any well-ordered set) of the general recursive definition will be given below.
8:
2393:
2295:
1988:
1906:
1421:
451:
584:
More generally, recursive definitions of functions can be made whenever the domain is a
2408:
2337:
2048:
1148:{\displaystyle {\begin{aligned}&a^{0}=1,\\&a^{1+n}=a\cdot a^{n}.\end{aligned}}}
251:
states that such a definition indeed defines a function that is unique. The proof uses
2205:
1566:
2261:
2242:
2232:
2209:
2177:
2119:
2040:
1978:
1691:
1571:
67:
204:{\displaystyle {\begin{aligned}&0!=1.\\&(n+1)!=(n+1)\cdot n!.\end{aligned}}}
2201:
2166:
2032:
1680:
471:
83:
51:
1354:
by this definition requires knowing the primality of every integer between 2 and
949:{\displaystyle {\begin{aligned}&0+a=a,\\&(1+n)+a=1+(n+a).\end{aligned}}}
1061:
959:
285:
79:
35:
2387:
2246:
2181:
2044:
1411:
unless it is obtained from the basis and inductive clauses (extremal clause).
593:
1333:
454:
and a recursive definition is that a recursive definition must always have
2192:(1977). "An Introduction to Inductive Definitions". In Barwise, J. (ed.).
2189:
2161:
1371:
71:
47:
2284:
2052:
1973:
1575:
1340:
585:
97:
87:
2196:. Studies in Logic and the Foundations of Mathematics. Vol. 90.
1993:
105:
94:
75:
2036:
420:
There are many sets that satisfy (1) and (2) – for example, the set
74:
1977:740ff). Some examples of recursively-definable objects include
851:
439:, then the property holds of all natural numbers (Aczel 1977:742).
1459:
are wffs and • is one of the logical connectives ∨, ∧, →, or ↔.
1336:
can be defined as the unique set of positive integers satisfying
601:
600:. However, a specific case (domain is restricted to the positive
39:
1694:
to solve goals and answer queries. For example, given the query
27:
Defining elements of a set in terms of other elements in the set
1687:
428:
of the natural number 0 (or 1), and the property holds of
2253:
30:
1560:
637:
mapping a nonempty section of the positive integers into
1170:
1073:
971:
863:
695:
651:
516:
398:
368:
335:
300:
266:
124:
42:, the stages are obtained via a recursive definition.
841:
607:
2114:(1st ed.). New Jersey: Prentice-Hall. p.
2165:
2107:
2023:Henkin, Leon (1960). "On Mathematical Induction".
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830:
678:
573:
406:
379:
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311:
274:
203:
1244:
1215:
1192:
1179:
214:This definition is valid for each natural number
2385:
442:
218:, because the recursion eventually reaches the
1478:is a wff, because the propositional variables
2269:
2225:Discrete Structures, Logic, and Computability
1287:
1274:
633:is a function which assigns to each function
1365:
854:is defined recursively based on counting as
800:
770:
416:is the smallest set satisfying (1) and (2).
2276:
2262:
660:
488:(i.e., base case) is given, and that for
400:
370:
337:
302:
268:
1574:can be understood as sets of recursive
495:, an algorithm is given for determining
29:
2105:
1561:Recursive definitions as logic programs
846:
679:{\displaystyle h:\mathbb {Z} _{+}\to A}
574:{\displaystyle f(0),f(1),\dots ,f(n-1)}
70:in terms of other elements in the set (
14:
2386:
2160:
2099:
2089:For a proof of Recursion Theorem, see
2022:
1415:
645:, then there exists a unique function
2257:
2188:
1382:of non-negative evens (basis clause),
34:Four stages in the construction of a
2222:
458:, cases that satisfy the definition
24:
1278:
1219:
1183:
422:{1, 1.649, 2, 2.649, 3, 3.649, …}
25:
2420:
2025:The American Mathematical Monthly
842:Examples of recursive definitions
608:Principle of recursive definition
1567:Definition § Logic programs
1374:can be defined as consisting of
1327:
1686:The logic programming language
2283:
2194:Handbook of Mathematical Logic
2141:
2132:
2083:
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2016:
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1161:can be defined recursively as
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1:
2206:10.1016/S0049-237X(08)71120-0
2154:
443:Form of recursive definitions
380:{\displaystyle \mathbb {N} .}
312:{\displaystyle \mathbb {N} .}
2404:Theoretical computer science
1999:Recursion (computer science)
1661:represents the successor of
1435:is a propositional variable.
