COMMENT ⊗   VALID 00011 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	.SS(Review and Reflection,,P85:)
C00022 00003	.<<c w mmorris>>
C00035 00004	.<<abstract syntax>>
C00044 00005	.<<lisp borrows>>
C00051 00006	.<<abstraction to function>>
C00082 00007	As we move again from procedural notions to  more denotational
C00094 00008	.GROUP
C00104 00009	To describe the evaluation of a function-call in LISP we must add
C00110 00010	One of the major difficulties in supplying models for applicative languages
C00121 00011	.CENT(Problems)
C00126 ENDMK
C⊗;
.SS(Review and Reflection,,P85:)
.BEGIN ep;
"I think that it is important to maintain a view of the field that helps minimize
the distance between theoretical and practical work."
.END
.pt24;
.BEGIN TURN ON"→";
→Saul Amarel, ⊗↑[Ama#72]↑
.END
.PT24;PT24;PT2
.FP
By way of review we
sketch the basic LISP evaluator of {yonss(P6)}:
%3eval%* plus the additional artifacts for %3label%1 and %3function%1.

There are two arguments to %3eval%*: a %2form%*⊗↓Throughout this section we
will say "form", "variable", "λ-expression", etc.  rather than "a
representation of a" ... "form", "variable", "λ-expression", etc. No
confusion should result, but remember that we %6are%* 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 an
application, that is, 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 LISP function, a list of evaluated arguments, and
a 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 binding9 in this case apply the function to the arguments
and use 
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keeKce of free variables calls the environment into play.
But still, we have no way 
to destructively change the environment. Constructs which do have such an
ability are said to have %2side-effects%1 and are discussed in the
next chapter.


LISP was the first language to exploit procedures as objects of the language.
The idea has been generalized substantially in the intervening
years. A concise statement of the more general principle appears in ⊗↑[Pop#68a]↑.

.BEGIN GROUP;INDENT 10,10,10
.P284:
"...This brings us to the subject of items.
Anything which can be the value of a variable is an item. %6All items
have certain fundamental rights.%1
.BEGIN INDENT 13,13;
.NOFILL;

1. All items can be the actual parameters of functions 
2. All items can be returned as results of functions
3. All items can be the subject of assignment statements⊗↓This issue will be postponed
until {yonsec(P274)}.←
4. All items can be tested for equality.  
.END
 ..."

.END
.FP
LISP  performs well on the first two principles, allowing LISP functions
to be the arguments as well as the results of functions. The fourth
principle is more difficult; that is, test for the 
equality of two LISP functions.  In can be shown (⊗↑[Rog#67]↑) that
no algorithm can be constructed which will perform such a test for arbitrary functions.
However in more selective settings, program equality can be established,
both theoretically (⊗↑[Man#74]↑), and practically (⊗↑[Boy#75]↑, ⊗↑[Dar#73]↑,
⊗↑[Car#76]↑).

.GROUP;
The language  POP-2#(⊗↑[Pop#68]↑)  has also generalized the notion of function appLication
in a slight, buT significant, way.  The generalization is called
%2⊗→partial application↔←%1. Given a function 
.BEGIN CENTER;SELECT 3;
.BOXA
f <= λ[[x%41%3; ... ;x%4n%3] %9x%3]%1,
.BOXB
.END
α.FP
we compute a new function of %3n-m%1 arguments by applying %3f%1
to %3m%1 arguments and obtain a function of the remaining arguments
%3x%4m+1%1 through %3x%4n%1:
.BEGIN CENTER;SELECT 3;
.BOXA
 λ[[x%4m+1%3; ...x%4n%3] %9x'%3] = f[t%41%3; ...; t%4m%3] 
.BOXA
.END
α.FP
where %9x'%1 results from %9x%1 by binding %3x%41%1 through %3x%4m%1 to
%3t%41%1 through %3t%4m%1 respectively⊗↓POP-2 actually binds the %6last%1 %3m%1
arguments.←
Further generalizations of partial application are imaginable (⊗↑[Sta#74]↑).
.APART;

We have pointed out several "procedural" differences. Our treatment
of conditional expressions differs from the usual treatment of
function application: our standard rule for function application
is "call#by#value" which requires the evaluation of all arguments
before calling the LISP function;  only some of the
arguments to a conditional expression are evaluated. Note that the whole
question of "evaluation of arguments" is a procedural one; arguments
to functions aren't "evaluated" or "not evaluated", they just "are".

