- Email: [email protected]
Descriptive set theory and Boolean complexity theory
C. R. Acad. lnformatique
Sci. Paris, t. 326, thCorique/Computer
Descriptive
SCrie
I, p. 255260, Science
set theory
1998
and Boolean
complexity
theory
Thtorie tlesct-iptiw rl(js ~tt.wttrltl~~s et Ih6orie de 10 cottrplexiti ttool&tttw
Version
frangaise
La thkorie (c’est-i-dire
descriptive
dcs
I’cnxemble
dcs
h-dire
I’ensemble
traite
de parties
dcs de
de circuits
bool&:na
infinis
alors
ont
ainsi
des
Dans par
ensemble\ suites
suites
des
infinis
rCsultals
le\
tie
II CI de
suitL*\
linies
i IY~;/.
d’idkcs.
Sipscv
de Bowl)
d’a~semhles
dCfinissahle~
le nombre de -- d~nombrable
disjonctions
et coii,jonct
-
polynomial
Note
pr6sentPe
dank dans
par
Ir Ic
Maurice
cas
us
dc
[ .‘I.
notre
que
(I et de
dc ‘fh&wie
I.
cas
m
type
de I’cspace
I’espace
Ia ThCorie
de
Les dc
preuve
de Baire
de Cantor
la complcxitC
II est possible
descriptive.
fink
cntrc
d’origine
“‘J!
(c’est-
boolCenne
de reformuler
analogies
-2
en termes Its
infink
cas
finis
ct
et d’obtenit-
If;])
10 1 propose
et enxcmblcs
projection
ou dan\
1 ). alor\
It I;I Gluation
Ggnifcatif‘\
projection\
so~~~-cnse~~~bles dCfinlssablr:s
Ic<
d’entiers),
cl66niti<~n~
d’appliquer
ordrc
tkdic inflnlc\
infinies
I’enaemble
permit
Ie mkne
(c‘est-h-dire
abrt!&e
par Ion5
LIIK corrcspondance
(non-llllif’~)rlrlc)h’P dck
at
circuit\
enw : dans
de profondeur\
enscrnhlcs
les dcux 2
L!t dc
cas.
analytiques ils sent
13 t’iillne
/y(W)
obtenus Oil
:
infini, lini.
NI\ \I’.
255
C. Sureson Sa motivation &it de cc deduirc 1) une d&nonstration de NP # Co-NP d’une preuve de I’existence d’un analytique non co-analytique (r&&at clasaique). Le mouvement infini vers fini (tie [ 21. [S]) procCdait en quatre &apes : - la donnCe de deux versions correspondantes tlinie et infinie) d’un m@me probltime. - I’Claboration d’une preuve cornbinatoire danh le cas inhni. - uric reformulation (finie et) probabiliste de\ GEmenta cl&s de la preuvc, - et finalement, une dGmonstration de ces assertion\ probabilistea. Sipser [6] pr&enta une preuve combinatoire de la non-analyticit de I’ensemble co-analytique WF des arbres bien-fond&, mais WJT n’a pas d’equivalent ~_ tini. Nous conservons son approche en proposant ici de nouveaux ensembles co-analytiques II- et II’ (tnspires de WF) qui. en revanche, PO&dent un equivalent fini Co-NP complex. Nous produisons cnsuite uric tlclnonstration combinatoire de la non-analyticit<; I& Ij7, ainsi qu’utle reformulation probabiliste de celle-l~i qui induirait NP # Co-NP. IndCpendamment d’une conclusion pcut-etrc inaccessible. ce type de dCtnarche nous semble intCressant en soi.
