PartC-Chapter5-4~5

7/28/2019 PartC-Chapter5-4~5

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32 0 V. Boundecl Arithmetic

1. Witnessingnnctions(a) XntrocluctionSupposea sou.ndbheory 7 proves a senLence Vr)(ly)p(*,y). If ? is weakand tp is simple, then we have some ailditional information on the functionswhich u'itness the existential quanbifier in Lhis sentence, .e . functions / fo rr,vhichwe have (Yx)tp(r,/(")). A classical result says that if ? is ID1 ancLis botrnded, then there is such an / which is primitiae recursiue) seeT'heorem-IV.3.6. Parikh fParikh 71] has notecl hat / is l inear time computable if ? is/Xs and ga s bounded. Buss [Buss 86 , Bounded Ariih.] has definecl a theorySi2 (a conservative extension of IDo + J?1), ragments S| ancl subsets D! otbounded formulae in the extended language. His theorems characterize theclassesof functions in the polynomial hierarchy using the above phenomenon.In parbicular, if p is a Xf formLrla ancl f is S| , then / is polynomial timecomputable. The converseof this special case s also true: if / is polynomial-time computable, then there exists " El fbrmula which clefines / and forwhich it is provable in 5] that / is total.We start this secbionwith the defi.nitionsof these classesof forrnulae anclfragrnents of bounded arithmetic. The next step will be positive results aboutdefinability of functions of certain complexity in these tragments. It turns outthat this is qr-rite mportant inforrnation about the fragments. For instance,knowing ftar T) defines unctions of trf*r, for i > 1, it is easy to prove Buss'theorem for Tj. The main part is sr-rbsectiond) where witnessing theorems ofBuss and of Krajfcek and Taheuti are proved. In subsection (e) we shall showa reduction of the problem of finite axiomatizability of bounded aribhmeticbo the problem of collapsing the polynomial time hierarchy.

Since he fragments of bounded aribhmeLichave been exbensivelystudied in[Buss 86 , Bounded Arith.], we concentrate on results and proofs not includeciin that booh. In particular, as Buss uses proof theory, we shall give proofsof the witnessing theorems based on model theory. We shali refer to Buss'sbook for basic facts provable in these fragmentsl holvever the reaclers withsome training in bounded arithmetic, for example those who have worlcedthrough Sect. 3, shoulcl be able to mal


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4. lVitnessingFunctions 32 1l t l - r1og2(r+ 1.)t: the lengthof the binary expansion f

r if.n ) 0, other:wise ,i .e. the sameas in Sect.3. l lere ryr denotes he least ntegerz ) ,y ; u,yLdenotes he integer parb of y, bhus he inberprel,ation >f.rfL-t is clear. Theinbencleclnberprebation f the smash unc-tion s

x #y: 2ltl*lvl : the po'wer f 2 whosebinary expansion aslengthl"l * lyl r t .Observe bhat the seLof the lengths of nurnJrersn a sound arithmeticzr,l heoryconbaining the smash function is closecl nclerrnr-Llbiplicabion,ence uncler anypolynomial increase (for standard polynomials). This is important, since itenablesus to formaiize many standard constructions, in particular polynomialtime computations. We shall denote this extension of Lo by L2. In th.is sectionwe shal.lwork in the language L2 and in an extension L2 of .L2.Now we recali the basic system of open axioms for the extendecl anguage.It is callecl simply BAS.IC and it plays a similar role Lo Q in bhe usuallanguage.4.1"Definition. BASIC is the following theory:( i ) v 1 n - - + y l S q(2) r I Sr;( 3 )0 ( " ;( 4 )n l a k x / y : S r S u ;(5) " -10 ,2 r l6 i( 6 ) y l r V r l y ;( 7 ) r l U k y < a - - + r - U i( B ) l U k y < z - - - + x 1 z ;(e) l0 l 0;(10) *6 - l2r l ,9( l " l ) 1.9(2r) l5 ' ( l r l ) ;(11)Tl r ;( 1 2 ) 1 a - , l " l S l v l ;( t 3 ) r # y l : 5 ' ( l r l * l y l ) ;( 1 4 ) # y : T ;(15) l6 -, t l f (2rc)2(r # ") k't l l (s(2n)) t(1 # ");( 1 6 ) r l f a : y # r ;(17) rl : lyl , x: z : u # z,( 1 8 )r l - l r l * l u l - + = t ' f y - ( u " = f f a ) , r ( u f t ' y ) ;( 1 9 ) z r * y i(20) < y kx * a -- S'(2r)


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32 2 V. Bounded Arithmetic(2,1)r+ v) + z : n 1- y -r z);( 2 5 ) r l - y < r + z = y 1 z ;( 2 6 ) x * 0 - 0 ;(2,7) r,(5y) - :ty -l n;(28)ay - y'"c;(29) rc(y+ z) : ry -l- cz;(30 ) r ) 1 - - r rA < xz : y S z) ;(3r.) #6 -, l" l : ,9(1,-nl2Ll) ;(32)a : ex12'= (2a r V S(2y) r),where T and 2 arc numerals as clefined n Sect. 3.

This system is probably no t cluite optimal. One can play with it in order:l,o ind a shorber or nicer system. For instance, if we redefine

l t l : ' rog2(r + 1)- . 1 i f r > o,0 otherwise,then r # y : 2l*l*lul becomes associaLiveancl distributive with respect tomultiplication, hence one would obtain a more homogeneous axiomatic sys-tem for such functions. We shall not do it here, as we want to concentrate onmore interesting questions.

We shall use the bounded quantifiers (Vr ( r) and (1 , < z); note thatr is a term in the language L2, nol just in Lg (not containing r, of course)'lVloreoverwe also need the so-called sharply bounded guantifiers. T'hey havethe form (Vr< lr l ) and lr < lr l ) ,i.e. the ourtermostunction in the bou.ndingerm is the length function. Theirrelabion o ordinary boundedqttantifiers n bounded arithmetic is similar tothe relation of bounded quantifi.ers o unbounded cluantifiers n Pzl. TheclassesE! and, A! of bouncled ormulae are defined as follows.4.2 Deffnition.(1) n3: Xf, consistof formulaewith sharplybounded quantifi.ers nly;(2) Xl,, ancl I!-., are the leasi seis satisfyingt t L [ - r L(") z!,n! e D!+t, nclut,n! rI!4;(b) a e E!-4+ (-u 1r)a e E!+r, Vz l"l) ' e E!+r,a e II!*1=+ Vr 1r)a e n!+r,(lr S l"l)" e II!-4;( . ) a ,g Elnr4 akg, av g e E! - r - t ,a,0 n!.- ,+ ak g,aV Be n!*1;( . t ) ae Dln1, e n!*r4-g, g -, * x, l-rr ,a e II!*r, g e D!-rr=*-,9,9 -r t Qn!-rr.


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(3 )(4\

4. WitnessingF\rncLions 323cv s in zli with respect o a l,heoryT,i:t cv s eqr-rivalenLo a D! formulaancl a l/rb formula in ?.D|@i) is definedas the closutre f E! uncler conn.ecbivesirclsharplybouncledquantification; nobe h.at this class s som.ebimesenobedalsoh., rtb r-t n'b, ' l " i - F . \ t t r ' r i _ l _ 1 .

L,ateron lve shall see.hat, fo r d ) 1, D! (resp. I/ob) ormulae clefineustLlresets n E! (resp. n).'th.e most important formalizationsof incLuctionfor bouncled formulae are the usual schemaof inclLrctionancl the foll.owineones.4.3 Definition.(1) PIND(a(x)) is the formula

ct(0)& (Vz)(a('-rlz-') , *(")). -' (Vr)a(z) ;(2) LIND(*(z)) is bheormr-rla

a(0)& (Vy< l"lX*(y)-, a(,S(y))). *(l" l) ;(3) LMIN (a(r)) is the formula

( l r )a ( r ) - * a (0 ) v ( t r ) (a ( r )sz(Vy < n l2 ) ( -a (z) ) ) ;(4) PIND? (LIND|, LMINT, respectively) s the schema (or the set)PIND(*(r)) (LIND(a(x)), LMIN(*(z)), respectively) or a(x) e- '.

Here we assume hat a(r) may contain other free variables, which arecalledparamebers.We have written LIND in the form lvhich contairrsonlyone additional quantifier and this cluanbifiers sharply bounded. Thus if a isA!-, th.n LIND(*(r)) is also ab;; t a(z) is in n! u i!, t]nen ,IND(a(z)) isEl,Q!)iVowwe are ready to define the hierarchyof subtheoriesof bouncled arith-metic.4.4 Definition.(1) For z> 0, St is BASIC+ PINDE?.(2) For > 0,r; is BASIC+ I D!,(3) 52 s ; Sig; z s l);T$.

The most importanL fragmenL of .92 is .9;1. L serves as a basic theoryrnoclr-Llo hich lesults on tlagrnents of 52 are provecl. Ibs role is similar to


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32,1 V. BouncledAribhmeticthe role of. E1 in .PA. n parLicularwe shall show in the next subsecLionthtr,tal l polynomial time compu.tabletincbionsare clefinableand provab.lytota,l n 5} . As fbr weaker ragments: 9! is booweak (it doesnob prove theexistence f the predecessor,ee Taheuti,Sharp]yBouncled]); he s trength of- n .Ti is noL quite l


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4. lVitnessing unctions J2bwhere tr is the code of the secluence, is its murimal element and ro is thelengtlr, f th,esequence, .e.

ma.r (p) u,lh(p) - u)

The fun.ction he i-tlr, element, f p is definecl y(p ) ; - n : r />uk(Vr S lu l ) (b i t@,) -b i , t ( t r , , iu t - j ) ) , i f i


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32 6 V. Bounded Alithrnetic

LIND(tp(r rsp( t jy) ) ) s ince nsp(r 0) : 0 , tmsp(r , S ' (y) )l2 ' - rnsp(r , y) andm s2 t ( r , l c l ) z . n4.6 Theorern.For i ) I, Sif tAbo.ProoJ'.Recaj.lbhemeaningof .tA!:'f.or every a E! and,B e n!

(Vr ) (a ( r ) = 0@)) - ' / (a ( r ) ) .The icleaof ihe proof is the following.The schema f induction telis us that wecannot definea proper subset f numberswhich conLains and s closedunclerthe successor. he schemaPIND implies that there s no subset of numberscontaining 0 anclclosed -Lnderhe successor nd the nnction 2u. We alreadyknow (Theorem II.3.5) thai for every cut we can definea subcut closedunderaddition, hencealsoclosed nder 2r.We shall use his construction o reduceinduction to PIND. There s a small technical complication,sincewe have ouse only bounded formulae,while the original consbruction sesunbouncleclquanLifiers.Let o and B be given. Put

,,h(a,o) af (Y* < a)(a(r) + B(min(n -l-Utc,)))Thus 1., s II!. Working n ,9$,assume (*) : 0@),,a(0) and (Vr)(c(z) -ra(S'(r))). Furthermoreassume hat a(a) fails for somea. We haveT/(0,a)tri.rially and r!(y,a) * ,h6@),4), since (Vz)(c(r) --+ a(,9(r))). We shallprove h@,n) r rh(2y,cl) . ssumeth(A,a)nd a(z) for some l a.By rh(a,o)we havea(rn in(n: y,a)) . Thus r *U ( a, since c(a). Flence (x *a,a).Applying ,h@,") onceagain with rf y insteadof :v r,ve et a(min(n*2y,a)).Hencewe have b(A,a) -->lt(2y,4).Since

the formula

implies

:c 2tr12: V r : S(Zrxl2L),

(Vy)(dfu,o) ,h@@),a)c{(2y,a))(Va)?h( f 2-', ) -> t[(y,a))

Hence singPIND(/) *. get t(a,a). Since (0), this mplies (n). We havederiveda(a) from its negation, ence t mtLstbe true. Thuswe haveshowninduction or a. nNote that, for i 2 2, a weakerversion of this theorem can be derivedalso rom Corollary 4.28belowusing he If*, conservabivibyf S'L1-1u"t T' i(Corollary 4.34).


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4.7 Theorem.f t ) F b r i > 0 . , 9 l c T l C\ /(2) Sz = T2 and they are

4. lVitnessing \rnctions

c, i+1.'J2 ,conservative ;rtensions f IEo * At.

2 r '7

.ProoJ. (1 ) Tb prove th.e first inclusion, observe th.at T; trivialllz provesLIND E! and apply 'Iheorem 4,.5 1). The seconcl nclusion is a corollary ofTheorem 4.6.'(2) I'he equivalence oL heories,S'2andl'2 follows:[rom (i) of the precedingproposition. In order to show that /Xg * fit is contained tn ,52we have only toprove axiom A1 in,52, which .followseasily from the boLrndu2(r) < (* =|t'da .Let us prove the conser:vabivityof.,92over -ftr'g At . We already know that;nf2L and lr l are definable in l'Eo+ J71.As we have a X9 definition of theexponenbiabionrelation, we car] define the graph of the function x {f y. Thisfuncbion s provably total in /Xs iQt, becauseof the bound r # y < ut2@*U).The axioms of BAS.IC are easy too. To prove indtrctior-r. or bounded formulaein language I'2in IEo l-.f21, recall that by Proposition 1.3 we can eliminateterms as bounds in the formulae used n incluction. Then we use the definiLionsof the new operations to translate the fbrrrruJae into the language .tg andapply /X6.4.8 Theorem. (a) For i ) 1, and a(u,r) e E|pb,

tr

S:$F-1ru)(V" lyl)("(",n) : bit(w, + T)- T)(b)For ) L,sir uwD(r8(rl)).