We have seen different algorithms computing the same function; for
example %3fact%1 ({yon(P21)} and 
{yon(P99)}) and %3fact%41%1#({yon(P21)}) all compute the factorial
function. If there are different algorithms for computing
factorial, then there are different alorithms for computing
the value of an expression, and %3eval%1 is just one such algorithm.
Just as the essence of %3fact%1 and %3fact%41%1 is the factorial function,
we should capture the essence expressed in %3eval%1.
Put another way, when  applications programmers use %3sqrt%1 or %7p%1
they have a specific mathematical function or constant in mind.
The implementation of the language supplies approximations to these
mathematical  entities, and assuming the computations stay within the
numerical ranges of the machine, the programmers are free to
interpret  any output as that which the mathematical entities would
produce. More importantly the programmers have placed specific interpretations
or meanings on symbols. We are interested in doing the same thing only
we wish to produce a %6freer%1 interpretation, which only mirrors
the essential ingredients of the language constructs. That is,
%3sqrt%1 represents a %6function%1 and %7p%1 represents a %6constant%1.
The %3eval-apply%1 pair gives one interpretation to the meaning of functions
and constants, but there are different interpretations and there are
different %3eval-apply%1 pairs.
The remainder of this section  will resolve some of the tensIon between
function and procEdure.


What does this discussion have to do with 
programming languages? Clearly the order of
evaluation or reduction iq directly  apPlicable. 
Our study will also  give insights 
into the problem of language specification.
Do we say that
the language  specification consists of a syntactic component and some
 description of the evaluation of constructs in that language?
Or do we say that these two components, syntax and a machine, are merely
devices for describing  or formalizing notions about  some abstract  domain of
discourse?  A related question for programmers and language designers
invoLves the ideas of coRrectness and equivalence ob pRograms. How do we
know when a program iq correct?  This requires some notion o@_ABAgβ#π;∪∂∪⊂4+∞;π';≥!β←#N≠!βSz↓βC↔≥!β?W⊂∧εNo
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L=→(∞,<≤[l}age. The Algol report (⊗↑[Alg#63]↑)
introduced a standard   for syntax specification: the
BNF equation. It also  gave a reasonably precise description of the
pragmatics of Algol statements in natural language.
The style
of presentation was concise and clear, but suffers from the imprecision
of natural language.
Regardless,
this style  of description is quite common and is very useful. A recent
report#(⊗↑[Moor#75a]↑) on the pragmatics of InterLISP  used this descriptive style.
If the language is quite complex, then a formal description can be
equally complex. In {yonss(p211)} we will see that our interpreter definition
will extend nicely to richer subsets of LISP.