0. Introduction Descriptive Set Theory deals with definable subsets of the Baire space “‘u (i.e. infinite sequences of integers), or in our case. subsets of the Cantor space “2 (i.e. infinite sequences of O’s and I’\), while Complexity Theory focuses on subsets of “2. for I/ < ti (i.e. finite sequences of O’s and l’s). Since one can reformulate Descriptive set theoretical definitions in terms 01‘ infinite Boolean circuits, analogies between the tinile and infinite cases have led Furst. Saxe. Sipxer [2] and Sipser 1.51 to interesting results concerning xct\ definable by bounded depth circuits of polynomial size (these sets arc the analogs of finitc ranh Bore1 \ets). Sets recog,nizable by circuits i)f polynomial _ si/c’ (with no consideration to cdepth) correspond to bore1 set\ in the infinite case. Sipser in [Cl. aiming at the prohltm iXP f Co-NP. proposed a similar link betueen analytic sets (i.e. projections of Bore1 sets) and (non-uniform) NJ’ \ets: both are projections (meant by the symbol 3) of sets definable by circuits of depth 2. and oi‘ the form A(W) where the number of disjunctions and conjunctions is: - countable in the infinite case. - polynomial in the finite enc. He then presented a combinatorial proof of the fact that a classical co-analytic set (the set %‘I~ of well-founded trees) is not analytic. lisual proofs were by diagonalization and non-combinatorial, In the previous moves (WC 121. ISI) from the infinite to the finite, the mecanisrn of the proof was always the same: some effective construction in the infinite case was replaced. in the finite enc. hy a probabilistic argument. There were four requirements: I) two corresponding versions (tinitc and infinite) of ;I given problem. 2) a combinatorial proof in the infinite case. 3) a (finite) probabilistic restatement of this proof. 3) and finally a demonstration of’ this probabilistic statement. The classical set WI? has no finite equivalent. hence one cannot proceed along these lines. But we keep Sipser‘s approach by proposing new co-analytic sets It’ and I? (inspired from WJ:) whic,h have a co-NP complete finite equivalent.
256
Descriptive
set theory
and
Boolean
complexity
theory
We then produce a combinatorial proof of the non-analyticity 01‘ I?. and show how a probabilistic restatement of the proof would entail NP # Co-NP. The result NP # Co-NP may be out of reach. bul this kind of proces. 0 seem5 interesting for its own sake. 1. The sets W, fi’
C -2 and their
finite
analogs
Let us first recall some notation and defnitions. originating from Set theory: - -‘I’ denotes the set of functions from *I- into J - ul is the set of integers N, - an ordinal (finite or not) is the sc%tof’ its predecessor\. - and I,Yl is the cardinality of .Y (i.e. #-\’ if .\- is finite). Hence &2 and “2 denote respectively the set of intinite sequence5 of O‘s and sequences of length II of O’s and I ‘\. We consider the product topology on “(*: (or on -.*? x Aa 1) The collection of of the closure under complementation and countable union of the open sets. properties. the analytic sets (which in- A>are projections of bore1 sets inaLC* X~ZI and the co-analytic
I ‘\. and the set of Bore1 sets consists Due to definability ale called 2: i-sets,
sets III-sets.
In order to detine 11’ Cy-‘2, we uill modify slightly the notion of tree: - Usually for a tree 7’ on w. on SEI~X:for ./ E -“L*:. 1 is an (infinite) branch of ‘I if and only it for any I) < W, .fi,, E 7’. WF is then defined as the set 01 well-founded lrces (i.e. without any (infinite1 branch), and is classically known to be 11: -complete. - For ;I set .A (whose styucture ha\ still to be defined). we will set f‘or .f E lk). J IS a branch of .4 if and only if for any 13 C ti such tltat / 131 = 2. 1’ !T E ‘1. l/I’ will be the collection of vets , I without any branch. Formally we set: - .S(rv./j) = (f : t function. domil‘ 5 0. ldotn(l)l = 2. range(t! C /j): - If A C: S( 0. /I), then we consider the set of branches of length o of A : :A],, = (,f t”:) : VI3 C o. IL31 = 2 implies .flu E :I}. We then consider some “natural” bi,jection h : U{ 5( II. /I) : II <: ti’) -- J. such that the image b), b of S(7),. II.) is r?(~/ - I)/‘2 (IS(r).. fj)I = ttZil I/ ~ I ),:“L). Any .A i S( LL’,~:) can thus be coded by a unique x4 E -2 such that, for i < Ld. x..~ (i) := I it’ and only if I/-‘( ;) E .4. In the same way. il 2 S’(I).. II.) is coded by x.1 E ’ ” “e ’ ‘l”, We consi,jer: DEFINHWN 1.2. 7t. < w. let VI;,, = ~~~fin,t,. := u { w,, : If we restrict the
Let IV = {x E “‘2 x codes a set ..\ C ,S’(u;. LJ) so that ,. I],. = E}. F~I {x E (71‘lJ,-1))/-‘2 : x codes a sct ..I <; ,S’(lr.s/) such that [.,lj,, = E} and let ‘// < w } . range of possible branches as in the li)llowing diagram w
ki--
0
m
m’
w
then we can define:
257
C. Sureson
DEHNITION 1.3. - For X, < LJ, IZ (A-) = (x E -2 : x codes .A C S(ti. A.) and [-lli = @}. For 1) < ~j. \I’,,(,(:) = {x E (‘J’ciim1r)/2’J x codes A C S(rr,. A:) and [A],, = a} q
Wfi*,i(,,(k)
=
U{ l’l,,(kI)
1 TI <
md
Cd).
