perm filename NOTES.APE[LSP,JRA]1 blob
sn#252733 filedate 1976-12-10 generic text, type C, neo UTF8
COMMENT ⊗ VALID 00009 PAGES
C REC PAGE DESCRIPTION
C00001 00001
C00002 00002 .SS(Review and Reflection,,P85:)
C00019 00003 .<<c w mmorris>>
C00031 00004 .<<abstract syntax>>
C00039 00005 .<<begin lambda>>
C00063 00006 .<<lisp borrows>>
C00073 00007
C00081 00008 In particular we can assume that the LISP primitives
C00091 00009 To describe the evaluation of a function-call in LISP we must add
C00099 ENDMK
C⊗;
.SS(Review and Reflection,,P85:)
By way of review we
sketch the basic LISP evaluator of {yonss(P6)}:
%3eval%* plus the additional artifacts for %3label and function%*.
There are two arguments to %3eval%*: a %2form%*⊗↓throughout this section we
will say "form", "variable", "λ-expression", etc. rather than "an S-expression
representation of a" ... "form", "variable", "λ-expression", etc. No
confusion should result, but remember that we %2are%* speaking imprecisely.←, that
is, an expression which can be evaluated; and an association list or
%2symbol table%*. If the form is a constant, return that
form. If the form is a variable, find the value of the variable in the current
environment.
If the form is a conditional expression, then evaluate it
according to the semantics of conditional expressions.
The form might also be a functional argument. In this case evaluation consists of
associating the current environment
with the function and returning that construct as value;
in LISP this is done with the %3funarg%* device.
Any other form seen by %3eval%* is assumed to be a function followed by arguments.
The arguments are evaluated from left-to-right and the function is then applied
to these arguments.
The part of the evaluator which handles function application is called %3apply%*.
%3apply%* takes three arguments: a function, a list of evaluated arguments, and
the current symbol table. If the function is one of the five LISP primitives
then the appropriate action is carried out. If the function is a λ-expression
then bind the formal parameters (the λ-variables) to the evaluated arguments and
evaluate the body of the function. The function might also be the result of a
functional argument binding; in this case apply the function to the arguments
in the saved environment rather than in the current environment.
If we are applying the %3label%1 operator,
recalling {yon(P192)}, we build a %3funarg%1-triple
and new environment such that the environment bound in the triple is the
new environment. We then apply the function to the arguments in this new
environment.
.P227:
If the function has a name
we look up that name in the current environment.
Currently we expect that value to be a λ-expression, which is then applied.
However, since function names are just variables, there is no reason that the
value of a function name could not be a simple value, say an atom, and
%2that%* value can be
applied to the arguments, etc. Indeed examination of %3apply%* on {yon(P69)}
will show that %3apply[X;#((A B))#;#((X#.#CAR)#...#)]%* %2will%* be handled correctly.
The obvious extension of this idea is to allow a %2⊗→computed function↔←%*. That
is, if the function passed to %3appl`dJTASLA]←h↓eKG←≥]SuK⊂ACfA=]JA←_AiQJ↓aeKG∃IS]N4∃GCg∃fXAi!K\AKYCYkCQJAiQ∀AMk]
iS←\5aCeh↓k]iS0AShA%fAeK
←O]SiKH\AQQkf~)oJAo%YXAC1Y←nAMkGPA→←e[f↓Cft~(]¬ ∂%≤Aπ 9) %∪Pw' →∃π(@fl~∀~∃|PQπβHFQ#+=)
FQ
β$FQ∧F\G∧$RRRF!#+∨)∀FQαF8G∧RR$~∀] 9λ_∧)t∩⊗≡&rα≡J>-↓nRε∀JQM! )1IQc→I%lhRS#∃ε ∪∪'&K?9β}1β∂?oβWS↔"β≠W;∨#'?;~β7?∪N3'↔Mα)OπCεce∃EαCog?rBAYeOq%βOfK∨#SgIh4)u~⊗2⊗≥!↓MlhP4+ππβ3e↓ci"n↑39oπ,w~f]nfO⊗⎇kPhUK=↔≡≡≡+66uTβ∩ε≡≡+6∂⊗tVC
*+<↔⊗?;[SXh+DεO≡=⎇g≥↑mkR↓J =vw≥<≡&:+F∀RU↑≡Yn;.x<Lt- I%+x<↓3\nn]FB.⊂⊂⊂λ⊂⊂⊂⊂ ↔↔⊂⊂λ⊂⊂⊂⊂λ⊂⊂⊂⊂λ⊂⊂⊂↔ .
\ islambda[fn] → eval[\body[fn];
\\\pairlis[vars[fn];args;environ]];
\ %Et%3 → apply[\eval[fn;environ];
\\args;
\\environ] ]]
.END
The subset of LISP which is captured by this evaluator is
a stong and useful language even though it lacks several of the more
common programming language features⊗↓It is "strong", both practically
and theoretically. It is sufficiently powerful
to be "universal" in the sense that all computable functions
are representable in LISP. In fact the %3eval-apply%1 pair
represents a "universal function", capable of simulating
the behavior of any other computable function. The
usual arguments about the halting problem#(⊗↑[Rog#67]↑) and related matters
are easily expressed and proved in this LISP. Indeed, the
original motivation for introducing the M-to-S expression
mapping was to express the language constructs in the
data domain. Only then was "S-expression#LISP" used as the programming
language. It was S.#Russell who convinced Mc#Carthy that the
theoretical device repreresented in %3eval%1 and %3apply%1
could in fact be programmed on the IBM704.←. This subset
is called the %2applicative subset of LISP%1, since its
computational ability is based on the mathematical idea of
function applicap %←]f\↓/JAQ¬mJAaIKgSgQK]iYdAeKM∃eeKH↓i↑A←UdA→∪M ~∃aI←GKIUeKfA¬fA→∪M AMk9GiS←9fXAKYK\Ai!←kOP↓oJAQ¬mJAg∃K\Ag=[J~∃⊃SMMKIK]GKLAEKi]KK\AQQJAG=]GKaPA←LA→k]Gi%←\AC9HAiQ∀AG←]
Kaif4∃←LAAe←GK⊃keJAIKMYK
hAS\↓→∪' 8A∪hA%fAiQ∀A[Ci!K[Ci%GCXA%IKBA=LAMk9GiS←8AoQS
PAiQ∀~∃ae=GKIkIKfA←_AC\A¬aaYS
CiSm∀AYC]≥kCOJ↓CeJAQesS]≤Ai↑A
CaikIJ\~∃IKOCe⊃YKgf↓←LAI%MMKe∃]GKf↓S\Age]iCp↓C]HA∃mCYk¬iS←\4∃gGQ∃[KfX↓iQJA⊃←[S]¬]hAG!CeCGQKeSgQSFA←_AC\A¬aaYS
CiSm∀AYC]≥kCOJASf@4∃iQCPABAO%mK\@ Mk]GQS←\D↓CaaY%KHAi<ABAO%mK\AMKhA←_ACeOU[K]iL@~∀JICYoCefJbAAe←Ik
KHAi!JAgC5JAeKMkYht↓KSiQ∃dAiQ∀AG←[AkiCi%←\~∃Ae←Ik
KfAC8AKee=dXA←HAShA⊃←Kg\≥hAiKI[S]CQJXA←HAShA¬YoCsLAae←⊃kGKf~∃BAMaKGS→SFAm¬YkJ\4∃)QJ↓CaaY%GCiSYJAgk gKhA=LA→∪M AI←∃g\Oh↓ckSi∀AgkG
KKH\4∃)QJ↓ieKCQK[K]PA←LA→eKJAYCeSC YKfA
CYYf↓iQJA∃]mSe=][K]PAS]i<AaYCd\~∃'<AS\A1∪' X↓oJA[UghAC⊃HABAAQeCg∀ACE←Uh@EB↓OSmK8AK]m%e←][∃]hDX↓C]HAQQJAa=S]h~)EKG←5Kf@AQQChAQQJAO%mK\A
←[akQCiS←8AS\AQQJAO%mK\A∃]mSe=][K]P~∃CY]CsfA!CfAi!JAgC5JAKM→KGh\↓)QChA[KC9fAoJ↓QCmJ↓]↑Ao¬r@~∃Q↑AIKMiekGQSmKYdAGQC9OJAi!JAK]YSe←]5K]h\↓→C]OUCOKf↓oQSG AQCm∀AgkG ~∃CE%YSir↓CeJAMCSHAQ↑AQCYJ@JeMSIJ[∃MMKGQfJbA¬]HACIJAiQ∀AEkg%]Kgf↓←LAi!J~∃]∃qhAG!CaiKH\~∀~)/JAQ¬mJAa out several "procedural" differenc↓Kf\A=kdAiIKCi[∃]h4V{→β∂}s∪'SN{;π1ε+cCK/≠O'?w→β∪'63↔KMε3K?5π##∃β/≠Wπ1π#K↔πMV.wD
v0h,nVv∨M≥vrε≡≤
M8x=
≥{L¬⊂≠zy⊂9]0w20\d rule for function applicap %←\~∃%f@EG¬YXGEdDπ[πg+∃ β>C'∂!π∪↔GWO∪↔Mβ&C∃β↔6 3WπM⊗}r
|bε∞MDε∂⊗}]V.wN1PV⊗\mw⊂ (_x-M4w3H:42P∪$ih function, whereas only some of the
ap¬≥k[K]QfAi↑↓BAG←9ISiS=]CXA∃qaeKMgS←\↓CeJA∀β[π3. S↔⊃rα;?S*βS#πDπ&FT∞vF`{→#!.=9<nM;{H
|HλY.l;≥8.M;{H
|H_<L};9;NNhH~.4_(≤∞-xy9∞↑X;λ
⎇Y.h ≡Y⎇;,]]≤c!.≠h→N]X⎇~-⎇\h_.,;I⎇∧∧Y=X-N8<α2Y⊃⊂7yλ⊃77zλ2{0v≥pz2rλ, they just "are".
We hap
∀AgKK8AISM→KeK]PACYO=eSiQ5fAG←5akiS9NAiQ∀AgC[∀AMk]
iS←\lAM←d4∃KqC5aYJ@∀gMCGPJbAC9H@Jg→CGhJPbJbF!ws←\! dbSxRAE←QPAG←5akiJ↓iQJA→CGi←ISCX~)Mk]GQS←\\↓∪LAi!KeJA¬eJAI%MMKe∃]hAC1O←eSQQ[fA→←dAG=[aki%]N~∃→CGi←ISCXX↓iQK\↓iQKe∀ACeJ↓ISMM∃eK]h↓CY←e%iQ[f↓M←dA
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βNsS↔Kπ∪↔Sπ&K?;LhS?Iβn+π;'v;Mβ?rβOg7⊗{3M9¬;∃βπ⊗)β';&+K↔O&+⊃β'rβ∪?'v9βS#*βOπ7*βS#'v9β?;gH4+←*β←'OBβS=βπ∪?∪W≡)β¬↓+∪≠K↔/⊃∃EβNsS↔Kπ∪↔Sπ&K?91π;#'∂Bβ?;3Jβ7'K⊗{KL4W##∃β/≠O↔;&Kπ1βNs∨K↔&K↔;S~β?→β&C∃β3∞s∨Wπ>)β∂?w≠SKW∨#M9α&CπQβO→04)+≠OGK")EβK/βK↔O.sSMβ
↓∃K≠.s∂S'}q∃Eβ∞s⊃↓∃?↓∃Eβ⊗+CK↔≡+;SMε ↓∃K≡{;OS∞sQ∃Eph*S#*↓∃O↔6 17ππβ3e∃
βCπ'∩β∨'[/→β?;*β';S/∪CK↔& S'?rβS=β&C∃β7. ;';:β?→β7+;∂SN{;L4V ;⊃β≡{;OS∞sSM1ε∪WQβ&C↔K∃ε K∃β&K≠≠↔⊗+;QβNsS↔Kπ∪↔Sπ&K?;Mε ;⊃β&C↔K∃ε K∀4V#'≠≠/∪↔;Qα)O↔[∞a7πCεce∃Eπβπ'K~p4(4U##∃β⊗+7π'v#↔Iβ}1βS#O→βO↔∨#'?9αβ←'3bβK↔O}c[∃β≡{7∃β}1βS#*βS↔;≡K?9β⊗+S←↔.p4+≠.s∂S'}qβπ;"βCK?≡+∪WK*qα←∃π;'31π≠SW∪JβO?7*β?→β&C∃β7∂##↔7∂#'∂πbβ'7CfK∂πSN{;L4V{→βππβ3'∂∂#'[∃εcπ;∨. ∨↔Mrα≠'K∨!βS#O→β←'faβ ∃ε#?;∃π+O';:βS#∃α)d!∃
k∂π3∨+3WMb4+¬ε3?K7∞a4+f ;∨W∞;∃β≠␈⊃βOS.#g';:βS#∃ε≠?;∂/βQβ?2β≠W;∨#'?9rαS#↔rβS#∃π∪↔OWg#Mβ?2βS#'~βOSW'H4+←Nc1β *βπCCfK↔⊃β&yβS#*βπCCfK∂πSO3∃α2M~AβO.∪O↔Qph(4(hR←#π"β∪?↔~βS#'~β∪'O∨+OO'}qβ#πlRπ&t Fzπ⎇≡FBQ.π⊗}},⊗nn≥lrεf≥lw.∞|↑3z∧=LV∂⊗O∀π&FT
w⊗&↑$ε}0Q,W6∞N\↔&N⎇dε␈∩∞,V'.>M⊗}r
≡2ε&≡,V∨&Nα(∧ <≤≠
≤x8[ UHβ"I}<H≤nN9≡(∞⎇;≠λ ≥≤{h∧ z=Y$
:β9tYt:8β
into the problem of lanfβkCO∀AgaK
SMSG¬iS←\8~∃ ↑↓oJAg¬rAiQ¬h_∧+Mε*εL≥f0≡89y$∧≤|→,=9Z8l≡~;{D {{\m≡⎇≤↓P≠s⊂0P≤|w:0Xz4qP_wvx7[2w:⊂_w2⊂9[vrFEλ22yq\4x:4[w⊂7sλ:he evaluation of constructpε↓S\Ai!ChAY¬]OkC≥J}~∃=dAI↑↓oJAg¬rAiQ¬hAiQ∃dπ∃β';=β∂}kC0≡l]g'~D∞7NwL∨αε∞lDε
ε\≤6FNlUBε∂,Tεn/,]GHh,LW6N<↑2ε6}H L<x|M≤Z;Yd∧≠|H m|[8-M>Z;Lt≠[⎇
≥{\h ≤[⎇=∧∧≤{{,T_8\nNX8⎇∧∧→≠{,≥;H≠la"Y~.<{⎇<N</hλλ∀≤Y; ≡→9λ∞≡9<⎇
≥{H→M}H≤≤M||X;-\<\h ≥Yλ≠ ≥Y⎇8,|(→→.=9{Y..c"Z-n[{≥L↑h≥~ T~9→,≡h≠yD {|\L\⎇≠Y.>h_;LD→<=-≡X;→ ≠1rP7Y⊂897Yy0vyK⊂$7{H27P;YFE5w≠{P;t→w⊂0P≤97sy_vP4yH1wy9→qz∨Pλ*44yH90xzZy2yP≤wvrP≠7z4w[⊂7s⊂_P9z0[20y2βE0spZw9z⊂≥t4qtλ:7P⊂≥2yz⊂≠zy⊂4[x62vYw:0z~ww9KQ*w6→yyFE≥42y2H4yP0H894w\⊂6pz~2vpz~qpv⊂→2s4w~z4wwλ7s⊂0H60w3]psrP_z⊂40[2⊗εE≥t7P4\P:7P≤p|P;Z2z42\⊂0P8≤7x7yYr⊂4v\62vr[:0z4[w⊂4yH∩Y1w\92qz X←Q⊃E/-iq[Q[Y.W↔/W⊂βE$s⊂≠zy⊂0[3wy4]46yP≤2pv6≡P0y2CE92s≠2qz4[3P3:[1z4w[0v⊂8≤7x2y≥4ryVλ:42wλ;rP9Z7zv2λ22{2[7x⊂6Yz47r≤FE37\⊂;2y~s<tw→P:40]⊂:47\rP89≠x2y:~ryP0\2P2|≤92yyYr⊂4wλ7zy⊂_v3wy~z46yK⊂ spZw9z⊂≥44yFB;rP⊂≠zyz⊂_0v0w_rP:4→P92p[4⎇0z~ww⊂:~0z⊂6Xw<P8≤7sy0[yP27[∪z⊂3~z⊂72Xz6<FB4w:7H:44yH6pz4→vpz4Xpv⊂3≤0vr{[y5W⊂∪pw<P≤97sy_vyP0\2P12\z⊂1t_y0qz→y4⎇2YεE0yH:42v\rv;2\W⊂$wλ:44yH1pyrH;rP9Z7zv2λ:42wλ12P4[:2y2\z2r⊂~w⊂εE≥2y3<Zw3P2\zt{0[2w1rH7s⊂8≤7sy0[yW⊂$Y⊂;rPλ22{2[7x⊂0H72{P_v3wy~z46FB;rP4_{2P9[vrP9→yx7w≤tq4v~z<P:≠P22v[w9z9_z2P:~0z⊂:~2P0v→wy4z~6yP0\2P2x]t{0v→w:εE~w⊂;2\<P;r[6⊂22Y4w2rλ;p|yEazy≤2w:⊂≥42wy≡P4yP~w0r2\zpz2H37y⊂→2pv4[3P;t]4εE;→y<P6Xw<P9→pv⊂8≤7sy0[vtw3H:0yu\W⊂$7]r{2yλ:42P≤2pv4↑0z4w[⊂:40]εE7w→P40yH0P92\x7w9Zq4v4]<P:7H1ww9Zr2y⊂≥42P8]ryz4[w9P∩L2{2wλ4w37\6pv6≡RXFE~yP0P≤wq2y~w3P7[2V⊂;Z4qt⊂≠wy2P≤97sy_vvry≤P9t7]v2⊂2↑82y4Yw1rW↔WεEεB*42P≤z:r<H7s⊂3≠y6pvλ9|yz→vyP4[⊂6pz~2vpz~qpv⊂≠7stqH7s32\9P4w≤tst: εE*4→y2V⊂≥rP0y→P892\rw:2Y⊂;tz~⊂0P9↑w:0|λ0w2⊂_P9|y]2vP7Y⊂0|4[vyP0[2⊂9:[2yP7Y⊂4w3→y2w1YWεE&[yz⊂:\zpv6≡P;rP_y2P0[9wP7Y32y2Y⊂0P⊃≠wr2vλ:42w\<Q⊂;Z4qt⊂→t{2yH:yPεB4w:2\892z_z4ww≤PεE7\⊂6wr→v9P3≠y⊂:4→P9|w≥0qz4XP37y≠pv⊂9↑yz2vNP:42H6wr2[⊂:42[y<P:\zpv6≡P9zx≤64ryCE0r2~z4ww_v⊂6rXw9P7Y⊂3t{~w3P1[w;4w_tw3P_y3zvYw:9P→7y⊂:~2P;0[4r4z≡P7s⊂≤z0z2[rw:9CE4w⊂≥42P3≠y6pvλ9|yz→vW⊂⊂∃42P0\3zvr[:9P6Xr2P;Zz44wλ:42P→7y6p[⊂9|y]2vFE_y2P1[zqt2Y⊂4w⊂≥2y6yH7s⊂⊃≤97{0X4v4z≡Q⊂;t→y2pyH0y3z[rw:9H7s⊂:~2P6wY2v⊂:~2wy<H0y2FB3t{2[⊂4w⊂≥2y6yH7s⊂⊃≥9:z4λ↔⊂εE⊃7y⊂0H24yq]yytw[⊂7s⊂→7y6p[⊂9|y]2vyP_w2⊂6[r2v⊂≥42wy≡P9rrH /-fYw⊃Xε4]↑.
