COMMENT ⊗   VALID 00044 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00006 00002	\input macro.tex[let,jmc]
C00011 00003
C00016 00004		It is not difficult to give sufficient conditions for general
C00021 00005		On this basis we shall say that an entity is  intelligent  if
C00027 00006	\display 61pt: 4.: The  right  way  to  think about the general problems  of
C00032 00007	\yyskip
C00037 00008	Such a program was first discussed in McCarthy (1959) and was  called
C00042 00009		A  representation  is  called  metaphysically adequate if the
C00046 00010	\yyskip
C00050 00011	The  interpretation  of  these  equations  is  that  the state of any
C00056 00012		We shall write
C00060 00013		We   claim   that  this  notion  of  $can$  is,  to  a  first
C00066 00014		The notion of "can" corresponding to the intuitive notion  in
C00072 00015		For example, suppose a pair of Martians observe the situation
C00078 00016		In  our  opinion,  this  explains  some  of  the   difficulty
C00083 00017		2.  FORMALISM
C00088 00018	Fluents
C00092 00019	            where we have adopted the convention  that  a  quantifier
C00096 00020
C00100 00021	Here, "p" is a  person,  "st"  is  an  action  or  more  generally  a
C00104 00022	In  the  above  program  the  external  actions  are  represented  by
C00108 00023		∃s0{at(John,office(John),s0) ∧ q1(O,result(John,face
C00112 00024	which  expressed sufficient conditions for the ability of a person to
C00117 00025	and with this, the possibility of opening the safe can be proved.
C00120 00026		The  present  approach  has  a  major technical advantage for
C00125 00027	The approximate character of "result(p,st,s)".
C00130 00028	write down that certain actions do not change the values  of  certain
C00135 00029		A formal literature is like a formal language with a history:
C00140 00030		1.  Any consequence of sentences of "s" may be added.
C00144 00031		Many  of  the  problems that give rise to the introduction of
C00149 00032		1.   L.  Fogel (1966) proposes to evolve intelligent automata
C00155 00033	examples.   The  task of improving GPS was studied as a GPS task, but
C00160 00034	axiomatization  of  a  fairly  simple  modal logic, the system "M" of
C00165 00035	By rule 2 of "M" (which is present in almost all modal logics), since
C00171 00036		This semantic theory already provides an answer to the  first
C00176 00037		∀x . F(x) → ∀x . exists(x,s) ⊃ F(x,s)
C00181 00038		Others of course may be needed when se add tenses  and  other
C00185 00039		∀t . ∀r . (cOhistorical(r,t) ∧ cohistorical(t,s)) ⊃ 
C00191 00040	which has reached a stage kf sophistication adequate to meet Simon`s
C00196 00041	actions  are  performed, "all" propositional fluents which applied to
C00199 00042	REFERENCES
C00204 00043	Green, C.  (1969)  Theorem-proving  by  resolution  as  a  basis  for
C00209 00044	Newell, A. & Ernst, C. (1965) The search for generality. "Proc.  IFIP
C00212 ENDMK
C⊗;
\input macro.tex[let,jmc]

\ctrline{\bf SOME PHILOSOPHICAL PROBLEMS FROM  THE STANDPOINT OF
ARTIFICIAL INTELLIGENCE}

\yskip

\ctrline{\bf by}

\yskip

\ctrline {\bf John McCarthy, Stanford University}

\ctrline {\bf Patrick Hayes, University of Edinburgh}

\yyskip

\noindent {\bf Abstract}


  	A computer program capable of acting intelligently in the world  must
have  a  general  representation  of  the world in terms of which its
inputs  are  interpreted.    Designing  such   a   program   requires
commitments  about  what  knowledge is and how it is obtained.  Thus,
some of  the  major  traditional  problems  of  philosophy  arise  in
artificial intelligence.

     	More specifically, we want a computer program that decides  what
to  do by inferring in a formal language that a certain strategy will
achieve its assigned goal.  This  requires  formalizing  concepts  of
causality,   ability,   and  knowledge.   Such  formalisms  are  also
considered in philosophical logic.      

	The first part of  the  paper
begins  with  a  philosophical  point  of  view  that  seems to arise
naturally  once  we  take  seriously  the  idea of actually making an
intelligent machine.   We go on to the notions of metaphysically  and
epistemologically  adequate  representations of the world and then to
an explanation  of {\sl can,  causes,} and {\sl knows} in  terms  of  a
representation of the world by a system of interacting automata.    A
proposed resolution of the problem of  freewill  in  a  deterministic
universe and of counterfactual conditional sentences is presented.