407:{\displaystyle \mathbb {N} }
344:{\displaystyle \mathbb {N} }
275:{\displaystyle \mathbb {N} }
238:and proceeding onwards with
7:
1967:
10:
2425:
1564:
1064:is defined recursively as
962:is defined recursively as
581:(i.e., inductive clause).
2291:
2092:On Mathematical Induction
1755:
1731:
1725:
1695:
1668:
1662:
1647:
1637:
1366:Non-negative even numbers
588:, using the principle of
450:The difference between a
2110:Topology, a first course
2009:
1911:
1909:, as in the definition:
1765:
1580:
1556:is a logical connective.
1494:is a logical connective.
115:is defined by the rules
62:, is used to define the
2223:Hein, James L. (2010).
2116:68, exercises 10 and 12
2106:Munkres, James (1975).
1754:it produces the answer
1724:it produces the answer
1984:Mathematical induction
1318:
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1052:
950:
832:
680:
575:
408:
381:
345:
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253:mathematical induction
205:
43:
2094:(1960) by Leon Henkin
2067:"All About Recursion"
1319:
1159:Binomial coefficients
1150:
1053:
951:
833:
681:
590:transfinite recursion
576:
435:whenever it holds of
409:
382:
346:
314:
277:
249:The recursion theorem
206:
38:. As with many other
33:
2229:Jones & Bartlett
2200:. pp. 739–782.
2004:Structural induction
1989:Recursive data types
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847:Elementary functions
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649:
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396:
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122:
60:inductive definition
56:recursive definition
18:Inductive definition
1907:negation as failure
1422:well-formed formula
1416:Well formed formula
1404:(inductive clause),
452:circular definition
2399:Mathematical logic
1730:. Given the query
1692:backward reasoning
1538:is a wff, because
1510:is a wff, because
1343:is a prime number,
1314:
1312:
1145:
1143:
1048:
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946:
944:
828:
826:
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571:
404:
377:
341:
309:
272:
201:
199:
88:Cantor ternary set
44:
2381:
2380:
2331:Genus–differentia
2238:978-0-7637-7206-2
2071:www.cis.upenn.edu
1979:Logic programming
1305:
1285:
1242:
1190:
813:
625:be an element of
616:be a set and let
479:recursion theorem
84:Fibonacci numbers
16:(Redirected from
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2168:Naive set theory
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1681:Peano arithmetic
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1420:The notion of a
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1403:
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1388:
1385:For any element
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1378:0 is in the set
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586:well-ordered set
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472:infinite regress
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943:
942:
888:
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812: for
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286:natural numbers
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261:
259:
239:
232:
225:
215:
198:
197:
143:
142:
125:
123:
120:
119:
109:
80:natural numbers
28:
23:
22:
15:
12:
11:
5:
2422:
2412:
2411:
2406:
2401:
2396:
2379:
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2340:
2335:
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2318:
2313:
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2303:
2298:
2292:
2289:
2288:
2281:
2280:
2273:
2266:
2258:
2252:
2251:
2237:
2220:
2214:
2186:
2156:
2153:
2150:
2149:
2140:
2131:
2124:
2098:
2082:
2058:
2031:(4): 323–338.