Throughout this paper, we will denote by 1.Y the complement of .Y. and \ve will often identify a set .4 and its code x-i (in the finite cast. II must be specified). Concerning analogies between the finite and intinite cases, we obtain:
(b) ] I/l;llri~<,T.Y/t’f-<~ot)ll>l~tr (/FI /irt’r 1 Ir’fi,,i,,.I,:i I is CI/WO~~ Nf-(~<)l~?l>/et<~). For technical reasons, we consider now a slight modification 6. of II,-: insttad of considering branches on the whole interval [(I. ti.~). we will admit branches on final segments 1X,,A), for k < ul. DE~;INITION 1.7. - For .4 2 ,S’(u-.. .,,lj. let [A] = (.f’ : ,J function, 3d* < r*l domain (j’) = [X. w’I. range (f’) C LL).and V.0 g [A..k.1). ;I11 = 2 implies .f’ 1j E -.I}. - Let r/T- =- {.I. E -2 : .I’ codes ;I set ‘1 5 S’ljiL!.;) such that [.+I] = :3}. For the finite equivalent, we propose the following:
DEHNITIOK1.X. - Let 0 < CY< I he fixed. For 11< ti, it’ ..I C ,S(II. lt,j. let: [A]‘“’ = {,/ : .f function. range ~,f) 1; II. 3.Y il 11domain(J) = .Y, S/ := II -~ 0” and Vlu c .I, 1~3 = 2 implies ./‘,I E A). G,J,,= {x (5 (~1‘(11~1III/22 : x codes a \et .I c ,s’i/I. ‘/I11such that [A]“’ ’ = ; .j}, IZ’filli(<~= ‘J{ 67,i : ‘r/ <: LCI}. (Rigorously. we should write IV,,I /I ).) One can deduce. from the following th;u 1G- is xi-(Bore]) *.a complete, and 1Il.ti,lic~, is NP-complete:
2. A combinatorial
proof of li. E- [I:\L~ c. -.
We shall use the characterization of !-,” sets in terms of circuits: -” FACT 2. I. -- For any 2:i-set S .~. ( 1“2. there exists a circuit C’(x, y) of the torm A II,,, where I#<-, Il,,jx.y) = W (I,,,, ,(x.y) with r/,,.l,!x.y) = .I’,(,, , ,. l:~~~,,~.,,~. !I,(,,,,,). or l,~,,,~ I’)’ ml for .I’ Ew2. ,II’& x E .Y if and only if‘ 3y cy-“:! (x. y) is accepted hy C’ To show non-analyticity. let us argue by contradiction: we supposethat i,i’ is ?:I, and hence thal there is a circuit (’ as above sati
x iI li- if and only if Sy t&Z (x. y) is accepted by (’
We want to find y and A C S(d.,;) 258
with a branch such that (x..~. y) is accepted hy (‘.
Descriptive
As in [ci]. we consider.
for s t “w. with
set theory
and
Boolean
complexity
theory
k < w’, the set
II, = {x : 3y ~“2 (x. y j is accepted by 6: and V’r <: k (x, y) satisfies the s(i)-th
element of L),. that is ~l,.s,,I}.