Note that (a) is a version of the :;$(l.f) "otnptehension schema in Siwhere $/e use bit(tu,r) as a convenient coding of 0-1 sequences.ProoJ. a) Consicler a more general schema for a D\@b:( b . 1 ) 5i F ( l t u ) ( V r t l y t l ) . . . ( V " ,< l y " l ) ( * ( " , x r t . . . , r n ): b i t ( u , , T ( " r , ( * 2 , . . , " , r ) ) ) :'We should shour that (b.1) holcls for a e D! ancl thal, tlle set of formulaea satisfying (b.1) is closed under boolean operations and sharply boundeclquanl,ificabions.

Suppose a Eo;.Take the formirla B(z,z) defined by( l tu)( l . r l lvr l lyz l , ,. . , r .y , lk nu,on(zu)t& (v r1< lv t l ) . v ' " i l ' v , ,1 ) (b i t (zu ,Tr (c r , ( *2 , . , r " ) ) )- , a ( u , , x r t .

T)


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328 V. Boundecl AritlimeticJ,etns lvork n 5i. T'his ormula s true for z :0 and fails br z : lyrl 'r lyzlr.. ' . ' , , lunl+ 1, sincemton(ro) l r . r . '1 .incea is E!, n lron s,d! anclwe canlrotrndu, say by yt#yz=t'|=. #yn, Ihe formr-rla is also X;o.Hencewe canwe LINDII! to fincl he smallesb for rvhichB(z,u) fails. Then take r.u r-r.cht,hat, (z - 1,tu). f z - t : la t l lyz l . , rlyn l ,Lhen all biLs of ?, are 1, hence we h.a.re b.1) lbr this tr.'.Now suppose that

z - I < lyt l lazl " ' , r lynl .I f (b.1) were not true for this tu, then.we could add another 1 in t he binaryrepresentat ion f u ; which codesan instanceof a(u,r1 t12, . . . , f in ) . But th isvrould contradict the condition that z - 1 is the last parameter for which.0Q - 1,T.o)s true. Flence o codesexactly the set determined by a.

The case of propositional connectives and sharply bor-rndedcluantifiers iseasy.(b ) follows immediately from (a). nIn BotrndectArithmetic there is a counterpart of the collection ariom for

Peano Arithmetic, see L2.I. We shall cal l i t the bounded col lection a.riom.Sometimes t is also cal led replacementariom.4.9 Definit ion. (i) BB(a(r,A)) boundedcoLlections the fol lowing formula

(V" S l t lX l y ( s )a (c , ) . t ( l u . ' ) (Vz l t l ) ( ( . ) * ( ska (n , ( . ) , ) ) .Gl BB f is the schema B(a) for a e T.

'Using (a.1) above, we can bound the variable tu in the bounded collectionaxiom; thus it has the following form (ecluivalent in 5]):

(Vr< lr lxfy s)a(r, ) .- ' ( 1 , ( r ( s , t ) ) (V r< l t l ) ( ( r ) , ( ska (x , ( - ) . , ) ) ,where r is a su i tab le erm (r(s , t ) : bound ' (s ,Zl t l+ t ; ; .4.10Tireorern.or > I, S'$V Afi .Proof, et a(r,y) be a trf forrnula. ssume

(V , < l t lX=y s)a( r ,y ) .Then we obLain

/ a| - t 4 I l ( r ( s , t ) ) ( V r l i lX ( . ) , ( sk a ( r , ( - ) , ) )


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by4. Wibnessing unctions 32 9

L . fND((1zu r (s , t ) ) (V r < lz l ) ( ( tu ) . , " k a ( r , ( . r ) , ) ) ) ,where the induction variable is z.r-rsehe elementary facts that theca,nbe ex1;encleclv a,n a,rbiLrarvsequence.

The forrnula s Xf , sincecocling s Al. Weempty seqllence has a code, each.secltrenceelem.ent and the estimaLe of the size of:r

LJ

Note thab the implication converse o that above is already provable in 5l(using only the fact that (u), is a defintr,ble unction in 5l). T'he bor:rndellforrn of the bouncled colleciion axiom enables u.s to intercb.ange sh"arplybouncled quantifiers with bounded cluantifiers. However we cannot push insicleall sharply bouncl.eclcluantifiers withor-r.t exbencling the langr-r.age, ince thecoding function is not clefinecl y a term in .L2. Therefore we need to extendthe language.4.1-t Definition. (i ) The language -t2 augmented with the.binary operation(r)r, wi l l be clenoteclby L2; the corresponding classesX!,n! eic. wi l l bedenoteclbv E!,II!, etc.

(ii) We say that a formula is strict\f ii it is of the fbrm( l r i < t1 ) (Va2< tz ) . . . (Qr ; 1 . t ; ) e ,

where Q is 3 if i is odd and V if i is even, and rp is s]rzr,rplybouncled.

We shall not introcluce new notation for theories ob'bained by extenclingthe language and aclding the definition of the coding relaiion. It will alwaysbe clear from the context which theory we mean.4.12 Corollary. For iformula, provably in S$ (augmenteclwith the clefinition of (r)").Proof. We can shi.ft all sharply boundecl qua,ntifiers beyond bounded (br-ttnob sharply) quantifiels using the bounded collection schema. Successiyecy-tanLifiers f the same lcind can be merged ttsing the coding function. tl4.13 Corol lary. For i ) 1 ancl *(r,y) in E!,

Si F (3 tu ) (V"< l t l ) t ( :y ( s )a ( r ,y ) : ( ( t r ) . , s ka ( r , ( . r ) . , ) ) l .(Tlris s sometimes alleds'trong eplacement.)Proo!.Let a(r,y) it f! be given.By T'heorem .8,we cancocle he seb f r'ssuch hab ( ly S s)a(r,y) using ome,uand fbrmula bi t(u,o): 1. Now take* ' ( * ,3 / ) to be t ( r , y ) V b i t ( u , n ) : 0 .


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330 V. Bouncled l i thmeticThen we have (V" S lt l)(:y ( s)a/(r, y), hencewe can apply bounded col-lection; the sequence{., ivenby iL sabisfieshe clauseof the theorem. n

(c) Definabilityof T'tuingMlachineCornputationsi.n [-'ragrnentr;sf Bounalecl rithmeticRecall ihat for i) I, l f clenol,es heset of functions compuLable n polyno-m.ial ime using oracles or EeO-.rsebs;n palticular, trf are ust the polynomizrltime computable funcLions, since Xfi oracles are sLrperfluous. n this sr-tbsec-tion we shall shor,vhat functions from n{ have suitable clefinibions n ,9}, an.i[the same holcls for nf a,nd. j,-I ,, :f. > t. ffris means that for functions fromJ! we haxe A!*., definitions in the corresponding theories and that basicproperties of the functions are provable in these th.eories. Ve shall only showthat the defining formulae have appropriate complexity and that they de-fine total ftinctions. Other properLies ollow from the construction and theirproofs are omitted.

We shall consicler delerminisbic Turing machines, possibly wiih oracles.Le t an oracle Turing machine fuI be given, lei e be the code of it, let A bean oracle. We formalize the computabion of NI on input a as two secluencesu and g, where ('u); is the instantaneous description of the state of. VI, theposition of the heads and the content of the tapes, and (q), is the queryasked in step i.1 The computation is determined by a constant f,rstq (let usset i t eqnal to 0) and five functions fi ,rstw, nertwg, nentw1,, ertqg, nettql asfbllows:

(tu)s : f,rstu(ct), (q)o : f,rstq;( r ) ; - r r - neutws(( - ) t , , (q) ; ) ,q) ; - r r ner tqs( ( . ) , ; , (q) ; ) ,f (q) ;4 A;( . r ) ;+ r - ne i l tw r ( . )1 , (q ) ; ) , ( q ) l + r - nen tq l (@) ; , (q ) ; ) , i f ( q ) ; A .

We can think of (r.o);as a sequencewhich encodes he current situation on thetapes and encodes the state of IVI. Then the functions above are only localtransformations of these sequenceslhence iL is not diffi.cult to write downa! aennitions for them using o.rr Al clefi.nitionof coding. If the oracle A isclefinedby a formula g(r), we thus get a formula

CornPIuI ,p61(ru Q, cr')explessing Lhat (tu, q) is a computa't ion of IuI on inpttt ct,.'It UI is a machinewithoLrt an oracle, .weshall wriie simply Compp1(ro,c) . We shall assume hat

1 !V.l.o.g. lve can assulrre hat M asks a query in each compuLation sbep; if it does no tneed information it asks some defaulb lixed qr-rery.


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4. lVitnessing runctions 33 1(ru); : (.r);-rr i:tr AI stops n the i-th sLep.Thus computaLionso.FengLh nexist for everly , anclwe havea simpleconclition or testing the termination.FnrLhermore ve shall assume hat th"is u); 'is tlte ou,tputof I\tI. We shallidentify oracleswith bheir defining formulael thus we shzr.il.erlk abouL XforacLesnsteacl .tEl oracles4.14Lemma.(i) Comp1r7(-u, t) s At', n S) .( i i) I f 9@) is D! , i )- I, then Com?M,e@)(ru, ,a) is ,8@!).(iii) There e:


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332 V. Boundecl r i thmetic(ii) Let ,K be in P. L'et fuI be a Turing machinep(lol) stepson inptLta anclor:bpubs iff a belongsx b v ( f tu)( /h(to)S 'p( lo l)& Comyt4(ru, t ) z .r)p ln l) 0) ,

or br (Vtu)(/h(tr) p( lo l)& Comp (u,,ct) -n (r)p1"1) 0) .By l,emm.a .7rL,he inner parts of the formulaeare A!; by the same etnmabhebound lh(u) S p(lol) impliesbhat u can be bouncled y a term in ct,hence he formtLlaeare ecluivalento El ancl /f formulae respectively, husthey are C!.

(iii) By Corollary 4.72 everyX,f furmula s equivalenbo a strict Xf formulain bhe anguageextendedby the coding elaLion; hus it can be written in thefbrm( l y r < t t ) ( V a z < t z ) . . . ( Q u ; I t ; ) c Iwhere @ s sharply bounded n the extended anguageand where Q is : if iis odd and V if f is even.Since he coding elation alsobelongs o P, formulaO definesa predicate n P. The bounds are just of the order needed or thedefinition o, tD! and the relationsa < t are n P as well. Thus, by T'heorem2.!!, each et clefinable y a Xf formtrla s in El .To prove the converse et X be in Epo. hus X is definedby

( 1 v t , l s l l p 1 ( l r l ) ) . Q v t . , l v ; lp ; ( l " l ) ) p ( t , u r , . . .u i )wlrere P is in P. By (ii) we can clefi.ne by a 4l formula. Similarly to abovewe can replace the bouncls lyi l Spi(l" l) bV bouncls of the forrn l l i ( ' t i torsti iLable errns tr,,...,f;. t 'hus we obtain a Xf formula defining -,(. tr4.L6 Theorern. (1 ) Let M be a Turing machine. Then 5l prov"s that for everyinput a and every b, there exists a unique computaLion of .&/ which has lb lsteps. Formally:

which always sbops afterbo X. T'l:en we can define

Sj F (l tu) (Compx,1(u, ) k lh(zu) l6 l)S) f Comp t(w, a, ) Comp r1(.utcr.)

k lh ( ru ) , lh (w t ) : lb l -+ J) l r '(2) Let i > I,Iet fuI be an or acle'furing machineancl eb p(u)be in Eb;.

