1334:, we would have to resort to our knowledge of multiplication and the number 3. Since any computation is simply a composition of the evaluation of suitable functions on suitable primitive arguments, this simple substitution principle suffices to capture the essential mechanism of computation. Moreover, in lambda calculus, notions such as '3' and '
4620:
The undecidable problems above (equivalence, existence of normal form, etc.) take as input syntactic representations of terms under a suitable encoding (e.g., Church encoding). One may also consider a toy trivial computation model where we "compute" properties of terms by means of combinators applied
4611:
is any combinatory term in which the primitive combinators that occur, if any, are not applied to enough arguments to be simplified. It is undecidable whether a general combinatory term has a normal form; whether two combinatory terms are equivalent, etc. This can be shown in a similar way as for the
5047:
However, note that it also immediately follows from this undefinability theorem that many properties of terms that are obviously decidable are not definable by complete predicates either: e.g., there is no predicate that could tell whether the first primitive function letter occurring in a term is a
1381:
is a model of computation equivalent to lambda calculus, but without abstraction. The advantage of this is that evaluating expressions in lambda calculus is quite complicated because the semantics of substitution must be specified with great care to avoid variable capture problems. In contrast,
1376:
It is perhaps surprising that lambda-calculus can represent any conceivable computation using only the simple notions of function abstraction and application based on simple textual substitution of terms for variables. But even more remarkable is that abstraction is not even required.
1390:
Since abstraction is the only way to manufacture functions in the lambda calculus, something must replace it in the combinatory calculus. Instead of abstraction, combinatory calculus provides a limited set of primitive functions out of which other functions may be built.
1358:' can be represented without any need for externally defined primitive operators or constants. It is possible to identify terms in lambda calculus, which, when suitably interpreted, behave like the number 3 and like the multiplication operator, q.v.
146:, which proved more popular than combinatory logic. The upshot of these historical contingencies was that until theoretical computer science began taking an interest in combinatory logic in the 1960s and 1970s, nearly all work on the subject was by
2794:
It is rules 5 and 6 that are of interest. Rule 5 simply says that to convert a complex abstraction to a combinator, we must first convert its body to a combinator, and then eliminate the abstraction. Rule 6 actually eliminates the abstraction.
2237:
as given is not a well-typed mathematical function, but rather a term rewriter: Although it eventually yields a combinator, the transformation may generate intermediary expressions that are neither lambda terms nor combinators, via rule (5).
5087:, that is, programming in functional expressions containing no variables, along with powerful tools for working with such programs. It turns out that tacit programming is possible in any APL-like language with user-defined operators.
2790:
The first two rules are also simple: Variables convert to themselves, and applications, which are allowed in combinatory terms, are converted to combinators simply by converting the applicand and the argument to combinators.
4984:
From this undefinability theorem it immediately follows that there is no complete predicate that can discriminate between terms that have a normal form and terms that do not have a normal form. It also follows that there is
1193:
The motivation for this definition of reduction is that it captures the essential behavior of all mathematical functions. For example, consider the function that computes the square of a number. We might write
5103:
corresponds to a reduction of a typed lambda term, and conversely. Moreover, theorems can be identified with function type signatures. Specifically, a typed combinatory logic corresponds to a
4139:
1712:
212:. It is easy to transform lambda expressions into combinator expressions, and combinator reduction is much simpler than lambda reduction. Hence combinatory logic has been used to model some
4069:
1101:
5256:
1545:
971:
1615:
5326:
230:
Combinatory logic can be given a variety of interpretations. Many early papers by Curry showed how to translate axiom sets for conventional logic into combinatory logic equations.
227:, whose sole primitives are the S and K combinators augmented with character input/output. Although not a practical programming language, Unlambda is of some theoretical interest.
5497:
4101:
4029:
2035:
if they always produce the same results for the same arguments. In contrast, the terms themselves, together with the reduction of primitive combinators, capture the notion of
5529:
587:
500:
320:
5710:
5669:
5361:
5617:
5413:
707:
368:
5565:
5643:
5465:
5439:
5280:
1182:
5591:
5387:
2316:
Bracket abstraction induces a translation from lambda terms to combinator expressions, by interpreting lambda-abstractions using the bracket abstraction algorithm.
1140:
905:
874:
843:
804:
773:
742:
653:
622:
543:
457:
426:
6323:
1330:
1297:
1228:
1354:
1255:
1017:
395:
279:
162:(1972) survey the early history of combinatory logic. For a more modern treatment of combinatory logic and the lambda calculus together, see the book by
1369:); that is, any calculation that can be accomplished in any of these other models can be expressed in lambda calculus, and vice versa. According to the
6670:
4754:. An analogue of Rice's theorem for this toy model then says that every complete predicate is trivial. The proof of this theorem is rather simple.
3732:
These combinators are extremely useful when translating predicate logic or lambda calculus into combinator expressions. They were also used by
6643:
6625:
1267:
of the function. To evaluate the square for a particular argument, say 3, we insert it into the definition in place of the formal parameter:
6482:
6656:
6379:]. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Translated by Bauer-Mengelberg, Stefan. Cambridge, MA, USA:
108:
6754:
6499:
6468:
6388:
3740:, whose name has been associated with their computational use. Using them, we can extend the rules for the transformation as follows:
209:
6597:
6546:
6439:
6309:
6205:
6102:
6061:
6035:
5942:
3737:
6689:
5934:
4608:
2101:
lambda term, and therefore, by Church's thesis, to any computable function whatsoever. The proof is to present a transformation,
76:
in 1920 with the idea of providing an analogous way to build up functionsâand to remove any mention of variablesâparticularly in
65:
213:
5969:
5740:
5104:
1365:
Lambda calculus is known to be computationally equivalent in power to many other plausible models for computation (including
2268:
holds. A very simple algorithm for bracket abstraction is defined by induction on the structure of expressions as follows:
2260:
built from variables and application and produces a combinator expression E in which the variable x is not free, such that
5725:
3094:â), so the recursion must terminate in a lambda term with no applications at allâeither a variable, or a term of the form
5096:
4294:
The reduction in combinator size that results from the new transformation rules can also be achieved without introducing
6446:
A gentle introduction to combinatory logic, presented as a series of recreational puzzles using bird watching metaphors.
5735:
5218:
is complete for the implicational fragment of the intuitionistic logic, which can be seen as follows. Consider the set
1624:
4621:
directly to the terms themselves as arguments, rather than to their syntactic representations. More precisely, let a
4034:
6744:
6475:
Chapters 17â20 are a more formal introduction to combinatory logic, with a special emphasis on fixed point results.
6197:
5080:
1041:
5229:
6142:
5066:
4106:
2752:
transformation is motivated by a desire to eliminate abstraction. Two special cases, rules 3 and 4, are trivial:
1370:
6749:
6380:
6301:
1478:
6695:
6524:
2031:. Extensional equality captures the mathematical notion of the equality of functions: that two functions are
918:
4228:
takes a value and substitutes it into both the applicand and its argument before performing the application,
1554:
6611:
6460:
6293:
6263:
5770:
5755:
220:
116:
104:
101:
5288:
1716:. This is the same general convention (left-associativity) as for multiple application in lambda calculus.
5750:
5223:
5076:
2245:. This definition is exhaustive: any lambda expression will be subject to exactly one of these rules (see
2071:
216:
5470:
1382:
evaluating expressions in combinatory logic is much simpler, because there is no notion of substitution.
6430:
5780:
5745:
81:
6072:
4002:
2508:(by feeding them in a queue-like fashion into the combinator 'from the right'), it reduces as follows:
6253:
6191:
5502:
6364:
6319:
5100:
4248:
2028:
556:
469:
289:
204:, in which lambda expressions (representing functional abstraction) are replaced by a limited set of
190:
135:
127:
85:
73:
61:
49:
5689:
5648:
5334:
4595:
Note, however, that this transformation is not the inverse transformation of any of the versions of
4076:
6368:
31:
5931:
The Lambda
Calculus, Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics
5596:
5392:
669:
330:
6574:
6566:
6431:
To Mock a
Mockingbird And Other Logic Puzzles Including an Amazing Adventure in Combinatory Logic
6346:
6275:
6175:
6159:
6007:
5975:
5538:
5072:
2047:
the expansion of primitive combinators. There are many ways to implement an identity function; (
97:
45:
41:
6677:
To
Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction.
6632:
A celebration of the development of combinators, a hundred years after they were introduced by
5622:
5444:
5418:
6639:
6621:
6593:
6505:
6495:
6464:
6435:
6394:
6384:
6305:
6201:
6098:
6086:
6057:
6031:
5990:(1930). "Grundlagen der Kombinatorischen Logik" [Foundations of combinatorial logic].
5965:
5938:
5764:
5730:
5532:
5084:
4244:
112:
100:
in logic, essentially by eliminating them. Another way of eliminating quantified variables is
5265:
3412:
There are one-point bases from which every combinator can be composed extensionally equal to
1149:
197:. Despite its simplicity, combinatory logic captures many essential features of computation.
6664:
6558:
6452:
6425:
6338:
6249:
6228:
6187:
6151:
6133:
6121:
6049:
5999:
5957:
5570:
5366:
5259:
1264:
976:
513:
182:
57:
6171:
1113:
883:
852:
821:
782:
751:
720:
631:
600:
521:
435:
404:
6716:
6706:
6683:
6607:
6589:
Combinatory logic in programming: Computations with objects through examples and exercises
6167:
5926:
5775:
5760:
5052:. This shows that definability by predicates is a not a reasonable model of decidability.
4646:
1359:
247:
201:
163:
143:
96:
Combinatory logic was originally intended as a 'pre-logic' that would clarify the role of
77:
6217:"On the Complexity of the Standard Translation of Lambda Calculus into Combinatory Logic"
1309:
1276:
1207:
142:
and his students at
Princeton invented a rival formalism for functional abstraction, the
6112:
Goldberg, Mayer (2004). "A construction of one-point bases in extended lambda calculi".
1724:
In combinatory logic, each primitive combinator comes with a reduction rule of the form
6660:
6216:
6091:
2718:
1366:
1339:
1240:
987:
380:
264:
4435:
combinator according to rules similar to those presented above for the conversion of λ
6738:
6617:
6490:. Studies in Logic and the Foundations of Mathematics. Vol. 149 (1st ed.).
6350:
6045:
6019:
5987:
4240:
3733:
1467:
147:
139:
131:
53:
6578:
6407:
5979:
6233:
6179:
5108:
3507:
715:. Applications model function invocation or execution: the function represented by
235:
194:
167:
591:
represents the function which, applied to an argument, binds the formal parameter
6137:
17:
6023:
5099:
implies a connection between logic and programming: every proof of a theorem of
4761:
By reductio ad absurdum. Suppose there is a complete non trivial predicate, say
1187:
186:
151:
399:
is a variable name drawn from a predefined infinite set of variable names, and
6710:
6125:
231:
170:
6398:
2079:
6587:
5331:
This definition obeys the conditions on satisfaction of â: on one hand, if
5179:
and function application corresponds to the detachment (modus ponens) rule
5961:
6491:
2107:, which converts an arbitrary lambda term into an equivalent combinator.
224:
5767:, other approaches to modelling quantification and eliminating variables
130:, published nothing on combinatory logic after his original 1924 paper.
6720:
6570:
6342:
6163:
6011:
2097:
can be composed to produce combinators that are extensionally equal to
155:
88:
and earlier defined combinators to define a result from its arguments.