2014:
2013:
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2008:
2007:
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1991:
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1981:
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1966:
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1572:Logic programs
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1523:
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1495:
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1407:Nothing is in
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1062:Exponentiation
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960:Multiplication
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741:
738:
735:
732:
731:
726:
722:
718:
715:
713:
711:
708:
705:
702:
699:
698:
675:
672:
667:
662:
657:
654:
621:
609:
606:
570:
567:
564:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
444:
441:
418:
417:
402:
389:
376:
372:
339:
323:If an element
321:
308:
304:
270:
212:
211:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
166:
163:
160:
157:
154:
151:
148:
146:
144:
141:
138:
135:
132:
129:
127:
36:Koch snowflake
26:
9:
6:
4:
3:
2:
2421:
2410:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2391:
2389:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2336:
2332:
2329:
2328:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2293:
2290:
2286:
2279:
2274:
2272:
2267:
2265:
2260:
2259:
2256:
2248:
2244:
2240:
2234:
2230:
2226:
2221:
2217:
2215:0-444-86388-5
2211:
2207:
2203:
2199:
2198:North-Holland
2195:
2191:
2187:
2183:
2179:
2175:
2170:
2169:
2163:
2159:
2158:
2144:
2135:
2127:
2125:0-13-925495-1
2121:
2117:
2112:
2111:
2102:
2095:
2093:
2086:
2072:
2068:
2062:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2019:
2015:
2005:
2002:
2000:
1997:
1995:
1992:
1990:
1987:
1985:
1982:
1980:
1977:
1975:
1972:
1971:
1910:
1908:
1764:
1761:
1693:
1689:
1684:
1682:
1645:
1579:
1577:
1573:
1568:
1552:are wffs and
1550:
1543:
1535:
1531:
1526:
1525:
1519:
1515:
1507:
1503:
1498:
1497:
1490:are wffs and
1488:
1482:
1475:
1471:
1466:
1465:
1464:
1447:
1437:
1427:
1426:
1425:
1423:
1406:
1397:
1384:
1377:
1376:
1375:
1373:
1363:
1345:
1342:
1339:
1338:
1337:
1335:
1334:prime numbers
1328:Prime numbers
1307:
1301:
1298:
1295:
1282:
1279:
1265:
1262:
1259:
1250:
1238:
1235:
1232:
1227:
1224:
1221:
1204:
1201:
1198:
1187:
1184:
1164:
1163:
1162:
1160:
1138:
1133:
1129:
1125:
1122:
1119:
1114:
1111:
1108:
1104:
1095:
1092:
1089:
1084:
1080:
1067:
1066:
1065:
1063:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1011:
1008:
1005:
1000:
992:
989:
986:
983:
980:
977:
965:
964:
963:
961:
939:
933:
930:
927:
921:
918:
915:
912:
909:
903:
900:
897:
892:
884:
881:
878:
875:
872:
869:
857:
856:
855:
853:
821:
818:
815:
806:
797:
794:
791:
788:
785:
782:
779:
776:
773:
760:
756:
752:
749:
747:
739:
733:
724:
720:
716:
714:
706:
700:
689:
688:
687:
673:
665:
655:
652:
620:
605:
603:
599:
595:
594:James Munkres
591:
587:
582:
565:
562:
559:
553:
550:
547:
544:
538:
532:
529:
523:
517:
503:
499:
492:
485:
480:
475:
473:
467:
465:
461:
457:
453:
448:
440:
432:
425:
390:
374:
357:
326:
322:
306:
291:
290:
289:
287:
256:
254:
250:
246:
242:
235:
228:
223:
222:
194:
191:
188:
185:
179:
176:
173:
167:
164:
158:
155:
152:
147:
139:
136:
133:
130:
118:
117:
116:
112:
107:
103:
99:
96:
91:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
41:
37:
32:
19:
2362:
2321:Fallacies of
2306:Coordinative
2224:
2193:
2190:Aczel, Peter
2174:van Nostrand
2167:
2162:Halmos, Paul
2143:
2134:
2109:
2101:
2091:
2085:
2074:. Retrieved
2070:
2061:
2028:
2024:
2018:
1904:
1762:
1685:
1643:
1635:
1570:
1548:
1541:
1533:
1529:
1517:
1513:
1505:
1501:
1486:
1480:
1473:
1469:
1462:
1451:is a wff if
1441:is a wff if
1431:is a wff if
1419:
1395:
1372:even numbers
1369:
1349:
1331:
1157:
1060:
958:
850:
618:
611:
597:
583:
506:in terms of
501:
497:
490:
483:
476:
468:
463:
459:
455:
449:
446:
430:
426:
419:
355:
324:
257:
247:
240:
233:
226:
219:
213:
110:
92:
59:
55:
45:
2373:Theoretical
2368:Stipulative
2343:Operational
2326:Intensional
2316:Extensional
2311:Enumerative
1642:represents
1576:definitions
1389:in the set
1332:The set of
48:mathematics
2394:Definition
2388:Categories
2353:Persuasive
2285:Definition
2155:References
2076:2019-10-24
1974:Definition
1565:See also:
686:such that
456:base cases
98:definition
86:, and the
76:factorials
2409:Recursion
2363:Recursive
2358:Precising
2348:Ostensive
2247:636352297
2182:802530334
2045:0002-9890
1994:Recursion
1667:, namely
1522:is a wff.
1445:is a wff.
1126:⋅
1036:⋅
1018:⋅
981:⋅
795:−
786:…
753:ρ
671:→
563:−
548:…
243:= 1, 2, 3
221:base case
186:⋅
108:function
106:factorial
95:recursive
2296:Circular
2164:(1960).