Regardless of the descriptive method used,
a language description should not attempt to explain everything about a language.
It need only explain what needs explaining.
You must assume that your reader understands something ... . McCarthy: %6`Nothing
can be explained to a stone'%*#⊗↑[McC#66]↑.
A  description of a language must be naturad and must match The e_pressive
poweR kf The lAnguage. That is, it  should  model how the constructs
are to be implemented.   What is needed is a description
 which exploits, rather than ignores, the structure of the language.
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.P6@/↓IBAG↓x¬Wε↓→8+ i@(ε@↔4λ⊂⊂@<β`h↓
@,↓⊂2FdhK@+%→1βαZ	∧εpw 9
+∀
@→-↓rp∞ds are  reXε@&∞@@⊃↓@
@∪\βP∨@/ λ2~qqz@spπSH/pπ ≡H	 2∞∧seh↓A`	xαP(∧πp∞ ↓hg>BJ(
≠@	}
λ¬d↔∃
V∞→(∞Oλ⊂⊃@l`C@H∧≥
8
 ↓QQKeJ↓SfA≠α{@⊗(≥⊂∀_t∧ s↓e]`∪∧π∞N@≥S@v≥0¬d\4⊂
H.T_m↓zpλdpπCr↓∞FF∞¬β"↓0∂r@∃IfJ↓!∧∩V _;LD	1∀d↓X∧2J@	β'd∧V#≤q|≤9_∞%*  t@Q⊂	β↔∂_V`,y(7↓`@EI8@'&X	∞8∧y"~+&ε4ε=Bs: Zu∧eHβSP⊃β$¬ε
ε0`7_β`ahβP∂λ"∞H
↓P9`%↓P[aQ↓(
w,αz )↓λ	β␈,H@.0¬d@πβπε↓ ⊃↓@
 8↓(G4αZ↓$∧b@1⊗%→+@∩X∂↔⊂ ∀@!`↔Mβ$¬εTλ[0∩Xtε`∪]α9∩ε|dαπ~⊂∩@≤β`h↓
@,↓⊂27@tp	@↔⊂∧≤≤∧y 3 is~∃α≠πCRεP≤L≤α⊂!<H!pp∩Tes`∪C8Aae↑ε#@.∨@∀↓p∧<@fα+cCIp∧2\_∩@* εp∪N∧P≡∞∞Lπ∩X↓(~(hQ 4⊂4SuT≥@
L``⊃≡ε4ε=αw0∀hl¬`↔∃∞FF*≥f∞fε0→@h¬`
β?1Ph↓Hπ!N¬1α$P¬d(αcdgλ!bgT@%& π0↓P¬ ∨ 9α≠↔cC⊃DbβRεH∞@.∧a`)↓x
Sb∧∂⊂∀∞∧seh⊂p∧|↓↓p¬`'≡]≤∞∨⊂	.
λ∧T∩
⎇B⊂hαK@"Pε@∩∀αrεBε⊂
@↓P¬⊂λFP~*←*βπK∃∞G/8∧w ' p	↑↓
⊗fα→8\∞0¬tAQQS@Mhat 
unless a domain %dC%* is a single element, then the number of functions
in %dC→C%* is greater than the number of elements in %dC%*.
This does %6not%1 say that there are no models for this LISP subset;
it  says  that our interpretation of "%d→%*"
is too broad. 