.8λy AnA[5←eeSL||~∀9∂%∨+@v~∃ε8A.\A5←eeSL@P-=m≠←dFTk;<R↓Sg←Y¬iKHAQQeKJ↓aKegAKGiSYKfA←8AYC]≥kCOJ0Ags]QCpXAAeCO[¬iSGf0~∃C]⊂AgK[¬]iSGL\@~∀4∀↓:
,:&9αLr∩⊗:"↓A1a\:J>Vπ0hPQ$S∃∨≥nF∂BU'"¬& Tπ∨NnMε/≡≡4ε∞vD ⊗v∞O≡6O~
|bπ≡]nF.v<↑2εNd∧ε
εL≥f0≡89y%d∃~~.4_<Y,∀~<c!.y;≠∧ ⎇;≥
≡X=→,D~;H∞∞[y|L≥;:;Lt≠_;L}89y$∧≤|→,=9Z8l∞~;{Ea"KQ)hβ"KH~⊂4Uπ1"KPHXr3H →Q⊃3JD &↔qtSjZ∞c"AQI,T∞,9{8.M8|i%'H∃~ T≤Y; ≡~;{D Y=≥l\;H≥
(≠_-l⎇89lT_;Y∧
=≤↓P≥yry↔λεE"{_v:pz≠y9V⊂≠4urP Yz3v[ps⊗⊂≥3vwpY9∩XP_w2⊂∩Lr{0v XV⊂1[vrP:[22y⊂βE:42H42pr~w3P7Y⊂890Y↓map %Gf\~)!eCO5CiSGLACeJ↓[←eJ↓G←[[=]YrAIKMKeIKHAi<ACfA=aKeCQS←]C0AgK[¬]iSGLAS\A⊃SgGkMgS←]L~∃←L↓ae←OIC[[S9NAYC9LπWπ>+M84Rr⊗:⊂hQ:
⊗<J9α&t"⊗:Qβ↓1EA\:J>Vπ0hPQ$S∃≡]\⊗w&≤>2*S$
FF*∞,Vf∂M≥vrε,↑G>.]dε≡}n>G↔.>N2ε}d∞FF*
L⊗v?\≤v*ε≥lBπ& Tε∞↔>N&∞∨AQ&}⊗,\7'~∞⎇εN≡∧∞FF/∀ F.v}↓→%d∃~~.4≤⎇8LM=Z<m≥{H~.4_{{-]{[≡$∞Y9Y..Y9λ∞Mh_<d →;[nL=~;ml;β"N<;8;NM8|kAQC"KHYQβ"AQT≥=∧ ~9YL↑Y;]
O+λ≤o≥]_>∧ →<xn-8Y<d <≤→,≡X;XlUλ≤≤L≤{8=
≤|h→ ↑x|Z,,<c"M≥<≠→-\;]_.M;{K∧∧_;Y↓Q\y;,≥]~8n4→→<l>Z8Y.4≠98-m;YkAQU~≠n\zλ≥
<Y(
≡h_(∞>≤[{Lt_{{L<;\⎇.4≠{H∞M→(≤o≥]_8nM8h≥
⎇{≤h m|H≤n 8z9O≥;Yc!-_;Yn\9y<b`0]=∧∞y9(αkVt≤L∀mlw+eWkλ∞M→<Y$
<h≠Mt_{{L<;\⎇.4≠{H↓QX9→.≡8=→$∞≤X9m\=~8n4_;Y∧ =Y;D∧≠→<n4_9|L\;9;NDions.
We will first outline the pragmatics questions and then discuss a bit
more of the semantics issues.
In this discussion we will use the language distinctions of Morris
even though the practice is not commonly followed in the literature.
Typically, syntax is studied precisely and semantics is "everything else";
however the distinction between computation (pragmatics) and tp¬UiP@QMK[C]QSGfR4∃SfA%[a←eQC]hA¬]HAg!←kYH↓]←hA JA[k⊃IYKH8~∀~∀4∃∨]J↓iQ←k≥QhASLAi↑A⊃KgGe%EJAi!JAae¬O[Ci%GfA←_ABAY¬]OkC≥JAS\↓iKe[LA←LAQQJAaI←GKgL~∃←L↓G←[a%YCiS=\\@AQQChA%fXAi!JAae¬O[Ci%GfASLAgaK
SMSK⊂AErAM←[JA
C]←]%GCXA
←[aS1Kd~∃Ae←Ik
S]NA
←IJA→←dAg=[JAo∃YX[I∃MS]K⊂AgS[AYJA[¬GQS]∀\@A)!JA[K¬]S]N↓←L~∃∧Aae←≥eCZA%fAiQ∀A←ki
←[JA=LAC\↓S]iKIaeKi∃dAS]QKeae∃iS]N↓iQSf~∃G←⊃J\~∃ khA]=nX~∃Q↑Ak]⊃Kegi¬]HAi!JA[K¬]S]N↓←LAB↓aCei%GkYCHAG←]MiekGQfXAi!K\Ai!SfAaI←a←g¬X@~∃IKckSIKfAi!ChAs=jAeK¬HAiQ∀AIKg
eSai%←\@~)←LAB↓G←[a%YKd@4∃C]H↓k]IKIgiC]⊂AiQJ↓gS[a1JA[C
QS]J8~∃)o<Aae← YK[f↓CeSg∀AS[[∃ISC`&+3e9∧≠?7CNc↔KMε Hε*
mw"π ≡'&N>]F∂⊗O∀hWN,⊗w∨ ≡&.wD∧ππ⊗|}&∞o5d¬≡.=⎇f"b ∀π6/.α(∞=;<≠ T≠88m
;Y(∧
8>(
m⎇λ_,L<<αp]2v<FB92s6→qz⊂:~2P0q]:pv⊂≠rqt0[4yvyH:yrrλ4w⊂0[⊂4vx≠2vrw≥0z4w[↔⊂(∧his aspect is
important if the semantic description is to be meaningful to an implementor.
There is a more fundamental difficulty her↓JAS_AoJA
←]gS⊃KdAi!JAae¬GiSG¬XAC`∨β↔∂S_h+?→π##'Mπ↓K?C␈≠π18hR←#↔rβπO/.!β0ε⎇⎇2π.lLW↔∨L≥f"α ∀ππ⊗|}&∞j∞}&O'L]bεNd∧ε
ε
≤vBnL↑f.b
L⊗v?\≤v*π;x
H:hink
about t↓QJJeEK!CmS←HJbA←_Ahβ#∂!βCK}w⊗∞T
⊗rε∀∞f/↔∀ FO⊗\>Bπ>∨∃b¬& Tππ⊗≤⎇V∂&≤>0hV≡4ε≡f}<Rπ ≠h≥
(≤y-\;]~,>kC"K≥⎇(≥
;Zh ≤[⎇=∧∞42P_wvx:Xz0x∧ional behavior as it executes; you
do not think about the code that gets generated and then think about the
execution of that code.
A more natural description of the meaning of
constructpε↓SfAO%mK\A%\AiKI[fA←_Ahβ#*βKW9o#'7∃ε∪↔#π6K?Iβ}1βS#/≠∃β∂}sOSK.≠SM8hRS#∃α)O↔[∞a∃)β7+;∂SN{9β∪/≠∂K'⊗+MβSF)β↔c.≠WS'}qβO↔∂+↔;∂*β?→βλh+K↔π∪↔O↔w#πS'}qβ?→ε 9βπ,&O',≡'J∧I~5αε←∞π⊂ <|z-⎇KHλ
M≥<h∧Vy=X-D*H~.4≥~→$∞y;8-n~8c! 22yq\4x:4[w⊂7sλ&$ih ⊂⊂)zXt⊂22\qy4x≥4ww9H⊂40{→P0P3≠0{7yλ7s⊂1Zy1zv_y4z<H;t4qZεE9w[rP34[2⊂24\x62p\tw3Wλ$7{r]2y⊂9[vrP1Zy1zv_y4z<H4yP4[2{4z_q62]H;rP6]yz⊂0\yzvrH:40zλεE9w[rz44[3P4yH5w7{H0w2⊂→7ryP≠7z⊂9→xzty→P3:y≥42y⊂→|864Xpz4w[↔εE$Y⊂<wzH22qtY2P:7H22yq\4q2P≥42P9Yvpw:~qyP7Y⊂60w→zpsrH&∩Z_IU⊂4wλ0FE9Zvx62\⊂60w→zpsrH&∩Z→ U⊂:4→w⊂2t]42y⊂∪∩Z→∩J⊂4yPλ9rv3λr{4r→w:⊃⊂≠y⊂<w]P6zy]⊂3t{→P0FE→2yqy~x:4w[⊂7s⊂≥42P6Ypw4w→P7s⊂∪∩Z→∩J↔εEεBεE)wK⊂92p[4yz4Xpv6<K⊂:42H1t7tXrP4yH;t2y→P:7P≤z7x⊗λ77z⊂≥t2z4→y⊂:7H9z7x εE*4→P&$iT⊂87yZz4wwλ4yP:~0z⊂:~2P60[3zpsYP0w2λ20z0H9z9:Xz:y2\P0y2H9zs3~qtrw≥6<FE≤tvx6→P:40]⊂9rv→⊗r2yXy4x:~ww⊂4\P9pz~ys0q]7y<WβE z:→vx:9H40{2H12rwλ6pr2H:7P3Z{2P7≠w⊗qt\1zv0\⊂4w:→y892]2y⊗q_yrrεB22yq\4x:4[w9P7Y⊂9rvXw:4q\P37yλ60w3]psryH7z42\⊂:40[⊂&$iT↔⊂⊂*~2y2P~yP0FB⊂+"&λ22yq\4x:4[w⊂7sλ(&↔XK⊂0w2λ0P22\qy4x≥4ww⊂λ7s⊂ S#gf⊂
≤εE1≡P0P&Xy5w{λ0v3w\4z46K⊂!7z~⊂:42\rP0z≥2vx:≤P92y]v:⊂4[⊂0P2→yqy4\:4wwβE;t4Xt⊂4yH67w3H0w2⊂≥w6pw_srpq≠2P37\⊂0v6λ1:z⊂≥42P6[yz⊂8≤2yty]2w:↔βEεEεB+rP9Z7zv2λ6rw:~ww⊂:~0z⊂:~2y2P~yP0w≠z42yλ1wvv[w⊂6r]47r⊂→7y⊂9\2qts≡tw3PβE:42H890s[pz4q\P7s⊂_P897Yy0vvZw3P6_w3zpYrW⊂*~2P7y~stw0[⊂ v3[v⊂92\7y:εB4w:9≠r:qrY⊂0P9]0w20\2⊂⊂⊂→7y⊂9↑w:0|λ9x2qZs4qp]4ww≥λ:42FB!'#⊂→xzpz~ww↔⊂∩z⊂0v≤wP⊂3X{2P0H92py[w0q6≡P892XtyrP→2yqy~x:4w[⊂7s⊂≥42FE≤90svXz4qyH7s⊂ [3wv⊂≤z0z2[rw:9H4w⊂7_z:y0[⊂60w→zpsrKεE*4→P9z<[2FE7Y⊂892\rw:0]4ww⊂≥pyP1[w1tyYP0w2λ1v2p\⊗⊂1:]⊂9zs→2y9P→7y⊂:~2P4v\92qt\twwεB7s⊂7_z:y0[⊂60w→zpsrH892yYw:0z~ww↔⊂∀r{2y_v⊂4w_ww9t\z2w1ZryP;Yy2P2~yqw{→y2r↔βE)2sXy262\yVεE≥44yP≤z<v2H⊂4yP→2yqy~x:4w[⊂4yP≤ztz2H1wvv[w⊂0w→⊂4yP≥2y<P≥yrs:[↔⊂ P≤2qrw≥εE92\7y:⊃J /-f[wy⊃[Mpno∀H7w⊂:~2P89_svpz~qyP7Y⊂$g*⊃i&$iT⊂⊂:yYr⊂:4~yP22\qy4x≥4{2P≤z<v2KεE$sλ:42P≠0w3zXsrP4\P8zt]2P1w[x62| ⊂:42[⊂0P3≠y6pvλ22yq\4x:4[w⊂1p[⊂12FB2xzp[6<P1[vx62↑↔⊂$wλ=|ww≤yT8→ XT␈⊂≥rP;t[6⊂9rYP:40]⊂7zyλ4w:2\892z→y⊂22Y4w4z~wwεE≥tv6⊂→|:2w→⊂74qYv<P:≠P94qZ2y⊂9]q9rz≤P7s⊂∪$ih↔βEεE*~2y2P_y2P9[vrP1[vx2v≠4w3Pλ92py[w9P3≠y⊂22Xtr4w→P7w⊂→4y2q]⊂1ty_zv0y~z<W⊂λ#4y9]⊗εE<[zP72Yr⊂7w≠<P62Xy7⊂7[2P60[3zpsYW⊂⊂$Y⊂:42H⊂9x2Xts4qXz4wwλ4yP3Z{2wεB:42P≤wzy1YP60w→zpsrK⊂:42[⊂<wzH62py≠⊂:42H897s\0vvt[3P60[3zpsYP0w2λ:42FB9x2qZs4qp]4ww⊂≠0w3zXsrP0]⊂:42H9pvrH:ime. Second, since the evaluator is
written in the language, we can understand the language by understanding
the workings of the single program, %3eval%*; and if we wish to modify the
semantics we need change only one collection of high-level programs.