	The  second  part  is mainly concerned with formalisms within
which it can be proved that a strategy will achieve a goal.  Concepts
of  situation, fluent, future operator, action, strategy, result of a
strategy  and  knowledge  are  formalized.   A  method  is  given  of
constructing  a  sentence  of first order logic which will be true in
all models of certain axioms if and only if a certain  strategy  will
achieve a certain goal.

	The formalism  of  this  paper  represents  an  advance  over
McCarthy  (1963)  and  Green  (1968)  in that it permits proof of the
correctness of strategies that  contain  loops  and  strategies  that
involve  the  acquisition  of knowledge, and it is also somewhat more
concise.

	The  third  part  discusses  open  problems  in extending the
formalism of Part 2.

	The fourth part is a review of work in philosophical logic in
relation  to  problems of artificial intelligence and a discussion of
previous efforts to program `general intelligence' from the point  of
view of this paper.
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CiS←9f@AC9H@AC⊃UCGK9GrAe∃YCiS=]f\~)'GK]∃fA[CdAEJAIKaeKMK]iK⊂ACfA1Sgifof bodies=tz4≡
|z=
≥{\k∧∧≤z_.<kβ!,;Y
π⊗≡LW~α
|bαε]}FN}ee↓αOO#WπSN{;M↓εkπe↓ε∪∃βK/βK↔O.sS↔⊃ε∪eβONk?3N_4+↔GβK↔O≡K?;Mπ;'S!ε33?>+⊃βK.c↔Mβ}1βSK∞sO≠?⊗kπS'}q9↓α/#S↔K∞s∂↔Mεkπe↓ε∪∀4+⊗+CK↔≡+;S↔#ErAI%OSiSiKHAMU]GiS=]fA←_AiS[∀XAErsequences;s⊂84≠w2vr\VεE0[2⊂80\9tw3\P7s≡<;]→-ly<kAQC"B)
⎇y=L↑Kλ∞}vTπK↔C⊗+O↔;&S'?rAaYCef@AB dominant<7v2\≥Yλ~-a"\z-↑≠→<D∞}<⎇]<h≠,∨(_Y$∞~→.}vO↔K↔C⊗+O↔;&S'?saeKg∃]h\@ This  is(0FE9→x92yYw:0z~ww⊂≤/(λ≤l↑≤h∞}2∧∞6.wL]f≡/4π'9βgkSi¬EYJA→←e[C0AY←O%GCX~)YC]OUCOJXAM←d example ${\omega}$-order  logic  with   function(⊂9|vX7v9VβE22yXy4x:~ww⊂7\2y0z≠y⊗⊂1[w24z~ww0vλ2|89→yytw[9V⊂9Yz9V⊂→z1W∀
z→=
<H≥lQ"[=.>λ~;L=≥9→$∧≠;y≥λλ≠n<X=
}\hλ∞⎇=~
π&]↔∩α∞,V6/,]g&N≥Eβ?C∞≠'SeβSf~∃U]IKG%IKH\ This representation dominates in the following sense:

\display 61pt:1.: All;z42yλ20z0H9z9:Xz:y2\P⊂40]2P⊂6~w3zt\z4qPλ22yq\4x:4[w9FE≥40z∀z=Y$∧≥~→$∞Y;_.M;{\dY=≥l\;H≥
(≤⎇∞.8⎇≥.,<h_-lλ≥z≡β"]
>(≥]≠λ_,-⎇=∂&FT∞v␈⊗LE`hULM↔∨εL∨∩β3≡∞CRβ%g"¬&UβOW↔∪?WSNs↔M↓εCπ[∃εc';∨.KOS'~β∪↔O∨∪'CSN{;Mβ&CπQ↓π#↔30hS←#π#iQKr↓I↑XA∃SiQKH@AS]QKe]C1Yr@A5C]SaUYCiS9N@AI¬iBX@or
externally;pw4x≥v0z4[3P:4→P;wy≠2↔εE↔24yx≠0|P≠x:≥⊗5NH∃
(≤]-L<h≥
=λ→/∞≤Y<n4∀T	n4_Y;
≤9\n∞⊗}↑Bαε
}rπ&Uβ←?⊗c⊂$4V∪↔#π6+M↓β∞s⊃βSFQβ∨O3∃βSF)β∂?w≠↔GW.s∂↔Mε{→βO'∪πS↔>K↔Mβ∂∪∀4+/CCK↔∨≠↔⊃βfK;∨WO≠S'∂∞c3e8hRs∪'∨β3πeβ1GCQRh\tAI Of@goals,  as  given by the9|82y~vrw:→y⊗⊂4]9P⊂2→{4yrYεE9zX3wpv≤V⊂4z≤P7x4[4ww≥⎇H~=∞4≤⎇_.L(≠yD∧≤≤[l}Y<|d∧_<Y$∧_;≠↓Q[~;L}:<⎇
≤x;≠∂∀→>≤∞,<|y,EC"WM<|≠∨(
L.∞∞H
%gH∃y$∞z_;
D≤x>$∞~_=∧
T	|d
;Y[n-8=~-⎇H~<d9→<.\=→/&z∞=vg6T⊂hW∞-v⊗f]Uβ'→βSh@A%f@AB logical 8ww9r\zrw1YP7s≤≥≠λ≥
<y#!.y;]]Xy<d∞~_=∧(_y.._:;D∞⎇≤X.L9}.}2∧⊗∨&≥⎇bαπ⎇≥Fbα∞=vg6QQ&O"aQ%f&≡>εf∂∀εc∂πG%Y9i¬∩Aβ'≠BAIK⊃kGiS=\Aae=OeCZthat=94ryH:7P≤m;Y
π∨N,↔&.⎇≤W_h-|bαε≤>FN}dπS#π"ASh@can <97{2H;tv6λ9wv;→P0P8≤7q62[]P7wβE34w→4w3]⎇Y+∞O"←ε.∨↑LW~ε≡E`hULM↔∨εL∨∩β3≡∞CRβug"¬∨N,↔&.⎇≤W~ε\∨∩εNnmvg6T∞7.⊗⎇|⊗g~∞⎇εN≡∧↔⊗*∞Mrε⊗T∞6}gl\Bε↔⊃Q%∃αDππ;⊃βaCeh or 8v6⊂≥|Hλ_$∧≤⎇≤L≡→9}$∧≠8>$∧_Y*ππ↑,VgHQ-⊗w&]MF.∨N\⊗bb∧∞FF∂Dπ'M1β[Cr@involve  the  search for8FE9z≤0z2s↑V⊂0↑∞[{yED≠|O≡}\W?S#/⊃β';&+33↔∨#Wπ1ε{+↔∨!βS#∂ 4+O∂#'O≠N+MβO}k∃β∂⊗KS↔KN	84(>↑ε/↔<\F."/∩π&
≡2πε≡W∩b∞|↔~π∞,W≡.nLV"r
MεO_Q.ε∂ε↑%β'Mαβ'9↓πβπKQβC\@A¬]goKH@Ai↑Y. Bar-Hillel's comment,=t2w≡M→#"M}Z9z-l;λ∂ε∂↑"απ|≡2απ∞,W≡.nLV"α≡BαπMRαβ↔⊗SBα∧
7No
}6O.T¬β?9α↓βS#(h*7↔≡Cπ;'VS'?rA←L@↓)Q←k≥Qh@AAe←GKMgKfX that  the paper involved some
philosophical presuppositions.

	Constructing()(⊂≥≥][{∞l<h∞⊗␈M¬βS#+KaSgQK[←Y=OSGC0AC]Hthe
heuristic<0y:9H7s⊂:~2P0y≥4s4qZpv⊂4[:2v6~srw1YP897X62v]λ:40zλ4yV∀∞~→#!-;Y[n-8=~-⎇Hλ~-dλ≠9-]|↑*εo↑>Bε⊗T⊗&/≡\↔&*∞Mrε&↑LW⊗n≥lRε
∞>G⊗∂L\wJεm} hV≤=εN/m≥f:πMRε>|≥BαGM
↔~π>N&∂&\␈∩αε\∨∩αε≥nf}glUβS#*ACGcUSgSi%←\@A=L~∃MUeiQKH@AS]→←e[CQS←\R and()(⊂≥↑<⎇
ε⊗Tπ∂3↔6+I↓β.s?W∨BAi↑A→S]HAQQJ~∃MieCi∃OrAC9HAiQ∀Aae←=LA←Lits correctness.  Of course, these<97q6→vyFE~w:2y_qz⊗∀]=
π≡≥l6*α∞MεO~∧∞ε∂ε↑%β'Mε3?∂W≡+⊃β?siQJA∃aSgi∃[←Y←≥SGCX4∃aCePXAoJmention the Missouri program (MP) that :w;7v≥2yP⊂≠w6<T∞~~<aQ\_<NEC"C!!5~→$∧∪:<n=⎇<Z$∧≤≤[l}X;*FON77?S&yβ'McA'Q←\A[JN$AI←KLA]←htry
to find strategies or proofs that =42P≡>≤X=\z9<d∧_8z
≤=Y*ε
∧v}∞EaPTNn>F.∞EEβ'QβCYY←]f@Ai!J@AKaaKeS5K]iKHAi↑AAeKgK9hAShproof steps and
checks their8wy92Xz72y\W⊂⊂&[y2w{→y⊗⊂;Z2w⊂4]⊂4y\{{]M≥Xy9∧tλ≥~≡λλ~.A"[⎇,⎇≥λ∂&z∞W⊗6}-Rε∞d⊗∨&≥⎇bε␈$WF.>↑F*ε∀∞7'⊗≡LV?J
≡Bε&|↑2π≡ueα←∃εkπd4W∪↔∨π⊗!βS#O→βCπε+Iβπ≠EKS]≤@AG←9GKe]∃H@Ao%iP@AQQJ@A
←]giIkGiS=\@A←_@AB~)≠Sgg=keRAAe←Oe¬ZAiQ¬hAGC8AEJAAKegk¬IKHAQ↑ACG!SKmJgoals.