1968:See also
1679:, as in
852:Addition
602:integers
598:Topology
292:1 is in
102:function
64:elements
40:fractals
2338:Lexical
2301:Concept
2053:2308975
1449:(p • q)
460:without
414:
392:
387:
362:
351:
329:
319:
294:
282:
260:
2245:
2235:
2212:
2180:
2122:
2051:
2043:
1897:))))))
1688:Prolog
1646:, and
1400:is in
493:> 0
464:closer
360:is in
327:is in
2049:JSTOR
2010:Notes
1900:.....
1757:false
1690:uses
1636:Here
1532:∧ ¬
629:. If
353:then
245:etc.
100:of a
72:Aczel
66:in a
58:, or
2243:OCLC
2233:ISBN
2210:ISBN
2178:OCLC
2120:ISBN
2041:ISSN
1953:even
1926:even
1914:even
1849:))))
1771:even
1736:even
1727:true
1700:even
1622:even
1595:even
1583:even
1545:and
1484:and
1455:and
1370:The
819:>
612:Let
288:is:
54:, a
50:and
2202:doi
2033:doi
1962:)).
1947:not
1721:)))
1683:.
1616:)))
1528:(¬
1500:¬ (
1439:¬ p
1398:+ 2
486:(0)
433:+ 1
358:+ 1
284:of
236:= 0
68:set
46:In
2390::
2241:.
2231:.
2227:.
2208:.
2176:.
2172:.
2118:.
2069:.
2047:.
2039:.
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2027:.
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1941:))
1923:).
1813:))
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1760:.
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1697:?-
1644:if
1639::-
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1547:¬
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596:'
510:,
474:.
255:.
140:1.
93:A
90:.
82:,
78:,
2277:e
2270:t
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2249:.
2218:.
2204::
2184:.
2128:.
2096:.
2079:.
2055:.
2035::
1959:X
1956:(
1950:(
1938:X
1935:(
1932:s
1929:(
1920:0
1917:(
1894:0
1891:(
1888:s
1885:(
1882:s
1879:(
1876:s
1873:(
1870:s
1867:(
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1861:(
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1840:s
1837:(
1834:s
1831:(
1828:s
1825:(
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1810:0
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1786:=
1783:X
1777:X
1774:(
1748:0
1745:(
1742:s
1739:(
1718:0
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1664:X
1658:)
1655:X
1652:(
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1625:(
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1601:s
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1512:(
1508:)
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1481:p
1476:)
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1468:(
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1433:p
1429:p
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1402:E
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1387:x
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1360:X
1356:X
1352:X
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1308:.
1302:n
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1288:)
1283:n
1280:a
1275:(
1269:)
1266:a
1263:+
1260:1
1257:(
1251:=
1245:)
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1236:+
1233:1
1228:a
1225:+
1222:1
1216:(
1205:,
1202:1
1199:=
1193:)
1188:0
1185:a
1180:(
1139:.
1134:n
1130:a
1123:a
1120:=
1115:n
1112:+
1109:1
1105:a
1096:,
1093:1
1090:=
1085:0
1081:a
1042:.
1039:a
1033:n
1030:+
1027:a
1024:=
1021:a
1015:)
1012:n
1009:+
1006:1
1003:(
993:,
990:0
987:=
984:a
978:0
940:.
937:)
934:a
931:+
928:n
925:(
922:+
919:1
916:=
913:a
910:+
907:)
904:n
901:+
898:1
895:(
885:,
882:a
879:=
876:a
873:+
870:0
816:i
807:)
801:}
798:1
792:i
789:,
783:,
780:2
777:,
774:1
771:{
766:|
761:h
757:(
750:=
743:)
740:i
737:(
734:h
725:0
721:a
717:=
710:)
707:1
704:(
701:h
674:A
666:+
661:Z
656::
653:h
643:A
639:A
635:f
631:ρ
627:A
622:0
619:a
614:A
569:)
566:1
560:n
557:(
554:f
551:,
545:,
542:)
539:1
536:(
533:f
530:,
527:)
524:0
521:(
518:f
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502:n
500:(
498:f
491:n
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437:n
431:n
401:N
375:.
371:N
356:n
338:N
325:n
307:.
303:N
269:N
241:n
234:n
229:!
227:n
216:n
195:.
192:!
189:n
183:)
180:1
177:+
174:n
171:(
168:=
165:!
162:)
159:1
156:+
153:n
150:(
137:=
134:!
131:0
113:!
111:n
20:)
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