But then, if it is clear how to code countably many well-founded trees rn one well-founded tree (one adds a root below all trees). it does not work for our notion of tree. Hence let us introduce some new notions: DEFINITION 2.2. - We often identify ;I set ‘-1 and its code x L. - If S C d x LJ. then let l‘Z;( ’ ) = (.-I E Ii’ : V/ E .4 graph (1) 2 S}. - Let E!, B’ 2 ‘Pi S(ti. d)). By D .’ ;. ” /I’, we mean that, for any A t 1). there exists -4 E LI’ such that :1 c 4’. We then construct inductively a set .Y !_ d x uI ,md .j E iti such that the following twofold hypothesis holds: Hyp (1’. j) : (a) V’k <: ti li’i”l “ C .’ Ij,i,, (b) ‘4; < LU’?I’,,~ = {; ,< J : (,;.,j) E .\‘} is nonempty. Finally, one can prove that Hyp (-‘i ~j J prevents I‘i- from being associated with the circuit 6”. More precisely, Hyp(x.j) allows the construction of y E”‘:! and of .4 C Y(d.uj such that. by (a). (2.y) is accepted by (‘, and by (b). A admits a branch. 3. A possible
application
to NP :f Co-NP?
A natural question is thus: can this proot’ of C I f I 1, be converted into a proof ot’ NP # Co-NP? As for analytic sets in the infinite case. we have Fhe following: FACT 3.1. - For any (non-uniform) VP-set .Y. there exists a sequence l;)f circuits ((1,, : 7) < u). X:. III < (*1, such that c’,)(z) is of the i’orm fl (tl,.,,Vrl,,, VCI,,~) where I/,,.~) := 2, ,, IJ or l~;(,,.;~, ;- r,i for ,j < rtk. 5 <: :1. z C ” ^““‘.L and x E X if and only if 3y lg1.11(y) = (lgt,h (x))“’
and x*-y is accepted by ( ‘,L,,i(X).
(As in passing from :&SAT to SAT, one can consider disjunctions of size :I.) If @h,lit,, were belonging to NP, then since NP X. E ‘Ir’,, if and only if 3y lgth (y, = (Igt 11(X))“! and x”y ih accepted hv (T(,,:r,,+1,,12. Our goal is thus to adapt the previous arguments with II (or (I?( II - I !jj:! or O,“(~J -- 1,)/Z)“‘) instead of J IO show that this situation cannot hold. A natural direction is to consider a tinite statement ll;l)(-Y. j). for X C_ II )(‘r/. j E”‘. 15.corresponding to the infinite Hyp (S. jj and to check that the probability (on -Y) for the existence of j such that Hyp(k7:,j) holds, is close to 1 (and hence different t’rom 0). This would entail that Ii’,, cannot be obtained by the considered circuit. for reasons omitted here, we shall have to inrroduce a second probability on S’(rc, rr). So let us consider two uniform dislributions: - f on r(.!5’(/,, II), such that for i c S’( II. I,), -4 C h’(n.fl). P(t E A) = l/2. E .Y) :x: l/Z. - r on P(r) :< 70, such that for i. ,: < TV, .Y C ‘r/ < ‘rt, I-‘((,i.,j) Formulated in terms of conditional probabilities. we propose the following statement which corresponds to this approach (with regard to I’. II’,, and I%‘,, behave similarly: li.1: ,r and “5” have their “natural” meaning):
259
C. Sureson
References [ 1] Boppana Science, (21 [3] [1] [S]
R.. Sip5er M.. The vol. A. North-Holland.
complexit! IWO.
01’ linite functwn\. pp. 75%X()-I.
111: Van
I.eeuwen
Furst M.. Saxe J., Sipw M.. Parity. clrctut!, and rhc polyrmlial ttrne hierarchy, Moschovaki\ Y.. De\ct-iptive Set Theor!, North-Holland, 19X0. Papadimitriou C.. Computational Complexity. Addiwn-Wrslq. lYY-1. Sipser M., Borel sets and circuit complexity. in: Proc. 15th Ann. ACM Symp.