6676:
6279:
4977:
6562:
6155:
6003:
4673:, transformed into combinatory logic; the combinatory versions have
3172:
Taking this simplification into account, the example above becomes:
1776:. It is in this way that primitive combinators behave as functions.
256:, which can be represented by the following three forms of strings:
6700:
6535:
5801:, The article that founded combinatory logic. English translation:
1456:
Applying a function to an argument. M and N are combinatory terms.
2044:
913:
to the old one. If a lambda term contains no subterms of the form
6355:
The article that founded combinatory logic. English translation:
1945:
itself is unnecessary, since it can be built from the other two:
5954:
Proceedings of the international conference on APL '91 - APL '91
4859:
Fixed point theorem gives: ABSURDUM = (NEGATION ABSURDUM), for
3119:
transformation can be made smaller if we take into account the
2320:
Conversion of a lambda term to an equivalent combinatorial term
134:
rediscovered the combinators while working as an instructor at
3663:
He also explains how they in turn can be expressed using only
6686:" A table distilling much of the essence of Smullyan (1985).
2701:)) is much longer than the representation as a lambda term,
5678:
be any formula which is not provable in the calculus. Then
4336:
calculus. The distinction corresponds to that between the λ
1466:, or functions that, when seen as lambda terms, contain no
223:. The purest form of this view is the programming language
6459:. Oxford logic guides. Vol. 27. Oxford and New York:
3226:
This combinator is equivalent to the earlier, longer one:
1855:. Or, following the convention for multiple application,
3041:â), it is sufficient to find a combinator equivalent to (
595:
to the argument and then computes the resulting value of
27:
Logical formalism using combinators instead of variables
5952:
Cherlin, Edward (1991). "Pure functions in APL and J".
5222:
of all deductively closed sets of formulas, ordered by
1473:
To shorten the notations, a general convention is that
1422:
A character or string representing a combinatory term.
3416:
lambda term. The simplest example of such a basis is {
173:
devised for combinatory logic in the 1960s and 1970s.
6549:(1979). "Another Algorithm for Bracket Abstraction".
6534:. World Scientific Publishing Company. Archived from
5692:
5651:
5625:
5599:
5573:
5541:
5505:
5473:
5447:
5421:
5395:
5369:
5337:
5291:
5268:
5232:
5044:) would have to be a complete non trivial predicate.
4458:
from combinatorial terms to lambda terms is trivial:
4232:
performs the substitution only in the applicand, and
4109:
4079:
4037:
4005:
2500:
If we apply this combinatorial term to any two terms
1627:
1557:
1481:
1342:
1312:
1279:
1243:
1210:
1152:
1116:
1044:
990:
921:
886:
855:
824:
785:
754:
723:
672:
634:
603:
559:
524:
472:
438:
407:
383:
333:
292:
267:
185:, combinatory logic is used as a simplified model of
5075:
used primitives based on Curry's combinators in his
4769:
is supposed to be non trivial there are combinators
4625:
be a combinator that, when applied, returns either
1373:, both models can express any possible computation.
847:, the argument, may be substituted into the body of
200:
Combinatory logic can be viewed as a variant of the
6481:SĂžrensen, Morten Heine B; Urzyczyn, PaweĆ (2006) .
5065:David Turner used his combinators to implement the
3161:to it; this is extensionally equal to the function
1821:) is the function which, for any argument, returns
1399:A combinatory term has one of the following forms:
6532:Randomness And Complexity, from Leibniz To Chaitin
6090:
5998:(3). The Johns Hopkins University Press: 509â536.
5704:
5663:
5637:
5611:
5585:
5559:
5523:
5491:
5459:
5433:
5407:
5381:
5355:
5320:
5274:
5250:
4133:
4095:
4063:
4023:
1706:
1609:
1539:
1348:
1324:
1291:
1249:
1222:
1176:
1134:
1095:
1011:
965:
899:
868:
837:
798:
767:
736:
701:
647:
616:
581:
537:
494:
451:
420:
389:
362:
314:
273:
6300:(Enl. ed., 2. print ed.). Cambridge, Mass.:
6268:Proceedings of the American Philosophical Society
6138:"Principal type-schemes and condensed detachment"
4446:combinators. See chapter 9 in Barendregt (1984).
3576:In fact, there exist infinitely many such bases.
3342:transformation transformed the identity function
3086:â each contain strictly fewer applications than (
1880:, which is a generalized version of application:
1758:is a term mentioning only variables from the set
252:Lambda calculus is concerned with objects called
6193:Lambda-Calculus and Combinators: An Introduction
4424:terms can be composed (modulo equality). Every λ
3517:Another simple example of a one-point basis is:
3029:Therefore, to find a combinator equivalent to
2713:construction may expand a lambda term of length
1707:{\displaystyle (...((E_{1}E_{2})E_{3})...E_{n})}
975:then it cannot be reduced, and is said to be in
777:as its argument, and the result is computed. If
6592:(2nd ed.). Moscow: "Center JurInfoR" Ltd.
5850:
4239:The modern names for the combinators come from
2246:
909:, and the result is a new lambda term which is
111:of combinatory logic typically exceeds that of
6525:"Binary Lambda Calculus and Combinatory Logic"
4064:{\displaystyle \mathbf {X} _{*}=\mathbf {XI} }
3596:included two combinators which are now called
3165:itself. It is therefore sufficient to convert
2849:is just the same as substituting it into both
2809:â) is a function which takes an argument, say
6696:Binary Lambda Calculus and Combinatory Logic.
6324:"Ăber die Bausteine der mathematischen Logik"
2324:For example, we will convert the lambda term
1096:{\displaystyle ((\lambda v.E)a)\Rightarrow E}
8:
6377:On the building blocks of mathematical logic
5554:
5548:
5251:{\displaystyle \langle W,\subseteq \rangle }
5245:
5233:
4971:, which contradicts the presupposition that
3153:x) is the function which takes an argument,
2063:to indicate extensional equality, reserving
2043:only if they have identical implementations
126:The original inventor of combinatory logic,
6633:
6373:Ăber die Bausteine der mathematischen Logik
6356:
5802:
5798:
4975:would be a complete non trivial predicate.
4134:{\displaystyle \mathbf {X} '=\mathbf {CX} }
3593:
2813:, and substitutes it into the lambda term (
2078:combinator, which can be used to implement
1814:, which manufactures constant functions: (
234:in the 1960s and 1970s showed how to marry
119:is identical to that of first order logic (
6056:. Vol. II. Amsterdam: North Holland.
5838:
5305:
5301:
4989:complete predicate, say EQUAL, such that:
2709:.(y x). This is typical. In general, the
6232:
6030:. Vol. I. Amsterdam: North Holland.
5874:
5691:
5682:does not belong to the deductive closure
5650:
5624:
5598:
5572:
5540:
5504:
5472:
5446:
5420:
5394:
5368:
5336:
5290:
5267:
5231:
4612:corresponding problems for lambda terms.
4123:
4111:
4108:
4084:
4078:
4053:
4044:
4039:
4036:
4015:
4010:
4004:
1695:
1673:
1660:
1650:
1626:
1601:
1582:
1572:
1562:
1556:
1540:{\displaystyle (E_{1}E_{2}E_{3}...E_{n})}
1528:
1509:
1499:
1489:
1480:
1341:
1311:
1278:
1242:
1209:
1151:
1115:
1043:
1021:represents the result of taking the term
989:
954:
941:
920:
891:
885:
860:
854:
829:
823:
790:
784:
759:
753:
728:
722:
690:
680:
671:
639:
633:
608:
602:
573:
558:
529:
523:
486:
471:
443:
437:
412:
406:
382:
351:
341:
332:
306:
291:
266:
6484:Lectures on the CurryâHoward Isomorphism
5886:
4603:Undecidability of combinatorial calculus
1784:The simplest example of a combinator is
1401:
966:{\displaystyle ((\lambda v.E_{1})E_{2})}
40:is a notation to eliminate the need for
5910:
5898:
5791:
5467:by modus ponens. On the other hand, if
4322:A distinction must be made between the
3338:Similarly, the original version of the
2059:) is yet another. We will use the word
1918:into each of them. Or put another way,
1610:{\displaystyle E_{1}E_{2}E_{3}...E_{n}}
6701:Combinatory logic reduction web server
5862:
5826:
1788:, the identity combinator, defined by
1025:and replacing all free occurrences of
64:and also as a basis for the design of
6296:(1996) . "Variables explained away".
6285:
6255:Foundations of Functional Programming
5814:
5321:{\displaystyle X\Vdash A\iff A\in X.}
5122:combinators correspond to the axioms
4750:has a normal form for every argument
4329:as described in this article and the
4303:
4251:'s original paper, what we now call
3106:Simplifications of the transformation
2070:A more interesting combinator is the
2039:of functions: that two functions are
1304:To evaluate the resulting expression
812:) is an abstraction, the term may be
56:, and has more recently been used in
7:
6406:Seldin, Jonathan P. (3 March 2008).
6266:(1960). "Variables explained away".
4428:term can be converted into an equal
4417:, which form a basis from which all
4369:has at least one free occurrence in
4352:calculus restricts abstractions to:
4302:, as demonstrated in Section 3.2 of
3443:It is not difficult to verify that:
878:in place of the formal parameter of
120:
6657:Stanford Encyclopedia of Philosophy
5492:{\displaystyle X\not \Vdash A\to B}
5083:. This enabled what Iverson called
5061:Compilation of functional languages
5032:If EQUAL would exist, then for all
4879:is supposed to be complete either:
3928:combinators, the transformation of
2067:for identical combinatorial terms.
158:. Curry and Feys (1958), and Curry
6457:Diagonalization and Self-Reference
3222:) (by η-reduction)
2717:to a combinatorial term of length
25:
6712:Combinators: 100-Year Celebration
5716:is not intuitionistically valid.
4871:NEGATION)) ⥠(NEGATION ABSURDUM).
4103:(traditional canonical notation:
4031:(traditional canonical notation:
4024:{\displaystyle =\mathbf {C} _{*}}
3604:, with the following reductions:
3115:The combinators generated by the
2677:The combinatory representation, (
138:in late 1927. In the late 1930s,
34:, a topic in digital electronics.
6530:. In Calude, Cristian S. (ed.).
5535:, thus the deductive closure of
5524:{\displaystyle X,A\not \vdash B}
4243:'s doctoral thesis of 1930 (see
4127:
4124:
4112:
4085:
4057:
4054:
4040:
4011:
3385:)). With the η-reduction rule,
2252:It is related to the process of
1810:. Another simple combinator is
1259:" to indicate multiplication.)
66:functional programming languages
6408:"The Logic of Curry and Church"
6190:; Seldin, Jonathan P. (2008) .
5992:American Journal of Mathematics
3502:} is a basis, it follows that {
582:{\displaystyle \lambda v.E_{1}}
495:{\displaystyle \lambda v.E_{1}}
315:{\displaystyle \lambda v.E_{1}}
6671:1920â1931 Curry's block notes.
6613:Combinators: A Centennial View
6234:10.4467/20842589RM.18.002.8835
6114:Information Processing Letters
6052:; Seldin, Jonathan P. (1972).