If we
wished t↓↑A¬IHA]∃nAYC9OkCO∀AG←]MiekGQfAi↑↓→∪' AoJ~)]KKH↓←]Yr↓[←IS→r@Jg∃mCXJ(Ag↑AQQChA%hAeK
←O]SiKfAC8A←GGUeeK]
JA←L↓iQCh↓]Kn~)G←]gQekGh0@AC]⊂ACIH↓B@AMU]GiS=\Ai↑↓IKgGISEJAQQJ~∃%]iKeAeKiCQS←\A=LAiQ∀AG←]MiekGP\~∃)!ChA[=ISMS
CiS←8A[SO!hAEJ↓KqiK9gSmJ0~∀AEUh@ASPAh∂'f`4+ *β¬β∂}sSK?fc↔⊃β⊗+['ON{9↓β⊗ S#↔∩βS#πrβ¬β∂}kC3↔&)βKππ∪?∨K∞k7';:β↔≠≠␈∪Q84Ph*¬β≡+7π;&K
β∪/≠∂K'π#'?9π≠#?Wf!β;?"βπSS.kCQβ&yβ↔cεcπ'9ε+[↔K↓≡FFNltε∞⊗}↑Bε
L⊗v?\≤v*`C"V-}(≠=.>λ_<n>;9(∞M_=λ∂≥⎇<H∞,89→.$≥;Y ↑\⎇_-l≤h≤m⎇9=~
≥Yh EeH H \px<NM≡.H∧VX∪[nM~;YaQXx;D Y(→/∞≠_:-l9λ≥
t_(≤nM{Y)dUKC"H∀≤y;,≥]~8d →<xn-<≥~-⎇H≠yD (≠_-l⎇89lT≠=<nD_Y(
l=≥<L≥ Hλ ≡λ≠=.>λ≠8.Lzλ≥
(→>∞∞Y<|m≡Y#"N
⎇y<D
yH≥
(≠_-l⎇89lUH⊂(∞<;8;NM8h→ ↑x|Z.∞~;{D∞z≠⎇-Lλ_;∞=h≠;lL;λ~
}h≥~ T_{{N>≤]8nNc"X.,(≥≠d Y(~-↑≠→;,]]→9∧
{H_$∞Y8<m⎇X8[ T≠88m
;Y+D∧∃z_.D~<h
l99→,D~<h ∀≤y;,≥]~8d↓"\Y.∞Y<y-n_=~-⎇H≥z
≤zλ→/∞≠≠z.Nkλ≤L≡~→<D∞~_;D
9{[n,<kλ∞M→(≤nN]8⎇∞↑Y(≠ld≥~→$
_;Yn\9y+AQR=λ <;H_.>⎇;9$ (_y.._:;D
→=Y-D≠yH∞]Y→<N>_;Y
≥Yh≠md≥~→$∞_<]∧
yH≥
#"\L\9→<Edλ∩=∧
Y99∧
{[≡$ >≤≠ ≥;H≥m =λ≠L\9≤h ←≤≠_-≥Z;Yea"S8.M→;8.M8x;∧
[⎇_.M;{H
≡h≠[d∞⎇8\nM=≥= T→[|D {→8.$≥~≠n\z≥ ↓QX]=∧∞y(_L]~9=LT_x<L\];λ∞>≥9≡$
yH≤l]8;]
≤|h≥m≥≠λ≠ \9λ≥
t_9→
≡~;{L≥β"Z-nz9z∞Nh~;D
_;Yn\9y( L<z9md¬@4Ed⊃ H
L;[Y-nλ~_.4~;]M⎇y9λ∞M~<h ≡≤≤[l≤zλ~-a"]~ T→→<m≤{H≠ld ,t*X4u %$j¬WK:→;HfvW7J%kkHβ!!"C"IL= |d
≠{zd =λλ∞M→(~.>⎇9(
|H≤}-n_>λ m|H_$
;{9-n Hλ↓QP(≤l≡~<yL≤⎇≠|O∀→→<l>Z<≥
≥{H≠ld≠=8m∧≠yC!.≤[yn,;;:-lh≠_-l⎇89lT≤};NL>λ~.4≤⎇_-l_<Y∧λSQKD∧∃~→$λSQH
≡h_( |;Y<L≡~=Y%D≠|H∞?;]~ ↑~8c!,|X;-\<KH∧
~_=∧
<kλ∞,=|Z.M;Yh∞.;→<d∞|→8m≤↑:;Lt~≠⎇d∞≠h→l]Y<X.L(λ≥l]≠λ→M}[99↓Q\⎇≤M≥Y|h ≡Y(→m≡Y;KD∧⊂(≤l]8;]
≤h≤| \z9Z,<=~;md≠{H∞M→(≠nM→<H
;Yλ <;H_LT_{{N=9→<L\β"]
t_Y( ∀→];L>~;{D∞z~8m∧≥_:l↑h_<d
;\≥.D_;H ≡XZ=∞,<↑(∞∞[y|L≥<h_-lλ→z.l<h_.4≠⎇=∞∞=β"L∀≤Y<∞,<y;NL=~;md≠yH∞M→(≥L≥≥9(
|H→> \⎇=~-lh≥~ T≤≤[l}X;+D∧∃~~.4~;<
M9<h∞M_=β!-⎇<H∞<;8;NM8h→N]X⎇~-⎇H≠=.>λ~_.l(≤{m\(≥x/∀≠yH ≥X;≡/-;Yh∞M→(≤nN]8⎇∞↑Y(≠ld≥~→$∞≤[yn,;+C!!"C"@ .<<abstract syntax>>
.BEGIN GROUP;
.P75:
In ⊗↑[McC#62]↑, John McCarthy introduces the concept of abstract
⊗→analytic sy.tax↔←. It
is analytic, rather than synthetic, in the sense that it tells how
to take programs apart#--#how to recognize
and select subparts of programs using predicates and selector functionrα,β/∀AoSY0~∃IK¬XAoSQPACEMieCGP@Jege]iQKQSFJb↓gs]i¬pAoQ∃\AoJ↓ISgGUgfAG=[aSY∃ef]>8~∃∪h↓SfAC gieC
h~∃S8Ahβ#*βO↔;≡)βS#∂!β'Qεkπ/↔~β;=β≡{77'MV.wD∞FzπM Rε/∞LW⊂
X;λ∞ 2x92\rw:0]4ww⊂≠s⊂:4→εE1w[9z4z≥z2w:≤P7s⊂≥42P8≤7sy0[W⊂⊂+YP72rY⊂7w6≡P12P_q62P≥7P92Xwsw4↑2P⊂:~2P7qXzy8αence
of a desired constp¬UGh\A→←dAKaC[aY∀t~∀]Q+%≤A=≤@E>λv~∀~(]¬ ∂%≤A∂%=+ w'∃→ π(fw)β ∪(bPDlRv~(~∃SgQKeZ@pz@⊃7mi:A←I79SgYCe7itv~∃9%gG←]Mi7i:l~∃9C9I7SgMk[7itwSgi∃e[7C⊃IK]Imi;:w%dπS↔⊗joπW>+;∩o%jvvvhh):⊗t 4(4R)Gπ;"βS#∃∧∩:→β/ WπSN{9h4Ph*⎇s&+K5yβQiu↓g3πIyπa↓s∂}sOQyπa↓sS/∪5y↓Z↓sS↔⊗iy84Ph)∃OO≠OW5bβπ∪∪.s⊃1∃
βπ;⊃α)OπW>+;⊃∃
aβ∪?r;QβK. 33eε≠πK∃π;#↔SF+IβSF)βOWjβ'MhSK↔C⊗+O↔;&+⊃βπ~4+a←I1β?∩↓∃M.←Aofu+ β?Iα)M"Be*M∞a≥I%∃)ε{I↓∃↔Ce-∃
q↓4U##∃β&K≠≠↔⊗+;Qβ/CS↔Kv 1βK/βK↔O.sSπSN{;L4V K∃β⊗+≠3↔∨#'?;~β?→β&K≠≠↔⊗+;Q1K∂?;∨∪↔S∃π≠g;S∂@↑␈↔≠YβS#*βπ ?6)α
:2β↔GW∂#'?;~β∨'[*β?;∃ph*'QεKMβC∂∪O';:β←#'≡Aβ3'v[Mβ∂}s∂K↔&)βOgw#πaβ>KS!β&C∃βπ↔≠SKπ∨!βOgw#πa8hQ:⊗: h(4*≡K;∂∃εKQβ'~βK↔π≡{;π f)βS=ε OOWn)βS#∂!βS#*β↔[πg+πS?∩β'Mβ&x4+?ε+KπS*β?9β≡{7∃βNsS↔Kv 1βK/βK↔O.sSπSN{9β?2βS#∃π≠?WK≡)βCK};Kπ5bβπ;⊃εK9β≠∞≠Q04V βK↔π∪↔O↔w#πS'}qβK↔6c↔∂SNs≥βSF)βOS.V∨'↑,Rε}d∞FF*∞∞&}?,≥RαF=⎇Vn}mO∩ε↑m}vph,≡2ε
∞ ↔↔≡T∞G⊗.U∃Bπ>T 6∞r LW≡∨-≤&*πM Rπ≡]\⊗w&≤>2ε}d ∩ππ-|w⊗∞T
⊗rπL↑&o~
|bε
nVv∨M≥vph.⎇εN≡∧
wε/,≡F/~
⎇bε
∞ ↔↔≡T∞G⊗.T∞W≡Nltπ&FT∞π⊗.M≤6∂&↑4ε∞vD∞6.f\>F␈↔4
v2πM PhV≥l⊗gOM≤2π∨≥nF∂Bdλ⊗↔∨NX,>λ≤}-n_>λ ={Xy.-\h~.Ny;→D
{[≡$∞z<α4λ:47yYP897\2y:4YyFE7Y⊂0P8≤7sy0[P;t4Xt⊂0y→P7s⊂~w:2y→yz⊂:≠P0w⊂→{0v:Xz7y↔βEεE,[zP6p↑P:42[⊂897Y4z0q≠<P:4~w5P7Y⊂:42H&rz0H2|89→yytw[⊂37y≠P7s⊂∪$ih⊂_yP0P_ww1y→z2P9↑w:0| εE,w]P6p|H:44w~P7s⊂≥42P&KP:7P∀Vr|8≤2yyt[w⊂:9_w9v0]7y⊂0\P:42H80y9Yy⊂;t~qt⊂6Xx9FE≥42P2↑:2y7_v⊂92\92yr[:0z4[w⊂7w≥7P0P≤0y9rH∀7y⊂_wvx:]0z4w[0v∀P≥92rWλ*42P≤rv2q]7y9FB0w2⊂≤92r4Xpz2yH7s⊂:~2P0w_v<z4XP9|w≥0|⊂0\2P9z≤0tst≥37y;Xy2≥P≤2qpv≠⊂:42H!'#⊂→7yεE∪$ih≥βEεE↔⊂"cdgλ* a$U_T_XJ]ci'Uh≥FEβE≡37\6←.≥∞≡P≡1[w9z0[:∨εE↔≥≥≡P∂;0y4Xq62←βE.≥≥∂↑3:w_z4ww∂-↑0y→←≥P↔ ↔⊂≥↑_y3←.CE.≥≥∂P-↑3≠y6←⊂εP≡37\6←≥P ↔↔⊂≥O37y6O⊂ P≡→7y6←↔FE.⊂λ↔↔↔↔βEεE↔⊃g"εE⊂vww3H7z42\⊂:44[3yV⊂≥42w⊗λ;rP7→rr⊂:≠P92q[sw4⎇→P1ww≤z0w:≤P∀∩YZyqww≤z∩U∀KεE;0\4pq6→yP∀∩Lty{0\∩U∀Vλ1ww2~z4ww_v⊂2|≤92yyZww9P
∩YtyXww2∩J∀V⊂0[2⊂3:[1z4w[9PεE
∩YtyY:w∩U
WεE+YP;wz[2⊂:4→w⊂72Yr⊂:7H;y4z→P0P3≥w1z4[w⊂4wλ9wvrH60w3]psrP→|892\ytw3H:42P≥0v:r\FE7sλ:42yYP1ww≤z9:q]9W⊂)Zw1rP≥42P8≤7x7yYr⊂2{_v:pz≠y⊂4yH:7P6Xw4x:[0z2P≤0y9rCE:92YyV⊂0[2⊂9t[1rP:~2P27[ptw⊂≠s⊂&$Th⊂3:[1z4w[9P∩Y~yRU⊂_4w0y≡P:92YyV⊂4]⊂9t7]v2⊂9YrvFE≠0z:y_v⊂:7H:yrP∪$ih⊂~z9rv→↔⊂⊂$Y⊂:44\P4yP≥42P1XyrV⊂≥42w⊂≥rP6z\z⊂92\92yr[:⊂:4→P80y≤rFE:≤2rP0\P0P&∩ih⊂)Kr|89λ0w2⊂≤2x92\rw:⊂≥42P9Yv2qz≠y9P0[2⊂⊂9→qwsw~⎇2y9H0yP&∩ih⊂3≥w1z4[w9P0[2εE8≤2r4qXz2yWβEεE↔⊂"cdgλ)bf"Pj⊂→]U a$j T_[∀Nci'jT≥FE∩Lh2y4_x9]∩LFEεE~yqww≤z⊂≡≡H∧-m|↔P7y-W7:vq→y8-|↔]FE.→xm|≥S$f.]CE.2xV|≥j.NFE.0[2-w7]-pz7[m|.nNrxms~y9z-↑.]hjSj"nFBεE4y]0y⊂≡∂P∧-m↑.P0w→-px∧om[x]; not[isconst[x]]]]
iscond <= λ[[x] eq[first[x];COND]]
.END
There are really two issues here: %2a%* representation of the analytic syntax of a language,
and a representation in terms of the language itself. In conjunction, these two
ideas give a natural and very powerful means of specifying semantics.
If this style of specification %2is%* really effective,
then we should be able to
specify other languages similap¬1r\@~)/QCh↓CeJAQQJAo∃CVAa=S]if↓←LA→%' ACL~∃Ae∃CXNAAe←Oe¬[[S]≤AYC]≥kCOJ|@A≠C%]YrAQQJAS9gSgi∃]GJA=\AES9CerAQeKJAIKaeKMK]iCQS←]f4∃←LA⊃CiB\A∪hA%fAck%iJAG1KCdAQQChA5C]rA¬aaYS
CiS←9fAG←UYHAaI←MSh↓Me←Z↓←iQKH~∃ICQBAeKAeKgK9iCiS=]f\@↓/QCh↓oJAo=kYHAQQK\A1SWJA%fABA1C]Ok¬OJAo!SGPA!CfAe%GQKd4∃ICi∧AgieUGike∃fAiQ¬\A→∪M AEkPAoQS
PASf↓IKgS≥]KHAQ↑ACY1←nA→%' [gQsYJAMK[C]QSFAgAKGSM%GCiS=\\~∃QQChA%fXAo∀AoSY0AEJA¬EYJAQ↑AOSYJAC\↓C]CYeiSFAMs]iC
iSFAMaKGS→SGCi%←\AM=dAiQ∀AYC]≥kCOJ8~∃/J↓oSYX↓EJAC YJAi<AoeSQJAC\↓KmCYUCi←d0ACYE∃ShA[=eJAG=[aYK`AiQC8@JgKYCXJT0AS\AQQJAY¬]OkC≥J~∃SQgKYL8@A)Q∀AKmC1kCi←HAoSY0A←aKICiJA=\ABAIKaeKMK]iCQS←\A=LAiQ∀Aae←≥eCZA¬fABA1KOCX~∃ICQBAgiIkGikIJA←L↓iQJA1C]Ok¬OJXA)kghA¬f@Jg∃mCXJ(A←aKICiKf↓←\Ai!JAgKaadAiIC]gY¬iS←\4∃←LAQQJA→%' AaI←OeC4\@A)!JAG←9GeKi∀Ags]QCpAo%YXAE∀AgaK
SMSK⊂ACfA∧AgKh↓←LA¬9 AKcUCiS←9fX~∃¬]HA←UdAaCIgKdA]SYXAQeC]g1CiJA1KOCX↓gieS9Of@Z4Aae←≥eC[f4ZAS]Q↑AaCIgJAiIKKf\4∀~∀] ∂∪≤↓)β¬∪PbPf`$w∂%∨U v~∃%\A←kQYS]J0Ai↑@↓gaKG%MrAB↓G←]gQekGhAoJA5kghA!CmJA¬hAYK¬ghAi!JAM←1Y←oS9Nt~∀4∃8JdD\JT@↓αAG←9GeKi∀Aae←⊃kGiS=\~∃8∀dd\J(@Aβ\↓CEgiICGhA⊃CiBAQsaJ~)8Jdf∀T\@A∧A[CaAS]NA→e←Z@∀dbJT↓i↑@JHdJT\4∃8JdP\JT@↓β\AKYCYkCQ←dAM=dAiQ∀ACEgQeCGh↓isaJ8~∀~∀9 ≥λ~)∪\Awe←]gK Q djXS|Ao∀AoSY0AISgAYCrA∧AeKG∃]hA→%' [Y%WJAY¬]OkC≥JXA 0bXAo!SGP~(JeI←∃fJbA→←YY←\AiQKMJAek1Kf,β
←[aCIJ@AgQKaf@JdbJDAiQe=kOP@∀dhJb↓oSiP↓iQ←g∀A←\@4∃ws←8Q bbXS|]>8~∃βMQKdAo∀AeKIUGJAi!JAck∃giS←9fA←L↓gs]i¬pA←L↓ae←OIC[[S9NAYC9OkCO∃fAi↑↓ckKgQS←]f4∃←LA¬Egie¬GhAgegiCp↓C]HAMieS`↓CoCr↓[C]r↓←LAi!JASeIKYKm¬]hAge]iCGQSFAI%MMKe∃]GKf0~∃Q←\A[C]d@@JeIKCXJDAISM∃eK]G∃fAEKQoKK\↓YC]OUCOKf↓CeJAQQKeJ|~∃'K5C]iS
fACI⊃eKgg∃fAiQ%fASgMkJ\~(~∀~∀4∀_\xy KOS\↓YC[E⊃B||~)'S]G∀AiQJ↓e←←iLA←LA¬aaYS
CiSm∀~∃YC9OkCO∃fACe∀AECg∃HAS\↓iQJ@∀r⊂Jb5GCYGUYkf@ -=7π!jFhcu<RXA]JAEK≥S\~∃]SiPA∧AEeS∃LAISMGkgg%←\A←_AiQJ↓akeJJr⊂JD[GCY
kYkf↓C]HA
←[aCIJASh↓oSiP4∃iQJ↓]←iS=]fAaIKgK]PAS\A1∪' \~∃∨kHAaeKMK]iCQS←\A]SYXA JAck%iJAEISKLA¬]HAS9M←e[¬XAC]⊂AiQJ↓iKqh↓oSYX↓dβ↔≠/⊂4)β&C∃β'w#↔K↔∨#↔⊃β⊗+π∪↔∩βS=↓εk?K∃ε≠?7Cf+S∃β'∪↔πSn+;SMph*'9π##∃βNsS↔K/≠Qβ?2βHε.≤L⊗⊗NM≡GJb∞|Rπ z;≠↓Q]|↓4]2P∧⊗Xpv1z[:yP2↑892y\tww9H4w⊂0H#wx∧hic bold-face type; e.g. %9λ%* and %dx%*
instead of LISP's %3λ%* and %3x%*.