[6] Sipser M., Alfiorithms.
260
A topological Colloquia
view of some pr~~blem\ in complexity Mathematlca Societatls Jnnos Boly~. vol.
J. (Ed.). Math.
Handbook Syhtrm\
on theory
theory. in: Lovaw L.. 11 IYXJ. pp. 3X7--301.
ot Theoretical Theory
Computes
I7 ( 1984)
of computing.
1983,
S/emewtli
I!. (Eds.).
13-27
pp. (,I-W. Theor)
01
Sci. Paris, t. 326, thCorique/Computer
Descriptive
SCrie
I, p. 255260, Science
set theory
1998
and Boolean
complexity
theory
Thtorie tlesct-iptiw rl(js ~tt.wttrltl~~s et Ih6orie de 10 cottrplexiti ttool&tttw
Version
frangaise
La thkorie (c’est-i-dire
descriptive
dcs
I’cnxemble
dcs
h-dire
I’ensemble
traite
de parties
dcs de
de circuits
bool&:na
infinis
alors
ont
ainsi
des
Dans par
ensemble\ suites
suites
des
infinis
rCsultals
le\
tie
II CI de
suitL*\
linies
i IY~;/.
d’idkcs.
Sipscv
de Bowl)
d’a~semhles
dCfinissahle~
le nombre de -- d~nombrable
disjonctions
et coii,jonct
-
polynomial
Note
pr6sentPe
dank dans
par
Ir Ic
Maurice
cas
us
dc
[ .‘I.
notre
que
(I et de
dc ‘fh&wie
I.
cas
m
type
de I’cspace
I’espace
Ia ThCorie
de
Les dc
preuve
de Baire
de Cantor
la complcxitC
II est possible
descriptive.
fink
cntrc
d’origine
“‘J!
(c’est-
boolCenne
de reformuler
analogies
-2
en termes Its
infink
cas
finis
ct
et d’obtenit-
If;])
10 1 propose
et enxcmblcs
projection
ou dan\
1 ). alor\
It I;I Gluation
Ggnifcatif‘\
projection\
so~~~-cnse~~~bles dCfinlssablr:s
Ic<
d’entiers),
cl66niti<~n~
d’appliquer
ordrc
tkdic inflnlc\
infinies
I’enaemble
permit
Ie mkne
(c‘est-h-dire
abrt!&e
par Ion5
LIIK corrcspondance
(non-llllif’~)rlrlc)h’P dck
at
circuit\
enw : dans
de profondeur\
enscrnhlcs
les dcux 2
L!t dc
cas.
analytiques ils sent
13 t’iillne
/y(W)
obtenus Oil
:
infini, lini.
NI\ \I’.
255
C. Sureson Sa motivation &it de cc deduirc 1) une d&nonstration de NP # Co-NP d’une preuve de I’existence d’un analytique non co-analytique (r&&at clasaique). Le mouvement infini vers fini (tie [ 21. [S]) procCdait en quatre &apes : - la donnCe de deux versions correspondantes tlinie et infinie) d’un m@me probltime. - I’Claboration d’une preuve cornbinatoire danh le cas inhni. - uric reformulation (finie et) probabiliste de\ GEmenta cl&s de la preuvc, - et finalement, une dGmonstration de ces assertion\ probabilistea. Sipser [6] pr&enta une preuve combinatoire de la non-analyticit de I’ensemble co-analytique WF des arbres bien-fond&, mais WJT n’a pas d’equivalent ~_ tini. Nous conservons son approche en proposant ici de nouveaux ensembles co-analytiques II- et II’ (tnspires de WF) qui. en revanche, PO&dent un equivalent fini Co-NP complex. Nous produisons cnsuite uric tlclnonstration combinatoire de la non-analyticit<; I& Ij7, ainsi qu’utle reformulation probabiliste de celle-l~i qui induirait NP # Co-NP. IndCpendamment d’une conclusion pcut-etrc inaccessible. ce type de dCtnarche nous semble intCressant en soi.