What is needed is an interpretation of functionality
which will allow a solution to the above domain equation; it
should allow a natural interpretation such that the properties which
we expect functions to possess are  true in the model.
D.#Scott gave one 
interpretation of "%d→%*" for the %9λ%1-calculus, defining
 the class of "continuous functions" (⊗↑[Sco#70]↑, ⊗↑[Sco#73]↑).
This class of functions is restricted enough to satisfy the domain equation,
  buT broad enough to acT as the denotations ob p@I←GKIUeKfA%\~∃CAaYSG¬iSmJ↓ae←OIC[KS9NAYC9OkCO∃f\
∃]JAoS1XAkgα)βS#*β;?S∂#'?9α↓	π∩X∧S"VF∩+
4\AJ+∀8B+#$XEj+∀$π&z
\V∞r∧.FF*∞8W"ε|aPV≡⎇nFNw]}W
εnYf∨&≥⎇g
εn-vjεM⎇V∞NdλB+#∀V∩π&tF}n≥≥b∧"V@DV(KC!$z⊂$\β the continuoupεAMk9GiS←9`
β←FK∂!β6KKOQπ≠WCCfK↔⊃β
β7?∪,¬@hRmwαπMRα+⊃	&∃8x;>;≥<d;Y⊂~z⊂4iH:42yYP3:`.ctionpεAoQ%GP@~(Agkaββ3eβλβπOO→β≠} λ∀≠8=
8εpz~qpv⊂≠wr2@, h∂LA¬aaYSα≠πS'4∧R∧d~:ααAhπVh⎇βy⊃[Lnk∀WβE
.GROUP;
W`
AG¬\ACgβ≠W7∃π##πQπ##∃αdJNAβ∧ε&Nn≡M↔6/1Q&&.β[uT≤|⊂∩Xts4qH1ww:~w:wz\β functions(A
=dAKq¬[aYJ0AiQJ4∃[Ci!K[Ci%GCXAα≠?W;&+KCπ↔!βS=π##∃αdJNAβ7+;∂SL{9↓∃≤∧6∂∩U$εO~∞Mε*ε\≡πεNlpλ∧\_x<DUH→TM⎇(β"D\α)RUλ:7P∩Y)RU⊂→2s4w→r⊂0yH37v ,ows 44∀~∧]	∂&p∧¬$)~C∩C⊗¬C∪α↔8u∀⎇Z↓PRt){∧λh+DV&≡≡ ∞Di9∀d\λε(∧\α)Rr↔RXFEβE..∩LtyP∩Na∩U⊂~pε %dt%* iq atomic
\%dcap∧QhSpJcSfKAhJPbJbA%L@KIPJTASL@KHQPJ`bJ(@\Ahα)QI∃RH4*rb)G'Mα)f	∃Rβ'2∧\G"*$	↔4	*0DUKC"EhSv⊂AQKQ3HA ¬↔ T i*;
.GROQP;
.F@ 4⊂
O'nK3πIπ≠SKπ&+∨eβ∨+≠≠'≤+@~πMtε.OlTε&.β[p~_z4ww≤β foR the otheR primitive LISP functions
and predicates(A
=`	β↔F7C3+Q4(hQ:
⊗<J9αR∩&QI@ε∪αc&¬∪\≥)zUhαKPI@l FE↔⊂r0z≠vX
 [%`	&K⊂@2@J)&KI:α)D4λhRra∃∞K@~α\\b*R
⊂	D∧9≥∩J⊂4yP≠5z⊂0]7vtqK∧E.∩Y0z7fJ:⊂n∩LtyP∩Yz∩U⊂~pε %dt%* ipεACi=[SF\4∃98JESf@JeJTA%L@KIPJTASβ→↓∃F∩))8Q%d∀⎇λ!PRtYh@hRhjhTmzFN≡T∧απ&≡Bπ&↑6*αnVv∨M→vw~≡&*π>N&N∨G$α.→J↓'_I9α$π(	.($,+C!%P4⊂**∞c"AQ@εEεB↔#i'Uh≥FE⊂wy92\x7w2~w3P:≠P∩Yj→vwps	U⊗⊂;YP;tf≠⊂40{→P0P3≥w1z4[w⊗⊂∩Nb∩Z:→RXV⊂βE;t4Xt⊂6p\9P2|≤92yyZww9FB7w:7H:42t\⊂22w≠z0z4[w9WεB)tw1YP⊂∩\Q∩Z:3IXP4yH⊂0w7]42y mapp@%]NAY%WB@@∀sdJbαaβ←∃∧εvNfAQ'/≡T∞FF*∧∧".2∧V∩∩ε≥lBα∩\e∩+
$'⊗∞=<W'~∞Mrε.βX{
}y(∪	~tλ_m⎇\⎇≤N\⎇≤kAQUy(
l99λ∞Mh~;NN[y≥,<αP9g[rP77]0z4w[⊂37iβE2v2[rw:9H5s⊂*~2P9r]9P≡9Y|89> and<fgrm>. 


Let %5s%* 
λbe a meta-variable ranging over <Sexprx~∃C]⊂@JkJ∀TAeC9OJA←YKd@yα3?K5p¬Bπ&]bh.|Rε≡≥`π?⊗≡LSPH!Q"t∀Xy∀rε<XNL<Z5π8tSu*πc"KH)v⊂#!+i*1∧V≥→`∩Yα(%5@LKLRJD@p∃↓,#M∃(hR⎇∃F ∧S'→i ∪
⊂¬3cap[%5@∀JU:Kα1%∃Eβi↓↔∪≡@∩BW_B+'Lp	,@∀∩Z`%%f)%D)
∀]	_∞bλhQ:ε: h):~α↓PW>≡Mαπ≡≥]⊗f∂$λ	-n≤Z1.P37`2 %3cdr, cOns _Aα+E1↓+π;⊃α)+πS|¬R*RaQ%&FT
7'↔\8
∞↑Y(∪l@⊂:44\P22@fin`∪i%←\ASβ→βK↔↔IβO'nK3πI∧εFzπM↔"ε|dα+∨L⎇V`<9I,%aα@
.APARTl~∀4Ts?@:∞|Rε≡⎇h