5741:Combinatory categorial grammar
5705:{\displaystyle X\not \Vdash A}
5664:{\displaystyle Y\not \Vdash B}
5483:
5356:{\displaystyle X\Vdash A\to B}
5347:
5302:
4096:{\displaystyle =\mathbf {I} '}
2241:This process is also known as
1720:Reduction in combinatory logic
1701:
1679:
1666:
1643:
1640:
1628:
1534:
1482:
1433:One of the combinator symbols
1171:
1165:
1156:
1153:
1129:
1117:
1090:
1078:
1072:
1069:
1063:
1048:
1045:
1006:
994:
960:
947:
925:
922:
696:
673:
357:
334:
208:, primitive functions without
1:
6551:The Journal of Symbolic Logic
6221:Reports on Mathematical Logic
5726:Applicative computing systems
4379:As a consequence, combinator
2086:Completeness of the S-K basis
551:of the abstraction. The term
5736:Categorical abstract machine
5612:{\displaystyle Y\supseteq X}
5408:{\displaystyle Y\supseteq X}
4616:Undefinability by predicates
4212:The motivation here is that
2256:, which takes an expression
1462:The primitive functions are
702:{\displaystyle (E_{1}E_{2})}
626:— that is, it returns
363:{\displaystyle (E_{1}E_{2})}
6586:Wolfengagen, V. E. (2003).
6284:Reprinted as Chapter 23 of
6080:. BirkhĂ€user. pp. 5â6.
5851:Hindley & Meredith 1990
5560:{\displaystyle X\cup \{A\}}
5206:The calculus consisting of
4705:if there are two arguments
4645:represent the conventional
4394:calculus. The constants of
3157:, and applies the function
2339:) to a combinatorial term:
2115:may be defined as follows:
657:, with every occurrence of
72:, which were introduced by
6771:
6690:Drag 'n' Drop Combinators.
6258:. University of Cambridge.
6215:Lachowski, Ćukasz (2018).
6198:Cambridge University Press
6136:; Meredith, David (1990).
5927:Barendregt, Hendrik Pieter
3510:programming language uses
3078:evidently fits the bill.
2772:is clearly equivalent to (
2247:Summary of lambda calculus
661:replaced by the argument.
245:
242:Summary of lambda calculus
115:, the expressive power of
29:
6755:Logic in computer science
6675:Keenan, David C. (2001) "
6143:Journal of Symbolic Logic
6126:10.1016/j.ipl.2003.12.005
6074:The Combinatory Programme
5638:{\displaystyle Y\Vdash A}
5460:{\displaystyle Y\Vdash B}
5434:{\displaystyle Y\Vdash A}
5067:SASL programming language
2991:By extensional equality,
2760:is clearly equivalent to
2023:) itself is not equal to
1914:after first substituting
6381:Harvard University Press
6302:Harvard University Press
6294:Quine, Willard Van Orman
6264:Quine, Willard Van Orman
5262:, and we define a model
5097:CurryâHoward isomorphism
4955:, again a contradiction.
4220:are limited versions of
3514:as its sole combinator.
2783:does not appear free in
2055:are among these ways. (
2027:. We say the terms are
2000:. Note that although ((
516:of the abstraction, and
150:and his students, or by
30:Not to be confused with
6717:Wolfram Physics Project
6461:Oxford University Press
5771:SKI combinator calculus
5756:Graph reduction machine
5686:of the empty set, thus
5275:{\displaystyle \Vdash }
4944:(NEGATION ABSURDUM) = (
4918:(NEGATION ABSURDUM) = (
4867:NEGATION) = (NEGATION (
4383:is not present in the λ
3169:to combinatorial form.
2243:abstraction elimination
2162:does not occur free in
1926:inside the environment
1780:Examples of combinators
1177:{\displaystyle ((ab)c)}
1108:By convention, we take
746:is to be invoked, with
238:and combinatory logic.
117:predicate functor logic
105:predicate functor logic
48:. It was introduced by
6093:Functional Programming
5751:Fixed point combinator
5706:
5665:
5639:
5613:
5587:
5586:{\displaystyle Y\in W}
5561:
5525:
5493:
5461:
5435:
5409:
5383:
5382:{\displaystyle Y\in W}
5357:
5322:
5276:
5252:
5077:J programming language
4344:calculus. Unlike the λ
4236:only in the argument.
4135:
4097:
4065:
4025:
3506:} is a basis too. The
3400:) is transformed into
2072:fixed point combinator
1876:A third combinator is
1708:
1611:
1541:
1350:
1326:
1293:
1251:
1224:
1186:(i.e., application is
1178:
1136:
1097:
1013:
967:
901:
870:
839:
808:(sometimes called the
800:
769:
738:
703:
649:
618:
583:
539:
496:
453:
422:
391:
364:
316:
275:
217:functional programming
121:Quine 1960, 1966, 1976
6331:Mathematische Annalen
6298:Selected Logic Papers
6046:Curry, Haskell Brooks
6020:Curry, Haskell Brooks
5988:Curry, Haskell Brooks
5962:10.1145/114054.114065
5781:To Mock a Mockingbird
5746:Explicit substitution
5707:
5666:
5640:
5614:
5588:
5562:
5526:
5494:
5462:
5436:
5410:
5384:
5358:
5323:
5277:
5258:is an intuitionistic
5253:
4963:ABSURDUM) is neither
4136:
4098:
4066:
4026:
2833:â). But substituting
1709:
1612:
1542:
1351:
1327:
1294:
1252:
1225:
1179:
1137:
1135:{\displaystyle (abc)}
1098:
1014:
968:
902:
900:{\displaystyle E_{1}}
871:
869:{\displaystyle E_{1}}
840:
838:{\displaystyle E_{2}}
801:
799:{\displaystyle E_{1}}
770:
768:{\displaystyle E_{2}}
739:
737:{\displaystyle E_{1}}
704:
650:
648:{\displaystyle E_{1}}
619:
617:{\displaystyle E_{1}}
584:
540:
538:{\displaystyle E_{1}}
497:
454:
452:{\displaystyle E_{2}}
423:
421:{\displaystyle E_{1}}
392:
365:
317:
276:
82:higher-order function
80:. A combinator is a
6523:Tromp, John (2008).
6383:. pp. 355â366.
6369:Van Heijenoort, Jean
6304:. pp. 227â235.
6250:Paulson, Lawrence C.
6071:Engeler, E. (1995).
5690:
5649:
5623:
5597:
5571:
5539:
5503:
5471:
5445:
5419:
5393:
5367:
5335:
5289:
5266:
5230:
5101:intuitionistic logic
4387:calculus nor in the
4107:
4077:
4035:
4003:
3736:, and much later by
2737:transformation": -->
2037:intensional equality
1625:
1555:
1479:
1371:ChurchâTuring thesis
1340:
1310:
1277:
1241:
1208:
1150:
1114:
1042:
988:
919:
884:
853:
822:
783:
752:
721:
670:
632:
601:
557:
522:
470:
436:
405:
381:
331:
290:
265:
191:computability theory
166:, which reviews the
136:Princeton University
98:quantified variables
86:function application
62:model of computation
6085:Field, Anthony J.;
4851:Define ABSURDUM ⥠(
4649:of true and false,
4599:that we have seen.
3943:) looks like this:
2729:Explanation of the
2254:bracket abstraction
2029:extensionally equal
1619:, denotes the term
1386:Combinatory calculi
1325:{\displaystyle 3*3}
1292:{\displaystyle 3*3}
1271:The square of 3 is
1223:{\displaystyle x*x}
32:combinational logic
6634:Schönfinkel (1924)
6365:Schönfinkel, Moses
6357:Schönfinkel (1967)
6343:10.1007/bf01448013
6320:Schönfinkel, Moses
6097:. Addison-Wesley.
6087:Harrison, Peter G.
5956:. pp. 88â93.
5803:Schönfinkel (1967)
5702:
5661:
5635:
5609:
5583:
5557:
5521:
5489:
5457:
5431:
5405:
5379:
5353:
5318:
5272:
5248:
5073:Kenneth E. Iverson
4929:, a contradiction.
4811:Define NEGATION âĄ
4759:
4450:Reverse conversion
4158:) does reduce to (
4131:
4093:
4061:
4021:
3594:Schönfinkel (1924)
1704:
1607:
1537:
1430:Primitive function
1346:
1322:
1289:
1247:
1220:
1174:
1132:
1093:
1009:
963:
897:
866:
835:
796:
765:
734:
699:
664:Terms of the form
645:
614:
579:
535:
492:
464:Terms of the form
461:are lambda-terms.
449:
418:
387:
360:
312:
271:
46:mathematical logic
6745:Combinatory logic
6709:(29 April 2020).
6684:Combinator Birds.
6682:Rathman, Chris, "
6661:Combinatory Logic
6644:978-1-57955-044-8
6627:978-1-57955-043-1
6453:Smullyan, Raymond
6426:Smullyan, Raymond
6188:Hindley, J. Roger
6134:Hindley, J. Roger
6054:Combinatory Logic
6050:Hindley, J. Roger
6028:Combinatory Logic
5933:. Vol. 103.
5765:Cylindric algebra
5731:B, C, K, W system
5533:deduction theorem
5282:in this frame by
5085:tacit programming
5079:, a successor to
4757:
4245:B, C, K, W System
1460:
1459:
1395:Combinatory terms
1379:Combinatory logic
1349:{\displaystyle *}
1250:{\displaystyle *}
1144:as shorthand for
1012:{\displaystyle E}
390:{\displaystyle v}
274:{\displaystyle v}
128:Moses Schönfinkel
113:first-order logic
68:. It is based on
60:as a theoretical
50:Moses Schönfinkel
38:Combinatory logic
18:Combinatory Logic
16:(Redirected from
6762:
6731:
6729:
6727:
6707:Wolfram, Stephen
6637:
6608:Wolfram, Stephen
6603:
6582:
6547:Turner, David A.
6542:
6540:
6529:
6519:
6517:
6516:
6510:
6504:. Archived from
6489:
6477:
6448:
6421:
6419:
6417:
6412:
6402:
6360:
6337:(3â4): 305â316.
6328:
6315:
6289:
6259:
6245:
6243:
6241:
6236:
6211:
6196:(2nd ed.).