T'lren Tj proves hat fbr every input a and every b, there exisbsa uniquecomputaLionof IrtI with oraclep(r) which has lb l steps.Formally:( i ) f ) f Qw,q ) (Comp n1, .p ( r ; ( tu ,l , cL )lh ( -u ) : lb l )( i i ) f ) | Compur,p(u)(zu,1, ) k Comptur, ,p(t : )(rotqt a)

k lh (w) , lh (zu t ) : l b l+ , , : , .u ' c :

r i )( i i )


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q q , l V. Bounded Ar i thmet ic

for /c : j - l 1, . . . , lq l - 2. T 'heexisLencef such ol anclg/ is again casi lyprovableby L.INDEl . Let d be cletermined yb i t ' ( i l , k ) : t , f o r : ! , . . . , 1 6 1- - ' I ,b i t ( u t, l b l j ) : o ,b i t ( i l , k ) b i t ( u , , h ) , . o rc : l [ , ] j - l \ , . . . , | b |

Tl ren ' 'a rehave V(u ' ,o ,b) , s inceut conespondsLo ot ,c1t,b t r l r r t


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4. Witnessing Fr-rnctions 335(1) if l(x) is a beL:rnrrd is the clefi.nitionf the corresponcling { frLnction,

then l(x) : g is ectuivalento r/(x, y) in S:);(2) nf functions are provably closed -rnder omposibion, .e. iI tp(x,.g) c\e-fines ./(x) and t!;(x,y) defineg;(x), then for somedefrnitionX(x, y) of/(s/r(") . . . , en(x)) T; (resp.Sl, l f i : 1) provesX G , y ) : ( 1 r t , . . , r r ) ( r l t t ( x , t ) & . . k r h n ( * , n ) k g ( 2 t , . . zn , y ) )

/ n \ t - D e(r/ Lr ; runccionsare closedunclerdefiniLions y casesdei;erminedby E!-,formulae (it is clear what is the corresponding ormula);(4) nf functions are closedunder Jrouncledecursion.The last conclition anbe formalized s ollows.Let /(x) ,9(z,x), [ '(y, x) befunctions n nf ; then the functionh(y,x) definedby the following ecursionmust also belong to nf :

h ( 0 , x ) / ( x ) ,h (y , * ) - m in (e (h ( 'y f2 ' ,x ) , x ) , k (y , ) ) .We recltrire hat this schemabe provable n Tj (resp. S$, it. : 1) for theclefining formulae of f ,g and h. To prove these conditions one has onlyto checlc hat the proofs in the standard model can be carried out in thefragments of Bounded Arithmetic, We omit the proofs since they are notdifiEcult and contain no essentiallynew ideas.So fzr,rwe are able to tallc about a single function form some class nf .Later we shall need formalizat ion of Turing machine computations such thatwe can talh about Tur ing machinesand oracles n the theory, t.e.we wanta formula defining the computation which has parameters also for Turingmachines and oracles. n the model theoretical anguage t meansthat we',vant to consider also nonstandard Turing machines and oracles.A simplesolution is to take a universalTuring machine,which is expliciblydefinedand thus has a formalization n Sr t by Theorem 4.77.Then the codeof an arbitrary Turing machinewill be the word which mr-rst e written on tlr.einput of the universalone n order to s imulate t. A natural universal Iuringmachine simulaLeseach machine IVI in such a way that the running time isat most polynomially longer than the running time of IuL lvlore precisely,the simulation time is boundedby a polynomial n the original rttrrning imeand the size of IvI' In particular, if . vI runs in polynomial time so does thesimulation of iVI.This was abouL Turing machines.Oraclescan be presented n a simjlartashion. Using a universaiTuring machinewe can representeverypredicateP(r) in P in the form l.Lo(e,,r,t"(x))or some ixed 4t fbrmul* po. Here e isthe code(the incle:r)of P and l" is a suitable erm determinedby the runnirrgtime of a Turing machine br P. This is still no t cluite what we need,sincethe dependence f f. on e is noL explicit. Thus we shal l make an adclitionalnatural assumpLionhab the predicate vith code e is deciclablen time n".


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336 V. Bottucled Alithmetic

Then our representation f preclicabesn P will be of the formpo(e,,21*f .

N'ote thab foL:e sl,anclarclhe exisbence f Zltl" is provable n 52t. Fbr therepresentabion f preclicales:r Xf we taveonly Loadcl. lui-r,nlifier:s.Lto l,hei:eis no need to use .wo ndices,one or the machineand one for the oracle.Weshall p.resen.tur conclusions s a theorem.4.18Theorem.There are formulae F0.,u0 Al, LLi E!, ru e l$(ff) to,:i > '1,such that(i) each preclicatewh.ich s in P, provably in 5l ) carrbe representecln S'2,

as plg(a,,zl*lur,for somenumeralE;(ii) for i ) 1 andfor everyV@) e X;b, here existsan e such hatS| V ,p(*1= t) i (e,r ,2 l*1",

(iii) for every function / which. s polynomial time cornpulable,provably in..9r1,he relation ("): ?Jcanbe represenLedn 52[byug(E,x,a,2 l*1"r, or somenumeral ;

(iv) for i >- 'I ancl or every function / which is in trf*r, provably in Ti,, therelation f (*):a canbe representedn Tit:l

u;(e,x.,y,,21'f) , for somenumerale .

Let us staLe i) more precisely.A preclicateP in P is determinedby a 0-1polynomial time cornputableunction /. For this function we have a definingior-:.,Ia V@,il which is A! and satisfi.esomebasicconditions n.921.ThusP(r) is defined by p(a,O). .Ihe precisestatement of (i) is that for sornenumber e -S) l ,p ( r ,0 ) : po (a , ,2 l *1 "The statement (iii) shouldbe und.erstoodn the sameway.

In order to simplifv notabionwe shali clefine{ " } ; ( t ) : ? l d f u ; ( e , x , y , T W f.

We should keep n mind thai iL is nob a bounded formula. However, f weprove the existenceof 2l*l' , then we can worlcwith ii as if it r,vere A!r-r,formula.


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cl. Witnessins F\rnctions 33 7(d) Wi il;nessingmnc{;iormIn this subsecLionwe shall clevelop he moclei th.eoryo:[ ragrnenl,sol 52 anclprove som.ebh.eoremsf the Lypemenbionecl L the beginningof Lhissection.We shall use bhe fact provecl above thaL 'Iuring rn.achinecom.pul,al,ions reclefinable in f::agments of 52.Firsb we shall show Lhat a substructure of a model IVI o,t?l. that is cl.oseclnncler the funcl,ions of nl,r,is I! elementary in IVl.ancl is a mo.lel of 1-j.The same holds for substructures of moclels of ,5) ancl the functions ol: nf .Tlris is appiiecl to prove witnessing Theor:ems 4.27 and 4.29 'tor fragmenl,s Zj .Then we prove a witnessing Th.eorem4.32 for:flragments 5l-F1. The theoremhas a sbronger conciusion than in Lhe original form used by Bussl it sbatestlrat (Vr)p(*,f (*)) is provable n the wealcer heory T'j , i f. ? > 0. T'his al lowsus to prove imtnediabely th.e well-lcnown conservation result for sr-rchpairs ofbheories.As the moclel-theoretical proof of Theorem 4.27 is rather diffi,cult,we first prove an aurxiliary ['heorem 4.31 abou.t extensions of moclels of 5l-l-1.4.19 Definition" Let IVI be a structui:e for the language of. 52, let A g IVI,i > 0. Assume that each {"};(") defines a function in .Att. The n! cLosu,re fA is the set

{ { " h ( o ) l c r e t r & e e 1 / } .We say thab A is n!, closed :f .A is its own closure.

We want bo use the fact that polynomi,al time prerl'icatesare a,bsolu't,eneaerA ! closedsubstructu,re f NI, a model of .9|. As'we clonob have symbolsfor all polynomial time predicabes n our language, we have to formalizethis statement. Before th.e next definition let us note that every formulacan be transforrned to an ecluivalent formula which has only &, V and -as connecbives, and ail negations occur only at atomic formulae. What isimportant is that this transformabion preserves the quantifier complexity.We shall say bhabsuch a formula is in negat,ionnormal form.4.20 Definition. Let i > 0, leL f be Slif i : 0 and itj otherwise.For a formularp in negation normal form and a modei IVI ? 7,, we define in.ch-rci;ive1y;hat ghas Jf,_r, Skolem uttc't ions n XI:(1 ) every open formula has nf_r, Skolem functions in M;(2 ) if g ancl {, huve nl;*l Slcolem uncLions i.n VI,l,hen Vzp, v k rh and gY {

t - T ) f lhave nf_r, Skolem unctions n IUI;(3) ii p(x,y) has Jli-rt Skolern nncbions n IuI, bhen ly) ,p1r,y) hzr,s f-r,

Slc.olemunctions n NI, if for some in nf*,.v/F (Vx)((3y)v@,a),

Flere/(x) is lepresenbecly {a} ; ( ( * t , ( *2 , .( " r , . ' . , r r r ) : c .p(" , / ( * ) ) ) ." , r r ) ' . . ) ) ,"vhere ly ' and


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4,2L Lernma.Let 7 be as above, Iet IVItet p(x) havenf-,-, Skolem r-Lnctionsn

338 V. Bouncled r i ihmetic

IUIF e@)+

F T' , Iet I ( C&1. Then lbrK F e ( a ) .

Proof. By inducbion on the complexit'y o'f.p.4.22 Cerollary. If p(x) and -?(x) have A'o*t Skolem functions in M F T, :['as above, then p(x) is absolule fo r !f*, closed substructures of IUI, i.e. il Ais a trf*., closedsubstructure of IVI and a is a string of elements of .A then

IvI re@) iff AF e@).The naburaldefinitionsof polynomial time computablepredicabes aven{Skolern unctions n M.In particular we shall need hat ihe formula * - (a),and ts negationhaven{ Skolem unctions n IuL Now iet P(r) be an arbitrarypolynomial time computablepredicate.Then it can be clefined y

( : .X l - l Sp ( l " l )kCornp (w, * ) k (T -u )011 '1 ;1 )

fuI be trf*, closed, ncla e I i ,

L

(d .1 )It is clear thab r.oand the existentially quantified numbers in Comp can becomputed in polynomial time, bhus (d.1) has nf Skolem functions in IuI.The same is true fo r the negation of (d.t), hence (d.1) is absolute. Thusin our formaljzation, polynomial time predicates are absolute in the situa-tion consicleredabove. The same argument worlts, of course, lb r polynomiaLtime computable funcLions. This enables us to work with polynomial timecomputable predicates as if they were present in the language.4.23 Lernma. Le t Iu I be a model of S$,leI K be a trf closed substructure.Then K tr LlND-striclib, + K tr Si .

Proof First we shall showusing LIND 'forstric'tEf forrnulae hat in /{ everyXnbormula is equivalenl,o a stri,cti'f one. thus we will baveLIND E! ^in X '..This ir proved. y indu.ction n the numberof quantifiers.Several uanl,ifiers fthe samekinclare eplacedby a singleoneusing he cocling elation.A sharplybolndecl universalquantifier s exchalgeclwilh the next existentialboundedquantifier using an lnstanceof BBi!,lor which LIND-stnct\! is sufEcient(see he proof f Th.or.m 4.10).We musbcheck hat the propertiesof thecoclingunction that we are using areprovablealready n LII{D-stridtbr. Weneed he followingproperties:


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(V "0 . . * i )Gy) ( ( y )o ro( : s ) ( /h (s ) 0 ) ;(Vr ,s) ( l l ) ( th( t ) : /h(s) F

4. WiLnessing fiuncbions

a ' " u(a) i : * i ) , fbr everY ;T & ( l ) / A ( , ) : 3 'k ( V z < / h ( s ) ) (t ) z ( ' ) , ) ) ;

3 3 9( 1 )(2)(3 )

(( i)rnf"i is the lasbelementof the sec;-Lence). Again it is clear^ f t n DlormuLlaeh.ave r1i Slcolem unctions in IuI. Since they ai:e rue inLrue also rn Ii by Lem.ma


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340or

V. BoundedAri lhmetic

p :: tIf 2(p -F lt).t, Q : (1, obherwise.By Theorem 4..|-6 he computation of length lt(")l always exisbs. In o::clerto prove its correctness we neeclonly to shor,v haL in l,he 7-Lh step of bhe^^.-.-",+^ r.;,,- r i .^ {^]1 " ' ;- ^. .-;n ^rcl i t ions arc sabii f icd:UI l fPLt t ,d, l , lUI I U l - l . (J L. r . r . I ! , JwI IJ .S t vvu uu l l

q(: 'Xp t(a)2r ;qk tp6, r) )- p s\ ; { , \

This is pi:oved y induction on 1. Once he computation s given, bheconciitionthat we ale pL:o.zings Xf , h.ence e neeclonly .L.[NDX!.tt the formu].a rasbhe brm ( l r < l t (a ) l )p (a ,) ,r,vhere i.s I!, we proceedsimilarly,but we use thorough search nstead ofbinary search.The caseof the universalboundedcluantifierand connectives&, v is trivial. lf4.25 Corollary. Let T be as above, Iet IVI F ?, ancl let K be a nr?*r closeclsubstru.ctureof IuI. Then I( is Es(Df ) elernentary..Proof.By l-,ernrna .2.1' nd Lem.ma .24. tl

SinceLINDT! is a XN(Xf) formula,weget that /( is a moclelot,Si. BuL'wecan provemore.4.26 Tlreorern.Let i > 0, Iet T be,l',) if. :0 anclT) oiherwise.Let l' { be aal*, closecl r-LbstrucLuref somemoclelNI oI ?. Then K is a moclelof 1')'ProoJ.Let i ) 0, let IVI F T, tet .I( g IvI be nf*r closed.Weshall showthat foreverycv(2, ) in D!, a formulaecluivalenbo t(a(r,b)) (inctuctionor a(r, b))has nf_,_,Skolem r-rnctions,houLghhis is not (lcnown o be equivalent o) a,t8@!) forrnurla.hen, by Lemma L22, t(a(r,,b)) must hold in If . We shallconsiclerhe fbllowing eqLrivalentormula

Again werepeat:

-a (0 , b) V (3r < a) (a ( r ,b ) 8z a( r + f , b) ) tza (c , b)shail use binary search. rVestart with p :- 0 ancl q :- a, alr-d

yt : '(p -t- )12', I t: Qt if a( '(p -t ct)l2L,lr)p t: p, q :-- ,(p -l q)12,t, othelrn'ise.