\yyskip

\noindent {\bf Representations of=42P;[y62?βEεE.≡yutxβEεE∧U42P3~y9z≡>→<λ
≥H≥~T→→<m≤{H≠ld∀T∞␈∩	ZαεO4πS=↓ε#↔∂'&)↓β←FP4+∨#KW∂'+K∃↓π##∃β>{K3⊃εKMβSzβ∃β⊗+∨πK&+⊃βπ≠QCmS9NXAC9HAQ←\AS]M=e[Ci%←\~∃¬E←khthe world8w2⊂4]9P60]yP7sλ1t0w→rP0y→P:7\,(≤Y.∞Y<y-n→9
εNdπS#∀hS7π∂FK;∃9ααS#'≠IKGSMS←\AQke]fout to depend on whether;w2P4\P:0v~tw3FB0q7z]⊂:42H2|89→yytw[⊂7s≤|;Y<L≥λ≠_.}h≠|D∞|→8m≤Z8n6∞>N2rα∧εS#W~a↓β?/⊂4+Wv#↔KO&;∪'v9↓β?3OCfA⊃s]C[%GfAI∃aK]ILA←\AQQJAe∃aeKg∃]iCi%←\A←_ABAO¬f~∃CL@AB@very(60y3YP⊂7:[q2y∀
yH∂ε∂.M⊗≡f↑5β7?6K;≥↓εK9↓↓π≠Cπ∂+Y↓↓β&C'L4W∪↔CK/≠↔;S∂#'?9βaYCsL@AC\ essential role in deriving the mechanical,
thermal electrical and optical properties of9pyryK⊂*42H9z0z→P7s∀∞~→#!,x<jε∂Dπ¬↓β>K[↔9βS]gi¬]hASLAeKO¬eIKHas determined by the position,
velocity and9|1tz_z4wwλ9z0z→yP7sλ2pqtλ80y:~qv2Wλ⊂$7{Y{2y⊗λ⊂;rPλ72{2\εE0q]:pv6≡P⊂22]2y6t[2P⊂:~2P⊂8≠ytz4[w⊗⊂≡l;≠xm≡≡(≠n$→>_m≡_=~-⎇H≠yD=Y;D#"\m≥Y{→$
;{→,Z¬dA *q↓α?/⊃βCK∞≠S'∂∞aβ/;␈;3↔∪>)β?→ε	βCπ↔#'∂WfI↓β≡7C3*↓β?_hS∨πMβSf@A∃qaeKMgKH@↓Er@AACeC[∃iKeflike=42P8≤2yyz\2V⊂:→vx2y_z:y2H0w2εB;2v7Xtz<\m9;→∞4λ≠|D∧→=Y-dλ≠;n,(λ→n-||{∂∀λ_↑$∧_=Y.,9y(∧∞≤Y<n><Y<d∧_;Y↓Q]→;.<X=∞↑Y<kD∧λ⊃\M⎇(≠⎇.$≤~~-M|{|

8x;∧∞≠z;ND≠yH∞m9=o&F≡7'Mβ.sS'K.cd4+v{K7πbaβπ;#oJACIJA]←PAS]G1S]KHto  deny  existence  to  entities =rFE1Xw77zλ⊂9rrK⊂7y≡Mh_Y$∞{h_-n~≤[n
xy;NNZ8n∂~∞MrεN\≤vNvT∞FF∂D∞FF*∞⎇w⊗fAQ&o/>Eβ∃π≠=↓β≡{;OS↔+∂S↔#iQChwe  have direct or even indirect access
to all of it.