0. Introduction Descriptive Set Theory deals with definable subsets of the Baire space “‘u (i.e. infinite sequences of integers), or in our case. subsets of the Cantor space “2 (i.e. infinite sequences of O’s and I’\), while Complexity Theory focuses on subsets of “2. for I/ < ti (i.e. finite sequences of O’s and l’s). Since one can reformulate Descriptive set theoretical definitions in terms 01‘ infinite Boolean circuits, analogies between the tinile and infinite cases have led Furst. Saxe. Sipxer [2] and Sipser 1.51 to interesting results concerning xct\ definable by bounded depth circuits of polynomial size (these sets arc the analogs of finitc ranh Bore1 \ets). Sets recog,nizable by circuits i)f polynomial _ si/c’ (with no consideration to cdepth) correspond to bore1 set\ in the infinite case. Sipser in [Cl. aiming at the prohltm iXP f Co-NP. proposed a similar link betueen analytic sets (i.e. projections of Bore1 sets) and (non-uniform) NJ’ \ets: both are projections (meant by the symbol 3) of sets definable by circuits of depth 2. and oi‘ the form A(W) where the number of disjunctions and conjunctions is: - countable in the infinite case. - polynomial in the finite enc. He then presented a combinatorial proof of the fact that a classical co-analytic set (the set %‘I~ of well-founded trees) is not analytic. lisual proofs were by diagonalization and non-combinatorial, In the previous moves (WC 121. ISI) from the infinite to the finite, the mecanisrn of the proof was always the same: some effective construction in the infinite case was replaced. in the finite enc. hy a probabilistic argument. There were four requirements: I) two corresponding versions (tinitc and infinite) of ;I given problem. 2) a combinatorial proof in the infinite case. 3) a (finite) probabilistic restatement of this proof. 3) and finally a demonstration of’ this probabilistic statement. The classical set WI? has no finite equivalent. hence one cannot proceed along these lines. But we keep Sipser‘s approach by proposing new co-analytic sets It’ and I? (inspired from WJ:) whic,h have a co-NP complete finite equivalent.
256
Descriptive
set theory
and
Boolean
complexity
theory
We then produce a combinatorial proof of the non-analyticity 01‘ I?. and show how a probabilistic restatement of the proof would entail NP # Co-NP. The result NP # Co-NP may be out of reach. bul this kind of proces. 0 seem5 interesting for its own sake. 1. The sets W, fi’
C -2 and their
finite
analogs
Let us first recall some notation and defnitions. originating from Set theory: - -‘I’ denotes the set of functions from *I- into J - ul is the set of integers N, - an ordinal (finite or not) is the sc%tof’ its predecessor\. - and I,Yl is the cardinality of .Y (i.e. #-\’ if .\- is finite). Hence &2 and “2 denote respectively the set of intinite sequence5 of O‘s and sequences of length II of O’s and I ‘\. We consider the product topology on “(*: (or on -.*? x Aa 1) The collection of of the closure under complementation and countable union of the open sets. properties. the analytic sets (which in- A>are projections of bore1 sets inaLC* X~ZI and the co-analytic
I ‘\. and the set of Bore1 sets consists Due to definability ale called 2: i-sets,
sets III-sets.