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23/76

342 V. BouncledArithmeticsinceotherwise,using W@,y) as an oracle,we could combine heseTurringmachines nto one prodrlcingy for a given r. 'By compactness, here s amodel M hor

T { -


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4. lViLrressirrg _,\rnctions 3,13Tlreorem4.26. we use p(r,y) asa II! oracreancl{a};11(r) as a subroutinein order to ask cluestions f the form vr(r,{e}*F1(")).Crr."."-p'Lation goesas follows:s lep0: 12 : : A , c l : : . c t , ) t r :_ {e } ;a1 (0 ) , u : : {e } ; . r r (a )s' tep- l -1: r : : {a};ar( l (p + i l12_,);i t pQ(p * c l l2L, r) b lren i : L(p* o) 2r , rL : re.[se ': r-(p f (Dl2-,, u :.: r.Tlre co:rrecbnessr; proved by L.\ND.tr.fshowing ihat in th.e7-t6 step we h.ave( 1 ) q - p < z - j a ;(2) eQt, ) & gkt, u) . fi

.A 1n! formula consisbs f a prefix of existential clu.antifiersolloweclby aZob brmula.4.29Theorern.LeL- > 0, rer cp(x,y,z)he a=II! lbrmula anclsuppose

T) ts (V )(zy)(Vz)tp(*,y ,") .Then for some o, . . . , f,, in nl*,( d . 1 ) , n / F ( V z , y t . . . , z n ) ( r p ( r , f o ( * ) , 2 0 ) VV @ , t ( * , " 0 ) , " r )y . . .. . V V ( * , f n ( * , z o t . . . , . ? n - I ) , z " ) ) .Moreover, his is provable n S| , if i :0 , ancl n 1',j, f i > 0.Proo f . eL? be 5 ] , i f . i - 0 , and j , i t i ) 0 , re t , (n ,y . ,z )bea l I I ! fo rmu la .strppose7' doesnob prove Lhe ormula in (d.1) for any choice"f ib ,. . . , fr.Take someentimeration of functions in nr?_pr tich that(1) b.he -ih lunction /,, depencls n ( n arg.Llmenrs;(2) each trr?-,-rccurs n the enumeralion nfinitely many times.(1, 'ornstance,{e};-r-r(((r i rz), . . . ,n")) , ,where (-, -) is the pair ing t 'unc-tion, has bhis properby, assr-rrning atr-rral coding of Turing maclr.ines.) BycompacLness,

' f ' -p ,9(c,fo(r),do) F 'v(c,f {r ,do),d) - i- . .wlrere c, cLg, r, . .. are ner,v on.stants,s a consistent theory. Let lVl' be a rnocleLof this Lheory and let

I ( : {Jo(" ) , t (c ,cLs) ,z( " ,c lg ,cL1) ,. } .


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34+ V. Bouncled.Al i thmet icSince al l pr:ojections occr,rr n the enumeration ancl each function occtLrs nthe enr:.merabionnfinitely many tirnes, we have(3) c, clg, l1, l2, . . I( ,( 'L) /( is nf ,. , closecl., a _ tBy (3), l l 'e have ya .K 1d e I( IUIF _,pG,o,, ) .Since -tpis VE!, ,usingCorollary .25we get,

/ f F (Ve)(12)-,e(c,y,)Now we woulcl like to strengthen Theorem 21.27 y taliing the weakerassumption Si-rr F (Vr)(:y)v@,,y) insteacl of Tj F (Vr)Qyjrp(r, s). Theproof above d.oes ot work for Sl+r, since we are not able to show that a nfn,

closecL ubstructure is a model of SiL-FI.t is worthwhile to reali ze lhe reason,then we shall better understand the forthcoming proofs. Let (!y St)rh@,y)be a E!*r formula, where t! is A!.In a nf*, closedsubstructure the cluantifierlg can be witnessed by different elements of the form {e}l+r(a). Since eruns over standard numbers, we can obtain a cut (for instance containedin some segment [0,b]) for which (ly < t)$(*,y) holds, hence the inductionfails. Therefore we shall talce closures under some functions wil]n nonstandardind i ,ces.

The basic idea of the proof is t he same as above: i f (=y)gQ, y) is notwitnessed by a nf*, function, we construct a subrnodel in which it does no thold. Thus we must be careful when adding {"};+r(c) for e nonstandard.We shall Lrse verspi l l . f ( ly)tp(c,y) is not witiressedby any {"}l+r(c), for esbaniLard, hen this must be also true for all e up to some small nonstandard11. Tlrtrs we ensure the failure of (1y)tp(c, y) by taking the closure ouly unclerfunctions with such small indices. However it is not so easy to ensttre theinduction.

Now we sketch the idea of the model-theoretic proof of this strengthening.IL is essentially the proof of Wilkie with some changes. Le t i ) 0, le t T beS$ it i : 0 and Ti oth.erwise.Suppose that for every e .Ay', does noLprove tp(n,{e};-rr(r)).Take a model fuI tr 7 with solne c }uI, such liraLM tr - - r1c,{E} ;+t( . ) ) .By oversp i l l h is is t rue a lso for a l l e - -11. We shal lconstruct sr-Lbstructuresonsisting of sorne elements of the forrn {eh-rt(c), for-'e 111, thus we shallhavg -(Vr)(1a)V@,y) i"the sttbstructures.So we haveonly to ensure LINDEh..We shall do i t inco steps (we use only countablemoclels) adding LIND for one formula and one string of parameters at zr ime.Let us talce he formula V(*) : (:y S t)rh@,y) consideredabove, ancl eL abe alreacly in our substrucl,ure K.'We want to extend /f so that it satisfi.estlre following insian ce of LIIttD D!-rr:

u

(d 2) -u7(0)( : j S l " l ) (v ( j )k -v , ( jf ) )v r ( la l )


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4. lVi inessincFuncl ions q , l t rd ' t t ,lVe take a suibable12 ( 11)and bry to find a witnessgs for 131n tZ(0) ofthe tbr:n {"hr-r("), r 1 rz. If we succeed,we try to find erwitness g.1 .or(1 ) o f the fo rm [ th+r ( ( . , (0 ,90 ) ) ) 1 r2 , and so on . The bound r2 i schosen o small tirabwe neverconsLruct{rh+r(.) wibh e ) rt.Sr-rppose,brexample, Lhat for some < l"l, we have fbr-rnci 1, bLrt bhere s no witnesstj+t o,f the form {t}l-rr((.,(r, gi))') for V(j -f T). Then we take the | lf*,clostrre f (c, (j , gi)) and -I( n N|' anclwe have(ci.2).We repeat his extensionprocess or other formulaeand other parameters. n order Lopreserve d.2)we use a sirnilarargumenb swe used or -(Vr)(=y)rp(r,A).In our par:ticularcase,V(j) is preservecl y Xob lementaryextensions, ince i is 3I{, while-V(j -FT) will be preserved ecausewe shall add only elementsof the form{r};+1(c), fo r e .--12. The following key lernma {brmalizesone step of theaboveconstruction.

4.30 Lernrna"Let i ) 0, let ? be .92rf. :0 and Tl otherwise.Let IuI be amoclel f ? , K sr-tbstructure,f.VI,,a,b /(, let Q(r,y)be stricttf-r , ancl etf(y) be strictl l!4. Supposehat(1) .K is a n'rn, closureof one element of IVI;(2) K is not cofinal n IVI;(3) /( F r(b).Then Lhereexistsa substructure.K'k uch hat.I{ g J(' ' g IVI, (1)-(3) holdsfor y'(* and(4) K' ' tr-@(0,)v (:j < lolXo(j, )& -,a(i+ T,b))v o(lal, ).Proof. Leb the assumptions e satisfied.Let J( be generaiedby an elementc. SinceK is nob cofi.naln IvI, there is some d e IUI such bhat

x K + l r l < 1 . / lSince errery r e I( is computed in polynomial time from c, and by overspill,we have

l " l " 5 ld l.t.3)fbr some rs nonstandard. Recall ihat

{t} ;-rr ") : I t = u;-Ft(",, U 21' l '7wlrere ;11is Abo*r. y (c1.3),e canwrite {t};+r(") - lJz,s Abo+,ormulawibh an adclitionalparameterd, since

f u IF u ; a 1 ( " , r , y , 2 l t l " ; ( V t < d ) ( , - 2 l x l " > ' u i - r r ( " , r , ' y , z ) ): (1 , 3 rL ) ( , 2 l ' "1 'r ; - t - t ( " , r ,y ,2 ) ) .


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J.,to

LetV. Bouncled Arithrnetic

@(r , ) : (1 , < t ( ' ) )q r ( r ,z ) ,! Z ( h ) ( Y r i s ) t | , , ( z ) ,',,2i:.ere(rc,:t) s llb, and l.,(z)s x;b; we shall. rnibbhepararnebersr.[romnowon..Consicler he follo''ring :fbrmula( .1.4)

( .1.5)

(Vt< l t lX{r} ; - r - r (c)s - , ,1,( [ r ] ; - r - r (" ) ) )Sin-ceK ? V., ( is L'! eLementaryinVI,ancle.rery lementof /{ is of the fbr.m{"}*vt(c), fo r e standard, his fbrmula s true in h'|, for every sl,anclard.\lsothis fbrmrr.las ecluival.ento u Ao;+t ormr-rlan AI. Since S:]. rA1, (Theorem4.7) and for i ) I, T; l- I A!-r, (Corollary tL.2B), ecanuseoverspill o cl.eclu.cetlrat (cl.zi) olcls n I'tl for some nonstanclard. ,et

1 :2 :

3 :-J .

r r : m in ( l "o l ,l r l l , . l )Now r,veconstruct a Turing machine with an oracle which searches forthe r,vitnessof the existential.cluantifier in condition (


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348 V. BoundedAlithmetic4.3LTheorem.Let i ) 0,Iet T be S'21f i : 0zt, ottn'tab/e oclelo, fT, K a substi:ticttLre f. . : . 4strtctl l i_,.1.Su.pposehat(1) K is a Jl,_r, closureof oneelementof fuI;(2) K is not cofinal n M;(3) r{ trv(h).Then there e:cisl,s substructure J{'F sr,rch ha t(4) K* F r(b);(5) I{' ' F S,i*'.

anclT$ otherwise Let MAI, h Ii, ancl et f(y) bebe

K C I('' C AI and

Proof. lVe shall construcb a countable chain of substructuresK : K o e I h e I { z c . . . c I v I ,

and s'trict,i !*, formulaei t r : V o , i L t , V z , . . . .

Take someenumerationof stric'tD!*. formulaewith parameters rom IVI. Inthe -blr step,7 ) 0, we ensureLINb-in Ki ror he first formula d(a,b) fromtlris enumeration for which LIND fails in K j-t, and whoseparameters b arein Ki-t. Supposewe consider-@(0, )v ( !c < la l) (@(r, l r )- -e(n FT, ))v o|a l ,b)

in tlre j-th step. We apply Lemma 4.30 to .Ki, Q andt?i. Thus we obtainI(ia1. Then we defineVj+t byViry = [, j k"-$(0),, i{ I{ia1F -rD(0);Vj_rt = tl/j&,--,O(k l-T) ,

i f K i a l F l c1 l a l k0 ( k ) k - $ ( k+ 1 ) , f o r so m e e K i a l ;tIi'4 = Vi, otherwise.EIerc Vial is not written in ihe strictll!-r., lbrm; however to show thabthey are ecluivalento such ormulae,we need only propertiesof the codingrelaiion that are expressible y equalitiesand hencepreserved rorrr MI Lon?,closedsubsbructures. inc. Vj+-t will be true in all Krr, f.or nt.> j + 1, alsoLIND@(a, b) ) will hold in thesemodels. Considel for instance the secondcase,wlrere {ia1F k < lalkaft) k- 'Q(lc r- 1). Then (tr(k)wlll be preservedby extensions, inceall |;hesLructures r,re f*, c1osec1,enceE! dement,ary,while -Q(lc *T) will be.preservedy the clefinitionof the extensions.Let Ii*be the ttnion of this If elernentary hain. Then all formuiae Vi are true in/'('*, hence b sabisfi.esonclition a) and

K* F LIND-sirictEb;a1By Lemma 4.23 this implies hat -I("' s zrmocleloLS'-"rt ft


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350 V. Bounded Al i thmet ic4.34Corollary.For i 2 t, Strl-ts a i'f-,-, conservaLivextension f Tj, i.e.for e@) ir i'f-p1,

s;-t-L (Vr)p(z) + r; F (Vc),p(c)

Proof. Acld a dr-Lmmye;


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.1..Witnessing Functions 351Then r,ve hallrecluce the finiLe axiornatizability of .S'2o th.enon.-collapsecrnessof Polynomial Flierarchy.4.36 T'heor:ern.For i > 1. , ach of Sl2ancL j. is finitely axiomatizable.Proof-sketch.T'he basic iclea of the proof is the sarre as to r fragrnents I Dn,seeTlreorem I.2.52. However,since his theorem s important, we give at leasba skeLchof the proof.In Theorem 4.IB we s.howed hat there ate L' ! formulae p; which are i1 asenseuniversal; more precisely, for every tp(r) in x. f there exists a1 e suchthaL

,5 ' | ,p( r ) : L t t (E,r ,Z l r l t !Furtherm ore, zl'lu p.rovably e:cists n sl; this fbllows from the bound

2lal,-vrS 2l*lu gr .For induction and PIND we need formulae which have an incluction variableand parameters. One parameber is suffi.cientsince the pairing function andthe clecoding unctions are A! definable. For the same reason, we can extendthe result above to formulae with two free variables:

s) F p@,y)= pi(z,* ,a) ,21* la;Anothereasymodificationof the resultabove ei'rclssfollows:

S ' ) z > 2l@,a)1"n ( rp(* , ,y) pt ; (e , (n , i l ,4)Now we only neecl to show that there exists a finite subtheory ? of Srl stichthal for al l tp(r,y) in x! the above equivalence s provable in ?. once wehave strch a T, we can finitely axiomatize Si by T pLus .B,A.g-IC lus

(V.,U, )P ND (z > 2l@'a)1". 1,,;(",*, y), )) . ,wlrere r is the induction variable (thus boLrnded n PIND), and e)a)z areparameLersof incluction. Similarly we can axiomatize Tj.To find such a T we have to anaryze ormulae pr ; undu; of .T'heorem4.1Bmore closely. The concept that we wa,nt to use is just Tarski's conclitions forthe definition of th.e satisfaction of E! formulae. Tarslci's conclitions d.escribethe satisfaction of forr.nulaeof a given classvia relating the truth of a fbrm.ulawiih the truth of its components. Fbr eaclr.closure condition of Lhe classof formulae there is one Tarshi conciition. The class s E! were clefinecl nDefinition 4,2; Larthermore tve have to use the inductive concliLions ol thevalue of terrns.