≥]9(∞MβP⊂≡M→(⊂↔→|:⊂ 3ubse@PA@∨→∧b&NAbβπ&M⊂L@P1w`.diti@=]C@0hS↔cC⊗#@∨≡≥xN∀β⊂ yH8πe 9←iKHαβ?)β\π⊗}rλπβBOeDε
εLX	n,9(⊂↔Xα care ne@∃H∀,,Rπ_:y-d≥z→-d≥y(≡≥→0⊗\8∧ to intep¬aeKβ!β∂∨v#'S'x
f∞bX∞∞∞Y<p→Zws9P~w⊂:2≤4¬s @=H	β7∂βC'l}2`!α@
Wd can↓`∂'↑≠
≤↑(⊂~~2P8rob`→KαiβO3L∧vG&O⊂∞D
9λ⊂∀\β easy to pπQ←↓8πS#π πS#∀hQβ∂?t∧FO&≥p≠L≥λ→ ≤≤92y`3io`≤Aα≠π)β⊗)β≠}-W.f≤LV"ε≥`λ∞L<[0→H0∂f ↓QQJ@Aβ≠'7Cd∧R@ ¬∩YZpε%bαk↔cC⊗#@∨≡≥pG∧¬
%3i`
 /α)QE∃S[∃∃Qλ∧RS↑TVC∩
+P	&∀β
We wi@1XAISβ≠C &_ε ≤∀→→ ↔≠βtatIohAMα{@∩π>Xλm∧	(	`)`∧@∃D∧∧WGπ,Xntp∂n@LX	α≤¬β!8πil↓0AEJAα	β .≡Mε.n≤¬~,pp ↓→kP≠∂&K79⊃∧∧⊗v"∞Iε/⊗\hn2P:4→P2{ !h	`↔παM⊗}@H≠`9→2y⊂6ill !CmJ@ε∪↔.αC"L≤\⎇≤L≤⎇→0∩λ5¬ut⊗↓ 
c%⊂≡∞α≠∞M→(1[βm`≠JεsAβ?2αP≡∞H∧⎇M@9:$εBα(`7rεs1D∧≤@1)p≤@%pε' 6L@→,Eλ≥⊂∀→P:y@d `∨L@ε≠0≡@→~ :~up∞al exp↓IKgg↓H
v`.h∩ 7λ8∧hE mh∂eJ4⊂πα0≥∞0qz<α`ae⊂@≡.β]_.M8πw 3 `∨LA1∪' @ε3W 6>M⊗`=β9Pfp¬Kc.#;@&Hε 
≡h≤d∩Xβh t@!Ch@~)KqCFπ##@J	p≠LT≠yH∞M→(8λ¬0	R↓*Q∨Mα¬_d∧αrr%* an@⊂AC@3bβS#∃
g&F↑.2ε∂,Tα\αs∩Uα
∃)!`↔MαH¬bπ~⊂∀@3 settih
N↓`πS#∃¬w,→<@≥\β⊂"@vah	kCQSW@9
vd≥∩→∧∞≤Y0∩~qpp∀D¬`
β&α4π↑y1@:Z⊂εE&lπd@E∃@≠≠≤0~,]β1|@" butn`h↓∧¬f.≡↑8λl≡\ ⊂tlεA[C%]`∪πH
bπ~→(∞rp∞s↓∀A@?→∞FF*L@,@4p∞i↓QS←@)ph ∃≡\Tπ@?8πw(P10bαK`∩tπkC!¬
∞↓	π&pεRε
M!IC⊗¬C∪∩↔8u∀@u0!&2z(¬3if$∧bAIJαs0∨&T∧V&NdVαπ≡P≤LV¬
∞@¬∨1α4⊂
`"\Mε 'H⊂
@%dTr¬*x I&∀QpK	LJT@2αC@∩M*Rt4!Q%e↓	(λt≤β%* @d@@∃GH∧b*R∂
+∞α8	RJ⊂∩r@4%*~∃p@↔∪'2Ca3∃H∞E≠	(:.P∩r=	XV⊂ )d∧@KI`JbASβ→↓.\dRRpβ"W↓⊂Xt`3  
`&⊂∧S
@λ≠`:~2y3t↓pπ@∀Q%d)v⊂C!%Q3Q↓QHπ#I∩πP @Xh!:~αh"S#Lε2εNnLG.≤Y0~_z4w`. `∂L@α≠?;∪H∞FN}βX;∧>≤9→yyt`/ns i@LACaaβ∪7CKL∧↔&*hn⊂ ISP; oQQKd~)S]iJπ∪CK↔&S'⎇hdπs⊂#ondi↓QS←@;∞c@~ε≡,Rπε|∧tm≤Xε2Wλ ε`∨dβKqC[ββ3∃@!Q h↓KPQ(@dg⊂*⊂a$j2(10,HhB`.=∩ ≥4β"E@!'d CE(%@⊃S@→∃α@$UL¬⊂-Ip∧T@$α)+a↔%→∃+a,∧E
* β∩α9∀i%*α@