The syntax of (well-formed) %9λ%*-expressions is simple:
.BEGIN TABIT1(11);GROUP;
.P252:
<wfe> \::= <function>
\::= <applic>
\::= <variable>
<function >\::= %9λ%* <variable> %d.%* <wfe>
<applic>\::= %d(%* <wfe> %d.%1<wfe> %d)%*
.END
For example: %d(((%9λ%dx.(%9λ%dy.y))a) b)%1 is a <wfe>. Notice that the
class of <wfe> only covers the unary functions. That is no loss of generality,
since an n-ary function can be expressed as a unary function whose
value is an n-1#ary function. Also it is apparent that the parenthesizing
of %9λ%1-expressions hampers readibility.
Therefore we introduce some abbreviational c will assume that
parentheses associate to the left; the application %d((M#N)O)%1 can be written
%d(M#N#O)%1 or even %d(MNO)%1 when there is no confusion. Also
the scope of %d.%1 will extend as far to the right as possible, allowing
us to write %d(%9λ%dx.(%9λ%dy.%7x%d))%1 as
%d(%9λ%dx.%9λ%dy.%7x%d)%1. Recalling our first
observation, we will abbreviate this as
%d(%9λ%dxy.%7x%d)%1.
Thus %d(((%9λ%dx.(%9λ%dy.y))a) b)%1 can first be written as:
%d((%9λ%dx.(%9λ%dy.y))a b)%1. This %9λ%1 expression represents the application
of a binary function to its two arguments; the value of that function is its
second ap¬≥k[K]P\A)Q%fASf↓[←eJ↓CaaCIK]hA¬MiKdA←kd↓eKIk
iS←\↓i↑~∀∃HPPJd⊂KIqd]rRA∧ADRJD\@~∀4∀] dTht~∃QQJAS9iKeaIKiCi%←\A←_AB@Jd⊂JT[∃qaeKMgS←\↓SfAO%mK\A%\AiKI[fA←_AgS[AYSMS
CiS←8~∃←d↓G←]m∃egS←8AekY∃f\A)<AaeKMK]hAQQKgJ↓ekYKLAoJA9KKHA∧AMKn↓IKMS9SiS←9f\~∃→Segh↓BAmCISCEY∀↓1↓⊗'A∃)1εKM↓β
↓∃HXN3K↔∃π3πK'∞∪3∀↑z)(Xε≡{7Cπ⊗(4+SFKMβ∪.3';'M⊗}r
|bε7,\Rπ>≡↓~∧∞~_=∧
;H≡o≥{\|e
,m%∨↓↔(∂ in a <wfe>, %dE%* if:
.BEGIN INDENT 0,10;GROQPl~∀KI∀JTASLAiQJ↓mCeS¬EYJXKIpJ(\~∀~(KI
J(ASfA¬\ACaAYSGCQS←\XKHQ∨@AαRJ(XAC]⊂@KIp∀TASf↓MeKJ↓S\@K⊃∨ JT↓←d@K⊃pJbA%fAMe∃JAS\KIαJ(\~∀~(KI
J(ASfA∧@Jr⊂∀T[KqAdβ↔O≡K?91α)d!↔'I:5∃
aβ'→α+∪a∃Rβ'Mβ7∪↔∃βNq↓↔∩j))βπv!↓↔∪B))β'_h+∪'∨#';∂"β≠K?j↓↔∪e*Q84(hQ:⊗: h*S#/→↓↔∪
)Eβ'~β≠K↔*β'9↓.!!!!+H!↔∪Bq!∃dB+∪e;JI'¬%ε⊃%∃Eph(4*
β[πKN 3∃εKMβ¬α)HXg⊗{W;⊃π3πK'∞∪3∀↑z))β'r↓↔∩∃*P4+'2β'Qβ}≠∂WK4εNr∧\D**$ ⊗v"
≡2εv}Dε7⊗\TεNr∧\D**%aPT6}$ε/F≥↑εf*D∧V'JV∀εO~ -w.vD
⊗rα\EαBBW⊃α.'¬eα+H∧\GJw∃∃⊗
J %∩+
aQ hPQ!PU&
≡2ε&↑<7⊗N,↑2π& Tπ∨NnL↔Bε|dπ&FT
F∞v}\⊗>*d
v*εl\V"π.]F/~ mw∩ε\≥fOπ]L↔&Nltπ&FQQ&}⊗,\7'~
|bπ& Tεf∞l}V∞>Ud¬≡Nl<Rπ& TεNwL]g"ε|dπ&FT 6∞f>]G/~
≡2π&t
VO↔-}"π ~→#!,<|y-ly(≠ld→];L>~;{L≥~=≡$ ;Yλ n;X⎇
≥{H_.∞≠~8l≡~;{ED≥~→$∞];→.4≤z≠n]→β"L<<≥≥.,(≥~
}y(_.>→8⎇∞5H∃~ T≤]; ↑h_<LT_x;
L9λ &,{{]L↑\z;md|]; ↑i,+AQU~→$
;]→-nλ~<d∞~_=∧∞~→(
\8;Z-lh≠yD (∂≥ll/H~.4≠[⎇∧ z_;L|9λ_O∀_;H ≡≤≠~,<=~;md≠yH
⎇Y#"NM→(_m⎇]Y<N=;{H∞.;→<ea"U~ T≤≤Y.<;]_.M;{H
|H≥~ T_{{Nl<\z-⎇H≤]-L<h~.4≤z;.
~9Z,\λ_9NL<H→.>_8[
≡z~;Lq"[{LT→]<NM→<H L9Z;M≡~;{Ea"P(∧VX{{NL>≥ &∀~<h ∀∂≥yLWH≥z.Mλ≠{LT≤⎇8L←≤≤Y.>z;{D
:<|m≥YkC!*y(≥.<(≥~ T≠[⎇ ≡~;{G$ 9⊃+4w)*D [|H ∀_{{NL>≥ Dλ[|H ←_;<
L(~9AQI9⊃+4w(o$49
'⊃ 9≡¬e ."∧\≡+Vd;**)&∃λ≥~ ]H 9λ[j .!∧9≡KE∧." ,N+↑J%≠),(
≡c"I,E ."∧\≡ J∧W" 9∂∃C"J∧W" 9∂%J .!∧9≥+O%*#"E∃),+D↓"C"EhtSu*πc"U
<Y( ≡Y(≥∞⎇h .!∧*K8m⎇]Y<N=;{H∞.;→<d∞z~8m∧_<Y$
yH~-n→<Y. z⊂42\2]εE !"cdS⊂$g"⊃g*⊂_ _Y]FBεE∩\@∩U⊗q[w;2y≤tww≥λ∩r"mI\D∩U≡↔&nRLP6p|H12P1[w;2y≥2r⊂:≠P∩r"VR\D∩J<W&SWRXP4Y⊂∩r<IU⊂4yH77zεB392rH4w⊂∩Y&RU⊂_w2⊂∩Y&SRUλ92yz[:9P3≤7vP∩Y&RU⊂_<P92\60qt[3P2{→y<P3≤2rP7Xqzy9→w1rFB7s⊂∩Y<∩U⊂_<P∩r≡RU↔⊂ \A∩U qww;→y9tw[⊂⊂0v≠7{yP_v840X2z4qH1t0w→rP7sλ392rH;0y4Xq62yKεEεE "g"εB#7y⊂→|0vx≠2]⊂∩Y∀∩\D r<↔<≡∀RXP [pRXKqww;→y:9P≥7P∩r
∩\D∩Y<W<|JRXWεBεE↔ T i*≥CE↔#i∪jh≥FB.BEGIN INDENT 0,11;
%9β%1-r↓KIk
iS←\h@KI lPJr⊂∀Up]~↓≤S:JD@Jrε∀T[eK⊃kGSE1JASL↓]↑Am¬eSCE1J@A←
Gke`⊗K;≥hS≠K↔*β'9↓."9∃(hS'Mβ⊗{W;⊃εK9↓↔$i∃)9α+∩⊗mB)d!∃WA:5∞rJu∃EεKMβK.#W∂'⊗c∃βSz↓↔∩⊗\i≡u∃Rβ←#↔⊗)↓↔∩j9∃)β⊗+OW3'_4+≠⊗{5βSF)βK↔εcπ∂↔n+;Qβ}1βπ3bβ≠K↔*β?∂∂/∪K↔;≡+Mβ?2↓↔∪a*Qβ'9α+∩5∃Rβ e↓."9∃)ph(αtYh@hTm}"ε/ ≥WεfUDα."¬¬α+H∧\GBr∧W⊂B.Nα+O∃*8*$ J),$∧-xI&∃<Y9∞\y<h∞MnC"D\
'⊃ 9≡%o*(_E∀,(λ ≥Yλ &|I,+.,9≥8l↑h≥≠g$ 9_DV+C"Eh4⊂4JGc"C!)=λ~.4≤≠|n=8[→$∞≠h~-l{≥9 T≠⎇~ ↑H~z-l≤h≠ld≤]; ↑h~;D ( .!∧,+8l≥_⎇;∞↑kH⊂$∞≡<~,<;β"L=_<|d <Y( <;≠→,A"I.!t,+<L\≥8⎇
≥{H≤N]→<kD
~→<lT_<Y$∞<y9∧ <h≤m≥<≠~,m8x=
≥{H≤N]→<h ≥Yλβ!.≠_>$∞~→(∞-{→(
|C"Xm⎇\⎇_-n≤hλ
≥H_(∧W" *E]_;Yn\9y+Dλ88z∧∞];→$
_<h∞M→(_L≡z8h
≥]→<N∞Y=_.M;{NAQHI9λT*H≠,∨(_;∞|><h ,(≤Y.
_8y,D_↑(∧\⊃)i%%HH∃
(_{mn⎇_;NNh_;LD .#dUK<Y,N8⎇~-⎇C"\N]→<h ≡Y(≥.<9λ≥
t_{{N>≤]8nD_( '⊃ *K,<;_⎇-N<h→M}H_(∞>→8z,m8h→
⎇8:;D
yH~-n→<Y.> C"AQKT Ff.C"JM→( '⊃ ,+,<;_⎇-N<h≥l≡h~;Nl;]→,D_<h ∀→[|M\;~<mT≥z~,=λ_{lM9Z9,D≥~→!Q\≤[n <]~,↑h≠yD ];XnM;{X-M=≡( ≥Yλ→N]X⎇~-⎇H_<∞
~8x.M;{KAQU~→$∧-x)&∀≤]; T→[|M\;~>L↑h≥~ T_Y;
≤9H≥
=λ_$ ];XnM;{H
≡h≥;L=_;Yl\β"Z,d≥y( =_;YlT~=≤d [|↓6Xv⊂80\0vrz→y9W⊂∃44yPβE897\2y:<H4yP1Xv62rλ∩Y \2s2y→w:4p[⊂:90[9x0y→w1|KWRXWεB$w⊂6[y2P2→z0tv ⊂0P6_w3zpYrP87\yryyYyFE9→s2y2[:4pvλ:90w≤x0y2[1|P4Y⊂:42H;0v:YP7s⊂_w⊂2|≤92yyZww⊂;Z4qt⊂_ww:0Zw9P9]q2|8≤2yyt[w9FE→2x2w→9P7w≠<P7wλ:42P≥0v:r\P7s⊂≥47yrH9zq2↑892y\tww9K⊂#7yλ2|0v\62V⊂~s⊂:4→P;0v≥rFE7Y⊂0w⊂→|892\ytwwλ1t0w→rr⊂1→qpzyYP;rP_t0w3Yr⊂:4→P70vYP7s⊂_P;0y~pq62H4wεE≠w2P7Y⊂4z9H9zq2↑892y\tww9K⊂:42[⊂:40]⊂;wz[2⊂;4[v0z2H92s2\2w:4Xv⊂:9_w9x0\2w1|KεE*4~yP6rXw9P&∩ih⊂4\P∩Y7≠z∩XP_P92s→y2w:_tv6<H:90w≤x0y2[:⊂60[3zpsY]PεE→7y⊂0H34xλed environment, the value
of %3λ[[x]#cons[x;y]][A]%1 need not be the same as %3λ[[x]#cons[x;z]][A]%1.
The difficulty again is the treatment of free varaibles: dynamic binding
violates referentially transparency. The %9λ%1-calculus is statically bound
and %2does%1 possess referential transparency. Referential transparency
is not simply a worthwhile theoretical property; its corollary, static binding,
is a very useful practical property. We will discuss some
implications of static binding in {yonss(P260)}⊗↓There have been
recent investigations (⊗↑[Hew#76]↑, ⊗↑[Sus#75]↑) of statically bound LISP-like
languages.←.
The %7b%1-rule expresses the intent of function application.
We would then expect that a model of the %9λ%1-calculus would hap
∀~∃BA⊃←[CS8AG←]MSgiS9NA←L↓Mk]GQS←]flAC]H↓oJAo=kYHAIKckSIJAiQ¬hAiQ∀@JoB∀bAC]⊂@JoD∀b~∃`↔+3↔Mε∪∃β7∞K;SπNs↔⊃β⊗+S←↔.qβ↔3.k↔;S4εNr∞Mε*ε]|F&bdλf␈∩ ←ε∞o
LRεNd∧V"Fd<∩J+⊃Q&/∂\≥G~α\Eε:≡∃∀S
ε≥dπ&FT
V}&]EBπ& ]bε∞o∀α+>∀V∩ε␈$∧S>∩V∀εn∞m≡π.f≡M⊗}w4↓PV}d∧π&F}<Rε/∞∞&/∨=≥vw~∞⎇⊗fb
mw"ε≤lf.∨D∞FF∂D W∂.≥M↔'JaQ hRh(T<Ldλu∀⎇Z∞aQU~→$
_;Yn\9y( My<{D}λ≠≠m⎇h≤_..~8⎇-L<[≡$∞Z8z¬D≤z;-≥_<↓⊂~w⊂87]ry⊂:≠P&$iTε↓
p∂%iPAUUghAMU]GiS=\AC`πβ3'∂∂#'?9ε∪WQβ>KS#?/!β∂?v#'S'}sπ1β/CCKπ∨≠'?;~p4*#␈;↔[↔∩βS#π"β'Mβ}s3eβ∞qβ'3g+O'?sYβ≠/$ε/F≥↑εf*∞|Rε≡≥dπ⊗/∞,W≡.nDε≡}lM↔&N⎇l⊗`h,←ππ⊗≡>6N}n4ε∂~ mvff}}3Ph*,V≡∞MDπ&F≡↓λ∧\
'⊃ 9≡∂∃↑*8$<J),$∧≤Y9∞\y<h∞Mh 9 $,+C!*z;:-L<[≡$∧9
'⊃ 9≡∂∃↑
8$<J),$∞↓tv6λ92r:XrP:7H∩r0RLWεE$Y⊂;rP→2s4w→V⊂:4≤7zstλ1ww;→w:4w[⊂7y⊂ \GRXKy:v2KεE∩r
∩\D∩Y<<W<JRXP:≠P12P rs∩XH0w2εB∩r∀∩ND∩r<≡W<∀RLP:7P_2P∩r]∩XV⊂_w2⊂2\z0q6~yt⊂0H1v0y\P7s⊂ \D∩XKr↓xpressions
which always reduce to %et%1 or %ef%1 then the %9λ%1-expression:
%d(p#a#b)%1 can play the role of
%3if[p;a;b]%1.