6183:
6129:
6108:
6096:
6081:
6079:
6067:
6041:
6015:
5983:
5948:
5914:
5908:
5902:
5896:
5890:
5884:
5878:
5872:
5866:
5860:
5854:
5848:
5842:
5836:
5830:
5824:
5818:
5812:
5806:
5799:Schönfinkel 1924
5796:
5711:
5709:
5708:
5703:
5670:
5668:
5667:
5662:
5644:
5642:
5641:
5636:
5618:
5616:
5615:
5610:
5592:
5590:
5589:
5584:
5566:
5564:
5563:
5558:
5530:
5528:
5527:
5522:
5498:
5496:
5495:
5490:
5466:
5464:
5463:
5458:
5440:
5438:
5437:
5432:
5414:
5412:
5411:
5406:
5388:
5386:
5385:
5380:
5362:
5360:
5359:
5354:
5327:
5325:
5324:
5319:
5281:
5279:
5278:
5273:
5257:
5255:
5254:
5249:
4696:
4682:
4647:Church encodings
4644:
4638:
4171:
4140:
4138:
4137:
4132:
4130:
4119:
4115:
4102:
4100:
4099:
4094:
4092:
4088:
4070:
4068:
4067:
4062:
4060:
4049:
4048:
4043:
4030:
4028:
4027:
4022:
4020:
4019:
4014:
3996:
3984:
3975:
3963:
3954:
3949:
3903:
3872:
3846:is free in both
3841:
3814:
3802:
3782:
3767:
3752:
3580:Combinators B, C
3232:
3178:
2745:
2744:
2740:
2216:
2189:
2177:
2157:
2142:
2127:
2114:
2106:
1775:
1750:
1715:
1713:
1711:
1710:
1705:
1700:
1699:
1678:
1677:
1665:
1664:
1655:
1654:
1618:
1616:
1614:
1613:
1608:
1606:
1605:
1587:
1586:
1577:
1576:
1567:
1566:
1548:
1546:
1544:
1543:
1538:
1533:
1532:
1514:
1513:
1504:
1503:
1494:
1493:
1402:
1357:
1355:
1353:
1352:
1347:
1333:
1331:
1329:
1328:
1323:
1300:
1298:
1296:
1295:
1290:
1265:formal parameter
1258:
1256:
1254:
1253:
1248:
1231:
1229:
1227:
1226:
1221:
1188:left associative
1185:
1183:
1181:
1180:
1175:
1143:
1141:
1139:
1138:
1133:
1104:
1102:
1100:
1099:
1094:
1033:. Thus we write
1032:
1028:
1024:
1020:
1018:
1016:
1015:
1010:
974:
972:
970:
969:
964:
959:
958:
946:
945:
908:
906:
904:
903:
898:
896:
895:
877:
875:
873:
872:
867:
865:
864:
846:
844:
842:
841:
836:
834:
833:
807:
805:
803:
802:
797:
795:
794:
776:
774:
772:
771:
766:
764:
763:
745:
743:
741:
740:
735:
733:
732:
710:
708:
706:
705:
700:
695:
694:
685:
684:
656:
654:
652:
651:
646:
644:
643:
625:
623:
621:
620:
615:
613:
612:
590:
588:
586:
585:
580:
578:
577:
546:
544:
542:
541:
536:
534:
533:
514:formal parameter
503:
501:
499:
498:
493:
491:
490:
460:
458:
456:
455:
450:
448:
447:
429:
427:
425:
424:
419:
417:
416:
398:
396:
394:
393:
388:
371:
369:
367:
366:
361:
356:
355:
346:
345:
323:
321:
319:
318:
313:
311:
310:
282:
280:
278:
277:
272:
183:computer science
109:expressive power
58:computer science
21:
6770:
6769:
6765:
6764:
6763:
6761:
6760:
6759:
6750:Lambda calculus
6735:
6734:
6725:
6723:
6705:
6653:
6628:
6606:
6600:
6585:
6563:10.2307/2273733
6545:
6538:
6527:
6522:
6514:
6512:
6508:
6502:
6494:. p. 442.
6487:
6480:
6471:
6451:
6442:
6424:
6415:
6413:
6410:
6405:
6391:
6363:
6326:
6318:
6312:
6292:
6262:
6248:
6239:
6237:
6214:
6208:
6186:
6156:10.2307/2274956
6132:
6111:
6105:
6084:
6077:
6070:
6064:
6044:
6038:
6018:
6004:10.2307/2370619
5986:
5972:
5951:
5945:
5925:
5922:
5917:
5909:
5905:
5897:
5893:
5885:
5881:
5873:
5869:
5861:
5857:
5849:
5845:
5839:Barendregt 1984
5837:
5833:
5825:
5821:
5813:
5809:
5797:
5793:
5789:
5776:Supercombinator
5761:Lambda calculus
5722:
5688:
5687:
5647:
5646:
5621:
5620:
5595:
5594:
5569:
5568:
5537:
5536:
5501:
5500:
5469:
5468:
5443:
5442:
5417:
5416:
5391:
5390:
5365:
5364:
5333:
5332:
5287:
5286:
5264:
5263:
5228:
5227:
5093:
5063:
5058:
4982:
4735:. A combinator
4697:). A predicate
4684:
4674:
4640:
4634:
4618:
4605:
4454:The conversion
4452:
4445:
4438:
4434:
4427:
4423:
4400:
4393:
4386:
4351:
4348:calculus, the λ
4347:
4343:
4339:
4335:
4328:
4320:
4317:
4313:
4169:
4110:
4105:
4104:
4083:
4075:
4074:
4038:
4033:
4032:
4009:
4001:
4000:
3987:
3979:
3966:
3957:
3950:
3947:
3888:
3857:
3826:
3805:
3794:
3787:is not free in
3770:
3755:
3744:
3584:In addition to
3582:
3410:
3408:One-point basis
3230:
3176:
3138:is not free in
3113:
3108:
2845:â) in place of
2821:â) in place of
2746:
2742:
2738:
2736:
2735:
2322:
2221:occurs free in
2201:
2194:occurs free in
2180:
2169:
2145:
2130:
2119:
2110:
2102:
2088:
1782:
1772:
1766:
1759:
1744:
1738:
1728:
1722:
1691:
1669:
1656:
1646:
1623:
1622:
1620:
1597:
1578:
1568:
1558:
1553:
1552:
1550:
1524:
1505:
1495:
1485:
1477:
1476:
1474:
1397:
1388:
1367:Turing machines
1360:Church encoding
1338:
1337:
1335:
1308:
1307:
1305:
1275:
1274:
1272:
1239:
1238:
1236:
1206:
1205:
1203:
1148:
1147:
1145:
1112:
1111:
1109:
1040:
1039:
1037:
1030:
1026:
1022:
986:
985:
983:
982:The expression
950:
937:
917:
916:
914:
887:
882:
881:
879:
856:
851:
850:
848:
825:
820:
819:
817:
786:
781:
780:
778:
755:
750:
749:
747:
724:
719:
718:
716:
686:
676:
668:
667:
665:
635:
630:
629:
627:
604:
599:
598:
596:
569:
555:
554:
552:
525:
520:
519:
517:
508:. The variable
482:
468:
467:
465:
439:
434:
433:
431:
408:
403:
402:
400:
379:
378:
376:
347:
337:
329:
328:
326:
302:
288:
287:
285:
263:
262:
260:
250:
248:Lambda calculus
244:
202:lambda calculus
179:
144:lambda calculus
94:
84:that uses only
78:predicate logic
35:
28:
23:
22:
15:
12:
11:
5:
6768:
6766:
6758:
6757:
6752:
6747:
6737:
6736:
6733:
6732:
6703:
6698:
6693:
6687:
6680:
6673:
6668:
6652:
6651:External links
6649:
6648:
6647:
6626:
6604:
6598:
6583:
6557:(2): 267â270.
6543:
6541:on 2016-03-04.
6520:
6501:978-0444520777
6500:
6478:
6470:978-0198534501
6469:
6449:
6440:
6422:
6403:
6390:978-0674324497
6389:
6361:
6316:
6310:
6290:
6274:(3): 343â347.
6260:
6246:
6212:
6206:
6184:
6130:
6120:(6): 281â286.
6109:
6103:
6082:
6068:
6062:
6042:
6036:
6016:
5984:
5970:
5949:
5943:
5921:
5918:
5916:
5915:
5903:
5891:
5879:
5875:Lachowski 2018
5867:
5855:
5843:
5831:
5819:
5807:
5790:
5788:
5785:
5784:
5783:
5778:
5773:
5768:
5758:
5753:
5748:
5743:
5738:
5733:
5728:
5721:
5718:
5701:
5698:
5695:
5660:
5657:
5654:
5634:
5631:
5628:
5608:
5605:
5602:
5582:
5579:
5576:
5567:is an element
5556:
5553:
5550:
5547:
5544:
5520:
5517:
5514:
5511:
5508:
5488:
5485:
5482:
5479:
5476:
5456:
5453:
5450:
5430:
5427:
5424:
5404:
5401:
5398:
5378:
5375:
5372:
5352:
5349:
5346:
5343:
5340:
5329:
5328:
5317:
5314:
5311:
5308:
5304:
5300:
5297:
5294:
5271:
5247:
5244:
5241:
5238:
5235:
5204:
5203:
5177:
5176:
5142:
5105:Hilbert system
5092:
5089:
5062:
5059:
5057:
5054:
5030:
5029:
5010:
4957:
4956:
4930:
4903:
4902:
4892:
4873:
4872:
4857:
4856:
4849:
4808:
4807:
4793:
4756:
4617:
4614:
4604:
4601:
4593:
4592:
4579:
4547:
4518:
4490:
4473:
4451:
4448:
4443:
4436:
4432:
4425:
4421:
4398:
4391:
4384:
4377:
4376:
4375:
4374:
4349:
4345:
4341:
4337:
4333:
4326:
4319:
4315:
4311:
4308:
4291:respectively.
4210:
4209:
4199:
4186:
4143:
4142:
4129:
4126:
4122:
4118:
4114:
4091:
4087:
4082:
4072:
4059:
4056:
4052:
4047:
4042:
4018:
4013:
4008:
3998:
3985:
3977:
3964:
3955:
3918:
3917:
3886:
3855:
3824:
3803:
3792:
3768:
3753:
3730:
3729:
3691:
3661:
3660:
3633:
3581:
3578:
3574:
3573:
3557:
3539:
3492:
3491:
3466:
3441:
3440:
3409:
3406:
3336:
3335:
3328:
3314:
3297:
3279:
3253:
3224:
3223:
3204:
3185:
3182:
3144:
3143:
3112:
3109:
3107:
3104:
3076:
3075:
3027:
3026:
2989:
2988:
2987:
2986:
2985:
2984:
2983:
2982:
2981:
2980:
2955:
2879:
2878:
2734:
2733:transformation
2727:
2675:
2674:
2673:
2672:
2669:
2662:
2651:
2637:
2619:
2597:
2572:
2498:
2497:
2496:
2495:
2468:
2441:
2416:
2397:
2384:
2377:
2370:
2363:
2353:
2321:
2318:
2314:
2313:
2291:
2282:
2231:
2230:
2199:
2178:
2167:
2143:
2128:
2087:
2084:
1994:
1993:
1992:
1991:
1985:
1968:
1922:is applied to
1901:
1900:
1874:
1873:
1847:for all terms
1845:
1844:
1806:for all terms
1804:
1803:
1781:
1778:
1770:
1764:
1752:
1751:
1742:
1736:
1721:
1718:
1703:
1698:
1694:
1690:
1687:
1684:
1681:
1676:
1672:
1668:
1663:
1659:
1653:
1649:
1645:
1642:
1639:
1636:
1633:
1630:
1604:
1600:
1596:
1593:
1590:
1585:
1581:
1575:
1571:
1565:
1561:
1536:
1531:
1527:
1523:
1520:
1517:
1512:
1508:
1502:
1498:
1492:
1488:
1484:
1468:free variables
1458:
1457:
1454:
1451:
1447:
1446:
1431:
1428:
1424:
1423:
1420:
1417:
1413:
1412:
1409:
1406:
1396:
1393:
1387:
1384:
1345:
1321:
1318:
1315:
1302:
1301:
1288:
1285:
1282:
1246:
1233:
1232:
1219:
1216:
1213:
1198:The square of
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1131:
1128:
1125:
1122:
1119:
1106:
1105:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1008:
1005:
1002:
999:
996:
993:
962:
957:
953:
949:
944:
940:
936:
933:
930:
927:
924:
894:
890:
863:
859:
832:
828:
793:
789:
762:
758:
731:
727:
698:
693:
689:
683:
679:
675:
642:
638:
611:
607:
576:
572:
568:
565:
562:
532:
528:
512:is called the
489:
485:
481:
478:
475:
446:
442:
415:
411:
386:
373:
372:
359:
354:
350:
344:
340:
336:
324:
309:
305:
301:
298:
295:
283:
270:
246:Main article:
243:
240:
219:languages and
210:free variables
178:
175:
93:
92:In mathematics
90:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6767:
6756:
6753:
6751:
6748:
6746:
6743:
6742:
6740:
6722:
6718:
6714:
6713:
6708:
6704:
6702:
6699:
6697:
6694:
6692:(Java Applet)
6691:
6688:
6685:
6681:
6678:
6674:
6672:
6669:
6666:
6665:Katalin BimbĂł
6662:
6658:
6655:
6654:
6650:
6645:
6641:
6636:
6635:
6629:
6623:
6619:
6618:Wolfram Media
6615:
6614:
6609:
6605:
6601:
6599:5-89158-101-9
6595:
6591:
6590:
6584:
6580:
6576:
6572:
6568:
6564:
6560:
6556:
6552:
6548:
6544:
6537:
6533:
6526:
6521:
6511:on 2005-10-16
6507:
6503:
6497:
6493:
6486:
6485:
6479:
6476:
6472:
6466:
6462:
6458:
6454:
6450:
6447:
6443:
6441:0-394-53491-3
6437:
6433:
6432:
6427:
6423:
6409:
6404:
6400:
6396:
6392:
6386:
6382:
6378:
6374:
6370:
6366:
6362:
6359:
6358:
6352:
6348:
6344:
6340:
6336:
6333:(in German).