λ¬0∩q+'M∃Rβe↓∃∞K∩*H<λ⊂¬1i@LJT@Kα+@"*!Q%b9~ ∪λX∧1%*(x →@Jci&@DV⊗O~∧X	∂$αXF⊂~pε %⊃p@∃D∧¬π4	9 ∪	U↔εE↔(%!ipε@J@L⊃∃EβHbα9~∩J⊂4yP	Xa∩Uλ0p∞d%`p∩@lAdJT4⊂
`%DV⊗O~∧P	∂∀, ⊂∀Yα %dx%* i@L@@∃@_ 	%$_;Y∧∧αr<P∂P8
¬*     ⊗↓% "	CgS]≤AG←]⊃SiS↑αsπ1hS↔cC⊗+@∨≡≥yg4≠{@⊂λ¬dif%41%!↓oWjYα!β∨'αh	$∧αYma_y(A\ ε2FBβY~α\P
∧Vhf @LnRP⊃
λ∧AmC1k@∃↓+→E∃Duy0⊂ ¬↔↓∪βXB
(¬⊗: h!:~αh*;↔H∞FF/ ∧V&NdU"ε@Xπy(¬`if%41%bβCeJ@β≠SK≤0≥∧
αpP⊂@ASP≠∨~p4Pβ"C!↔ h T*εEεBα.GROUP;