In order to detine 11’ Cy-‘2, we uill modify slightly the notion of tree: - Usually for a tree 7’ on w. on SEI~X:for ./ E -“L*:. 1 is an (infinite) branch of ‘I if and only it for any I) < W, .fi,, E 7’. WF is then defined as the set 01 well-founded lrces (i.e. without any (infinite1 branch), and is classically known to be 11: -complete. - For ;I set .A (whose styucture ha\ still to be defined). we will set f‘or .f E lk). J IS a branch of .4 if and only if for any 13 C ti such tltat / 131 = 2. 1’ !T E ‘1. l/I’ will be the collection of vets , I without any branch. Formally we set: - .S(rv./j) = (f : t function. domil‘ 5 0. ldotn(l)l = 2. range(t! C /j): - If A C: S( 0. /I), then we consider the set of branches of length o of A : :A],, = (,f t”:) : VI3 C o. IL31 = 2 implies .flu E :I}. We then consider some “natural” bi,jection h : U{ 5( II. /I) : II <: ti’) -- J. such that the image b), b of S(7),. II.) is r?(~/ - I)/‘2 (IS(r).. fj)I = ttZil I/ ~ I ),:“L). Any .A i S( LL’,~:) can thus be coded by a unique x4 E -2 such that, for i < Ld. x..~ (i) := I it’ and only if I/-‘( ;) E .4. In the same way. il 2 S’(I).. II.) is coded by x.1 E ’ ” “e ’ ‘l”, We consi,jer: DEFINHWN 1.2. 7t. < w. let VI;,, = ~~~fin,t,. := u { w,, : If we restrict the
Let IV = {x E “‘2 x codes a set ..\ C ,S’(u;. LJ) so that ,. I],. = E}. F~I {x E (71‘lJ,-1))/-‘2 : x codes a sct ..I <; ,S’(lr.s/) such that [.,lj,, = E} and let ‘// < w } . range of possible branches as in the li)llowing diagram w
ki--
0
m
m’
w
then we can define:
257
C. Sureson
DEHNITION 1.3. - For X, < LJ, IZ (A-) = (x E -2 : x codes .A C S(ti. A.) and [-lli = @}. For 1) < ~j. \I’,,(,(:) = {x E (‘J’ciim1r)/2’J x codes A C S(rr,. A:) and [A],, = a} q
Wfi*,i(,,(k)
=
U{ l’l,,(kI)
1 TI <
md
Cd).
Throughout this paper, we will denote by 1.Y the complement of .Y. and \ve will often identify a set .4 and its code x-i (in the finite cast. II must be specified). Concerning analogies between the finite and intinite cases, we obtain:
(b) ] I/l;llri~<,T.Y/t’f-<~ot)ll>l~tr (/FI /irt’r 1 Ir’fi,,i,,.I,:i I is CI/WO~~ Nf-(~<)l~?l>/et<~). For technical reasons, we consider now a slight modification 6. of II,-: insttad of considering branches on the whole interval [(I. ti.~). we will admit branches on final segments 1X,,A), for k < ul. DE~;INITION 1.7. - For .4 2 ,S’(u-.. .,,lj. let [A] = (.f’ : ,J function, 3d* < r*l domain (j’) = [X. w’I. range (f’) C LL).and V.0 g [A..k.1). ;I11 = 2 implies .f’ 1j E -.I}. - Let r/T- =- {.I. E -2 : .I’ codes ;I set ‘1 5 S’ljiL!.;) such that [.+I] = :3}. For the finite equivalent, we propose the following:
DEHNITIOK1.X. - Let 0 < CY< I he fixed. For 11< ti, it’ ..I C ,S(II. lt,j. let: [A]‘“’ = {,/ : .f function. range ~,f) 1; II. 3.Y il 11domain(J) = .Y, S/ := II -~ 0” and Vlu c .I, 1~3 = 2 implies ./‘,I E A). G,J,,= {x (5 (~1‘(11~1III/22 : x codes a \et .I c ,s’i/I. ‘/I11such that [A]“’ ’ = ; .j}, IZ’filli(<~= ‘J{ 67,i : ‘r/ <: LCI}. (Rigorously. we should write IV,,I /I ).) One can deduce. from the following th;u 1G- is xi-(Bore]) *.a complete, and 1Il.ti,lic~, is NP-complete:
2. A combinatorial
proof of li. E- [I:\L~ c. -.
We shall use the characterization of !-,” sets in terms of circuits: -” FACT 2. I. -- For any 2:i-set S .~. ( 1“2. there exists a circuit C’(x, y) of the torm A II,,, where I#<-, Il,,jx.y) = W (I,,,, ,(x.y) with r/,,.l,!x.y) = .I’,(,, , ,. l:~~~,,~.,,~. !I,(,,,,,). or l,~,,,~ I’)’ ml for .I’ Ew2. ,II’& x E .Y if and only if‘ 3y cy-“:! (x. y) is accepted hy C’ To show non-analyticity. let us argue by contradiction: we supposethat i,i’ is ?:I, and hence thal there is a circuit (’ as above sati
x iI li- if and only if Sy t&Z (x. y) is accepted by (’
We want to find y and A C S(d.,;) 258
with a branch such that (x..~. y) is accepted hy (‘.