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352 V, Bounded Ar i lhn-ret ichlow we shall describe the conditions more explicitly.(1) Fbr each unction symbolof L2,,ve shall have an integer consbant,e5,

-1,e* , ebc. ,such thatS') us(esr t a. ,2 lxlzs= y :,5(r)S) r - us(e+," r ,12 ) ,u ,21* , ' ' " ' ) lu - t

1

- J r I t T 2 ;etc.(2) For relation symbols and ( wee< such hat

,91.s po(e:(e1, 2),,2 la l"=(" ' ' " t ,

consLruct binary functions e- ancl

= (1yt ,yz) (uo(et ,,U\ ,2 lx l " 'k ro("2, , yz,2 l '1" 'k n : az)and similarly fo r (. The functions ur" Alt

(3 ) For each of the conditions of the definition of the class Xf we have afunction and a condition. Consider fo r instance the condition

A\a e D ! + ( V z < l t l ) ' E ! .

Then we have a binary definable unction elvl such thatS') pt ; (e lv l (e1,z) ,,2 l ' l " lo t ,

= (Vy , ) (us (e1 , r ,U ,21 '1 " ;z l l y l* t t ; ( ez , ( r ," ) ,21G ' ' ) 1 "1T'he flurnction ompuied by the machine with the code e1y1 oes the following.If e1 is the code of a machine tor a term t and ez is thei code of a machinefbr a formula p(r), then elvl(er,ez) is the code of a machine which decidestlre formula (Vz < ltl);o(x, z).Let ? consist of the above condibionsplus finitely many senbences eededbo ormal ize u.nct ions - , " ( ,e lv l , . . . (ex is tence, n ic luenest ,A?. ef in i t ion) .Then we can easily show, using induction on the depth ot tp X;0, that thereexists an e such that T ts


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.J DZI

(".2)V. Boundecl Arithmetic

l V F ( V r , 2 0 t . . , z n ) ( t p ( r , / o ( " ) , o )V p ( r , f 1 ( n , z o ) , t ) V ' ' '. . . Y tp(n,. f . r . ( r , o , . . . t z ,n , - l ) , " ))

The resi; of the pr:oof s only comple,.ribV heory. We wz'r,nL,o shor,,r li.abstrchfurn.ctions annoL exisL or al l polynomizr,lLime predicates C'.II n were allvays0 (as i.n-Buss's theorern) bhenor-irgoal would be sirnple. In such a czr,se,orexarnple, we cotild construct a maximaL clicl'uen a graph.using a polynomialtime com.putable function /s . This is impossible r-tnlessP : A/P. In fact wecan talre any /VP problern, nob only optimization proltlems. Let C'(r,y) bedefinecl, say, by

lJ : 0 or y is a Hamillonian circtLit in graph rT'hen foki would be 0, if z is not Harriltonian and /s(r) would be a,lfamiltonian circuil in s otherwise.Bub in general rz need not be 0. So we rnoclify otr.r probiem asj fbiloi,vs:rwi!.I be a string of graphs and y will be a string of 0's and Flamiltoniancircuits in corresponding graplr.s. \ow we assume that lbr Lhis particular Cwe obtain (e.2) with n - I, (this is suffi.ciently nstrucbive). Consider an n' abwo element string * - (G;,GZ), where both graphs are llamiltonian. Thenthere are two possibilities:(1 ) /o ( " ) i s no t a feas ib le o lu t i ono r . fo ( " ) : ( n ,yz ) where y r :1J2 :0 ;(2) /o(") : (uy y2) ancl y1 is a llarniltonian cycle in G1 or y is a Hamiltoircycle in G2;In case (Z) fo has prodr-rced ontrivial inforrration. Iir case (1) it is not so ,bu t /1 :must prodtice inlbrmation: if we take zg : (y',0), where yt is someHamilLonian cycle in G1, Lhenwe have -g(r,/o("),zs), sinceeither i6(r) isno t tr, easible solution or z0 is a better olle. Flencewe get

(Y z1)9@, f {*., zo) zt ) ,i . " . , f i ( r , r i l i s an op t ima l so lu t i on .Th i s means tha t f t@,zo ) : (A r ,az )where both yl ancl a2 ate Hamilton cycles. Let us draw an arrow G1 -, G2in case (1) and an arrow Gz -- G1 in case(2). f'his indicates that by havinga Flamiltonian circuit fo r the tail \ /e can construct a Hamiltonian circuit fbrthe head. Consider al l Hamiltonian graphs of size m. Tlne above argumentshows that there is an arrow at leasb n one direction between any pair olthem. An easy counting argumenbshows that there exisbsa set of poiynornialsize of such graphs which covers he rest. For each of bhesegraphs, we choosea Hamill,olian circrrit in it. Let [n be the set consisbing of these pairs. [1lhas also polynomial size. Using lFllwe can decide in polynomial time r,vhebhera graph Ci is Hzrmiltonian: try il a,nc[ fi ott ali pzrirs (C|,II) and (/'I,G),where l' 1 r.uns ,hror-rgh ILCompul,aLionswhich useadditional polynornial size


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4. WiI;nessing Irunctions ' ) D Jirr.formaLionare called coltrpuLaiion.s ''riLh olynom,ial afluice. T'her.e s :r wr:lllcnow-na'r,rgurnenLsee fl(arp-l,ipi;on 80]) siro,,vingbhat if every /\/p problemcelnbe compr-ttecl sing polynomizr.l ime cornpr:.tzrtionwith polynorniai ,rc[uice,thenEl - tItr.Nor;v;T prove Iheorem4.38 n clebai.l.Ve sh.all livicleh.eploof irrl,o everzr,llerrrLrras.,eb ) 0 begiven. \. or.mLrlaC'(n,y) i l l becalleda.lI! op.ii,mizat,iortp r o b L e r n i t c ' ( r , y )s a r { f b r m ' - r l a s e r c h . t h a L c ( 2 , 0 )' c l c ' ( r , " y ) - - > y 1 . n i spro'zable n-predicate ogic. Ve shall say that y is a length opti:rn,aLsoluri;io1Lo r if the followinq formu.la lolcls:

(Vz)(C(r ,y) (M < lr l - , - -C '(n,z)))4.40Froposition" For every I! oprimizaLion roblem C' , Sr+r proves her,t;has a length optimal solution o every r.ProoJ. 'et,fi(x,y) bedefined y

t l t (x ,u , ) (1a< x)(C' ( r ,y) ly l >_") .By LIND E!-r, we have-t!(r ,0) v ( lu < lr lX/(",tr ,)k.-.t f . t(n,S(")))t l t(r ,, l r l )

since ',h(r,6) is always true, iL ust e,rpressesh.eexistenceof a lengthsolution. optirnalnIt is an easyexerciseo plove thab, n fact, the existence f length op{,imalsolutions to tI ! opLirnizationproblems is eqtti,aalento PIND i!a-1 oyer asufficientlystrong base heory, sayS$.We aregoing bouseTheorem4.29.It is convenienLo refer o the conclusionof the theorem, see formula (e.2) above, as a kind of interactive way ofcomptrt inga witness o ly in (Vr)( fy)(Vz)tp(r ,y,z).Let r be given andstlpposewe want to construcLy. Assume (e.2) holds true. First consider/b("). If (Vz-)p@,,f0(r), ), then take y : .fo("), othe:rwisehere s some zssuclr hat -tp(r,fo(*),26). V/e shall call this z0 a co.LLnterexumpleo ft(r).By (e.2), t must be n ) 0 and we have

/V F (Vz,zrt . . . , , n)( tp(n, (*, z0), t ) V . . .' . Y ,p ( " ,t " ( r , z0 t . . , zn -L ) . , n ) ) .

f lence vvecan repeaL bhe reasoning above and we get that either J(r,to)wil,nessesy or the disjunction calr be redr,Lcecltrrther by taliing a countere,v-arnple z1 Io h(*,zg). I{o.,veverhis processcan be repeated at mosb rz birnes,since when we obbain

r,vecatt sbop.(Yr , zr r ) tp(x , n(r , zot . . . t zn-r ) , r r r )


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J D U V. Bouncled Arithmet,ic

.Let C(r,y) b" a .trf opbimizzr,tion roblern. Assume tha.E i : ,g;.-". T'hcn.Ii j proves thab C' has always length optimal solr,rtions.Tbis staternent canbe e;rpressecl:.r. he form (VcXfyXVz)g(r,y,z) with rp in E!-rr, see (e..1).i.Ience, by Theorem 4.29, we get tirat an optimal solution can be cornputeclusing n,?-r, firncbionsancl a fixed number of counLerexamples.This is our firstIJ.emma.2tr.4-{-,emma.f TA: SL-",;length optimal solutions cannumber of counterercamples.

) 0, |;henor every tI! optirnizaiionproblem iLsbe computecl -Lsing r?-n, unctions and a fi;recl

'.['hecompr-riations ith countere:rar rrpleslonobclefine function comple,r-ity class,sincewe allow the askin.g f counterexamples nly for the particularoptimizatior: problern hat we want to solve. f we allowecl slcingarbitrarS'clueries f the same ogicalcomplexit.f,we woulclobtair: ust the class rf*r.But it is easy o prove hat optimai solutionscan be computedby such u.nc-Lions.Thr-isour ne:it steD s a leduction bo a different unction class.4.42 Definition. A function / belongs o Jllpolg if bhere xisbs omeg inand a polynomiil p(r) such hat for every fo here is o6 ( p(k) stich thatevery r, lnl : ft , l @ ) : g ( r , a n ) .

Here cr7,s called a Ttolynomialaduice; it is some extra inlormation givenfor free for each size of input; the dependence on the size of input is cluiLearbitrary.4.4i3 Lernrtra. LeL i > 0. Suppose that for every I/f optimizabion problem itslength optimal soluLions can be cornputecl using al+, functions ancl a fixednumber of counterexamples. Then for every ,h@,y) in .A! there exists an /in trf-r, poly such hat

N F (Vz)((1a r)r[(r ,a) - ,h@.,("))) ..ProofLet { be given.Defi.ne" flf optimization roblemp(u,,u)byp(r ,u) Ih(u)< lh( t t ) (Vt< lh(u) ) ( t1 . , ( ( . , ) r ,r ) , )& ( r ) r < (" ) r )l ,eL s(u) , . . . , n(nru0, . .,un-1.) e someunct ionsn nf- r , which nterac-l,ively compube length optimal solutions 'f.orp. Let k be given. lVe shall con-sbruct a polynomial zr,dvicebr inputs of size /c. Once we clescribe he aclvice,|;he clefinition of the funcl;ion o will be clear. l,et

r1Pu ifor

V 1 { r ; l * l : I tk ( 1 y < " ) r ! @ , y ) }


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4. lViLnessingnunciior-rs 35 2Choose ome unction tu(r) such hat, for r V1,

r [ ( x , u ( n ) ) .we shall trse he followingnoiation. If rrz //z(u) trre'

w(r, I m)denotes he sequence,r.r . ' ( ( t , )o), .. , lu(( t) , )) . To eachn * 1-tup1ezo[ e le-mentso,f.V1,,ve ssign pair (l,u).,0 < / ( n, r-ising hefollor,vingpi:ocec1t,.re:Step 0: compute o(") ; i f p(u,, /o(")) atd th(Js(")) > 0, then p' r / : Q

::O , : (/o("))e and stop,otherwise o to step 1;S tepm (m ,< n ) : co m p u t . * ( u , , zu ( u0 ) , . . . , u 1 ( u l m - 1 ) ) ; f

p ( n , * ( u , , u ( u 0 ) , . . , t t ( u | - - t ) ) )ancl I h ( f r " ( u , u ( u l 0 ) , . . ., z u ( u l m - i ) ) ) > - ,t l r e n p u t l - m a n c l

a : ( J * @ , r . u ( u| 0 ) , . , u ( u I m - 1 ) ) ) , , ,otherwisego to step nz 1;Step n: If we have reached his step, then it necessarily olcls hat

p ( u , n ( u , - ( " l 0 ) , . . , u ( u | " - t ) ) )and I h ( f " ( u , , - ( " | 1 0 ) , . . . , , u ( u l n -) ) ) n I . I ,t h u s w e p r - r t / : n a n d

U f n ( u , r o ( " 0 ) , . . . , w ( t t ,"- 1 ) ) ,

and stop.Let trs caIIy a uitness br e if ,1,,@,A). The rneaningof bheaboveprocectr,rreis the tollowing. Let ,u be an n * 1-tuple and let l,y be assignedo it. T'heirIraving 'uvi tnessesor (u)g,. . . , (n)t- l r wo can comptrtehe wi l ,ness tor (u)1.For an n-elemenb nbsetQ "f V1 and r e.Vt\Q, *" shail say thaL paii:( Q , " ) t s good f f o r so m e r r a r l g e m e n t{ r g , . . . t r l - 1 x : l + 1 . . . , r n } o l Q , I i sassignedbo r.rby the aboveproceclure,vhereu is the secluence

( " 0 , . . t i c l - l r x , r l + ' 1 , . . , 0 r ) .