6332:
6325:
6321:
6317:
6313:
6311:9780674798373
6307:
6303:
6299:
6295:
6291:
6288:
6287:
6281:
6277:
6273:
6269:
6265:
6261:
6257:
6256:
6251:
6247:
6235:
6230:
6227:(53): 19â42.
6226:
6222:
6218:
6213:
6209:
6207:9780521898850
6203:
6199:
6195:
6194:
6189:
6185:
6181:
6177:
6173:
6169:
6165:
6161:
6157:
6153:
6150:(1): 90â105.
6149:
6145:
6144:
6139:
6135:
6131:
6127:
6123:
6119:
6115:
6110:
6106:
6104:0-201-19249-7
6100:
6095:
6094:
6088:
6083:
6076:
6075:
6069:
6065:
6063:0-7204-2208-6
6059:
6055:
6051:
6047:
6043:
6039:
6037:0-7204-2208-6
6033:
6029:
6025:
6021:
6017:
6013:
6009:
6005:
6001:
5997:
5994:(in German).
5993:
5989:
5985:
5981:
5977:
5973:
5967:
5963:
5959:
5955:
5950:
5946:
5944:0-444-87508-5
5940:
5936:
5935:North Holland
5932:
5928:
5924:
5923:
5919:
5912:
5907:
5904:
5900:
5895:
5892:
5888:
5887:Goldberg 2004
5883:
5880:
5876:
5871:
5868:
5864:
5859:
5856:
5852:
5847:
5844:
5840:
5835:
5832:
5828:
5823:
5820:
5816:
5811:
5808:
5804:
5800:
5795:
5792:
5786:
5782:
5779:
5777:
5774:
5772:
5769:
5766:
5762:
5759:
5757:
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5732:
5729:
5727:
5724:
5723:
5719:
5717:
5715:
5699:
5696:
5693:
5685:
5681:
5677:
5672:
5658:
5655:
5652:
5632:
5629:
5626:
5606:
5603:
5600:
5580:
5577:
5574:
5551:
5545:
5542:
5534:
5518:
5515:
5512:
5509:
5506:
5486:
5480:
5477:
5474:
5454:
5451:
5448:
5428:
5425:
5422:
5402:
5399:
5396:
5389:is such that
5376:
5373:
5370:
5350:
5344:
5341:
5338:
5315:
5312:
5309:
5306:
5298:
5295:
5292:
5285:
5284:
5283:
5269:
5261:
5242:
5239:
5236:
5225:
5221:
5217:
5213:
5209:
5201:
5197:
5193:
5189:
5185:
5182:
5181:
5180:
5174:
5170:
5166:
5162:
5158:
5154:
5150:
5146:
5143:
5140:
5136:
5132:
5128:
5125:
5124:
5123:
5121:
5117:
5112:
5110:
5106:
5102:
5098:
5090:
5088:
5086:
5082:
5078:
5074:
5070:
5068:
5060:
5055:
5053:
5051:
5045:
5043:
5039:
5035:
5027:
5023:
5019:
5015:
5011:
5008:
5004:
5000:
4996:
4992:
4991:
4990:
4988:
4981:
4980:
4979:
4974:
4970:
4966:
4962:
4954:
4950:
4947:
4943:
4939:
4935:
4931:
4928:
4924:
4921:
4917:
4913:
4909:
4905:
4904:
4901:
4897:
4893:
4890:
4886:
4882:
4881:
4880:
4878:
4870:
4866:
4862:
4861:
4860:
4854:
4850:
4847:
4844:
4840:
4837:
4833:
4829:
4825:
4821:
4818:
4814:
4810:
4809:
4805:
4801:
4798:
4794:
4791:
4787:
4784:
4780:
4779:
4778:
4776:
4772:
4768:
4764:
4755:
4753:
4749:
4746:
4742:
4738:
4734:
4730:
4727:
4723:
4719:
4716:
4712:
4708:
4704:
4700:
4694:
4691:
4687:
4681:
4677:
4672:
4668:
4664:
4660:
4656:
4652:
4648:
4643:
4637:
4632:
4628:
4624:
4615:
4613:
4610:
4602:
4600:
4598:
4590:
4587:
4583:
4580:
4577:
4574:
4570:
4567:
4563:
4559:
4555:
4551:
4548:
4545:
4542:
4538:
4534:
4530:
4526:
4522:
4519:
4516:
4513:
4510:
4506:
4502:
4498:
4494:
4491:
4489:
4485:
4481:
4477:
4474:
4472:
4468:
4464:
4461:
4460:
4459:
4457:
4449:
4447:
4442:
4431:
4420:
4416:
4412:
4408:
4404:
4397:
4390:
4382:
4372:
4368:
4364:
4360:
4357:
4356:
4355:
4354:
4353:
4332:
4325:
4309:
4307:
4305:
4301:
4297:
4292:
4290:
4286:
4282:
4278:
4274:
4270:
4266:
4262:
4258:
4254:
4250:
4246:
4242:
4241:Haskell Curry
4237:
4235:
4231:
4227:
4223:
4219:
4215:
4207:
4204:
4200:
4197:
4194:
4191:
4187:
4184:
4181:
4178:
4175:
4168:
4167:
4166:
4164:
4161:
4157:
4154:
4151:
4148:
4145:And indeed, (
4120:
4116:
4089:
4080:
4073:
4050:
4045:
4016:
4006:
3999:
3997:(η-reduction)
3994:
3991:
3986:
3983:
3978:
3973:
3970:
3965:
3961:
3956:
3953:
3946:
3945:
3944:
3942:
3939:
3935:
3931:
3927:
3923:
3915:
3911:
3907:
3901:
3898:
3895:
3891:
3887:
3884:
3880:
3876:
3870:
3867:
3864:
3860:
3856:
3853:
3849:
3845:
3839:
3836:
3833:
3829:
3825:
3822:
3818:
3812:
3808:
3804:
3801:
3797:
3793:
3790:
3786:
3780:
3777:
3773:
3769:
3765:
3762:
3758:
3754:
3751:
3747:
3743:
3742:
3741:
3739:
3735:
3727:
3723:
3719:
3715:
3711:
3707:
3703:
3699:
3695:
3692:
3689:
3685:
3681:
3677:
3674:
3673:
3672:
3670:
3666:
3658:
3655:
3651:
3647:
3644:
3641:
3638:
3634:
3631:
3627:
3624:
3620:
3617:
3614:
3611:
3607:
3606:
3605:
3603:
3599:
3595:
3591:
3587:
3579:
3577:
3572:
3568:
3565:
3561:
3558:
3555:
3551:
3547:
3544:
3540:
3537:
3534:
3531:
3527:
3523:
3520:
3519:
3518:
3515:
3513:
3509:
3505:
3501:
3497:
3489:
3485:
3482:
3478:
3474:
3470:
3467:
3464:
3460:
3457:
3453:
3449:
3446:
3445:
3444:
3438:
3434:
3430:
3426:
3423:
3422:
3421:
3419:
3415:
3407:
3405:
3403:
3399:
3396:
3392:
3388:
3384:
3380:
3376:
3372:
3368:
3364:
3360:
3356:
3353:
3349:
3345:
3341:
3333:
3329:
3326:
3323:
3319:
3315:
3312:
3309:
3305:
3302:
3298:
3295:
3291:
3288:
3284:
3280:
3277:
3273:
3270:
3266:
3262:
3258:
3254:
3251:
3248:
3244:
3240:
3236:
3229:
3228:
3227:
3221:
3217:
3213:
3209:
3205:
3202:
3198:
3194:
3190:
3186:
3183:
3181:
3175:
3174:
3173:
3170:
3168:
3164:
3160:
3156:
3152:
3148:
3141:
3137:
3133:
3129:
3126:
3125:
3124:
3122:
3118:
3110:
3105:
3103:
3101:
3097:
3093:
3089:
3085:
3081:
3073:
3070:
3067:
3063:
3062:
3061:
3059:
3055:
3051:
3047:
3044:
3040:
3036:
3032:
3024:
3020:
3016:
3012:
3009:
3005:
3001:
2997:
2994:
2993:
2992:
2978:
2974:
2970:
2967:
2963:
2960:
2956:
2953:
2949:
2945:
2941:
2937:
2934:
2930:
2929:
2928:
2927:
2926:
2925:
2924:
2923:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2893:
2889:
2885:
2881:
2880:
2876:
2872:
2868:
2864:
2860:
2859:
2858:
2856:
2852:
2848:
2844:
2840:
2836:
2832:
2828:
2824:
2820:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2786:
2782:
2778:
2775:
2771:
2767:
2763:
2759:
2755:
2751:
2741:
2732:
2728:
2726:
2724:
2720:
2716:
2712:
2708:
2704:
2700:
2696:
2692:
2688:
2684:
2680:
2670:
2667:
2663:
2660:
2656:
2652:
2649:
2645:
2642:
2638:
2635:
2631:
2628:
2624:
2620:
2617:
2613:
2610:
2606:
2602:
2598:
2595:
2591:
2588:
2584:
2580:
2577:
2573:
2570:
2566:
2563:
2559:
2555:
2552:
2548:
2544:
2543:
2541:
2537:
2534:
2530:
2526:
2523:
2519:
2515:
2511:
2510:
2509:
2507:
2503:
2493:
2489:
2485:
2481:
2477:
2473:
2469:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2439:
2436:
2433:
2429:
2425:
2421:
2417:
2414:
2410:
2406:
2402:
2398:
2395:
2392:
2389:
2385:
2382:
2378:
2375:
2371:
2368:
2364:
2361:
2358:
2354:
2351:
2347:
2346:
2345:
2342:
2341:
2340:
2338:
2335:
2331:
2327:
2319:
2317:
2311:
2307:
2303:
2299:
2296:
2292:
2290:
2286:
2283:
2281:
2278:
2274:
2271:
2270:
2269:
2267:
2263:
2259:
2255:
2250:
2248:
2244:
2239:
2236:
2228:
2224:
2220:
2214:
2211:
2208:
2204:
2200:
2197:
2193:
2187:
2183:
2179:
2176:
2172:
2168:
2165:
2161:
2155:
2152:
2148:
2144:
2140:
2137:
2133:
2129:
2126:
2122:
2118:
2117:
2116:
2113:
2108:
2105:
2100:
2096:
2092:
2085:
2083:
2081:
2077:
2073:
2068:
2066:
2062:
2058:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2011:
2007:
2003:
1999:
1996:for any term
1990:
1986:
1983:
1980:
1976:
1973:
1969:
1966:
1963:
1959:
1958:
1956:
1952:
1948:
1947:
1946:
1944:
1940:
1936:
1931:
1929:
1925:
1921:
1917:
1913:
1909:
1905:
1898:
1894:
1890:
1887:
1883:
1882:
1881:
1879:
1872:
1868:
1865:
1862:
1858:
1857:
1856:
1854:
1850:
1843:
1839:
1835:
1832:
1828:
1827:
1826:
1824:
1820:
1817:
1813:
1809:
1802:
1798:
1795:
1791:
1790:
1789:
1787:
1779:
1777:
1773:
1763:
1757:
1749:
1745:
1735:
1732:
1727:
1726:
1725:
1719:
1717:
1696:
1692:
1688:
1685:
1682:
1674:
1670:
1661:
1657:
1651:
1647:
1637:
1634:
1631:
1602:
1598:
1594:
1591:
1588:
1583:
1579:
1573:
1569:
1563:
1559:
1529:
1525:
1521:
1518:
1515:
1510:
1506:
1500:
1496:
1490:
1486:
1471:
1469:
1465:
1455:
1452:
1449:
1448:
1444:
1440:
1436:
1432:
1429:
1426:
1425:
1421:
1418:
1415:
1414:
1410:
1407:
1404:
1403:
1400:
1394:
1392:
1385:
1383:
1380:
1374:
1372:
1368:
1363:
1361:
1343:
1319:
1316:
1313:
1286:
1283:
1280:
1270:
1269:
1268:
1266:
1262:
1244:
1217:
1214:
1211:
1201:
1197:
1196:
1195:
1191:
1189:
1168:
1162:
1159:
1126:
1123:
1120:
1087:
1084:
1081:
1075:
1066:
1060:
1057:
1054:
1051:
1036:
1035:
1034:
1003:
1000:
997:
991:
980:
978:
955:
951:
942:
938:
934:
931:
928:
912:
892:
888:
861:
857:
830:
826:
815:
811:
791:
787:
760:
756:
729:
725:
714:
691:
687:
681:
677:
662:
660:
640:
636:
609:
605:
594:
574:
570:
566:
563:
560:
550:
530:
526:
515:
511:
507:
487:
483:
479:
476:
473:
462:
444:
440:
413:
409:
384:
352:
348:
342:
338:
325:
307:
303:
299:
296:
293:
284:
268:
259:
258:
257:
255:
249:
241:
239:
237:
233:
228:
226:
222:
218:
215:
211:
207:
203:
198:
196:
192:
188:
184:
176:
174:
172:
169:
165:
161:
157:
153:
149:
148:Haskell Curry
145:
141:
140:Alonzo Church
137:
133:
132:Haskell Curry
129:
124:
122:
118:
114:
110:
106:
103:
99:
91:
89:
87:
83:
79:
75:
71:
67:
63:
59:
55:
54:Haskell Curry
51:
47:
44:variables in
43:
39:
33:
19:
6726:26 September
6724:. Retrieved
6711:
6631:
6612:
6588:
6554:
6550:
6536:the original
6531:
6513:. Retrieved
6506:the original
6483:
6474:
6456:
6445:
6429:
6416:17 September
6414:. Retrieved
6376:
6372:
6354:
6334:
6330:
6297:
6286:Quine (1996)
6283:
6271:
6267:
6254:
6238:. Retrieved
6224:
6220:
6192:
6147:
6141:
6117:
6113:
6092:
6073:
6053:
6027:
6024:Feys, Robert
5995:
5991:
5953:
5930:
5911:Cherlin 1991
5906:
5899:Engeler 1995
5894:
5882:
5870:
5858:
5846:
5834:
5822:
5810:
5794:
5713:
5683:
5679:
5675:
5673:
5330:
5260:Kripke frame
5219:
5215:
5211:
5207:
5205:
5199:
5195:
5191:
5187:
5183:
5178:
5172:
5168:
5164:
5160:
5156:
5152:
5148:
5144:
5138:
5134:
5130:
5126:
5119:
5115:
5113:
5109:proof theory
5094:
5071:
5064:
5056:Applications
5049:
5046:
5041:
5037:
5033:
5031:
5025:
5021:
5017:
5013:
5006:
5002:
4998:
4994:
4986:
4983:
4976:
4972:
4968:
4964:
4960:
4958:
4952:
4948:
4945:
4941:
4940:ABSURDUM) =
4937:
4933:
4926:
4922:
4919:
4915:
4914:ABSURDUM) =
4911:
4907:
4899:
4898:ABSURDUM) =
4895:
4888:
4887:ABSURDUM) =
4884:
4876:
4874:
4868:
4864:
4863:ABSURDUM ⥠(
4858:
4852:
4845:
4842:
4838:
4835:
4831:
4827:
4823:
4819:
4816:
4812:
4803:
4799:
4796:
4789:
4785:
4782:
4774:
4770:
4766:
4762:
4760:
4751:
4747:
4744:
4740:
4736:
4732:
4728:
4725:
4721:
4717:
4714:
4710:
4706:
4702:
4698:
4692:
4689:
4685:
4679:
4675:
4670:
4666:
4662:
4658:
4654:
4650:
4641:
4635:
4630:
4626:
4622:
4619:
4606:
4596:
4594:
4588:
4585:
4581:
4575:
4572:
4568:
4565:
4561:
4557:
4553:
4549:
4543:
4540:
4536:
4532:
4528:
4524:
4520:
4514:
4511:
4508:
4504:
4500:
4496:
4492:
4487:
4483:
4479:
4475:
4470:
4466:
4462:
4455:
4453:
4440:
4429:
4418:
4414:
4410:
4406:
4402:
4395:
4388:
4380:
4378:
4370:
4366:
4362:
4358:
4330:
4323:
4321:
4304:Tromp (2008)
4299:
4295:
4293:
4288:
4284:
4280:
4276:
4272:
4271:were called
4268:
4264:
4260:
4256:
4252:
4238:
4233:
4229:
4225:
4221:
4217:
4213:
4211:
4205:
4202:
4195:
4192:
4189:
4182:
4179:
4176:
4173:
4170:
4162:
4159:
4155:
4152:
4149:
4146:
4144:
3992:
3989:
3981:
3971:
3968:
3959:
3951:
3948:
3940:
3937:
3933:
3929:
3925:
3921:
3919:
3913:
3909:
3905:
3899:
3896:
3893:
3889:
3882:
3878:
3874:
3868:
3865:
3862:
3858:
3851:
3847:
3843:
3837:
3834:
3831:
3827:
3820:
3816:
3810:
3806:
3799:
3795:
3788:
3784:
3778:
3775:
3771:
3763:
3760:
3756:
3749:
3745:
3738:David Turner
3731:
3725:
3721:
3717:
3713:
3709:
3705:
3701:
3697:
3693:
3687:
3683:
3679:
3675:
3668:
3664:
3662:
3656:
3653:
3649:
3645:
3642:
3639:
3636:
3629:
3625:
3622:
3618:
3615:
3612:
3609:
3601:
3597:
3589:
3585:
3583:
3575:
3570:
3566:
3563:
3559:
3553:
3549:
3545:
3542:
3535:
3532:
3529:
3525:
3521:
3516:
3511:
3503:
3499:
3495:
3493:
3487:
3483:
3480:
3476:
3472:
3468:
3462:
3458:
3455:
3451:
3447:
3442:
3436:
3432:
3428:
3424:
3417:
3413:
3411:
3401:
3397:
3394:
3390:
3386:
3382:
3378:
3374:
3370:
3366:
3362:
3358:
3354:
3351:
3347:
3343:
3339:
3337:
3331:
3324:
3321:
3317:
3310:
3307:
3303:
3300:
3293:
3289:
3286:
3282:
3275:
3271:
3268:
3264:
3260:
3256:
3249:
3246:
3242:
3238:
3234:
3231:
3225:
3219:
3215:
3211:
3207:
3200:
3196:
3192:
3188:
3179:
3177:
3171:
3166:
3162:
3158:
3154:
3150:
3146:
3145:
3139:
3135:
3131:
3127:
3120:
3116:
3114:
3099:
3095:
3091:
3087:
3083:
3079:
3077:
3071:
3068:
3065:
3057:
3053:
3049:
3045:
3042:
3038:
3034:
3030:
3028:
3022:
3018:
3014:
3010:
3007:
3003:
2999:
2995:
2990:
2976:
2972:
2968:
2965:
2961:
2958:
2951:
2947:
2943:
2939:
2935:
2932:
2919:
2915:
2911:
2907:
2903:
2899:
2895:
2891:
2887:
2883:
2874:
2870:
2866:
2862:
2854:
2850:
2846:
2842:
2838:
2834:
2830:
2826:
2825:, yielding (
2822:
2818:
2814:
2810:
2806:
2802:
2798:
2797:
2793:
2789:
2784:
2780:
2776:
2773:
2769:
2765:
2761:
2757:
2753:
2749:
2747:
2730:
2722:
2714:
2710:
2706:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2676:
2665:
2658:
2654:
2647:
2643:
2640:
2633:
2629:
2626:
2622:
2615:
2611:
2608:
2604:
2600:
2593:
2589:
2586:
2582:
2578:
2575:
2568:
2564:
2561:
2557:
2553:
2550:
2546:
2539:
2535:
2532:
2528:
2524:
2521:
2517:
2513:
2505:
2501:
2499:
2491:
2487:
2483:
2479:
2475:
2471:
2464:
2460:
2456:
2452:
2448:
2444:
2437:
2434:
2431:
2427:
2423:
2419:
2412:
2408:
2404:
2400:
2393:
2390:
2387:
2380:
2373:
2366:
2359:
2356:
2349:
2343:
2336:
2333:
2329:
2325:
2323:
2315:
2309:
2305:
2301:
2297:
2294:
2288:
2284:
2279:
2276:
2272:
2265:
2261:
2257:
2253:
2251:
2242:
2240:
2234:
2232:
2226:
2222:
2218:
2212:
2209:
2206:
2202:
2195:
2191:
2185:
2181:
2174:
2170:
2163:
2159:
2153:
2150:
2146:
2138:
2135:
2131:
2124:
2120:
2111:
2109:
2103:
2098:
2094:
2090:
2089:
2075:
2069:
2064:
2060:
2056:
2052:
2048:
2040:
2036:
2032:
2024:
2020:
2016:
2012:
2009:
2005:
2001:
1997:
1995:
1988:
1981:
1978:
1974:
1971:
1964:
1961:
1954:
1950:
1942:
1938:
1934:
1932:
1927:
1923:
1919:
1915:
1911:
1907:
1903:
1902:
1896:
1892:
1888:
1885:
1877:
1875:
1870:
1866:
1863:
1860:
1852:
1848:
1846:
1841:
1837:
1833:
1830:
1825:, so we say
1822:
1818:
1815:
1811:
1807:
1805:
1800:
1796:
1793:
1785:
1783:
1768:
1761:
1755:
1753:
1747:
1740:
1733:
1730:
1723:
1472:
1463:
1461:
1442:
1438:
1434:
1411:Description
1398:
1389:
1378:
1375:
1364:
1303:
1263:here is the
1260:
1234:
1199:
1192:
1107:
981:
910:
813:
809:
713:applications
712:
663:
658:
592:
548:
509:
506:abstractions
505:
463:
374:
254:lambda-terms
253:
251:
236:model theory
229:
205:
199:
195:proof theory
180:
177:In computing
159:
125:
107:. While the
95:
69:
37:
36:
6240:9 September
5863:Turner 1979
5827:Seldin 2008
4609:normal form
4439:terms into
4249:Schönfinkel
3976:(by rule 7)
3908:is free in
3877:is free in
3819:is free in
3121:η-reduction
3111:η-reduction
1464:combinators
1453:Application
1029:in it with
977:normal form
711:are called
504:are called
206:combinators
187:computation
152:Robert Feys
74:Schönfinkel
70:combinators
6739:Categories
6515:2017-04-22
5971:0897914414
5920:Literature
5815:Curry 1930
5787:References
5593:such that
4777:such that
4765:. Because
4713:such that
4703:nontrivial
4224:. Whereas
3006:â) = (
2376:))] (by 3)
2300:) :=
2233:Note that
2061:equivalent
2015:) for any
1549:, or even
911:equivalent
232:Dana Scott
214:non-strict
189:, used in
171:Dana Scott
164:Barendregt
42:quantified
6434:. Knopf.