.END
Notice that the conversion rules make no commitment as to op¬⊃KdA←_ACaa1SGCi%←\\~))QCh↓[KC]LAiQJ↓GCYGUYkfAIKMYK
ifA]<AG←[5Si[K9hAM←HA←eI∃dA←L↓KmCYUCiS←8A←LA¬eOk[∃]if~)S\AMU]GiS=\AC`πβ3'∂∂#'?;~qαS#∂!β'Mε Mβ'"βO#?.c⊃β *q↓α'rβS#∃π∪↔∪W∨#'?9ε{_4+
↓∃d!+ 7↔cπ∪↔OON{9βSF+Hε*
\↔Jε,Tπ≡/l↑&∞b∞,V'.>M⊗}w4∞ε␈∨=≤&f* ≡Bε∞o∀ε}vT∞FNnUaPU& ↑&*ε≡,Rπ'⎇tε≡}]]vrπ>N&∂&\⎇⊗/~ mw∩ε=
w≡Nltε
π,\G.∨M≥vrπ>LWαpQ!PT∂∞
FN≡≡M↔6*
}&&/$ 6␈↔,↑7ε}lN2π&t 6∞fE\'Jol≥G.*D∞&.'\=⊗v:∞Mε*εe function. Normal order reduced the function applicap %←\~∃→SeghlAiQSLAG←eIKga←9IfAi<AGCY0[Er[9C[J\↓βfAo∀AW]←\AMe←4A→∪'@XAiQ∀A←eI∃dA←L~∃Km¬YkCi%←\AG¬\AS]→YkK]
JAiQ∀AiKe5S]Ci%←\A←_ABAG=[aki¬iS←\8A)QCPA←Eg∃emCi%←\~∃!←YHA→←dAi!J@Jr Jb[G¬YGkYUf\A)<AakeMkJAo!ChAi!ChA[∃C]fAAeKGSMKYrX4∃eKcUSeKf↓BAESPA[←e∀AiKe5S]←Y=Or\~)/JAo%YXAg¬rAiQ¬hAB@Jr⊂J([KqaIKggS=\ASf↓S\@~(Je]←I[CXA→←eZJ(ASL~)ShAG=]iCS9fA]↑↓Kqae∃ggS←8AeKIUGSEY∀AEr@JrεJ(\~∀Jd∧JT[YCeSC9ifACIJASO9←eKH8~∃≥←PACYX↓Kqae∃ggS←9fAQCYJA]←I[CXA→←e[fh@KHP Jr⊂K⊃p\Qq`RRPJd⊂KIr8QsrR$RJbA⊃←KfA9←hv~)]←hA¬YXAe∃IkGi%←\Ag∃ckK]
JAiKI[S]CQJtAG¬YYS]≤AiQJ↓CE←m∀@Jr⊂∀b[KqAeKgg%←\~∀∀oλJb0A]←e5CXA←IIKdAIKIkGQS←\AQKe[S9CiKf~∃oQ∃\ACaAYSKH↓i↑@@∃HPPJd⊂KIp9rR@J]λKHR∀bAoQ∃eKCf4∃Caa1SGCi%mJA←IIKdA]SYXA9←hAi∃e[S]¬iJ\~(~∃β\↓Kqae∃ggS←8ASfAIKIkG∃HAi↑↓]←e[¬XAM←IZAEr↓CaaY%GCiS=\A←L↓iQJ~)dβ↔∪.≠S'?rβKW3/→9αSF)βOg∨#↔7π&K
βππβ3'∂∂#'?9ε{→βSF)βK↔'+∂S'}qβKWf+M1β≡ d4+Nqβ↔'Mε/∩ ≡πεf≤<↔&OlTε␈∩
mw⊗n≥Dε␈⊗L↑"bε<≥bε⊗T 6}w=≤F/⊗\Dπ&z ,Rε
↓Q&≡}↑∞W&∂M≥vrπ<=ε.nUd∧Nr∞MεO~
≥g&/.∞&/&≡M⊗}r∞=vn* =voπ↑L↔&N⎇n2π&↑-VNv≡LRbε≥l@hW=⎇V*εM⎇b?"dλ∩βg|lSrε ≡2ε
mw⊗n≥Dε6␈-TεW/>DεNr∞Mε*ε<≡6*α∞Mε∂" ∀π⊗.N\7&N⎇dπ≡/≡\Vv≡T↓PW&↑-VNv≡LW~r∧
FF* =voπ↑L↔&N⎇l⊗bε≥nF/↔∞,W&∂M≥vrε|dπ&FT∧SHBV∃V≡∞L>Vg/4∞'.f↑4ε≡∞aQ&⊗* ←π&.lLV"πMtπ&FT F/6]Mwεn]nBε}d∧SHBV∃V≡∞L>Vg/4
V∞≡
≥f/~¬αeu\L≥b≠3K[bJpQ*FF*∧W⊂B+∃\6∞f>]G/~
\⊗≡F≥lW~ε<≥bπ≡≥↑Vf∂LTπ&FT∞&.'\>FN}d∞'.f↑4εNr∞<W6/,≥Bπ>∨≡3Xh.=⊗v≡T∞FF*∞.Vf/4
V∞↑T
fzε=⎇VnOM\Vw" mw∩ε},F/∩
|bε∂∞
FN≡≡↓~-⎇Kλ≥
=λ_m
z8y$
<h≠n ;KC!(;≤{eD≥~→$∞Y9≥,>~;{D∞];→.4_<Y$ →<xn-8Y9∧
;H≥ ↑[<h
|H≤⎇,.⎇~=∞↑~;{N7h_(
\8z~-l#"[-≤z≥λ∞=;=; ≡→(≥
<h→L≤y=λ M<Y8nM≡(≠n$≠:9m∞λ→;.
≠|∧P_P6rqZ0w4y[P64uYP:42CE&$iT⊂9|vX7v⊂:_q62WβEεE*~4yP2~yqzy\twwεB9t7z[2⊂9zYsryzλ9wvrHεE4w≥2y2y]4w3P≤97q6→vyWεB#4y9]⊗⊂εE≥42y2H6p|P≥rv6⊂_2P:;[P7y⊂≠wy2P≤rxzr[1ryP≠s⊂92Y:qz4[w9P3≠y⊂0P \D∩U r|89→yytw[≥FE0\yzvt[3P:4→|P:2\6tw0]2VεE~yP:4→y2P0[<P92Xyww⊂≥7P12[4r{2H:40zλ:42yYP92r≥qz4w[⊂9rx]rw1r\P;tv≠⊂<tr[2⊂εE≥42P9XvrP7≠y6pvλ37xαmrα}@@4∃'KG=]HXA%LAoJ↓QCmJ↓io↑@∀r⊂JT5Kqae∃ggS←9fAoQ%GPAe∃IkGJ4∃i↑A⊃SgiS9GhA]=e[CX↓M←e[LXASf↓iQKe∀AC]r↓eKCg=\Ai↑↓EKYS∃mJAi!ChAi!KrACIJ~∀E%]QKe∃]iYr↓ISMM∃eK]hλ@Jr⊂∀T[KqAeKgg%←]f}4∀~∀~))QJA→Segh↓ckKgQS←\A%fAKCMSKdAQ↑AC]MoKd\4∃∪LA ←iPAIKIkGQS←\AMKckK9GKfAQKe[S9CiJAQQK\AQQKr~)dβ↔O.cQβ'rβS#∃π≠π7∃εs?K7∞aβ≠?⊗i9αSFKMβ'~βWOW∞c3eβ≡ 33↔"βS#∃∧≠#WK≡A6K?∨≠↔H4W##↔?⊗+59hP4*SF)βO↔≡{;⊃β∂+↔OSN{9↓β⊗+GW'⊗+MβO}k∃β↔Gβ3π;∂#'?9ε{→↓ Ns#↔K.sS3eε#'≠≠/∪↔;Q∩p4*π"β?;∃εc↔[↔bβ←∃βnK∨#Qπ≠πeβ&CπQβ↔Iβ∪↔6K;'SN{91β/CCK↔∨≠'?;~β←'SBβ∪'≠6+K↔; h+;?⊗kπ1β6{K7Mε K∃↓⊗K;#↔⊗+;S3Jβ∪'≠6+K↔;"⊃94T∪WQ↓π##';↑K;≥β}1↓∃dB))7↔GβK↔O≡K?;Mε Mβ≠.s∂S'}sM04VKMβSz↓βOπJβS←=α)d!∃Rk↔cC⊗+OO'}sMβπ⊗)β∪'∨#';∂"β'→β>)β∂πrβ↔c#N∪'Qβ∂∪∨W7.sSL4W≠W∂!π##πQπ##∃β6 3W∃ε≠?7C/#↔⊃β↔Iβ?;*β↔cC⊗+OO'}qβπCεc'↔⊃π#=βSF)βπK?+7↔;"β'L4V#'≠≠/∪↔;Qε3K?5π##∃β6 3W∃ε≠?7C/#↔⊃β↔IβS#*β?S#/⊃9α'v#↔↔⊃bβ≠?Iα)d!∃Rk↔cC⊗+OO'}sL4+>C'∂!εCπ[∃εs?K7∞aβ≠?⊗kM1α~q∞ ?.C5β#∂→β↔O& 3'≡C↔⊃β&C'M8hP4*←F Qβ∂∞qβ←∃π≠πeβ∞∪?WQα)d!∃Rk↔cC⊗+OO'}sMβ←FK∂!β&y↓∃Kv{Q∃)εCπ[∃εs?K7∞aβ≠?⊗kM|4T K∃β∞c1↓β/CCK↔∨≠'?;~β←'SF{WQβv{K7πbβ≠?Kj↓ ↔G.K[π3.sQ ⎇∧s=mβO!β∂πrβ ∃β≡C?←8hSS#π"βOW∂Bβπ9βN#↔;SN3'∂π&K?9β>{W3⊃εc↔π⊃π#=β∂}sSKπ&K∂S'}sM84U∪↔∂πfc';≥π##∃βNsSW'&K[∃↓π∪↔3π&K?;OFKAβ /#←↔↔rβ;?9o#↔K7NsπS'}qβπ;"↓ ;=εs?K7∞aβ≠?⊗i 04V≠π9β>)β↔O& 3'≡AβS#∂!↓∃dB)Eβ↔GβK↔O≡K?;Mπ;'S#␈+Qβ;␈∪7π1ε3?K7~βπK∃α∪∪'≠6+K↔;"⊃β≠K}h4+↔GβK↔O≡K?9β>C'∂!εCπ[∃εs?K7∞aβ≠?⊗i⎇α;zβπ∨πNq1ZuZ←π⊃≠9Fvyε+c#'⊗KSL4W#←=β/CCK↔∨≠'?;~a↓↔∩J)Eβπv!↓↔∩R)E↓β>KS!β∞s⊃β←O##?W"β;?Kn 1β≠␈∪51β⊗+OC↔≡KS[↔gI84*F{←↔[/⊃βS#/≠∃βS>yβ↔cπ∪↔OON{;Mβ∂∪∃βSF)↓ O∞k∃ βNqβS#*βO↔;≡)βS#∂ 4)Mε ;⊃↓∩qeeeKI
99rβπK∃π##∃↓↔≠π7∃∩qαS#∂!β'Mα+∩)∃
β'Mβ&C∃β3Nk'Qβ}1β¬β≡+GW↔v≠∀4+}1βπCπ∪?c'n S'?w→βS=α+∩%∃
p4*πg≠=β←*β∂π9ε+c#'⊗KQβS>yβOW≡A↓∃dB))7↔GβK↔O≡K?;MbβOπeα+∩e∃# ∃Eβ∞s⊃↓↔%I∃QI+ 04)π;#'∂BβπK∃ε#'OSNs∂P4VK9βSF Qβ;zβK↔∪.≠S'?rβO↔G.+;∂∃π;'31π∪↔∪W≡)β?;*βS=β&C∃β?&C↔I1ε∪WQβ␈+H4+NsSW'&K?9β≡ gMβ&CπQβ&C↔eβ∂∪∃↓ &C∃βO∞k∃β≠.s∂S'}q 9α&CπQβO→1β≠␈⊃βπ;Hh+πK?+7↔;"a↓↔∪
))β←*βH∂/∞
GJb∧ &␈&∧∞&.'\>FN}n4π⊗/>]G"ε≥dπ&FT∞6∞nT WGπ,↑7≡N⎇e`hUM ↔"ε≡4α." ∀S#
\Dε
JπTαEJVF"." ∃∩+
aQ hUM Rπ⊗\NV∨&≥⎇bπ↔]LW~ε|dπ&FT∧SHBU%V≡∞L>Vg/4 6∞vm}BεF]Nαπ/5aPU& Tπ⊗/=⎇G/&≥⎇bε}d∧π&FT
⊗&.∀
v2α.<⊗n*]lW∨~$∞&/∂]≡&/~∞>G⊗}l|W∩ε≥l⊗gO=≡10
M Rh-≥g&/.∞&/&≡M⊗}r
|bα↔<≥V*nl↑7~∩∞⎇εN≡∧∞v*ε ≡f*ε,\Vrπ↑=⊗v:
≡2απM ↔"ε|dε/GL]g≡N⎇l⊗`h,↑↔.∞M≡GJr
Mε∂"
≡2bπN⎇rε7]l7&N⎇n2ε∂,Tε≡}n=⊗&/,\Bπ& Tπ≡∞\Tε7.l>FN}d
⊗0h-mrε&≤lf/⊗]l6/~ <⊗rε,Tε&/L\7&.D∞Vv&↑$ε∂π
M⊗≡∂M≥vrε|dπ&FT g.v>M⊗}w4∞Fzε≥o∩ε∂,}Vn.nN2uxQ*v*ε<≥`hV⎇≡f*ε≥dεNwL↑'π⊗↑L↔&N⎇dπ&z∞Mε*αW⊃α*R\<⊗f∨]NW~π>\6BπM ↔"ε≥dhWM ↔"ε≥nF/↔∞,W&∂M≥vrε}↑"πε≥≡'~Q,WGπ,↑7≡N⎇n2α.I∀S
ε≥lBα.I$S
b
}"α.K∀S#
V∀ε∞vD∧V%JVF"+
AQ&F∂lTπ&FT∞6∞nT
V.∞m≥f:r
MεO~∞=ε␈.LDε⊗* ∀ε≡}nm⊗v≡≥lphV≡,w.n]nBε6}$εn∞≥nF∞Nm≥f:πM ↔"πM WJε≡,Rπ& Tα↔≡≥\Rε7]l7&N⎇d"ε/l]bπ&
}V>@Q.FF*∞,V'.>M⊗}r∞.Vf/4 ↔⊗*∧V&v␈DU"π∨\lfN≡≤]g"πMtε&O>
F∂J∞Mε∂" ↑↔.Ol≥F.v<ThQ`~FFO4 F.n⎇n7'⊗≡M⊗}r ≥G≡z ⎇↔6/4 7⊗.L]f≡*∞Mrπ& Tπε␈=≡FN}d∞FF∂D∞FF(Q-V.∞m≥f:α∞N&∞w<<Vv'4∞FF*∞,V'.>M⊗}r∞.Vf/5dα∧≡⎇↑ε∂⊗T∞FF*
≥f≡}↑
F/&]lW∨~∞,W∨.NN0hV|d∧Zrλ⎇v&.Ekrph$λBr¬<=w'"∧V&F∂4U"ε/
⊗⊗OL\Bε
]v&.D
w∩ε≥nF/↔∞,W&∂M≥vrε|aPW& Tα+H∧U"n≡≥L7.g↑5Bε∞lD∧"r
↔⊗Z
↔~π=
w>r∞Mε*ε↑≡VO6≥LP