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35 8 V. Boundecl Arithmetic

.Defi.n.e sequence of sttbsets o,f.V1V t ) . V z )V s " '

having /V1, {2, . . .? espectively, l.ements.he j l- 1-stelemenLn the secluenceis cle{inecl s follows.Find an n-elerrentsuLrsetQi 9I,'i suchthatN; - nl { " e V i ; ( Q i , x )ss o o dl > ; + r ,

ancl Lake Vj+, t : Y, { " e Vi ; (Q1,* ) is good} .We mtr.stshow thai it is alwayspossible to choosesuch a Q .BV the procedureabove, for each n * l-eiement subset {ro r...rrn} of vi we can construct agood pair (Q, r) such that

{ , 0 , " ' , t n } : Q u { r } ,by taking u to be the secg-Lencers, ...,rn). FIence here are at least (/lr)goocl pairs. On the other hancl there "t " (f ) n-element subseis Q of V5',soat least one such Q must form good pairs with at least

( ui \.1t,)\ " - l - t ) \ " ) n * telements.Hence

- l f ; - n nNi+rS iVi : ;* ,trui-F ),from which we get

f f i . , ' (*) ' ' , * h* (#)' * (*) ' - ,\ n * 1 /

Hence we get lft < n * 2 after f steps, fort < log2(N1).(los2((nF )/n))-t < ollog2Q\) - o(k)'

trVe take bhe polynomial size advice to be the seclueuceof pairs (r , tu(r))'wh.ere tr runs trough all elements of

Q t uQ z " ' u Q t u V


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4. lVitnessing Ifunctions 35gNow let r be such that

l ' l : k k ( 1 y< d t @ , y ) ,i.e. e e Vt . T'hen either n Vt ancl so a witness lbr o is in . the aclvice, or(Q5,t) is a good pair for som.e St. But then we can consbruct a witnessfbr r from the wibnesses or the eJ.ements f Qi (using nf_r, functions in theproceclureabove). n4"44Lernrna. Le t i > 0. Suppose that fo r every ,b@,y) in III there exists anf in lle.-rrlpoly uch hat

/r F (Vr)((:vS x)rh@,y), ,!@, (*)))ThenElt-r:nl+2.Proof. Let .4(6) be n il!*r.Withor-Lt lossof gen.erality e can assume hat itcan be representedn the form

(Vr ) ( l " l lb l ( ly < b)t f t , r ,u)) ,where 1 is II!.'L,eltft(:c,y)be " n! formula suchthat, h ( ( b , * ) , , v )l " l : l b l +@S U B c I ( b , r , a ) ) .I'et f be a function in z't-rrlpoLaguaranteedby Lemrna4.48 f.or{, ancl etfunction 9 in nr?*, anclpolynomial p be guaranteedby Defini tion 4.42 for /.Then we can write .4.(6) n the following D!*, fbrml

A ( b ) : ( 1 a , ) ( a p ( l ( 6 , b ) l ) & ( V " X l r l l b l j ( b , t , g ( o , ( b , , ' ) ) ) ) ) .The right to left implication is trivial, the left to right implication fbllows bytalcingan advice a. nThis finishes he proof of Theorem 4.38.

There is a similar reduction.of the cluestion5i : T;? to a problem, ncomplexii,y theory. F'irst we sirall define tha,'r,or an opt\rnizalion problemC(*,y), U is an oqt'ti,maloluLtiono r if(Y )(C r y)k (a { z -+ -C (r , ) ) )

Secondly, we define interacbive cornputations with an unbounded number ofcounterexamples as the Lhenal,ural extension of the concept above. Of course,1 1.lo{r"bhaLg is tIl*., defirrable b;, fh6slsln z}. lJ.


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360 V. Bor-rncledrithmeticther:e is an implicit polynomial bound to the number of counterexamples,since we consider only polynomial tirne oracle computations. Note that thenttmber of feasible solrtLionsLo r may be exponential in lzl, thus polynomialnumbel of corinberexarnplesapparently cannob help to solve some diffic1ltoptimization problems (such as the TRAVELLING sALESPERS9N), r,vhi leit gives tL:ivial algorithrns if we ash only f.or length optimal soiuiions (as inCLIQUE for in.stance).We irave Lhe ol.lowing counterpart of llheore.m.4.2g.21..15heorem. Leb - > 0, Iet S'!, (Vr)(ly)(y, < t)p(*,U,2), for g in E!*,anill' a term. Then, fo r a given tr we can compute y using al+, cornputationswit.ir (an unbounded nr-rmberof) counterexamples.

T'his theorem can be proved using the proof theoretical method whichliluss used fbr h.iswitnessing theorem. We are not going to prove it here. Thecor-rnterpartof Lemrna 4.41 is the following.4.,16 Corollary. llbr i > 1, if SL : Tj, then the optimai solutions for ev-?ty nlq problemcan be computed ,ri.rgnf-r, computations ith (an un-bor-rndedumberof) counterexamples

We conjecture that the conclusion of Corollary 4.46 is false, hence alsotharS$# r;.t

The research o.n witnessing fr-r.nctions s still going on and we can look fornew nice results. The task of proving that Bouncled Arithmeiic is not finitelyaxiomaLizedwith,ou,t ssu,mptions n compleai,ty lasses eemssti l l too h.ard.What might be more accessibJ.es to prove nonconservation results for .93andZj using such assumptions.

5. Interpretabil' ity nd Consistencu(a) InfinocnuctionIn the last section we were rnainly interested n the strength of theoriesobtained by restricting the cluantifiercomplexity of bounded formulae inLhe schemeof induction. The main theme of this section is investigationof the strength of theories obtained from I Es by adding functions of adifferentgrowih raLe. These unctionshavegraphsdefinableby fo formulae,so Lhe strengLh s increasedby assuming hat they are toial.) We sirall


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362 V. Boundecl AribhrneticWe shall Lise he notal,ion introducecl in Sect. 3. Thr-is br insLancernax(s)is bhe naxirnal element of seclttence . In this subsection we use only the s,ran-itarcl aritlrrnebical language -t9. T'he forrnalizabion of synLax is as clescribeclin . Sect. 3; thi-rs f r is the Godel nttmber of an expression, lr l is it s length.Our approach is to consiclera botLnded ormula literally as a program. The

cornptttation of Lerms needs no e;cplanai;ion;Lhe inierpretzr,tion of boundeclciual:tiiiers is as search.proceclures,Tlr-ecrr-rcialbl:ing is Lo esLiinate the sizeof the numbers th.at ',,yi11occur in su.cha computabion.Let r-rs onsicieran example firsi;. Leb cz(z) be a boirncled ormula( ly < *2111, y2l,(1u 22) G),71, rt)

where B is also a bor-Lndedorrnula. To compu.te i;he bruth .zalueof a(o) wetr-se ,hree F0R Loops)olre fbr each bor-rndedvariable. hlote that the thircl looplras the range 0 to r2u.5,1Froprosition /to). (1) For a term I ancl a string of nurnbers s,

f (t ) < (max(s) -t -}v t .(2) The maxirnal value needed to comr.p'utehe truth of a bounded formulacv on a string s is bor-rnded rom above L.y

(max(s) -F 21zt"thlote in passing thar,l,, V (1), the rrzrlueof .a cLoseiLerm f is bouncled by2ltl


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5. Interpretabiliby ancl CorlsistencyWe shall esbimate the e;rponent:

(lrl-r D2lpl< 2ltl-rl-Flal2lol,which proves bl-re::clr,LcLionl;ep.

FirsL we define the graph of Lhe aul'f'unction '',shich defines the 'yalue ofarithm.ebical erms. A.lsohere we n.eeilan adclitional pararneter:.The meaningof. Valu'e(a,ttz,r. l) is: y is the valr-re f term r computecl on string z, y istlre parameter. Leb Var(r,, i) be a formalizabion of the conce'pb f bhe i-thvariabie.5.2 Froposition, T'here e;ristsa 16 formtra vulu,e(y,*, z,u) and a polynomialp such that it is provable in /Xs that:

J O J

n

(2 )

(1) T 'er rn(x1)zTerm(n2)e*t l , , l *z lS , & l " l > p( lmax(z) l ,) -- + ( V a r ( r r , f ) . * V a l u e ( y , t r t z t u ) : ( z ) ; - - y )k (Value(U,6u z, u) - -+ : 0)k (Va Iue (A t ,5 ' ( x t ) , z ,u ) k Va lue(y2 tn1 t , t t ) - -+n : S ' ( yz ) )8 t (Va lu ,e (y , r y l - . 2 ,z ,u , ) Va lu ,e (y1 , ,n1 ,, z t )

k Value(y2tn2t ,u , ) - , A : lJr * az)k (Vulue(U r7 ' tu ? , z , u) k VaIu,e(yI171 1u)

k Valu ,e(y2tn2t,u , ) - -+ : A ' r , rAz) ;Term(r)&" l < uk lu l l , luz l p( lmax(z)1,)8cValue(y t r t .z u,1) Value(y2, , z, u,2) -+ r : Az

Proo f . 'Va lu ,e ( ' y ,n ,z ,u )i s de f i nedas ( l s 1u ) tp (y ,x ,z , s ) , r , vhe re ( y , r , z , s )is a Xg formula expressing the following conclitions:(1) s is a sequence f pa i rs (oo, bo) , . . , (nr r ,brn) ;(2) for every i 1m, either a; is a variable, or:ai is a;-F"cr,1r, here 0 < j,

h { i , etc. lbr S, 0 ancl ,1.;(3) i l a ; is the T- th -zar iab le ,hen b; : (z) ; i f o ; is a i - l - 'a6, l ,henb; : b j -Fbr ;ebc. or S,0 and 'r.;( 4 ) A : b r n a n d r : c L r n .

Clearl.y, f s saLisfying Lh.eseonditions e,rists, i:.en t determines y unicluelyand the conclitions of the proposition a e pr:ovable. irlamely, using induction,one carnprove that every secluence f pails for a su.bterm of tl s a subsequenceof some secttleirce f pairs for /.) Th.r-rs e only have to fi.nd an upper boundto the size of s. Assume s is given. By Proposil , ion 1 (1),

b; ( (rna.r') -t-Z1@;l


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5. Interplebability ancl Consisiency 36 5Thus the a;'s are Godel numbers of subfornrulzreof r r,vith some variabiesreplacecl by nr-LmeralsD for

n ( (max(r ) r z) r ' " tThe engthof such ormulaes estima.tedy

c1 r .l (max(r ) r2) r ' ' 'l r . " l ,wherec1 is a consbanb,ince h.e engthof a numeral s linear n ibsvalue.The engthof (a,;, ;) s thus

c 2+ l ( m a x ( " )2 ) r ' " ' , * " l ,for some constant c2 . Now we estimate the number rn of. uch pairs. We haveat most lr l variables in r, and each variable can be replaced by a numeral Dwith the bound above for n. This gives ((max(z) * z\zt' t * r) l ' l possibi l i t iesand there are ab most lr l subformulae, hence

m 1 lrl 'r. (max( ) + 21zt"t t)l ' l .Thus we can estimate the length of s byca rl(maxr) + 2)' ' " ' l * l r l2,r (rnax(z) -r z1zt'r r ; l* l ( (max(z) 2)" ' ' '

where ca and c are suitable constants. Hence if u is larger than or eclual tothis bound, then there exists some sequences that witnesses the truth of e.The provability of Tarski's conditions is shown as a'bove. n

For a str ing of var iables r, . . . , ,znwe clenote y (q,. .the seqtrerrcert. . . t zn.5.5 Corollary. (1) For every' ounded ormula d(2t,. , . , z.n)a constant k such that

I Do l - . , > 2 (* t * (2 t , " ' , zn )*2 )on (* ( r t " , , . . , 2n )= .1 - (a ,r t , . . ., rn ) ,u ) ) .(2) Let IUIbe a model of IEs,,Iel, a,b,d e IvI,Iet a(z) be a Xo formr-rla.Sr-rppose is nonstanclarcl ncl rs 2kt-12)'t. hen

I U I sa ( a ) : T ( a , ( " ) , , b ) .