6399:503886453
6367:(1967) .
6351:118507515
5630:⊩
5604:⊇
5578:∈
5546:∪
5484:→
5452:⊩
5426:⊩
5400:⊇
5374:∈
5348:→
5342:⊩
5310:∈
5303:⟺
5296:⊩
5270:⊩
5246:⟩
5243:⊆
5234:⟨
5224:inclusion
4855:NEGATION)
4623:predicate
4340:and the λ
4314:versus CL
4046:∗
4017:∗
3420:} where:
2494:)) (by 4)
2467:)) (by 3)
2440:)) (by 6)
2369:)] (by 4)
2362:)] (by 6)
2287: :=
2275: :=
2080:recursion
1344:∗
1317:∗
1284:∗
1245:∗
1215:∗
1073:⇒
1052:λ
929:λ
810:applicand
561:λ
474:λ
294:λ
6638:(eBook:
6610:(2021).
6579:35835482
6492:Elsevier
6455:(1994).
6428:(1985).
6322:(1924).
6252:(1995).
6089:(1998).
6026:(1958).
5980:25802202
5929:(1984).
5720:See also
5697:⊮
5656:⊮
5516:⊬
5478:⊮
4932:Case 2:
4906:Case 1:
4875:Because
4741:complete
4318:calculus
4117:′
4090:′
3912:but not
3881:but not
3357:) into (
3060:â), and
2415:) (by 3)
2396:) (by 6)
2352:] (by 5)
2249:above).
1906:applies
1419:Variable
1235:(Using "
225:Unlambda
221:hardware
6721:YouTube
6571:2273733
6371:(ed.).
6180:6930576
6172:1043546
6164:2274956
6012:2370619
5531:by the
5499:, then
5441:, then
5226:. Then
5186:: from
5159:)) â ((
5040:(EQUAL
5012:(EQUAL
4993:(EQUAL
4959:Hence (
4822:) then
4633:(where
3494:Since {
2671:= (y x)
2542:) x y)
1714:
1621:
1617:
1551:
1547:
1475:
1356:
1336:
1332:
1306:
1299:
1273:
1257:
1237:
1230:
1204:
1184:
1146:
1142:
1110:
1103:
1038:
1019:
984:
973:
915:
907:
880:
876:
849:
845:
818:
814:reduced
806:
779:
775:
748:
744:
717:
709:
666:
655:
628:
624:
597:
589:
553:
547:is the
545:
518:
502:
466:
459:
432:
428:
401:
397:
377:
370:
327:
322:
286:
281:
261:
156:Belgium
102:Quine's
6642:
6624:
6596:
6577:
6569:
6498:
6467:
6438:
6397:
6387:
6349:
6308:
6280:985250
6278:
6204:
6178:
6170:
6162:
6101:
6060:
6034:
6010:
5978:
5968:
5941:
5712:, and
5645:, and
5363:, and
5214:, and
5198:infer
4978:Q.E.D.
4815:.(if (
4365:where
4287:, and
4247:). In
3920:Using
3621:) = ((
3538:) with
3486:))) =
3123:rule:
3082:â and
2898:) = ((
2869:â) = (
2857:â, so
2853:â and
2837:into (
2764:, and
2664:= (y (
2661:x) y))
2653:= (y (
2650:x) y))
2639:= (y (
2621:= (y (
2556:) x (
2482:)) (
2455:)) (
2430:)) (
2383:(by 1)
2051:) and
1933:Given
1754:where
1405:Syntax
375:where
168:models
160:et al.
6663:" by
6575:S2CID
6567:JSTOR
6539:(PDF)
6528:(PDF)
6509:(PDF)
6488:(PDF)
6411:(PDF)
6375:[
6347:S2CID
6327:(PDF)
6276:JSTOR
6176:S2CID
6160:JSTOR
6078:(PDF)
6008:JSTOR
5976:S2CID
5167:) â (
5091:Logic
4826:else
4758:Proof
4401:are:
3734:Curry
3648:) = (
3461:)) =
3245:))
3218:))
3199:))
3184:= ...
2779:) if
2636:x y))
2618:x y))
2596:x) y)
2571:x) y)
2411:))
2225:â or
2065:equal
2057:S K S
2049:S K K
2045:up to
2041:equal
2033:equal
2021:S K K
2008:) = (
2002:S K K
1962:S K K
1951:S K K
1891:) = (
1889:x y z
1450:(M N)
6728:2023
6640:ISBN
6622:ISBN
6594:ISBN
6496:ISBN
6465:ISBN
6436:ISBN
6418:2023
6395:OCLC
6385:ISBN
6306:ISBN
6242:2018
6225:2018
6202:ISBN
6099:ISBN
6058:ISBN
6032:ISBN
5966:ISBN
5939:ISBN
5763:and
5674:Let
5415:and
5190:and
5118:and
5114:The
5095:The
5016:) =
4997:) =
4967:nor
4951:) =
4925:) =
4830:) âĄ
4802:) =
4788:) =
4773:and
4724:and
4709:and
4683:and
4661:and
4639:and
4413:and
4298:and
4267:and
4216:and
3924:and
3904:(if
3873:(if
3850:and
3842:(if
3815:(if
3783:(if
3667:and
3600:and
3588:and
3569:) =
3528:.(x
3508:Iota
3431:.((x
3381:)) (
3134:(if
2957:= ((
2748:The
2739:edit
2689:)) (
2632:)
2614:)
2592:)
2567:)
2527:)) (
2504:and
2217:(if
2190:(if
2158:(if
2093:and
1970:= (
1960:= (
1937:and
1869:) =
1851:and
1840:) =
1799:) =
1767:...
1746:) =
1739:...
1408:Name
549:body
430:and
193:and
52:and
6719:on
6659:: "
6559:doi
6339:doi
6272:104
6229:doi
6152:doi
6122:doi
6000:doi
5958:doi
5175:)),
5151:â (
5147:: (
5133:â (
5107:in
5081:APL
5042:x A
5038:λx.
5020:if
5014:A B
5009:and
5001:if
4995:A B
4936:= (
4910:= (
4834:.((
4792:and
4743:if
4739:is
4701:is
4688:= (
4629:or
4584:= (
4201:= (
4188:= (
4165:):
3988:= (
3892:â (
3861:â (
3830:â (
3774:â (
3759:â (
3726:K K
3724:) (
3720:))
3714:K S
3696:= (
3684:K S
3678:= (
3556:and
3465:and
3414:any
3383:K I
3375:K K
3369:) (
3367:K S
3332:y x
3330:= (
3325:x y
3316:= (
3311:x y
3299:= (
3283:S I
3281:= (
3261:S I
3255:= (
3250:x y
3243:S I
3216:S I
3206:= (
3197:S I
3187:= (
2975:â)
2931:= (
2922:))
2910:) (
2894:â)
2725:).
2695:K K
2687:S I
2668:x))
2646:x (
2603:y (
2599:= (
2574:= (
2545:= (
2488:K K
2480:S I
2470:= (
2461:K K
2453:S I
2443:= (
2428:S I
2418:= (
2409:S I
2399:= (
2386:= (
2262:E x
2205:â (
2149:â (
2134:â (
2099:any
2074:or
2019:, (
1987:=
1910:to
1897:y z
1893:x z
1202:is
1190:).
181:In
154:in
123:).
6741::
6715:.
6630:.
6620:.
6616:.
6573:.
6565:.
6555:44
6553:.
6473:.
6463:.
6444:.
6393:.
6353:.
6345:.
6335:92
6329:.
6282:.
6270:.
6223:.
6219:.
6200:.
6174:.
6168:MR
6166:.
6158:.
6148:55
6146:.
6140:.
6118:89
6116:.
6048:;
6022:;
6006:.
5996:52
5974:.
5964:.
5937:.
5671:.
5619:,
5216:MP
5212:AS
5210:,
5208:AK
5194:â
5184:MP
5171:â
5163:â
5155:â
5145:AS
5141:),
5137:â
5129::
5127:AK
5111:.
5069:.
5036:,
5024:â
5005:=
4987:no
4891:or
4841:)
4832:λx
4813:λx
4731:=
4720:=
4678:=
4667:λy
4663:λx
4655:λy
4651:λx
4607:A
4578:))
4564:.(
4562:λz
4558:λy
4554:λx
4552:=
4546:))
4535:.(
4533:λz
4529:λy
4525:λx
4523:=
4507:.(
4505:λz
4501:λy
4497:λx
4495:=
4484:λy
4480:λx
4478:=
4467:λx
4465:=
4441:CL
4430:CL
4419:CL
4409:,
4405:,
4396:CL
4389:CL
4359:λx
4331:CL
4324:CL
4310:CL
4306:.
4283:,
4279:,
4275:,
4263:,
4259:,
4255:,
3980:=
3974:)]
3967:=
3958:=
3936:.(
3934:λy
3930:λx
3914:Eâ
3910:Eâ
3883:Eâ
3879:Eâ
3852:Eâ
3848:Eâ
3809:â
3798:â
3748:â
3728:))
3716:)
3686:)
3671::
3659:))
3628:)
3592:,
3567:X'
3564:X'
3560:X'
3552:=
3550:X'
3548:)
3546:X'
3543:X'
3526:λx
3524:âĄ
3522:X'
3498:,
3429:λx
3427:âĄ
3404:.
3393:.(
3391:λx
3387:λf
3377:)
3350:.(
3348:λx
3344:λf
3327:))
3313:))
3292:)
3274:)
3263:)
3149:.(
3147:λx
3130:=
3102:.
3096:λx
3090:â
3054:λx
3052:â
3046:λx
3037:â
3033:.(
3031:λx
3025:â)
3019:λx
3017:â
3011:λx
3002:â
2998:.(
2996:λx
2969:λx
2966:Eâ
2962:λx
2950:â
2944:λx
2942:â
2936:λx
2918:â
2912:λx
2906:â
2900:λx
2890:â
2886:.(
2884:λx
2877:â)
2873:â
2865:â
2841:â
2829:â
2817:â
2805:â
2801:.(
2799:λx
2787:.
2766:λx
2754:λx
2707:λy
2703:λx
2697:)
2538:)
2490:)
2463:)
2379:=
2372:=
2365:=
2355:=
2348:=
2332:.(
2330:λy
2326:λx
2310:Eâ
2308:)(
2306:Eâ
2298:Eâ
2295:Eâ
2264:=
2229:â)
2184:â
2173:â
2123:â
2082:.
2004:)
1984:))
1957:)
1953:)
1949:((
1941:,
1930:.