∃)↑↓C@∪⊃α)O'→+	β↔c∞&/∨=⊂m@9P:'H⊂¬ 3λα)S@&pV∩bπα:0∩Z24w#H⊂¬0∪λα)S@&x∧@∩LF@
ge i`≥Fεc@.→ 
εB↔!"cRdε @)¬
&Qλ¬β∩↔0⊃I_α  64∀]¬∨aα~*`∧SL"T¬≥↓y∩s∀	Yts-H¬0JJPbJf↓X¬+9)-ε$αY@; 
`↔∃+!M∃NβP	,∀R@1=
∀Jε#'	∧S∀DαZ0∀gp∧@↔→@∧S.) ~1 @→%(∧BbαTε1∧↓X∧tgr @→↓++∃∃Q⊂∧Vd)1⊗λ⊂¬ ∪λα)S@&x@∩Yα(% *α)+i ∪
Rr∀R↓1
∀\	_∞BλhQ ∀3Qβ!↔ h T*∧EεBα~*&C∃β;(∂παε=p≠Ntr2@ratio8ASf@π##∃β&+; ?L≡FN}βX;∧→<`⊃\αipt@%←P→βxb∧dα4t∧
9→ ↔≥4s$`%n whh`πPA[¬afAC8ASIJεsS'~L∧W! ¬2itheroh
i↑↓BA&[α+cCH∧λn⊂7w*≠β an↓\[C@↔Iβ≠↔t∧7&Nβ{@≤90∂m↓"P↔↔Dεπ.h≥⊂↔H)P
e↓aaef\4⊂
S"¬_d
<h:~2P2@3pπ@↔;_Rε↓yH:~2P0y→βu`≠JεsQβW_V"ε≥`λ
≥X
97Y8¬cI9@≥↓∀ε8λ.yp∂c$∧T@@#\π⊗}@\|j
π,J ?
P∞∩∃9P∂@&TπS#ε ¬+≡≤∧`9[qm|;h	 ∃πPλd↓p∧l(`RBb↓∧¬↔4_;SNM→<@≥≥β0∪talce ↓=@→β∧∧↓P@-βx2@ratio↓9CX@7&+; /L≡FN↑βX;∧∞Y ⊗ !pi`∨]α≠#'Aph ⊂↓ ¬↔BAG@∪≤β)#%≤β_∞$d∧'@4Hε⊃?λα;~∃→X∂Iβ(∂ε∞N∧≠2K
∃O%iK@9∧∧∩∧dα4t∧
9→ 7≥4s$`%p∧@@JIpJ@)βC]H↓∧∧∩ε9;8L↑H≠`3λ⊂¬de↓9l@∃(`π@≡∂⊂λβ!8∧hE `¬`↔;_∞FN}βH	,Om⊂≤,28@1dπ∩c#h∂I%%λ≥⊂∀→u⊂εE	X	DJ(AgA←β+ &"
Xλ.∧	(	@8%* o↓9i@≥↓(C∩*C@~M~8	P~yP0wλ4p∞tuiti`-α)β@>≤ε 
|β⊂9`!yi`≥N4⊂@&≡Bα,∧b$* shnuld @5C`A↓λπ'.β8Y\α of <ide↓9iSMSα+@∪↓H≠`↔≥4π A!0
Mkαs∨&≥p≠DUHπ⊂*~4yP&ujction~∃β;'30εkπAβλπ7.β8Y∞⊂7s⊂	p∧en@XJTA↑εsS :≥bε.β→0⊗Yw:⊂ /`∧@KILJT\~)∪]@S⊗{∪@.=_L↓P∩P∃@$JTACLABA[∃`∪¬74∧↔-88[T≥∪h∞0p∞ge h∂mJβ⊃↓c'&+;@&≤`~,↑Lπ⊗εBαp	QKαqβ≠} 
\≠λi'⊂β∩`$
K]lJ@A`∨∃
ε∂6PεC!¬PSvλ⊃ ¬↔↓QdεTER9
∀Je⊃.e	&\∧Rc∀Hp∧ ⊃X$BbFZα→.εE
αJ+⊃Q"`(Sp!βA∪"g⊃ε@
.↓→ 4*Iε*αT¬@∩→w0∂tati`∨\α)Eβ?2βπ)βHF.WM⊂→M≤αy @%`
β¬g.@X⎇~-⎇H
FE≥βherdas tHe % ,α3π3W*))β?2βπ)βHF.WM⊂→M≤αy @ @~ε≥`λ]→;9-nα⊂7@f  I&∀b&KH∧39+∃aP@ αE*4→P:92↓atme@9hA←L↓S@∪↔w#'6α9 9≤β le@¬IfAI C↔∂&ceβ'pεF`4≥~→↓PY→0↔≠βtio↓9CPACMaKGiLA@?→g.V>M⊗}@H_<∞
α4q`!ti`∨\αp4*←*β@≡F≥@≠∧
αpt@.pa`∪\βiQJ@ββπKπ`
F.g4λλL↑≥y ∩[⊂2{ !h	`↔παI⊗}@H_;LD→→ ↔~βtat@%[\@1∧'Jε⎇⊂
M≥β3FE	X	D¬4e!1Aα;⊃↓)J"-_ ∩LP1w@rp¬Kgaα{;∪'t9β@&t∧C≡=X;∧TH_;LD	 →`!ppl@2∀T\4Tc↔Q↓++2, ⊂⊂_2P0@:¬`@7⊗ ↔$≠yHπLX
w #ti`∨\βqβπl@
VαrRX↓ be @∧A@↔↔n∪↔Iβxbβ↓→S`9≠←⊂ ↓QQK@(hS⊂⊗↓|@⊂ ! ga`,α+1β↔`V@\αp∞to`@Jε#↔+Y+	1↓∃β_BVα0RXλ4¬aaβ→↓+Xd@&∀≠{P~≠β an↓KYK[∃]`⊃βx`h↓I9#≠⊂¬1ah
H⊂s⊃∃αLR

Xλ.∞h	-,T, ⊂ /np	↑↓α0∩ε]@→-↓p¬`≥hβ←@→↓(E
⊂C!¬
∞↓@dpdt4⊂	:
,:&rλ8@)J⊃0	4↓t;~∃→X∂Iβ(∂ε∞N∧≠∩Nα ←Je⊃+L⊂	(@∀∩@∪@
CdK$@↔"	@
$⊗(∞@(¬`	GCHJbALεs@∩ε≥@≠∧∧αr6∩↓1 in @∪↔tεb
aQ"@hαg"εBα*P@(β⊂4RβQ`)∩πP ↓Xh ∩th¬P@*x∧vdZ0y @∃ckC@$K? 74λ

@t2⊂flπdAiαC∃β?αMε/∩	I*h⊂8rim`∪IαK[¬β0∞Vf∨M⊂m@9P0w→⊂892Y4qpz→yP∞
IhAO∃]KeC0XAiQ∃X	hQ H ¬GIN CEN↓Q⊗H↓1Q"@(Sp C@
% 3⊂@∃@&∀P→E∧αZ`&%`@%∀∧@