	From =42P≤≡]~9M≤z8;∧∧~;]]≠~9l]Xy(∞
z;]∧
yH≥M≤=h≥lT_x;D∞~→;AQY→9M≥Y(≥
∞Y9.↑NlN2ε}d⊗&/≡\⊗∨Jmw∩π,↑π⊗/<]g&∂M≥vw~
|bπ&W←?Kf!84(;{H∧
<hλ<;≠→,Dλ≠9.L<~≡.=8x;
O(_9↑=8=T~9O&FQQ'>␈-LBε≡}]F"ε≡f*πM↔"εm}&jπ⎇≡FF␈↑G∂?;'∪π∪'∨#';≥βiQJ@facts  of(:42FB0yx2Xz⊂7sλ92pv~z<P:~0z⊂4[:2y2\z9P:\W⊂⊂_←_;<
L<h≠ld≠9=≡~≡<m≤x;≠∂⊃"X9↑=8=T≤Y<∞,<y;NL=~;mnh→[n$→~9Ll<Y;ND_<|\⎇≤n}2∞,V∞f≡O∩ε∂,W hPQ+F&O>
F∂Kf≡π#Rε∃cR¬MRαπ,↑π⊗/<]g&∂M≥vrα
|bαπMRαπ⎇}&f"∧↔~α↔∂?3f+∂S'}q↓β?0h+Cπ↔#'∂3/→β';&+Kπ∂&K;≥β&CK?W>Aβ≠?⊗≠↔Mβ⊗+S←↔.qβ↔π≡AβCπO⊃β?_hSCπK&K∂3↔~p4*s&KOC3∂I↓YGπ!i↓IsQαK↔π∪↔O↔w#πS'}qβ?→π##∃β>{K3⊃εMβ¬ε;'π;#ckC]QkZ[[∃GQC]%GCX~)oCmJfunction.
\display-_x:≥λ→W≥~,<≤Y.<;]_.M;{Jε∂4π¬↓β∨KOS↔jA←L@↓S]iKICGiS9N@@A⊃SgGe∃iJ~∃¬ki←[¬iB\A]JAgQ¬YXA[¬WJAkMJA←Lthis<2x92\rw:0]4ww↔βEεE&Yz0x4≡ytqp[6<P≤≤→<=,≡→(∂⊗/∞,W≡.nL↔&N⎇n2αε≡,Rαε\≥⊗vg∀¬βWO.3W1↓αβ≠?HhS∂?;∨#KW∂&K;≥β>+;↔K∞aβS#.{K'↔~q↓α∪/∪'['v9β?≡+K[π⊗c∃β∂}sO↔G.+;∂↔≠Me←Z4∃iQJtheory is8P3:y≥42y≡>→<AQC"B(∀λ≤Y.∞Y<y-n_=~-⎇Hλ~.4_x;
L9λ→.
<⎇→-]{≠ym≤x;≠∂∀_9→.≡8=→$[|Nλh.W↔≡⎇g?Iβn∂#'v)β'→εKQβ∂∞qβ∃π+O↔⊃πβKπ∂&K∂π3gIβS=ε+cCK/≠MβSF)↓β≠∞≠SL4W##πQβ←]J@actually :0yP0X7zz≡M→(_.>→8⎇∧
yH≥
(≥{n-→J¬&∞↑2εv⎇lRε}aQ'&FT⊗⊗␈lUVn.nM⊗}v\GK↔C⊗+O↔;&S'?w→βπK*ACIKEkCiJ to 9|892\yP⊂3_qz9FB64urH⊂0%7Z7⊂⊂4\P⊂0zλ⊂47vYSP⊂7\⊂027YyP1t_yrP1Xz9S]}H_∩M⎇≠I|d∞→;→.
≠{Y!Q[];,,<H~.4lL%Vm.∧uHλ∪n,~;X./(λ≠≥Y⎇8,|(λ~.4λ≠xNm;⎇<mO(λ_,L<=8.L(λ≥
q"Y>∞∞Y<|d∧≥~→$∧→X8nNh≥~≡λ≤→-}≠→.≡}]↑VvN<≡F*πMtε.∞=π?S#/⊃β'9ε{K∪'vKd4Vcπ;∨.∨∃9β∪hASLA]←h0AM←dinstance,8r2xzXz2P:≠P2|8≤2yyT∞z_=∧∧≤→;n
→#"M=[⎇jε∞-}W"α