1899:))
1836:)
1829:((
1470:.
1445:.
1441:,
1437:,
1362:.
1085::=
1001::=
979:.
816::
6730:.
6679:"
6667:.
6646:)
6602:.
6581:.
6561::
6518:.
6420:.
6401:.
6341::
6314:.
6244:.
6231::
6210:.
6182:.
6154::
6128:.
6124::
6107:.
6066:.
6040:.
6014:.
6002::
5982:.
5960::
5947:.
5913:.
5901:.
5889:.
5877:.
5865:.
5853:.
5841:.
5829:.
5817:.
5805:.
5714:A
5700:A
5694:X
5684:X
5680:A
5676:A
5659:B
5653:Y
5633:A
5627:Y
5607:X
5601:Y
5581:W
5575:Y
5555:}
5552:A
5549:{
5543:X
5519:B
5513:A
5510:,
5507:X
5487:B
5481:A
5475:X
5455:B
5449:Y
5429:A
5423:Y
5403:X
5397:Y
5377:W
5371:Y
5351:B
5345:A
5339:X
5316:.
5313:X
5307:A
5299:A
5293:X
5240:,
5237:W
5220:W
5202:.
5200:B
5196:B
5192:A
5188:A
5173:C
5169:A
5165:B
5161:A
5157:C
5153:B
5149:A
5139:A
5135:B
5131:A
5120:S
5116:K
5050:K
5034:A
5028:.
5026:B
5022:A
5018:F
5007:B
5003:A
4999:T
4973:N
4969:F
4965:T
4961:N
4953:F
4949:B
4946:N
4942:N
4938:N
4934:T
4927:T
4923:A
4920:N
4916:N
4912:N
4908:F
4900:T
4896:N
4894:(
4889:F
4885:N
4883:(
4877:N
4869:Y
4865:Y
4853:Y
4848:)
4846:A
4843:B
4839:x
4836:N
4828:A
4824:B
4820:x
4817:N
4806:.
4804:F
4800:B
4797:N
4795:(
4790:T
4786:A
4783:N
4781:(
4775:B
4771:A
4767:N
4763:N
4752:M
4748:M
4745:N
4737:N
4733:F
4729:B
4726:N
4722:T
4718:A
4715:N
4711:B
4707:A
4699:N
4695:)
4693:I
4690:K
4686:F
4680:K
4676:T
4671:y
4669:.
4665:.
4659:x
4657:.
4653:.
4642:F
4636:T
4631:F
4627:T
4597:T
4591:)
4589:L
4586:L
4582:L
4576:z
4573:y
4571:(
4569:z
4566:x
4560:.
4556:.
4550:L
4544:z
4541:y
4539:(
4537:x
4531:.
4527:.
4521:L
4517:)
4515:y
4512:z
4509:x
4503:.
4499:.
4493:L
4488:x
4486:.
4482:.
4476:L
4471:x
4469:.
4463:L
4456:L
4444:K
4437:K
4433:I
4426:I
4422:I
4415:S
4411:C
4407:B
4403:I
4399:I
4392:I
4385:I
4381:K
4373:.
4371:E
4367:x
4363:E
4361:.
4350:I
4346:K
4342:I
4338:K
4334:I
4327:K
4316:I
4312:K
4300:C
4296:B
4289:T
4285:Z
4281:I
4277:C
4273:S
4269:C
4265:B
4261:I
4257:K
4253:S
4234:B
4230:C
4226:S
4222:S
4218:C
4214:B
4208:)
4206:x
4203:y
4198:)
4196:x
4193:y
4190:I
4185:)
4183:y
4180:x
4177:I
4174:C
4172:(
4163:x
4160:y
4156:y
4153:x
4150:I
4147:C
4141:)
4128:X
4125:C
4121:=
4113:X
4086:I
4081:=
4071:)
4058:I
4055:X
4051:=
4041:X
4012:C
4007:=
3995:)
3993:I
3990:C
3982:T
3972:x
3969:T
3962:]
3960:T
3952:T
3941:x
3938:y
3932:.
3926:C
3922:B
3916:)
3906:x
3902:)
3900:T
3897:T
3894:B
3890:T
3885:)
3875:x
3871:)
3869:T
3866:T
3863:C
3859:T
3854:)
3844:x
3840:)
3838:T
3835:T
3832:S
3828:T
3823:)
3821:E
3817:x
3813:]
3811:T
3807:T
3800:I
3796:T
3791:)
3789:E
3785:x
3781:)
3779:T
3776:K
3772:T
3766:)
3764:T
3761:T
3757:T
3750:x
3746:T
3722:S
3718:K
3712:(
3710:S
3708:(
3706:K
3704:(
3702:S
3700:(
3698:S
3694:C
3690:)
3688:K
3682:(
3680:S
3676:B
3669:K
3665:S
3657:x
3654:g
3652:(
3650:f
3646:x
3643:g
3640:f
3637:B
3635:(
3632:)
3630:g
3626:x
3623:f
3619:x
3616:g
3613:f
3610:C
3608:(
3602:C
3598:B
3590:K
3586:S
3571:S
3562:(
3554:K
3541:(
3536:K
3533:S
3530:K
3512:X
3504:X
3500:S
3496:K
3490:.
3488:S
3484:X
3481:X
3479:(
3477:X
3475:(
3473:X
3471:(
3469:X
3463:K
3459:X
3456:X
3454:(
3452:X
3450:(
3448:X
3439:)
3437:K
3435:)
3433:S
3425:X
3418:X
3402:I
3398:x
3395:f
3389:.
3379:I
3373:(
3371:S
3365:(
3363:S
3361:(
3359:S
3355:x
3352:f
3346:.
3340:T
3334:)
3322:K
3320:(
3318:y
3308:K
3306:(
3304:y
3301:I
3296:)
3294:y
3290:x
3287:K
3285:(
3278:)
3276:y
3272:x
3269:K
3267:(
3265:x
3259:(
3257:K
3252:)
3247:K
3241:(
3239:K
3237:(
3235:S
3233:(
3220:K
3214:(
3212:K
3210:(
3208:S
3203:)
3201:T
3195:(
3193:K
3191:(
3189:S
3180:T
3167:E
3163:E
3159:E
3155:x
3151:E
3142:)
3140:E
3136:x
3132:T
3128:T
3117:T
3100:E
3098:.
3092:E
3088:E
3084:E
3080:E
3074:)
3072:T
3069:T
3066:S
3064:(
3058:E
3056:.
3050:E
3048:.
3043:S
3039:E
3035:E
3023:E
3021:.
3015:E
3013:.
3008:S
3004:E
3000:E
2979:)
2977:a
2973:E
2971:.
2964:.
2959:S
2954:)
2952:a
2948:E
2946:.
2940:E
2938:.
2933:S
2920:a
2916:E
2914:.
2908:a
2904:E
2902:.
2896:a
2892:E
2888:E
2882:(
2875:E
2871:E
2867:E
2863:E
2861:(
2855:E
2851:E
2847:x
2843:E
2839:E
2835:a
2831:E
2827:E
2823:x
2819:E
2815:E
2811:a
2807:E
2803:E
2785:E
2781:x
2777:T
2774:K
2770:E
2768:.
2762:I
2758:x
2756:.
2750:T
2743:]
2731:T
2723:n
2721:(
2719:Î
2715:n
2711:T
2705:.
2699:I
2693:(
2691:S
2685:(
2683:K
2681:(
2679:S
2666:I
2659:I
2657:(
2655:K
2648:I
2644:K
2641:K
2634:I
2630:K
2627:K
2625:(
2623:S
2616:I
2612:K
2609:K
2607:(
2605:S
2601:I
2594:I
2590:K
2587:K
2585:(
2583:S
2581:(
2579:I
2576:S
2569:I
2565:K
2562:K
2560:(
2558:S
2554:I
2551:S
2549:(
2547:K
2540:I
2536:K
2533:K
2531:(
2529:S
2525:I
2522:S
2520:(
2518:K
2516:(
2514:S
2512:(
2506:y
2502:x
2492:I
2486:(
2484:S
2478:(
2476:K
2474:(
2472:S
2465:T
2459:(
2457:S
2451:(
2449:K
2447:(
2445:S
2438:T
2435:T
2432:S
2426:(
2424:K
2422:(
2420:S
2413:T
2407:(
2405:K
2403:(
2401:S
2394:T
2391:T
2388:S
2381:T
2374:T
2367:T
2360:T
2357:T
2350:T
2344:T
2337:x
2334:y
2328:.
2312:)
2304:(
2302:S
2293:(
2289:I
2285:x
2280:y
2277:K
2273:y
2266:E
2258:E
2235:T
2227:E
2223:E
2219:x
2215:)
2213:T
2210:T
2207:S
2203:T
2198:)
2196:E
2192:x
2188:]
2186:T
2182:T
2175:I
2171:T
2166:)
2164:E
2160:x
2156:)
2154:T
2151:K
2147:T
2141:)
2139:T
2136:T
2132:T
2125:x
2121:T
2112:T
2104:T
2095:K
2091:S
2076:Y
2053:I
2025:I
2017:x
2013:x
2010:I
2006:x
1998:x
1989:x
1982:x
1979:K
1977:(
1975:x
1972:K
1967:)
1965:x
1955:x
1943:I
1939:K
1935:S
1928:z
1924:y
1920:x
1916:z
1912:y
1908:x
1904:S
1895:(
1886:S
1884:(
1878:S
1871:x
1867:y
1864:x
1861:K
1859:(
1853:y
1849:x
1842:x
1838:y
1834:x
1831:K
1823:x
1819:x
1816:K
1812:K
1808:x
1801:x
1797:x
1794:I
1792:(
1786:I
1774:}
1771:n
1769:x
1765:1
1762:x
1760:{
1756:E
1748:E
1743:n
1741:x
1737:1
1734:x
1731:P
1729:(
1702:)
1697:n
1693:E
1689:.
1686:.
1683:.
1680:)
1675:3
1671:E
1667:)
1662:2
1658:E
1652:1
1648:E
1644:(
1641:(
1638:.
1635:.
1632:.
1629:(
1603:n
1599:E
1595:.
1592:.
1589:.
1584:3
1580:E
1574:2
1570:E
1564:1
1560:E
1535:)
1530:n
1526:E
1522:.
1519:.
1516:.
1511:3
1507:E
1501:2
1497:E
1491:1
1487:E
1483:(
1443:S
1439:K
1435:I
1427:P
1416:x
1320:3
1314:3
1287:3
1281:3
1261:x
1218:x
1212:x
1200:x
1172:)
1169:c
1166:)
1163:b
1160:a
1157:(
1154:(
1130:)
1127:c
1124:b
1121:a
1118:(
1091:]
1088:a
1082:v
1079:[
1076:E
1070:)
1067:a
1064:)
1061:E
1058:.
1055:v
1049:(
1046:(
1031:a
1027:v
1023:E
1007:]
1004:a
998:v
995:[
992:E
961:)
956:2
952:E
948:)
943:1
939:E
935:.
932:v
926:(
923:(
893:1
889:E
862:1
858:E
831:2
827:E
792:1
788:E
761:2
757:E
730:1
726:E
697:)
692:2
688:E
682:1
678:E
674:(
659:v
641:1
637:E
610:1
606:E
593:v
575:1
571:E
567:.
564:v
531:1
527:E
510:v
488:1
484:E
480:.
477:v
445:2
441:E
414:1
410:E
385:v
358:)
353:2
349:E
343:1
339:E
335:(
308:1
304:E
300:.
297:v
269:v
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.