Descriptive
As in [ci]. we consider.
for s t “w. with
set theory
and
Boolean
complexity
theory
k < w’, the set
II, = {x : 3y ~“2 (x. y j is accepted by 6: and V’r <: k (x, y) satisfies the s(i)-th
element of L),. that is ~l,.s,,I}.
But then, if it is clear how to code countably many well-founded trees rn one well-founded tree (one adds a root below all trees). it does not work for our notion of tree. Hence let us introduce some new notions: DEFINITION 2.2. - We often identify ;I set ‘-1 and its code x L. - If S C d x LJ. then let l‘Z;( ’ ) = (.-I E Ii’ : V/ E .4 graph (1) 2 S}. - Let E!, B’ 2 ‘Pi S(ti. d)). By D .’ ;. ” /I’, we mean that, for any A t 1). there exists -4 E LI’ such that :1 c 4’. We then construct inductively a set .Y !_ d x uI ,md .j E iti such that the following twofold hypothesis holds: Hyp (1’. j) : (a) V’k <: ti li’i”l “ C .’ Ij,i,, (b) ‘4; < LU’?I’,,~ = {; ,< J : (,;.,j) E .\‘} is nonempty. Finally, one can prove that Hyp (-‘i ~j J prevents I‘i- from being associated with the circuit 6”. More precisely, Hyp(x.j) allows the construction of y E”‘:! and of .4 C Y(d.uj such that. by (a). (2.y) is accepted by (‘, and by (b). A admits a branch. 3. A possible
application
to NP :f Co-NP?
A natural question is thus: can this proot’ of C I f I 1, be converted into a proof ot’ NP # Co-NP? As for analytic sets in the infinite case. we have Fhe following: FACT 3.1. - For any (non-uniform) VP-set .Y. there exists a sequence l;)f circuits ((1,, : 7) < u). X:. III < (*1, such that c’,)(z) is of the i’orm fl (tl,.,,Vrl,,, VCI,,~) where I/,,.~) := 2, ,, IJ or l~;(,,.;~, ;- r,i for ,j < rtk. 5 <: :1. z C ” ^““‘.L and x E X if and only if 3y lg1.11(y) = (lgt,h (x))“’
and x*-y is accepted by ( ‘,L,,i(X).
(As in passing from :&SAT to SAT, one can consider disjunctions of size :I.) If @h,lit,, were belonging to NP, then since NP X. E ‘Ir’,, if and only if 3y lgth (y, = (Igt 11(X))“! and x”y ih accepted hv (T(,,:r,,+1,,12. Our goal is thus to adapt the previous arguments with II (or (I?( II - I !jj:! or O,“(~J -- 1,)/Z)“‘) instead of J IO show that this situation cannot hold. A natural direction is to consider a tinite statement ll;l)(-Y. j). for X C_ II )(‘r/. j E”‘. 15.corresponding to the infinite Hyp (S. jj and to check that the probability (on -Y) for the existence of j such that Hyp(k7:,j) holds, is close to 1 (and hence different t’rom 0). This would entail that Ii’,, cannot be obtained by the considered circuit. for reasons omitted here, we shall have to inrroduce a second probability on S’(rc, rr). So let us consider two uniform dislributions: - f on r(.!5’(/,, II), such that for i c S’( II. I,), -4 C h’(n.fl). P(t E A) = l/2. E .Y) :x: l/Z. - r on P(r) :< 70, such that for i. ,: < TV, .Y C ‘r/ < ‘rt, I-‘((,i.,j) Formulated in terms of conditional probabilities. we propose the following statement which corresponds to this approach (with regard to I’. II’,, and I%‘,, behave similarly: li.1: ,r and “5” have their “natural” meaning):
259
C. Sureson
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01