\ r l I r :. , z n ) t n e c o o e o t

in trg Lhereexists

atomicd use

nProof. (1) FirsLapply Proposition5.2 to prove he statement or ausing inducbion on the cornplexity of Lerrns rr it. .fhen tbr generzr,lin.iluctionon the logical complexiby f a anclTheorem 5.4.(2) This is an immeiliateconsecluencef (1).


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366 V. Bounded Ar i thmeLic5.6 Theorern. .IXs * Erp is finitely axiornatizable.Proof l,eL T' be a finite fragment 1X6 such that ? proves Tarsiri's concliLio.nsof Theorem 5.4 for -f. Then for every a bouncled, 7' proves the formula ofCorollary 5.5 (1). LeL tp be clefinedby

V(n., l ,, , u) : | (r, (y, p), LL),(Here we are misusing notation: bo be c1-riLe recise, we should distinguishbetween zt" ai,rand a i 'wo elementsequence,bat we do not r,vantbo nbrodricenevr nobal,ion.)Let Tt heT plus Lhe easi nr-rmber rinciple for V(rc,y,,p,r:),wlrere y is the induction variable and z, p and u areparameters, plus Erp. Leta(y,p) be an arbitrary 'bouncleclfor:mula for which we want bo prove the leastn.umber principle (clearly it suffices to have just one parameter). We shallalgue inTt. Let p be given and sr-rpposehat for some y0 we have a(ys,p).Since we have Erp. we can take

u t 2!nax(vo,R))owhere fu corresponds to cv.Flence, .or y 1yo,

a ( a , ) : f ( o , y ,p ) , r ) : V ( * , a ,p , ? r )Thus we can tahe he leasty for g(cv,U,p,Lt).(c) An Interpnetation of J'56 i* QQ is a very q/eak theory, e.g. the associabivity of -F is not provable in ib .Elowever i:om the point of view of interpretabiliby it is cluite strong. We sha1lshow that J'Xg * 9n is interpretable in Q for every n. Moreover th.e form ofinberpretation is very naturzr,l:we take the same operations and only restrictthe dornain. 'fhe domain of interpretation is, roughly speaking, an initial seg-ment of the original numbers. This suggestsan attractive finitist's program,which was pursueclby Edward Nelson fNelson 86]. Namely, the consistency ofQ seemsqtr.iteevident; further it is also evident that interpretation preservesconsistency, bhus f we develop mathernatics only in theories inLerpretable inQ, we are saf'e-gtLarcledgainst inconsistencies. Later on we sha1l see thabrve can even interpret in Q some consequences of I Eo -l Erp which are notavailable in -IXs -F fln (Theorem 5.27 (i)).1

1 L"t us rernark, for those ,vhoare nterested in minimal founda.l,ions f mathemalics, thalon e can use an extremely weak systern fol set theory inslead of Q, since Q is intelpletablein ii. The sysLem has only two simple axioms:

(3c) (Vy) ( -y 6 e) ;(Vc) (Vs) ( :z ) (V" ) (u z =(u o v r ' : u ) ) .

n


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5. InLerpretabiliLync lConsistency J6,rThe main reason why we consider interprebation of Boun{ecl AriLhrneticin Q is , however, different. The interprelLability of systems of BouncleclA.rith-metic gives tts er1'Lticonsistencyf them 'with Q, ancl this ecluiconsistelcy isprovable in Bor-rndecl .::iLJ:meticoo . FIenceGoclel theorem implies bhz-r.Lttcha sysLerncloesnot ar,lready rove consisterrcyo, tq.T'he rest of Lhe sttbsectiorr s alinosbenbirely devotecl.Lo the pr:oof of th.efollor'ving theorem. In the .rhoJ.e ubsection we shall rrseonly inberpreba[ionsgiven by restricting the clomain. Such an inl,erpretation is cleterminecl by aformula r,vith one free variab]e.

5.7 Theorem. For: every n, bhereexists a global inLerpretation of ILg | ,enin 8.By o global nierpretation we mean the usual concepLof interpretabion.The word global s useclhere to stress he fact that we have one translationof the language (but in fact only the domain will be different) sr-rch hatthe translations of eachone of the infinitely many axioms are provable. Onthe othe:: hand, local interpretation means that we can choosea differenttranslabion or every inite subsetof axioms.hlelsonuses ocal nterprebationsin his book. lVe shall usea local interpretation as an inbermediate tep.lVe shall say that a formr-rla(r) is inducti,uen a theory T' if

7 F /(0)& vr)(/(,)* /(s(r)))Recail that / is a cut in 7 if it satisfiesmoreover

I ( rc) ka I x - , t (a) ,prouably in 7. We shall say thab /(z) is a su,bctt'tf /(r) in 7 if moreover

7: sJ(x) -+ .t(r) .Let Q+ b" Q atrgmented with the following axioms:

(associat'iai,tyf a)(Ieft d'istribuiuity)(associutiuiiy f .r)

( r - l a )* z : t * ( a* r ) ;* Q + z ) : r U n z ;@u)" r (az ) .

5.8 Lernrna"There ex ists an inductive formula /(a) which determines aninLerpreLationf Q-F " Q.We shall not provethis lemrna. t can be provedusing similar triclcsas'weshall irsebelow, but iL is technicallymore complicatecl, ince ,ve l,art wibh avery weak l,heoryQ. A conrpleteproof can be fbuncl n lNelson86l.


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368 V. BoundedArithrnebic5.9 Lemryra. et .I(c) be an ncluctive ormulain Q. Then bhere xisbs sr-rbcutJ(x) of I(r) h Q.Proo!. We sha[].tse he following three easy heoremsof Q:( i ) r ( 0 ,f i i ) n 1 u : 5 ' ( e )< ^ 9 ( u ) .v / ,( i i i ) 0 < r.LeL I be inductive in Q. Let, I( be clefined by

K ( t ) : / ( t ) k ( Y a , r ) ( t < a k y ( r - > z . - r ) .We shall show that r'( is inch-rcbive.f(0=) ollo,,vs asily from (i) andfrom 1(0).Suppose Ii(r). Then, clearly, /(S'(z)) and it remains to prove the seconclpalLof t l re concl i t ionforK( .9(r ) ) . Let z I y < S(") . I t z :0, then, by ( i i i ) , z 1 x.Otlrerwise, by (Q3), , : S(" '), for some z/. Then, by (i), (Q5) and (Q1), ycannot be 0. Thr-rs : ,9(yt), or sorneg'. Thus we get z' 1y' ( ir, using (i i).By /((c), we have z' I r. Using (i i) once again, we get , : S(r ') < S(").Hence /( is inductive.Now define

J(r) : (Vy < r)Ii(y) .First we shall show that .I is inductive. /(0) follows easily from (i). Assume./(r) anct et y ( S(r). We neecl o show that /f(y). If y : 0, then I{(y),since y' ( is inductive. Suppose that g : S(y'), for some y/ . Then, by (ii), wegel At ( r, hence ((yt). SinceK is inductive, K(S(A'D, which is /f (y). ThLrsK is indr-rctive.Now we shall orove that

J ( r ) 8 t u S n - J ( y ) .Strppose(r)ky l x,Let z ( 9 be arbitrary.We needLoprove {(z).ByJ(r), we have in particr-r lar {(n), hence z 1r. Thus we have K("), sinceJ(") . n5".10X,ernrna.Let /(r) be an inducti'rre formula in qr-. Then there exists acu t "/(r) such that J is a subcut of -I closed under 5, -l - anit +, i.. Qr proves

il;.1,;:,';:'-(s(.))(, ,y) (.y)Proof. By Lemma 5.9 we may assume that 1 is a cut. We shall use the sameconstruction as in the proof of Theorem 3.5, Chap. III. Take

/o(")= (Vy)(r('u), J'(y ")) ;J(x) = (VsX o(y)+ /s(yc) ) .


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5. Interpretability ancl Consisbetrcy 3 6 9Then -Is conbainsT, since / is inducLive; ,16 s closecl ncler * by associativibyof -l-; .I is conta,ined n ,,/6 (since ,/s contains T) ancl hence also in 1; J isclosed trncler"J- by left distr ibutivi ty ancl since./6 is closecluncler -l ; J iscloseclunder 5', since t contains T ancl t is closeclun.cler-l ; J is c.loseduLnc].er'r by associabirrity of 'r.. To sho,,v ha t J is closeclclownr,varcls, e use t,heer.ssrurtpl,ionher,l, is tr cuL, le.fl,disLribr-rtivibyzr,nil .I 9being closecl rncler -F.

5.11 remrna. f6 is locaily nLerpretablen e..Proo f . y l ,emma .8 ve an ahe8-F nsteado fe .Le t ,g7 r ,p ) , . . . , ,pn ( r ,p )be given bounded formul.ae.We want to interpreb Q plus inclr-rctionorp t ( n , p ) , . . , p n ( x , p ) n Q + . L e t

I j ( r ,p) = pt(0) (Vy < 4(vi(a,p) * ,p iTfu),p)) , e @,p)f b r T : 1 , . . . , n . L e tI ( " ) : (VpX/ r ( r ,p ) . . . k l " ( r ,p ) ) .

Sinceeach /i(r, tr) is inductive with respect o z in O*, so is /(r). By Lemrna5.10 we can find a stibcut J(z) closeclunder * and *. Since J is cioseclunclerS, -F and 'r and it is closed downwar:ds, we have, for: any borrnclecl formuia* ( z t ' , . . . , zk ) ,Q - F - J ( " i A . . . k J ( z p ) - - + a ( 2 1 , . . . , 2 n ) : ( * ( r t , . . . , r k ) ) . r ) .

SinceJ ( r ) - -+ I ( r ) - , I i@)

provably in Q+, for e'uery , J determines an interpletation of incluction forevery t1:i. AII buL two axioms of Q (even Q+) are open, hence they hold in theinLerpretation. The remaining two axiorns are (Q3), " #6 -+ (3y)(r : 5'(y))and (QB): the definition of the relation (. The first one is provable usingincluction for bounclecl ormulae from the others (hint: first prove r 15(r),then r *0 -, (!y S r)(z - S(y))). Hence we can simply suppose that wel rave enough formulae amonEstet , . . . ,en to prove b. The last one causesno problems, since we can trivially interprel, Q in the theory obLained fi'omQ bV eliminating ( from the language and deleting (QB) nProof of Theorem 5.'7.By'I 'heorern 3.5, Chap. I II, we know that each cut ina sufficiently strong bheorycan be shortened to a cut closed under un,for agiven n. One czr.n asily checlc hat IXg is sLrongenough for that. So, by bhelasL lemma, lve only have to interpreb 1tr'6 in a finite fragment ? of J'Xs. lVetzrlte? so stronp. hat


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5. Interplebability and Consistency 3'7

is only one place w.here \,vehave to modify Llr.e nterprel,atioir given. above.When we prove the leasbnr-rmberprinciple tbr:a boundecl formula a, we defineparameter u by

u : 2 ( ^ o * ( y o , p ) ) " .Now we have to take into accounL ho.v k clepencls n cl, since a is nol; [-irer].Thus insteaclof the double exponential frinction in the definition of f(r.),.,vetake triple e,'(ponential.Flence

Q r I ( r ) - , ( l y ) r A l k a : 2 " ' \ ,\ /where cr-Lt I(y) determines an in.terpretation of a sufficiently strong finibefragment T o,f. L'g in Q. Then we replace number k in the defi.nition of u bya more explicit bouncl given by Theorem 5.4:

u : 2 ( ^ u * ( y o , p ) ) " 1 ' lwhere c is the constant indepen.dentof a from Theorem 5.4. No'w, using theestimate

2(max(a0,il)"' ' S rzzmax(vo'n)* la l

one can show thab u exists and "f(u) holds true, and the proof's of bhese acbsin Q arc of polynomial length in the length of a. llence the induction axiomsare replaced also by polynomially long secluences. n

(cl) Cut-Elirninatiom.anclHerl:rand's'Xheorernin Bounded ArithrneticThere are two basic measures of the complexity of proofs: the length of proofsand the cluantifi.er cornplexity of proofs. Cui-eliminabion and Herbrand'stheorem implies that each irst order taubology a has a proof whose cluantifiercomplexiLy is not larger than the cluanLifiercornplexity of a. The cost for lhereciuction of the cluantifier complexity is an increase in the size of Lhe proof.T'his is the reason why this theorem is not provable even in /Xg { Eap inful.l generati|;y.Never:theless h.ere s a restricted version of this result whichis pro.zable in lXo + Erp and which has several applicaiions. We shall proveit in this subsection. The proof will be just an analysis of a classical proof ofcub-elimination. T'here is an alternative proof based on Flilberi, style caictLli,bub we shall use a variant of Gen.tzen'sapproach since t is simpler and moretransparent. The transformations used in the pr:oof will again be some simplemanipulations with secluences, ence clefinable by Io formttlae. Thus we onlyhave to estimate the size of constructed proof's.

lVe shali 1rse a slighLly simplifiecl version of Schr,vichtenberg'ssysl,em[Schurichtenberg77 ] and follow his proof of cut-elimination.