ly(~ ≠⊂:44\P6wr→vεE7Y⊂∩r,IZ_RXH0w2⊂ r,RZ ∩XV⊂_w2⊂!K⊃kpr≤{wy:~⊂0yP≤t7{wλ:42P→xzt{_v2w1YP7sεB∩r$RLP0w2λ∩r%∩LWεEεB*47zYt⊂:4→P9|w≥0|⊂0[2⊂89_svpz~qyP7Y⊂:42H∩\D∩J⊗qpv_zv:yH4yP8]tz2P≤tvx6→V⊂εE≥42yrH92yz[:9P9Z7zv2λ44w:λ0z⊂:~2P87]ry⊂7Y⊂:42H∩\D∩LVqpv_zv:yH4w⊂:~2FE9→pv6P≠s⊂9r[pw:4XyWεEβE yP≥rP9pZr⊂0zλ:42P_2stw≠4w3P≠s⊂:4→P9rq]4ww⊗λ:44yH1pv1]v:yP≥pyPεB4w:2[22r⊂≥7P2|≤64qp]2P:4→P4r2XP7s⊂λ3:w1]4ww⊃λ0w2⊂λ3:w1]4ww⊂_x864Xpz4w[⊃↔εE∃42y2H4yP0H92py[w0q6≡P9zq≥62P2~yz4w_z4wwλ12z;Yrw⊂!Z:y1t yP1w[1rx:~wwεE≠s⊂0P→:w1z~ww⊂0\P0P9≥v2P7Y⊂1wy≤2yx7[22w1YV⊂0w→⊂:42H:yzp[⊂9rz z42w\2z4qCE;4r]P7s⊂_P3:w_z4wwλ0yP0H9rz⊂≠s⊂7y→2y2rλ80ty≤KirYP=|w[9yT(∞Y∀␈/KεE$wλ:42P≠0z:2\⊂9rz≥4w3P≥rP90]42y⊂≠0z:y_v6<P≥44w5H7s⊂:~2P2v→vrw:≤P7s⊂≥42P2≠vptwβE0w2λ90w3YP7s⊂_P3:w_z4wwλ0yP2↑4yz4[3P∩Y≤94wy U⊂:7H:42P_ww9z≤:qz4[w⊂7sλ:42P→:w1z~ww≥εB↔!"cRg⊂**T'⊂'c⊃⊂⊃=Q ⊃?⊃≥CE⊃&2]⊂∩Ys U⊂12H:42P→:w1z~ww⊂=O<⊗_← ⊂≡<V ∨⊗⊂↔ ↔?⊃↔βE↔"g⊃εE*4≥yP4wλ:44yH390vYP7s⊂≠tw2⊂≥rP27H77z⊂≥44w5H7s⊂3≥w1z4[w9P;Z4qt⊂_pw⊂:_urFE≥42vyYv;2yH0yP0\3zvr[:9]⊂ Ys-s↔RU↔⊂∀zqt⊂→:w1z~ww9P_y2P1Xv62rλ∩Y \rv3⊗Xx864Xpz4{→KoRU εE)r]2y0vλ60w3]psryK⊂4w1[:r4w→P&$iT⊗⊂0v≠7{P:~2P89≠qrr:\0v⊂0[0v7sH7sεE≤rv3⊂_x864Xpz4{→P3:w_z4ww≤W⊂*4→|P0y→P0P9\2qtp[⊂1pyYP7s⊂→:w1z~ww0vλ0y3z[rw:9HT=|w[9yT(
[∀␈∀KεE*4→P&$iT⊂3:w_z4wwλ24yq]yyrrλ4w⊂:~2P89≠q62vH7w⊂=↑ww∀(∞\T␈⊂~yP0wλ4w9z_w1rFB7s⊂0H9rv3 px86~qpz4]2P&$Th⊂3:[1z4w[↔εE)Zw1rP≥rP1p[⊂22p[⊂;tz~⊂9rv→⊗px8≠4qpz~ww⊂0\P0P8≤7qrr≥y0v⊂_ww1r\:εE0]⊂62p\z⊗⊂⊂λ⊂82y~0x9P≤wvrP≠s⊂:4~yP⊂:[22y9]0w24[3P;t[6⊂42[8εE;Zz4⊂:~2P6p]42vp]4qpvλ8zry]4ww9K⊂ spZw⊗⊂;YP::y≠⊂:7P≥42P∩ND∩XVXpv1z[:yWεBεE*4→P1pv_zv:yH4yP0[⊂0x8≤7x94Xz2P:≠wv⊂3≠y⊂9z≥r<tw→P9rv→⊗px8≠4qpz~ww⊂0[2εE3≥w1z4[w⊗{0[syntax allows such c↓←]giIkGi`~qα';&+↔⊃↓+∪↔[↔↔KS#'v9∃D4VK9βSF)β∂πf≠W3W~β'Mβ
βK↔C⊗+O↔;& S'?rβ?→β
β≠W;∨#'?80
∂?7ε K∃β&C'L4W;'S!π##∃β≡{;∪'&K?9βNqα2&≥↓β←#.qβ←∃π##';Zβ?→β&C∃αMn+cCK/≠O'?ph+K↔π∪↔O↔w#πS'}qβ?→εcπ;∨. ∨∃β∂→βS#*β3π;?+π∨∃εKSO↔f19αSF+9βSF)β∪'∨#';∂&K?84V∪↔S←.+9βC⊗{∨Kπj↓βπ;"β∪πS
β∪'O∂βC↔π↔→1β+/≠Qβπ~β'Qβ&{↔MβNqβS#*↓∃d!+ 7∂πf≠W3W~r⎇84U##∃β≡{;[↔↔≠'?9π∪W3↔~β?→β&C∃β∂∞c∂W3/→βπ3f{]β¬α)d!∃Rk↔cC⊗+OO'}qβS=ε∪∀4+∂βC3'.!βPεt ⊗wJ∧∧SHBU%V/G∞,W∨≡≥⎇bεNl=G.&≥lrπ& Tε/G∞,W∨≡≥⎇bα+-≡G≡.LdRRpQ*FF/,Tε∂⊗T∞v.fE]6v␈⎇dεf}⎇≤6∞b M⊗66≤>Vg&≤↑2↓0~.W∨≡]MB?~∞ ↔⊗∞M␈αuz
≤bπ>T ⊗ff}qPW.n,W⊂≡≤Z8nL9λ≤l]→K8.∞≠~8l≡~;{Ea"U~ ↑Y(_. 2P0zλ62py]⊂:8πo questions. Can we characterize a useful subset of functions
for which self-application is tp¬¬GiCE1J}~∃UgS]N↓iQSf↓gkEg∃hXAG¬\AoJ↓MS]H↓BA[←⊃KXAM=dAiQ∀@Jr⊂∀T[GC1GkYkL@~∃o!SGPAIKiCS9fA←kHAS]iUSiS←9fACE=khAMU]GiS=]f\@4∃)QJ↓ISgG=mKe`Jβ?→β&C∃βππβK?C⊗KπS∃ε≠3πO~β?→β7+;∂SN{;Mβ∞s⊃βSF)β7?&+04+≡{;OS↔+∂S'}qβπK*βS←=ε{→βSF)β'7ε{KSπw!↓βπ∨β↔∂S~β?→α"qαO∂␈#Q∨Mπ;?K-ph(4(hP19qfc'OAε∪?KK␈;MyxhR2&Nαβ#πLhS ?K⊗{←↔⊃εC↔π[Nce1β↔+Qβ'v3?K7∞c3e1ε3K?5εkπS#.kπS'∨→βπ;"β3?∨N→84*O 4+W≡+MβSF)β3πv;Wπ∨*β?→β7+;∂SN{;Mβ∞s⊃β≠.s∂S'}sπ1β≡{7C?≡KS'?raβ W"↓β←∃εCπ[∀hS;?S.!βS#∂!βS#O→β'Mε βO3␈βC';/≠Mβ'rβS↔KnK;?3};eiβ&C∃βW∨+π04VK;S↔↔βK↔S∂#'?9αβ?→αdJNAβ/CCK↔∨≠'?;~β'Mβπ∪?∂↔'+Kπ1ph*←∃π;π;Qπ#=βK.cπS∃π##πQπβK?∂.#WKπbβ/;?>c↔∪∨*βS=β&C∃β≠␈∪7π3O≠5βπv!β7π&C↔7π&K∂L4W;#'∂Bβ←∃βF [∃β⊗+↔9β&+[↔3␈β';≥r4(4Tk?OQε{→βSF)β∪↔≡≠K'C&K?9β}1α2&≥↓β←#N≠!β←*β#π[*β∨'[.qβO=ε3πIβO→β∂3∂≠O'≠N+⊃βπ~Xg?ε+KπSN{;π0=y84*&CπQβn+π;Mε{WIβNs≠?Kn 1β∪/≠∂K'π#'?9ε{→α2M~Aβπv!β?W∩β3πS/⊃β∪↔≡≠K'C&K?9β}1βS#*α2&Nh+';&+KCK/#↔Iβ∂∪∃β∂␈+∂#↔"β'9β&+K7Mε{→↓ F{]β∪}+Mβ'"β←?KZ↓β?Iε{C↔K∂#∃ 9∧K;∪↔. 4+SF)β←#}c∃βC/∪C?O*β?→↓+≠↔[πb))β←∂→βS=ε#↔O∂⊗K ∃β/CC3'≡KS3eπ;#πQεKMβSzβ#πCε+84+>C↔9β
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w:ε↑\6Bε|dα+≡↑l⊗b*$
↔~α.,V∞fO∀"∧d~:αε∞lDεF␈t
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)QE∃R↓999∧ ∃S9*Q&u0hQ:⊗:#X4+←*βπK∃h+K↔f S';:β¬β?ε+KπSN{;π1εs?S'}qβ←'&Aβ¬β&+;?S∂#'?;∞`4+;␈#'?9rαS#∃εc↔≠QεCπ;⊃π≠'∪∃ε{→βSFKMβ↔∂+πS'}p4+'~β∪↔;␈#πS'}sπ1mπ##∃β⊗K∨#QεCπ;⊃π≠'∪∃εKMβ?ε+KπSN{;π1ph*S#O→β∪↔v{SπSN{;π1n{C↔K∂#'?;∞aβ∪'∨#';∂&K?9βO→βπCπ∪?CKN S∃βNqβ¬βn{K∃β>+;↔K∞aβ∂?w#↔cQph*←#.qβ←∃ε K∃βπ∪↔O↔w#↔⊃β>KS!β≡{7↔?v)∨Mβπ∪?∨K∞iβπ;"βπO/.!↓ ←F Qβ∪}+Mβ'"β∂?7π+S∃⎇⊂h+←∃π+OWπfceβ .;'9β␈+Iβ'w3↔OSN;πS'}qβ?C/∪πS'}sπ33Jaβ∪'≡≠?[↔⊗K;≥↓↔;#πQε#?↔MεKQ↓∃⊗#=∃)z⊃84(0
π;?Mε/∩ =vnn⎇dεn∞m≤f/∨L≡FN}d
v2πM
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someone else's."←
Frequently, by tracing itpε↓KqKGUiS←\0AoJA
C\AI%gG←m∃dABA⊃K]←i¬iS←]¬X@AI∃gGeSAiS←\h~∃
]≤\XFE¬PBGSPGG←[AkiKf
iQJGMckCe∀Ge←←PD\~∀9 ≥λ~(~∃/Q∃\@Jg≥eKCh↓[←iQ∃dJTA]CfAaIKgK]QKHASPAoCf↓OSmK8ACfA¬\A←a∃eCiS=]CXA∃qKeG%gJX~)oSiP↓iQJAAeS[CIrAS]QK]hAQ↑AS]Qe←Ik
JAiQ∀A→∪'@AKmC1kCiS=\Aae=GKgf↓oSiQ=kh~∃%]m←YYS]N@↓G←[a1SGC`&+⊃β;∞k↔Mβ∞s⊃β∂}s∂↔C'→9α≠␈∪7MβNs[?36K;≥↓+≠∨K↔∂!β7?&C↔I∃Rβ←↔K*4+↔6 3WπLV"π ↑&F∂∞4π>OM
w/"∞]f&/.>F∞vM≥f:πM Rε&]mw&∂M≥vrb .W"ε≤dε∂≡<\Bα↔⎇ ↔"εM|W_h$V6?⊗\≡Bεn}Mε/∩U$ε&{t$πN␈T∞p
};→λ ,(~_. 2⊂89→yyrrλ:o given a comprehe↓]gSE1J~∃aTdβ↔3Jβ?C↔⊗ S'?v 1β∪.3';'M⊗}rd ε␈>↑lW∩b∂≥w*ε]≤vG"
↔6* M↔≡≡}lW⊗.D∞FF*
≥g&.lLV h-l↔'/,Tε}2∧V6?⊗\≡Bεn}Mε/∩U$πN␈↑.6.ffαh∞M→;H
≡λ≥{n]→λ_LT≤Y; ≡~=Y-O(→8.?(≥≠d →<xn-8Y#!-=≤↓P
42y∀H12t0]4wy↔λ$w22Yr⊗⊂7[1rP:~2P22[7z0z~ww⊂7Y⊂∩Ys\eat mother%* has been discover↓KH~)ckKgQS←]f↓YSWJE/QCPASfAQQJAm¬YkJA=L@JgQO[←C→6QπβHFQ#+=)
FQ∧F\G∧$RS:J(}@D@4∃CeJ↓kgkC1Yr~∃¬]goKIKHAEdAkgS9NAiQ∀AIK]=iCiS=\A←LJgiO5←CLJ(t@Eo!ChASL@~∃i!JAmC1kJA←_@JgG¬e6Qα \G∧StJT}D4∃)QJ↓ckKgQS←\A=LAQ←\A←]J↓OSmKLABAG=]mS]
S]NA¬eOk[∃]hAi!Ch@~(JgKm¬XH¬)α)K∪?/→∃Eβ6 'S#7+33eπ∪↔CK/≠↔;Q∧b&NAε+[π3. S'?rβ'Mβ&C∀4+∨+ +↔∨!β?→↓2zn∨␈⊃
]Nmq84(hP04*>)β←'faβCπ⊗ 33↔b↓β?W∩β∪↔[.c?C7.sQβ?2β';S/∪CK↔&+KMβ6{Iα2M~AβONs∂∃β. ∂!β∨+ O↔"`4)∃∨#∨7?∞11βS>k?π≠∩))1β∞s⊃↓∃≡+[π1*Q04+>K31β∞cO=βNsSK?'+∂∃βv+]βC⊗{ 3↔o→β≠?∩β?WIεkπS#.kπS'≡ 1βO.kπ;SN≠M84Ph(∧␈↑$ε6O.>B∧d~:απ∨\.6/" =vw≡≤LW↔~ nVv∨M≥vw~D 6}o
}7&N⎇eBε∞lDε≡}n>F∞wN5`hT=⎇g∨&≥nG~π⎇≥Fbε,Tε.f]\Vw'4
v2ε}↑"ε&⎇\⊗Nr
|bεNnLW↔π,↑F∂&≥⎇bph)}W∩εM⎇V∞Nd∞vNfD
⊗v≡N\F(h.Mε*¬5\WGπ,↑7≡N⎇n2π≡≥l6*αV-V␈∨DU"∧d~:αε/∞∞&/∨=≥vw~∧V&&.m}F**$
6/G∞.2bε≥lBπ≡≥l6(h$
V∞w∀
v2ε}↑"∧d~:αε7]l7&N⎇n2ε∂,Tπε∂.M⊗∞b nVv∨M≥vw~D↓PVOD
↔~ε=⎇g6.m≤Vw"∞Mrπ&≥M2ε∞-}W"πM Rπ.lLV6Nl\Bπ6≥NV*b∧W∀∩+∃d∧Nr
}FF/$∞v␈⊗Nk∧∞y#"N⎇<zλ∞Mh→>∞L;Yλ
}<H≤ ≡]~8-D→];L>~;{N4λ≥≠d Y( &.≠|α0[∩↓* functions on
an extended dlπ[C%\\~∃¬fAEK→←eJ@!ws←\! djb%|RX~)oJAg!CYXA
CYXAQQSfA∃qiK]⊃KHAI=[CS\KC&JD\~∀~)∧β↔≠␈∪∃β←*β∂π9π#π3-π3↔KeπβK↔∂O≠↔3eε ?W"βS#∃αβCK?ε+KS'/→4+n S#↔n S'∂∞aβ≠Wv≠S'?w→β∪↔v{S↔⊃ε∪eα2M~Aβ≠.s∂S'}sM04W;∃β7/≠Qβ∨O3∃β7␈∪∃β∂∂∪↔≠WbβOSW'IβS=π##∃βv SWK*β?→β&{7π'w→84*v{]β?2β?WIπβK'7O#'[∃ε#?7πNsL4+O→↓↓↔#cπS?kq∃)8hR'Q∨~β';S.KS'[*βOSK.≠SWK*β'Mβ∂+'S∃π≠'7Cf)1β ∂≠'∂πfceβ+/≠Qβ¬π≠↔Qβ}1βπS}kL4+␈⊃β;πn+Mβ←O#!β;zβ';#/∪↔;Qπ∪↔3π&K?;OFKCM↓ε 7?;:βS#∃ε+3↔7.sSM8hRπ;?&C↔Iβπ∪'7'M↔6* Mvn∞≥n2εO4∧V
'$V∩bπM Rε&⎇\⊗Nr
|bπ'.↑FBπl≥G./5aPPh*Mε*εM⎇V∞Nd∧V#g<←ππ∪dU"αα
≡2εn},Rε≡⎇↑εf/π4εO"
≡2αε∀∞6/"
|bε.L]V.wNk∧ ]=λ
\:β<FB2v2vYw:8β are related.