D∧R@18
@KI0QL@%*Q0⊂hαK@!∪βP⊃∧~(P~,hAP@%P4⊂**α"@↓
→`⊂$escrhb`
AiαC∃β↔0⊗@Nαp`∀@%X∂ 2
p→D(3ulcti↓9\[GC1P⊃β'pε2&Nαπ@≡
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`↔≠⊂≡FN@y¬⊗~βb~∀β≠0≡@9(⊂\αguM@∃]`∪M∞F`4_Y(≡≤⊂⊗ie`λ@1⊗v"≥bε.β]@4≤α`∨@;l∧Vwλ≥p∀~qp⊂cmj	iC%XMhCS#∃F.X∧w )ti`∨\β←@→↓+≠2(¬⊂!n!HAC1PAiQ⊂~+lP_l↑|x<O∀≤⎇0⊃≤βidi@¬`eβ& ⊗ -8π4@tio`≥f↓]KKI∃HAERαJgLJ(XAiQ∃X@J↓≤∧↔πεH∞)% ε@
c@¬X	βN¬_.]_8
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 λZ¬0∂bvβ8∞∧εαP0&P_]@%et%* →  ∃7βA`≡X8`:⊗βx ~@
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 PαIT2 b`!D 
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 ¬RO↓@A0↓%β	$4+KBE"¬∃	_εb`∪ X  ⊃
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8πg,λ
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 PαIT1 b`@%Xh ∩t){∧λhαW⊂∪Zxεes  @↔P∧Q"εNd∧S?	*@≥≡h⊂⊂@<αeprd¬`∂↔nL↔&Nα{@≥\β⊂ !p	@↔S∞aβ@.l≡'I@5∩
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bARεqβπ
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(Q H'h!εBα.END	α ≥αAβ%(Xh ∩th¬P@)Xπp∀i`πBAiαCπQβHbα,⎇∪@]0p@71Sgi7α1∃@E 	,gQP∞.9;0∃∀ε⊂∞O~∞N'∞+λ:~2p∞ ↓→X∂Iβ∂∪'↓L,↔/ 	,l∀(¬εE≥β`
AoαK3b
λλ.l(	,l≡≤⊂⊗ 9XπP≠πl∧U@<β∩P_∩ 
g:@↑c'OR\∧∃KZ9H
.z-`&$8R%fh	~@Hπ↔]X
Bb↓∞F/⊗β: 7_z4w#H0p∞d
s the  problem of
representation again.
 LISP %6is%* more than a programming language; it is also a formalism for discussing
computation.


.CENT(Problems)
.GROUP;
.P271:
.NL
1.##Recall the problem on {yon(P99)}, dealing with the following
factorial algorithm:
.BEGIN CENTERIT;SELECT 3;GROUP;
.PT2
←fact <= λ[[n] f[function[f]; n]]
.PT2
%1where:%3←f <= λ[[g;n][n=0 → 1; %et%* → *[n; g[g; n-1]] ]]
.pt2
.END
Rewrite %3fact%1 in terms a unary function %7t%1:
.BEGIN CENTER;
.pt2
%7t%3 <= λ[[x] function[λ[[n][n=0 → 1; %et%* → *[n; x[n-1]] ]]]]%1.
.END
.pt2
Show that %3fact%1 = %7t%3[fact]%1.
.APART

.BEGIN GROUP;
.NL
2.##The language  described by the %7a%1 and %7b%1 rules 
doesn't look particularly rich, similar in power to LISP
with just function application bqt without conditional expressions.
That is only an Illus@%←\\AMQ←nAQQChA]JAGC8AeKaβ∪↔O↔w!β¬β≤¬⊗oεLQPR+=≤b+
nVv∨M≥vrαV=⊗5←π>FF.g=w&F↑.vO≡[P	&∃@εE$~w:≥⊂≤t7{P≥40zεB∩\D∩Y∀∀<⊃↑TQ|TIXP4yH0P3w[r⊂92\92y`%ntation for %ef%1 and
%9λ%d((x#y)#x)%1is a good representation for %et%1.
¬
.END
∃␈p∨@∩N&:¬@0GJ}