The proofs will


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37 2 V. BouncleclAritb'mebic

be on1.y riefly slcebchecl;o.arever e do nol, presupposecnowleclge f pr:oofl,5eory. or more clelaiis he reaclershotLlclonsulL Takeuti B0]' It shodd bestresseclere hat the branslationsi:om JlissysternLo he systern ntrocluceclin Chap. 0 anclbaclclvard"s].ecFriteelementary. n parbicular (assuming ef-ficient cocling,which was ntroclr-rcecln Sect.3), it is possill lebo prove iir/Xo -l-CI1 haLa sen'cences proval:le n onesystem ff its translation is prov-ahle n the oLhe.r ne.)l i z > " ' > j ^ ) ,-t ,wher-e 2'" clenotes2,t, (2,F ' ' :F )) . Think of t; as a modifi-eil ntrmeral. Then

follow the stanclrrra ofgo'rinms fbr -l-' and ,r to obbain Lhe proofs of (e.1-3)for the rnodtf,edn.umerals.Finally bransform the modified numerals back intoor ig ina l nu,mera l (S(" r ) ) ,numera l (n t1-x2) a;nc| 'nu,meru l ( r1 ' r .12)espect ive ly .As is easily shown by induction, the maximal of these three bounds (the


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38 2 V. Boundecl AlithmeLic

bouncl fo r the lengbh of the proof of (e.3) is asymptotically maximal) is abouncl to the pi:oof of the eclualiLyo. [general terms. To prove Lhe neclttality,balrea suitable term tr ) Prove

t ( i t , . . , i " ) f - 5 ( u ) s ( z : 1 , '. , i n ) ,o1' o ( t r , . . , i n ) * , 5 ' ( T . r )t ( i t , , . . . . , i n ) ,and r-rse he act that

Q t s ( r ) U t ( a ) - + x - r S ( Y ) / " . n

5.23Lemma.Thereexistsk such haL5; F (Vy)RPr"q(8,,(vc)(r nu,meral ' (yr 1) = ( r l nurnercr l ' (v)V

n : n u r n e r a l ' ( y + 1 ) ) ) ) .

Proof. We consbruct a cu t / in Q such that( " . a ) Q | ( Y r , z ) ( I ( z ) * ( r < S ( r ) = ( r < z Y r : S ( ' ) ) ) ) 'First take "I definedbY

J(* ) :a f 0 + r : x k(Vz) (S(z)* r : S(z + t ) ) .One can easilyshow hat ,/ is inductive. Now take I to be a subcutof J anclverify (".a).trVeshallwork in 5']. Let anumber g be given.By Lemr''aS'2'Lwe haveQ I l(numerat(y)), henceby (e' )

Q l * 1 S ( n u m e r a l ( y ) ) = ( *< n u m e r a l ( v ) V r : S ( n u r n e r a l ( v ) ) )B v L e m m a S . 2 2 Q l m r ' m e r a t ( y + 1 ) : S ( n u ' m e r a l ( g ) ) ' T h u s Q p r o v e s

( V r ) ( c n t t m e r a t ( A - , r ) ( , l n u m e r u l ( y )V r : n \ L r n e r u l ( y+ 1 ) ) ) .All that we ha.zeusedwere proofs of boLrncleclut-ranlc. n

For the lbllov,ringemma, 'weshall fi.xsome ranslation of L2 into tro which.clel,erminesn inter]p::etation ,fBASIC 01]solnecut in Q. Sucha translationexists,since IDO f fi t is interpretable.in Q by I'heorerr 5.12.We shall saythat a formula 4; o, fLg is essentzalty! (resp.n!r!,it it is the bra-r.nslationfa Xf (resp fIP) formr-ila ! Lz .


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5. Inierpletabi lity and Consistency 3835.24Lemrnzr formalized completeness f 8) .( i) '.Let1.,(r,g) b" abottnded ormul.ain rs. Then there existsk sLrch hat

I Eo r Enp (Vr)((rv)@,v) , RPrfi(E,1=v1,1,1,t,/)))( i i) LeI $(x,y) be esseniiallyXf . T'henbher:existsk such hat

s} t- 1vr;((1y),b@,y), RPriT(E,(1a),h@,y)))

Proof First we observe thab in both cases t suffi.ceso prorzeonly(Vr)(t!@,,v) EPr'q(E,rh@,ilD

since (lilrh@,y) will tollow by an applical,ion of the rule (1).(i ) The task of prolring ,b@,il is similar to the taslc of decicling its truth:'weshall prove ,b@,y) by proving numerical instances of the open patt o;f. f; .

Thus v,/e an bound the size of the numerals needed n this proof in the sameway as in ihe proof of Theorem 5.4. The numerical instances of the openpart of ,r/ follow frorn the nurmerical instances of abomic ancl negated atomicsttbformulae. The existence of bounded cut-rank proof's of atomic sentenceswith - and their negabions ollows from Lemma 5.22.'Ihe atomic sentenceswith ( and thei:r negations are reduced to atomic sentences with : in thesame way as the negations of atomic sentenceswith : had been.We must be a little careful when pasting the proofs of the numericalinsbances into a proof of a bounded sentence, since we n.eed a proof ofbounded cut-rank. A sentence of the fbrm (ly < t)*(A) follows from sorrle*(tt), where n is less than or ecltLalo the value of f . A proof of (Vy < t)t(y)is constructed from the proofs of a(0), t(T), . . . r*(m), where rn is the valtteof 11, s fol lows. We prove successively Vy < n)*(A), for n - 0,.. ' ,rn' Forn : 0 iL is an immecliate consequenceof *(0); to obtain it fbr n + 7 fi'orn aproof lo r n, use Lemma 5.23 ancl a(rz * 1). F'inal1yuse Lemma 5.22 to showQ ts vv< t)*(y)= (Vy


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5. Intelpletability and Consist,ency ;lBs

By l,emma5.21 I Eo j- ErTt (Vr)fi Prft(F, r(r)) ,for some constant /c .Hence

r Eo + Erp F (Vz)R/,r [ (T,{(r)) .By T'heorem5.20, IL'O * Erp proves he con.sistency ,f.Q with respecl,boproot,sof cut-ranlc /c.Thus we haveIEo l ExpF (V,)- ;P"h(T,-rh@D.

By Lemma5.2a i)rLlo -t-Enp (Yr)(-n[@) t RPrq(m,'-{(i)))

fo r some m. We can asstlme that k : nr , (otherwise talce max(k,*) insteadof /c and rn). Flence (3 ) follows from the last two stateme.nts.(3) + (1 ) Suppose "IX6 * Eap | (Vr)rl',@). BV the lemma above we havesome k such Lhat

IEo F (Vr)((ry)(a 2f) - ,h@DTake ctrts -I, J in Q such that ,.I determines an inberpretabion of I\s in Qand -I is a strbcut o, f J such that

Q I ( x )* ( 3 y ) ( y2Xk t ( a ) ) .Then we have Q F (Vr)(I(r) --' ,h@)). tr

5.2? Tlreorem.Let {(n) be a boundeclormr-tlan t rs. Then the following areequivalent:(1) IEo -r Exp | (Vr)/(r) .(2) For some c sr1 _1vr;Rpr"q(E,r l r@)).I:t{,,(r) is the translationof a.trf forrnula p(r) into -[,s, hen (1) a,nd 2) areequivalent to:(3) For somek Si +-RCon'q(k)F (Vr)e(r) .

ProoJ.Q) =) (2) Suppose D1 t Ex)p - (Vr)/(r), r lt botrnded.Their bvT,heo|em .26we haveQ r (vr)( .1(x) rh@D br sorne ut - I i r . Q.By Xrcompleteness s') nprb(L (v"x/(r) -, /("))) ,


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38 6 V. Bouncled Alithmetic

for some .BV Lemma5.21,s$- vr;RPr$(m,/(r))

for so:me n. Talce& to be the m.aximumof 7 and rn, then we ha,ie (2).(Z) --- (t) Bv definil;ion

(".5) s').r RCon$(hF (vr)l iPr'q(T,rb@D' (vr)- 'Ji rf i(T, , . / ,( i ))By Theorem5.20,RCon'q(E)s provablen /Xe "+'Erp, hus (e.5) s provablein. [Eto - Erp. Sincewe assum. 2), we have

IEo l Exp s Vr)-.g.Prb6,-.rh@DBy Lemma5.2a( i )

I Eo l Eap s Vr)(-ftPrb(T,-r/(")) , ,h@DFIence 2) implieu 1).Now assttmehat ,r/ is the translationof a l7f formula cp(r) into ,ts.(Z) =+ (3) BV Lemma 5.24(ii) we have

5',21(Vr)(-RPr$(E,-,rh@)), ,!@))Then using (e.5)we get

s) -vBCon$(E) (vz)/(z)from (2). But 9@) is ecluivalent o {:(a) in S}, thus we have (3)-(3) + (1)'Ihis is because )lo* Erp "contains" S$ and,provesnCon"q(E).n

We wouldlike to axiomatize I/1 consec{ttences! IEo* Enp over' say, S'}(put oLherwise,we want to find some nice basis). Tlie ecluivalenceof (1 ) with(3) gives only a partial answer:since RCon$(E) can be written ai,s formula ofthe fbrm (vr)/(z) with { in n!, the s,etof senbencs RCon"g(T), lc : 0, 1, . . .axiornabizes he Vj?f consecltlences f I Es + Etp-

(f) Incompleteness heorernsIt wouicl be paraclo,ricalf the incompleteness heorems did not hold inwealcsysbems.Of course ormalization of syntax in weaker theories is amore clifficuli l,aslc,hereforewe cannot apply classicalproofs of the secondGodel incompletenessheorem cluitedirectly. IL bur-ns ul , that Lheseconcl


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5. Inlelpretability ancl Coirsistency 387

incompleteness heorem an be even sLrengLhened. typical exatnple s theunprovabiliLyof the consisbencyf Q in IEo -FEap. The resultspresenbeclnthis subsectionwill be corolla:ries f bh.eheoremsproved aboveand of Lhefollowing theorem.Iiobe bhab n the following heore* (i) holds for every iniiely a.riomal.izedsecltreniialheory containingQ, SNc hall no t pr:ove his generalbheorem ere.For finite theories n bhe anguage f arithmetic exbenclingX1, staternent i)'followsfrom Theorem 3.20 1) Chap. III. Statement (ii) gener:alizes lreo-rem 3.1..1, hap. III, to weaker heorieslhowever he assumption bonL irea,riom.abizabilitys sLronger ere. t is possible o use here he sameassump-bion about the ax,iornatizabilityas in Cher.p.II, but we shall not pro're thisstrengLhening.Recall hat -BCont' (T) is

(Vz)(t(z)kc-ranl , : ( t )ST, -n -,Proof| (" . ,0=T)).

5.28 Theotr'em.(i ) For every k there exisbs a cttt I in Q sttch thate I RConSD;

(ii) Let ? be a consistent theory containin"gQ ancl having a Xf axiornati-zation. Then for every cut, J in 7 there exists rn such that-(? l- ncon?f@)) .

ProoJ. i) Bv Theorem .20 ii), for everykIE o + E u p l RCo n ' q ( E ) .

By Tlreorem 5.26,, his implies the fi.rsbsbatement.(ii) The proof of this sLaternent will be similar to the proof of Theorem3.11, C[ap. I11. RoLrghlyspealcing,we only have l,o talcern su'ffrciently arge.Since bhis resurlbplays zr, ey role in the rest of this section, we shall pr-ove tin detai l .Because of the shortening techniclues,we may stlppose thal, ../ has sorTleactditional properties. Thus 'we shall assume that J is a cut in Q and itcleterrninesan interpretation of S$ in Q.In particular, i t is provable in Q,lrence also in T r Lhat for any tr,vosequences n -I, th.eir concatenation lies zr,lsoin J.'We'wouicl lilce to use the provability condibions of 2..16, Chap. III, fo r.RPr.y(m,,r). We shall prove the first tr,vo.Th.e third one, which is folinerls'zecl'.tulorl,usonens, is clearly false, since Modu,s Ponens increases he cr-rL-ranlc.Thus .,ve hall first take a sLriberblez, and Lhenwe shzr,llollow the usuzr,l roof


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38 3 V. Boundecl Alithmetic

of the second n.completeness lr.eoremand check that ihe thircl condition isactu.ally appliecl only Lo formutlae of ranlc at mosl, rn . \vVeshall clivicle theproof inbo several. laims.Clalnr 1" For ever:y' entencep, if T F tp, bhen T F CFPr'{ f'a'.Proof. Su.pposeT I p. By cut-eliminatj.on vre ha.re a cLtt-free proof of cpinT. By the I-'-completenessf Q, Q F CFPr+(A).Since .Icleterminesn

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