In our discussion of %d<sexpr>%* bedπS]9S]NA=\Aws=\Q h\S|~∃]JAi`⊗K↔⊃β&yβ7π↑)β'Qε≠3↔π∩βS#πDπ&F↑,RεO4
V␈⊗T∞FF∞d∞7NwL∨αεNnmvg6\E`hU|Tε≡␈]LBπ≡≡α(∞M_=β!,[|H∧\≤i-ε∀,(_-lλ 9∞ RZ→∩LP4w⊂ r<sexpr>%* then the essence of "dotted pair"
is contained in the concept of the set-theoretic ordered pair,
<%ds%41%1,%ds%42%1>. P !kfAi!J@E[∃C]S]≤DA←LAiQJ↓gKhA=LAI←QiKHAACSef↓Sf~∃
CaikIKHAEdAπC`↔#↔O'∞qβCK}#W∂Qb↓↔⊃s≡+cCIr→↔πa*Qλ g<←ππ∪dV∩ph!Q"pλtSu*πc"S ↑ |h ={]~-n9(≥
(_;L≥≡<z.4≠yNAQC"KH(1r3Dλq3UλZR5∞aQWo≤l←≤≤ODπNO(πL=≠{'d∨λ
πNy>≤∞'H HπNy>≤∞'J+C! ↔"g"βE↔ h⊂i*εEβE+rP_y2P:≤<tw3H:7P4[:2y8≤2z⊂:~4yP!∪#⊂2x]pz4w[⊂0xβ a definip %←\A←_AiQJ4∃I←[¬S\@K⊂ygKqAd|JT8A%KCM←]CE1JAS]QKeae∃iCiS=]fA←_@DtttDAC]⊂@ExD↓CaaK¬dAi↑4∃EJA∃ckCY%irAC9HAgKP[iQK=eKiS Ak]S=\XAe∃gaKGQSmKYd\A)Q%fAeKMkYif↓S\Ai!J~∃KEkCiS=\t~∀4∀]¬ ≥∪≤Aπ∃≥) %%(w' 1 π(A⊂v~∃>qgKaqH|@z@qCi←Zx@KB&∀T@yg∃qad|KCpJ(@ygKaad|@∀b\~∀9 ≥λ~(~∃)Q%fAY←=WfAY%WJAC8ACYO∃EeCS AKck¬iS←\0AC]H↓CfAgUGPX~)[CrA=dA[CdA]←h↓QCmJAg←YUiS←]L\~∃)!SfAa¬eiSGUYCd@ I←[C%\AKcUCiS←8DAQCLABAg=YkiS=\tAi!JA&[∃qaef8~∀~∃QQKeJ↓SfAB↓]CikICXA[¬aaS]≤A←LA ≥ AKEkCiS=]fA←9i↑AgUGP@E⊃←[CS8AKck¬iS←]LDX~∃¬]HAi!JAg←1kiS←9fAi↑↓iQJA⊃←[CS8AKck¬iS←]LACeJ↓gKif↓gCiSMMsS]≤@~∃i!JACEMieCGPAKgg∃]GJA=LAiQ∀A¬≥ 8@~∃)!JAeK
K]hAMikIS∃fAEr↓λ\G'
←ihA¬]HAε8G'ie¬GQKr4∃gik⊃rAiQ∀AEKQ¬mS←d↓←LAgUGPAKEkCiS=]fXA¬]HAO%mJAKaSgiK9GJAG=]ISi%←]fA→←d~∃M←Yki%←]fA→←dAgUGPAKEkCiS=]f\~(~∃≥←\AG←]MSIKd↓iQJA→←YY←]S]NA ≥ t~(]¬ ∂%≤Aπ 9) %∪Pv~∃>qoMJ|ttz@qmCeS¬EYJ|↓x@Jr JTym¬eSCE1J|KH8JTyo→J|Ax@KHP∀TyoM∀|@yo→J|KH$JT~∀9 ≥λ~))QSf↓SfAC8ACEEIKmSCQKHAM=eZA←_AiQJ↓¬≥ A→←d@AQQJ@Jd⊂Jb[
CYGk1kf\~)/JAo=kYHA1SWJAQ↑AIKMGeSE∀AiQJ↓IK]←QCiS←9fA←L↓iQKg∀AKck¬iS←]LAS\A∧~∃gieYJAg%[SYCHAi↑AQQChAUgKHA→←d@yMKqadxOf\~))QJ@↓IK]←QCiS←9fA←L↓iQJA∃qaeKMgS←]LX@yo→J|X~(A←LA¬aaYS
CiS←9fX@K⊂PJby]MJ|FqoMJ|∃HRJb0AC]H↓←LAi!J@Am¬eSCE1KfX~(ymCe%CEYKL|XACIJAUkMhAG←9giC]QfA←L↓iQJA1C]Ok¬OJvA
CYXAQQSfA⊃←[CS8@KIε∀T\~∀4∃ qaIKggS=]fA←_AiQJ↓M←eZDJr⊂∀bymCISCEY∀|KH\∀byoM∀|DACIJAgkAa←gK⊂Ai↑AIKaeKMK]h~)Mk]GQS←]f8A
SeMhAoJ↓G←]g%IKd@4∀AiQ∀AgKh↓←LAMU]GiS=]fAMI←Z@K⊃εJTAQ↑@KI JT\A]eSiJ↓iQCh↓gKhA¬f~∀K⊃εF2G JT\AQQK\A=kdAI=[CS\↓KckCQS←\A%fAKqAeKgg∃Ht~∀4∀]¬ ≥∪≤Aπ∃≥) %%(w' 1 π(A⊂v~∀~)?ε@z↓ε3εK∧&KIε4∀↓:⊗t 4*SFKMβ↔∂+πS'}qβ#π_h+;=α)K';&+K↔O&K;≥∃
βO?3/#'?;~qα¬β≡K7C3*β∂?Ww#';≥ε K∨Wn+;Qβ>K31β/≠Sπ fKO!β&CπQhSW;3/≠MβSF)β∪?n '9↓."
∃)εKMβ¬π≠';∨f)β↔3.k↔;QbβS#↔rβS#∃εsW7 /⊃β?→ε3W;∂&K?;LhS'9↓."
e≤→∃)βO→β∨K. S↔Iπ##π9π##∃βw+7 ↔∩β?→β.c↔7↔w#Mβ'r↓↔∩
*Q84*&C'Mβ&{↔M↓+∪;?Q+ βOπ↓∀π&F≡↓λ∞M→<Y$ <Y(
mh≠;lL;≤↓P≠s⊂:4→P∧⊗qXv1zv≥yWεE∩z⊂⊂9X|yP4\P:40]⊂7zyλ4w:2\892z_z4wwλ7s⊂⊃ r RUλεE4yH:7wP_97pr ⊂εEεB+t0zλ4yP7→rr2rλ4yP0[⊂4w:→y892]0z4w[⊂7s⊂→:w1z~ww0v~z<FE≥t4qtλ;tv6λ0v67]P0P9[v:z4[w⊂:7H:42P_q7{2H27vpZw⊂2x]pz4w[≥P4w→2rr⊂~zεE9Z7zv2λ0v67]P0P7_z:y0[⊂4w:→y892]0z4w[⊂9zqZ⊂:40]⊂:42H897x→y:4r\P;t4XtεE;YP2|8→qz⊂3≥w1z4[w9P:≠P87y\ryP0\2V⊂4[⊂30q]⊗⊂:9≥rP4wλ:42P≠wr2v εE)q[z:⊂3X{2P7[2P9zXt⊂εE~w:2y≤92z0]4ww⊂≠s⊂⊃∩Y RU⊃λ22v4[tz4w→P⊂;t_z⊂42H1pv6≤FE⊂:~2P1v_yyP7Y⊂⊃1w[:4w:[zyP3≥w1z4[w9Q↔βE*44\P1v0\yP7sλ3:w1]4ww9H4yP9→yz94Xz2r⊂→w7zsZ⊂:7P≤pz4yY<P:4→P92x]ty2vYw:9P→7yεE→7y6p[⊂9z:Y<V⊂1≥z⊂19≠pr⊂2[7zstλ:7P0Xz⊂0yH:42P→2w7z_z4ww≤P7s⊂≤97qrY:y2yH4wεE_x864Xpz4{→P897Yy0vvZw3P6_w3zpYryWεB+rP;Zv6⊂:\rP:4→P77z_z4wwλ⊂⊃∩r⊗RXb∩M_RXQIr RXHb∩Z→ r.RXH⊂:7P≠rpw⊂λ:42P≤rz⊂7YεE1w[:4w:[zyP3≥w1z4[w9P3≤7vP2≠vptwλ"∩Z_IXP:7H27vpZw⊂"∩M→∩XQ εE$zλ4yP:~2P1w[:4w:[zyP3≥w1z4[w9P;Z4qt⊂→4y8βt supplied a model
for the %9λ%1-calculus and it is these functions which we wish to
exploit in supplying a map !K[Ci%GCXA5←IKX↓M←dA1∪' \4∀~∀ In particular we can assume thapλ↓iQJA1∪' AAeS[SQSmKf4∀β∪↔v{S∃β∨β↔∂'6K
β∂}sS';.{WMβ7+;∂SN{;M9∧3?Iβ/Cπ7Cf)1βSF(4+7∂##↔7∂#'∂πbβ∂?Ww#↔KC∂∪QβSzβS#∃∧b&NAε3W;∂&K?9↓+≠∂πI*Qβ'Mπ##∃βn CC'v9↓↔∪≡ I∃)ε3K?5h)↔ε~))βSz↓↔εM*Qβ∪↔6K;↔⊃ε Mβ≠}c3?←≠P4(4Rr
⊗≡LqαRε∀JQI! ↓1IAKZ≡J>-4(4Ph*q↔&≠πIi¬Y↔εM.!e↓*RM↔∩j)D4(hRrq↓+ qβ'~↓∃f *Qβ'→α+∪Q∃Rβ'Mβ∂#?7'_h*q↔&≠πI#"Jq∃Eb↓↔∪Q+!E∃EεK→↓↔'!∃)βO→↓↔⊃G!∃QE*Q↓9β")QI∃RH4*rb↓∃GqεKM↓∃L⊃∃)βN1↓↔∪"))β'~↓∃f *Q84(hQ:⊗: h*O'nK3πIπ≠SKπ&+∨eβ∨+≠≠'≡+MβSzβ∨'[*β∪↔;␈#πS'}sMβ≠␈⊃βS#*β?S#/⊃βCKNk'S'6)α2&≥↓β≠Wv≠S'?w_4+πv!βCK.#'∂π&+M9α6{Iβ↔F 7C3+Q4(hQ:
⊗<J9αR ∩&QIC A1IαIn≡J⎇*@4(hP4*q.#πS?kQαm↔
→↔⊃J↓∃*M."u∃DhP4*rb↓∃GqεKM↓↔.1∃)βN1↓↔∪"))β'~β;?Qε S?7N→84*b+∪πS}i#Q&b)EqβO→↓↔↔"))β'2↓↔∪Q*Qβ'Mε S?7N→84*ea↓∃Gbβ'M↓+J ∃)εK→↓↔'!∃)βO→↓∃f∩))84Ph):⊗t 4*;␈#'∂∃α↓βS#∂!βS#/≠∃↓β7+;∂SN{;Mβ∂∪∃βO'∪'∂Q~Cog?rBAEa~Ky%8hP4(4Ph*∂?↔∪↔OC}s∪';:βS=↓+≠S∨7} →∃)bβ←∃β>K31βF [∃β
β≠W;∨#'?9b↓∃f⊃+#S≥∃
aβ7ππβ';≥ε+cCK/≠O'?w_4+?w#=βSF+'Iβ&+;?S∂#'?;~p4*ONs∂∃↓α)f⊃∃'#≥∃EεKM↓β∞s?S#/⊃β7ππβ';≥εc'/∃α↓∃gI+ 1β←*β←'3`h+WO*βS#∃α↓ ↔→B)E β∞s⊃↓ .1%∃E∩β Kπ≡[↔SMπ#=β↔v≠3?O*α2&Nαβ∂?;∨#KW∂'→84*>)β;↔.!βS=εK;SK}#W∂∃π≠?7∃εs?Sπ&K?9β6{H4+.c↔7↔w#Mβ?2βS#∃π≠↔SMβcO↔cπ⊃yβπv!↓s≠␈∪5y9∧c↔Q↓. M∃)π∪π;∨*β?[↔∩↓sO↔GβIx4V ;⊃↓. →∃)π∪π;∨*β?[↔∩↓s≠?⊗iy1β&C↔9hS←∃β≡ 9β←⊗KS∃hhP4):∀*≡&9ε≠↔;S/∪'Qn=∩>VAXh(4*z)f⊃∃'#≥↔→B+πM↔2I∃E↓j↓↔∪M*P4*⎇+J⊃∃S&9↔→!+≠∂πJZ+π→∃Ui↔→%+ ↓u↓.#∂πI*Q!∃f")SS≥.1!↔π2+→%∃
H4(4U{←'SBβO'7NcπIβ.sSK'/→β≠?∩↓∃O∂'⊃1β∂}sM1β/ 1↓∃∞ ;⊃↓*SπS?j))84Rr⊗:⊂hP4*SF)βOS↔+∂SW⊗)β?→π##'Mε#↔≠'vKS'?rβ'Mβ6+Keβ≡K7'3∂⊃βS=π##πQε{→↓∃∨#∨7?∞1∃E8hR3↔Q?→β;?:β∂?;&K;W∃π;'S!π##∃βv+cQβ∨+ O↔"β?→αdJNA1ε ∪∪'v9β∂?v#'S'}sπ04V+cCK/≠O'?w→βS=ε{WIβ&KO∂W∨≠'?9rαπMβ>)β;?&+⊃β?rβog?rBAaaOq1β¬ε#↔∨K.)β?→ε≠πK∃εs↔↔⊂hS ∃β& /↔9π;#↔9π;∃βπ'#↔7C"βS=βNsS↔Kπ∪↔Qβ≡{;∪'&K?;πbβ↔cC⊗+OO'}sMβ'rβS↔Ko→β?→εkπCCNs∨M8hR←∃β≡ 9βONkC3'7IβS#*βCK?⊗c↔5β≡c'∨#&ceiβO!β'Mε+πOeπ#=βOF{]βSF Qβ¬ε;↔;↔⊗ 04*dJNAβ≡{;∪'&K?;πbβ∂π9ε∪∃β↔GβK↔O≡+⊃β'rβS↔Ko→β?→π##∃βn{K∃β≡K7C3*4)∃≡K→∃En+cCK/≠O'?r`4)∃≡K~oA+!E∃)↑)∃QE*Qo∃∃#⊃∃*u+ 84*>)β←'≡AβS=ε#'OCf eβ¬ε#↔;?& S'?rβ≠?Iπ≠W∂!α)O'→+ β↔cπ∪↔OON{;M9∧KP4+>K31β⊗)β¬βn S#↔n S'∂∞aβ≠Wv≠S'?raβπ;"βS#W~β↔[πg+πS'}qβ?K&+Iβ←Nc1β#∂3∃β .+84+∞∪OSK∞≠S↔⊃ε{WPX
)FK↔≡ 31β&C∃β∂}k7↔;"β?→α> ∪O←␈∪S 4RCog?rBAeEOq%9αNs∪↔↔"aβS#*βWO∃ε{→β∂}s∪'SN{;π1ε+cCK/≠O'?w→β'9π##∃βn{K∀4V OS⊗ ∂Qβ⊗+CK↔≡+;Sπ&K?;Mε{→α2M~Aβ≠.s∂S'}sMβ≠⊗+GW↔w#3eβO→βOW≡AβS#∂!4+/Cπ∂SgIβ?;*β?→β&C∃βA+#%∃)?→β'Mα+↔Q∃Rβπ;⊃ε 31β&C∃β?&C↔KMε K∃↓.+→∃)ph*S#/→β'9π##'Mπ≠↔SSNs≥1β&C∃β?⊗#↔Iβ}1β↔[∞cWπSN{9β?2βS#∃πβK↔∪N≠πS↔~β'Mβ/≠↔≠Wb4+≠␈⊃↓ ↔63'∂'.s∂e ε∪WQβv{Qβ;.≠↔OO∂∪eβSzβ7π'w#π'9π##∃β≡+;O∃ε{→βSF)β∪↔6K;'SN{984U≠↔∃β←K?9"β AE'rr⎇84Ph):
,:&9α$
&Q∩AEA1∪⊃%n≡∀zV@4Tc↔Q↓+≠'→∃
β∪↔;␈#∃↓↔&K→∃Eπ;#↔K+P4(4Ua↔∪'3Qαm↔
#I∃+B+εM∃WA↔εM*Qe↓.