w:α∞Mrπ⊗\=v>v∨,Rε
∞↔↔&≤>Vf∂$f∞≡UeαS#+gKG←9HAaCIhA←L4∃iQSLAaCa∃dASfconcerned=tz4∀;Hλ↑~<⎇];{≠l⎇8x;
O(λ_,L<=8.L(λ→M}[8;↓Q\Y<∞,<y;NL=~;mdλ≠yD∧_{{-]{K<l]\y(∧X8⎇∞4λ≠yD∧_x=.<;~=∂∃λλ_,-;~=∂∀λ_;LA"Z{M}{→9|+C"AQB0*π⊗↑∞&/≡]nF∂&≥⎇bαε≡5β∂πfc↔⊃↓εC↔WKO≠S'∂∞c3e↓ε∪↔G.S∃βN1βS#(h+K↔∂≠?;'v9βCK}≠↔OO/→βπ∂'+π33KO←]Jthrough in solving 8P⊂89≠q62vH⊂0y2CE2|8≤2yytX62P⊂~w⊂⊂:~2P⊂6_w3zpYrW⊂∀
y(∂≡F≥MBαεm}BαπN,V∂"∞MεO~∞=vn/⎇↔ h.LVw&≡M↔6.O↔CK?ε{O↔⊃ε≠?;∂/βQβ≠/∪S#↔∪S\Ai!SfAa¬aKd@except  to  point
out  later=40z≥⎇Y(≤≡]~8n]_<O⊗/∞,W≡.nL↔&N⎇gO↔↔o→β↔CO≠S↔7}c?∨'≡33dhSWQεs?QβF+WK'∨#'∂πfceβπ&+GWπ&)84(hP&'9βiQJ@remaining<rqz4[w9P7Y⊂:42H34y9]⊂80y≥⊂7s≡M→(≤≡→<O>(Q.6F∞MGWO∃π##∃β⊗+CK↔≡+;Sπ&K?;Mε{→βSF)β←?⊗c⊃βπ≠BAgsMiKZA=LAS]QKeCGQS]N~)Cki←5CiB@to  explicate ;7z4w[9P⊂7Y⊂1pz\pv4z≡V⊂0q~v4z<H0w2≥=[⎇{\→y#!%~;XmN9~;Lt≤y;e:{[n⎇→9→lU+C"@kivEβE.77Zw22w≥⊂=n1→⊂*42H0zz7[pz7wλ92x9→yrw:_z4wwλ0w2≡M→(≠M}~;{D
yH≡≡∞d␈`hPQ+GO≡=≡hPQ)F/"∂;G≡b
?bε⊗T∩π∨≡>F.j
|bεNnLW⊗∞>M⊗v:M↔≡∨,↑F*εm≥fO&T↔/&⎇\↔&
∞>V≡B≡0hWM↔"π=
w>r
≥bε6≤}W⊗*ε⊃PPh!Q hR∧¬↓↓↓αZZZZ4ZZZZ4~∀@@    ?⊂⊂⊂⊂λ⊂⊂⊂>βE⊂⊂⊂λ⊂⊂⊂⊂∨⊂⊂⊂⊂λ⊂⊂⊂>βE⊂⊂∀∧λλλ∂Dλλ
α∧∂BjjUURjjUUPhR∧¬↓↓↓βx@@@@@@Ap@@@@@@@kp~∀@@    ?⊂⊂⊂∀∧λλ∨∧∧λλ
αα∧∂@hR∧¬↓↓↓αZZZZ4ZZZZ4@@@@@@@Ap@@@Ap~∀@@@@@@@@A< |       (⊂⊂⊂>λ⊂⊂⊂>βE⊂⊂⊂λ⊂⊂⊂⊂λ⊂→>⊂λ>⊂⊂⊂λ⊂⊂⊂∀∧λλ∨∧∧λλ∨↓QHλ
αα∧¬↓↓βb↓βq↓α↓↓↓↓α↓↓↓↓πa↓↓↓π`4)↓α@@@@@@AxAxf@    (⊂⊂⊂>λ⊂⊂⊂>
εE⊂∀∧λλλ∧∧λλ∨∧∧∀∧ααα∧¬↓↓↓α