M∃*hh(4*ea↓∃Gbβ'M∃Rβe↓∃≥⊗2*$∂αα+≥≡2*R∧\W"*!Q%b.M≤bGBO∃GRMDV∪bε≡4α.'$V∩bε≤dα.'∧V∩εO4∧V.2U%`hUKDα+∂D
↔~αW_"+
D
w&F↑.vO≡QQ"t,hAPRtz)u-β1Q hUM
↔~ε≥nF/↔∞,W&∂M≥vrε|dε≡}lM↔&N⎇l⊗bε←∞π⊗/>=⊗}w4
↔~ε≡∞π⊗␈∞-⊗∂&T f␈∩ I∃≥β4
w&F↑!PVNnLW↔π,↑F∂&≥⎇g~ε|dε≡}lM↔&N⎇l⊗g~ ≡&*π
}7≡N-LRr∧m}"ε/ ≥WεfW!PPh%h$,<→d¬$ )~C∩C⊗¬C∪"↔8u∀⎇Z↓PPh+DV&NdVC
*'$¬P∀85≤DU↑ 8*4*↑ ,~i*Hβ∀ 84dUW#"AQW↔λ∧V?λ~.4*H≡$∧,:9DUH≡λ∧V:<i%$ 9=∧UC"W∧\~9I&F)*J∂¬≡+≡E≠ ,/∧
<h ,OI,+∧
9H ,O ,(
≡h 9,d*KC!+↔λ &∨λ~<d∧.0I&∀~9H∧\≡ *D
<h '_I*H ≥Yλ ,O(εh∂$*C"KKλ ,/D~<h∧\≡),$
9H ,O *H
≡h .($*H_-lλ 9∂∀∂(≡DUC"KHYQβ"Im⎇~8lT≠Y:.M→<H∧\~9I%$≠[|D∧9~9DV ),$ <Y(∞>≤Z8nD≠8<∞
;Y|ea"KP*λ4Uβ!!"KQj)u4≥CEεE*≠P0r2λ∩Yts XP2|≤92yyZww9P≥7P∩\Q∩Z:3IXV⊂<Zrv24[3P∩\Q∩Z:3\∩XFE≥rP4w_v:r2NεE↔!⊃cdg⊂∃ a$j T_Y∀Nc$f&∞FEεE↔∩\b∩M:3y∩Y∀∩YtY-Rps Z_RYH≥P∩pY∩Z→∩L]P∩pY∩Z→RLnRs∀IXP≡FB∩r4s
∩\b∩M:3y∩Y∀∩ps Z_Rs
Rr⊗∩Nb∩Z:→y∩s∀ ps∩Z ∩s∀RY⊗∩\b Z:3y s∀∩pY∩Z→RY∀Rr∀IXFE↔⊃g"εE h i∃εEεEβE*42H72|:λ1ww9Zr2y0]4ww⊂~yP:4→P22w≠z0z4[w0v⊂→2yqy~x:4w[⊂7s⊂∪$ih⊂~r2w:~s4ry≤WεE$Y2w:4Y4ry9H4w⊂&∩ih⊂9→x92yYw:⊂2Zz42yλ)Vr|≤99P7\⊂3:w_z4ww≤WεE*~:yFE_w⊂4r→w:4s~ry⊂2Zz42yλ22w7]2yP0[⊂7q5→qz⊂7[⊂7zyλ27vpZw⊂∩pTRXP7\⊂22w≠z2yP_P3:w_z4wwβE7q5→qz↔εB&2z⊂ pc7∩J⊂70vYP:42H9rz⊂≠s⊂1w[:4w:[zyP3≥w1z4[w9]⊃I\iRZ≠≡X∩r⊗RpiRN7∩r⊃FQRpiIr.RXKεE0w→⊂∩pdY∩XP1→P∩r≡~r2w:~s4ry∂∩XHβ%9B%1.
We know that the value of a LISP <identifier> ({yon(P66)}) depends on the
current environment.
Then we might characterize the set of environments, %denv%1, as:
.BEGIN CENTER
%d[%aId%1 → %aS%1 ∪ %aFn%1%d]%1.
.END
That is, an element of %denv%* is a continuous function which maps an identifier
either onto a S-expr or onto an n-ary function from S-exprs to S-exprs.
This is the essence of the argument used in introducing %3assoc%* ({yonss(P92)}).
Note that %3assoc[x;l]#=#%dl(x)%1 is another instance of a
operational-denotational relationship.
.BEGIN TURN OFF "{","}";
Given a LISP identifier, %3x%*, and a member of %denv%*, say
the function %d{<x,2>,<y,4>}%*, then
%9D%* should map %3x%* onto %d2%*. This is an intuitive way of saying
that %9D%* should map a member of <identifier> onto a %2function%*. This function
will map a member of %denv%* onto an element of %aS%*.
Introducing %ai%* as a meta-variable to range over <indentifier>,
then for %dl#%9ε%d#env%1 we have:
.CENTER;
%9D%f(%ai%f)%d(l)%1#=#%dl(i)%1
.END
The %2meaning%* or denotation of a identifier is a function;
whereas the %2value%* of an identifier is an element of %aS%1∪%aFn%1.
The treatment of identifiers leads directly into the
denotional aspects of function application.
We shall maintain the parallels between evaluation and denotation, by giving
%9D%4e%1 and %9D%4a%1 corresponding to %3eval%* and %3apply%*.
Let %ag%1 be a memeber of <function> and %af%1 be a member of <form>, then
for a given element of %denv%1, %9D%4a%1 maps %ag%1 onto an element of
%aFn%1, and %9D%4e%1 maps %af%1 onto an element of %aS%1.
.BEGIN CENTERit;
For example: ←%9D%4a%F(%3car%f)%d(l)%1 = %dcar%1
.END
.GROUPl~∃'S5SYCd↓KckCQS←]f↓Q←YH↓M←dAQQJA←QQKdA1∪' AAeS[SQSmJA→k]Gi%←]fA¬]HAaIKISG¬iKf\4∃∪\A≥K]Ke¬XXAi!K\t~(]¬ ∂%≤Aπ 9) $v4∀~∀JeλJiB∃LPKC_KLRK⊂QXRJD@z@K⊃XQLR∀TXAo!KeJ@∃CLJTJr @∀byMk9GiS←8|\~∀9 ≥λ~(]β!βI(~∀ To describe the evaluation of a function-call in LISP↓oJA[UghAC⊃H~∃C8AKck¬iS←\↓i↑@JeλJiJ∀bt~∀9¬ ∂∪8@A)β ∪(bPDjRw
%→_w)U%≤A∨8@DFDl~∀~∃pJsλJQJKLP∃CLJclKCfJPbJbX \\\X∃CfJ`&q∃Fu.1%↔⊃Fa%∃E≠i 4)+J⊃∃S
+→!↔∞1↔→%.!#1%B)f⊃∃&)↔→!. M∃Q
+→%↔"C1%∃
a
99ra∃f⊃+#∃↔→B+πM∃&q↔→%.!#1%J)D4)t*:⊂4U;∃β7/≠Qβπg≠=β7∞[∃β∂}sO'O&+;Qβn{∪'≠N≠πS'}sMβSzβS#∃πβK↔[N{WMβ≡cπWO/→β?→α)f⊃∃'#∨I∃
βS<4V ∂∂?.sQβ≠␈⊃β↔;6KK?;n+;SMph*S#∂!β'MbβS#∃π3π3W*β?→β
β∂?;∨#π;QεKMβ'v#↔C↔v#↔;Qε{→βSF(4+Oε+∂'≠N→β↔;6KK?;n+;QβNqβ←#N≠!β'"β'Mβ/3π3W∂#↔⊃9h):
,:&9α%*J9α|2→↓ Z⊃1 y∪ZRVJrα>9↓∩→ o∂.sS↔IXh)∃f")S∃↔2A↔πM.1%↔⊃Fa%∃E≠i
↔∪~)E9hP4):,r⊂4*
βO'7NcπI↓εk?∪'6K∂πSN{9β7/≠Qβ *β7π∪*β≠?HhS∂?;&KS'?v 1β↔GβK↔O≡K?;Mph(4*⊗+≠?K*β←∃β>+Qβ[/∪eβ≠∂⊃β'9ε CC3NK;≥β7+;∂SN{;Mβ&yβ[πg+↔L4W;∃β7/≠Qβ∨O3∃β¬εkπS#.kπS'≡ 1β∂F Kπ∂&+K'k∂#'?9ε{→β≠.s∂S'}qβ∪↔6K;'SN{;M8hR'9β&C'Mβ≡+∂S'}qβ←∃π;'31εCπ;∪f(4)βF ;∪3*!7;␈#πS'}qβ←'&C?WQε3K↔∃π3πK'∞∪3↔MbβC?O'β?;'v9β7?⊗)β∂?oβ3↔`hS∂πO/→βS=π[g?;∨→"AeαKy84Ph*πO∨+7';:βS#∃ε{;3eε3K↔∃π3πK'∞∪3↔MεK9↓∃OA∃)β∂∪∃βπn{;≥β&C∃↓∃∨A∃S%+ ∨L4W##∃β&+;?S∂#'?9ε{→↓∃_Bnoa+!E∃)b↓999bβa∃Sr)*u↓+Ka∃Fjβ'9β
βOC↔≡K≠'↔ h+↔;6KK?;n+;Qβ≡C?W3"β ∃β
β≠W;∨#'?9h+≠K}i↓↔ε~)c9∃
βS=↓.
M∃)π≠W∂!π##πQPh):
,:&9α$
&Q
AEU%\2&21]"VJ9∧z9↓ ~⊃l4(hRq∃f")S¬↔2A∃L"]Y↔πY+!E∃Eb→999~+πY∃&q∃Fu. M∃Fj+→%↔"C1%∃
→u 4R+⊃∃dB+⊃#a+!E∃)b→999b→↔∪a+#9∃)J)f⊃∃&)↔→!. M↔→J+⊃#1≠Q
sa+!E∃)b+∪Y∃# ↔⊃yb→999b→sa∃&q∃)1.#Y∃Sr+⊃y%+λ4):,r⊂4(hS←#↔⊗)!βO→βS#*α2&Nα!7;␈#πS'}qβπ;"↓∃d!*Qβ'MεKSMβn S#↔n S'∂∞aβ∂?.sS↔Kε KQβ∞s⊂4).#Y∃SJ)Eβ'~βS#∃ε#↔;?& S'?v 1β∂␈+;S↔↔βπKQε{→↓↔∂1∃S%+ 84(hR'9βn{K∃β&+Sπ'cP4)∃HA↔⊃#B)QE∃Ra↓99r↓3a∃&q∃)'*Ca∃Q
))1↓rq9↓3B)S5∃RI↓∃GO→β¬β7+;∂SN{9↓↔&1∃)iα+εM∃Fq∃EJ↓↔εM*QβOW≡AβS#∂!h4(hQ:
⊗<J9αR ∩&QEC U%n=∩>VAXh(4*bβqβ'~↓↔∪∃G!∃QE*Q1↓9rq↓3Q+#9∃)J↓∃G'2β5↔ j)+9β∞s⊃↓↔→!∃+%.#Q∃SJ)E↓↔→Y∃)↓+J ∃Eph)↔∪2CQ∃Q
))1↓rq9βQ+#5∃)J↓↓∃Fc`4*qπaβ'Mα)f ∃Rβ?S#/∪←'O(h(4)t*:⊂4T 3O=b↓↔⊃#b→i
sB)QE∃Ra↔∪Y+!E↔⊃ra
99ra
sa+#9∃)b+∪Y∃&q↔⊃yJ)Eβ'~β¬β7}#'≠'≡ S'?ph+?→α+∪1∃RβOW∂BβS#π"β↔π∂B↓↔∪Y+#%∃EεKMβ ␈+;⊃β&y↓βSF)β∂?↔∪↔OC}s∪';:↓↔∪a+#%∃Eph(4)t∩⊗≡&rαRε
M!I!IαaQA ↔8u∀⎇Z∞aQU~_.D~<nKD9
≠∧7Ho≡¬NOJ)&∀~<nD∧." ,E≥I-ε∀*J)&+~9H∧\≥H∂$∧.0I%$ ,↓:~2w⊂∩Na∩Y⊂βE..2[9rP4Y⊂∩r; Z_RUλ≡P∩\P∩Y⊂:~2w⊂∩Na∩YεB..2v≤rP4sλ∩r8ε%41%* = x%2 t↓QK8@KI`2)H4*ec↔3O*↓↔∪1G1∃QE*Q%∃Eph(4+>C↔K∃↓$α+⊗≤dV"π∧U"π ~→;D∧9≤)%$→;≤lT 9≤DV(λ~.4λ 9
≤J≤ ∞∃≤J)&∃C"KHYQβ"AQP9] ↑H_;
D≥~~.4≥{|M4≥~→. 2P92Xv6<P≤t7zv→⊂12P_P1wv\0y0q≠2P92]:y7⊂≠wεE7[⊂7zyλ4w;2\z6rw≥↔⊂ wλ4vvrY4px∧e benefit is
clearer understanding of the differences between mathematics and programming
languages and a
clearer perception of the role of definition and computation.
It should raise questions about the class of objects called function
and the class of objects called procedures.
Analysis of our study will show that one of the larger difficulties
was caused by our insistence in dealing with type-free languages. Self-application
is one indication of this. We can show that imposing a type structure on our language
will also solve many of the questions of non-termination. In a
typed %9λ%1-calculus
an expression will always have a normal form#(⊗↑[Mor#68]↑).
Computationally this means that all the programs will terminate.
Also models for typed %9λ%1-calculus are much more readily attained.
However the type free calculus is a stronger system, and requiring all expressions
to have a consistent type structure rules out several useful constructs;
in particular, the %9λ%1-calculus counterpap¬PAi↑AQQJA→%' @JMYCEK0JbA←AKeCi=d~∃G¬]]←h↓EJAG=]gSgQK]iYdAisa∃H\~∀4∃
e←4AiQJ↓aeCGQSGCX↓gSIJ0ABAieaKHAMiekGQkeJA%fACYM↑ABA5SqKH↓EYKgMS]N\4∃→⊂ε≥↓βCK};Kπ7n+KMβ7∪↔GW.sS3eεk'OMπ##∃β&+∂3π⊗ S'?w→β?→ε≠?77}qβCK};Kπ7nK;≥βf ;∨W∞;↔M8hR3π;?+π∨∃ε#↔3π∂∪πS'}sMβπ⊗)β¬β6{K5β}1βSgεK;≥β∞s⊃β∂∞qβ ∃π W'S*β#↔3ε3W1βNp4+CNsC?'w#';≥πβK?∨⊗ 77'v9β↔K-w↔≠4 F.≡L≡&∂&≥⎇g~ε<≥bε∞N=rε⊗T∞W≡.D 'Jε=⎇WεNL↑'~πMthV ]Gαπ∞-v'.<Tε␈πM≥VOV\Dε≡}LUb∧F}|W6/$∞FF␈<Tπ≡∞\Tππ⊗|}&∞n\↑'~π|≥g"πMtπ∨..lW↔ Q.FF*∞O↔ε*∞>G↔.>NW⊗*D∞GOε≤<⊗fg∀
⊗rπM Rεv≥\Rε}d V66≤=⊗.v?∀ε␈∩ ←πε.M≤Vv∨∃aPT∞N=rε
∞O↔ε*∞>G↔.>NW⊗* <⊗rε,Tε
π,\⊗bεn]↔≡∞l<Rπ> ]bπ'/≥⊗v:∞Mrε&\.V:ε∀∞π⊗}},⊗kXQ-↔"ε≡4ε7⊗↑≡V.wMO∩ε&↑=↔⊗.≤-F*πMtαε/ ≥VNvT ⊗v"↓Q&n}M≤gJπM Rπ⊗↑∞&/≡]nF∂&≥⎇g~ε|dε∞↔>N&∞∨D F∂&∀∞7'↔\>G/⊗↑5b¬&
}6*ε=≥f'~
|bε␈ ↑&∂&≥⎇g_h-≥Wεg∀∞FF* ≤&Nf≡O∩π&t
⊗>v},Rπ& Tπ'O TεNvm}&n∂M≥vrpQ!PRR%%&↔⊗≤Lv*R%$αεf≥]↔&.D∞GOε≥ls{{wqPPh!Q%>*
=f␈:∞Mε∂"∞Mε*∧I~5ααV=F∞⊗]DS
ε} W⊗∂M}"εO4∞6Nn≥L↔∩πMtα∪cT%Bε↔↑APR+=L⊗⊗.DV∩ε↔]≥F'~ ∀π&.↑
w⊗∂/∀ε&.m≥fO&≥⎇bbπ⎇
⊗f*∧'Cj∩
]v&Nm≤W~πM PhV]nfO⊗⎇mV.wEd¬π⊗|}&∞n]≥f:εL≥f?.≤|Rε≡⎇n7'↔\>G~π⎇
⊗≡B
]v&No∀π&FT Vw6≡-vvn]n@hV≡,Rπ≡≥≤Bπ&t
ε∂6T∧S↔≡≤LRn.llV∨'4V∩ε∞lDαε∂,Tε∞r
≥g∨&≥l6(h-|bπ> ≡BεO4 6∞fL\Bε
≥Wε/,≡FO6T 6}w>N'.∨EaPU& Tεv/∞Dε≡F≡∞F/∩
≥g'⊗|NV≡/4∞FF*∞∞&}≡\NW⊗∞D ↔∨ε\>G~ε|dhV≥↑ε/⊗≡M↔6* =vw∨N.V∨'4 ⊗v"∧
⊗rπ?≥vw∨5
βKα∨aPW>T∞vNfD
⊗w6↑>FN>≡LRπ≡⎇\Rε}d∞FF*
\↔&F]\↔&N<≥Bε∂> V∨'4
v0h$'Cj∩ ≥f"αV=F∞⊗]DS
pQ!PPh