Alan Skelley

A t hesis submit tecl in conforniity mit h the recpirements for the degree of Alaster of Science

Grachate Department of Compiiter Science C'niversity of Toronto

Copyright @ 2000 by Alan Skelley

National Library I * m of Canada Biblioth&que nationale du Canada

Acquisitions and Acquisitions et Bibliographic Services services bibliographiques

395 Wellington Street 395. rue Wellington Ottawa ON K i A O N 4 Ottitwa ON K I A ON4 Canada Canada

The author has granted a non- L'auteur a accordé une licence non exclusive Licence alïowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sel1 reproduire, prêter, distribuer ou copies of ths thesis in microfonn, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de

reproduction sur papier ou sur format électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts f?om it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.

Abstract

Relating the PSPACE reasoning power of Boolean Progranis and Qiiantified Boolean

Formulas

Alan S kelley

Ilaster of Science

C;raduate Depart ment of C'onipiiter Science

Ilni\.ersity of Toronto

2000

WC present a nrw propositional proof systeni basecl on a recerit new charaçtcrizntiori of

polynoniial space ! PS P:\(l'E) callecl Booleari Progranis. due to Cook niid Sol tys. 1\é show

tliat tliis rieir iJvsteri1. BPLli. is polynoniially ecpi\.alrnt to the systmi G. wliicli is Iiased

ori i l l e faniiliar arid wry tliferent q~1antitic.d Boolean forniulii ( Q B [: ) ctiaracterizatiùn of

PSPAC'E dite to Stocknicyer ancl lfeyer. \\*e conçliitle w i t h a clisciirsiori of sotiic closely

relatecl open probleiris and t heir implications.

Acknowledgement s

Thanks to my parents for being not so bad after all. A nod to NSERC for greasing

the wheels mi t h PCS.-\-2OS'26-4- LWY. SI!. officemates Steve 5 tevenator" Slyers. Iannis

..,'\sioni" Tot~rlakis. .John .*.\Toonman" L\at kinson. Jonathan .-.-\ninial" Shektec. ,Vatasa

-5 t ash" Prziilj and Eric .*Do. .J" Joanis For many helpftil discussions and productive

distractions. Iileoni Ioannidou for moral support.

Tsuyoshi ltorioka for helping ivit h sonie protliiction cletails. Uictiael Soltj-S. for t kit.

topic. .\[J. second rraciér. Tmiarin Pi tassi. for readiiig iincler duresu.

Esprïiall!. rlia~iks to rri?. super\-isor. Steptien Cook. [or coiiiitless helpfiil (lisciissioris

arid crucial iileas. nut to iiieiitiori a lot of restling and corrcctirig.

Contents

1 Introduction 1

1.1 Background and Slot ivation . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 O\-ervit'lv of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Preiiminaries 5

1 Propositiorial ProofSystenis . . . . . . . . . . . . . . . . . . . . . . . . . - r )

2.2 L I i and Qiiantifird Propositiorial Logic . . . . . . . . . . . . . . . . - - . li

2 .3 Boolrnn Progran is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s

2 . - S o t n t ional C'onvent ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . il

3 BPLK and C; 11

3.1 BPLIi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . i l

3.2 Basic Results on BPLIi and Ci . . . . . . . . . . . . . . . . . . . . . . . . 12

4 BPLK P-Simulates Ci' 1'7

4.1 Special Sotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . - LS

4.2 -4 Translation from the Language of C; to that of BPLK . . . . . . . . . 19

.Pl 4 .-\ Simulation of G' by BPLIi . . . . . . . . . . . . . . . . . . . . . . . . . --

5 C; P-Simulates BPLK 29

-5- 1 -4 Translation from the Language of BPLIi to that of G . . . . . . . . . 30

5.2 A Simulat ion of BPLIi by C; . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Future Work and Conclusions 42

6.1 -4 Technical Irnprovement . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6.2 Kitnessing and Search Problems . . . . . . . . . . . . . . . . . . . . . . 42

6.3 Subsystenis of BPLIi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - t3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Mscellaneous 43

Bibliography 43

Chapter 1

Introduction

1.1 Background and Motivation

\\E o l ien argile t tiat a part iciilnr mat heniat icnl concept is in iportant i f it is riat

nicaris rhat i t siirfaces iri nian! places ~ v i t h ilitferent origins and (Irtiriitiuns. arid rol)ti';t.

siicti t hat a variet?. of disparate formulations u l it end iip heing zqiii~-alent ut- at lrast

closely relatecl. Liketvise. the applicabilit?.. mat iiri t'.. a n d iniportarice of a body of r~si i l t s

are greater when t hat field is foiind to have a st rong conriect ion to anot her. Tlirer areas

of stucly intricately connected in stich a ~iseful ~ v a y arc çompritational cornplesity. the

proof t heory of arit hmetic and propositional proof coniplesity.

Computational complesity is the study of conipiitation and the resources requiretf to

perforrn it. .A staggering nimber of different kinds of computation ail fa11 into the domain

of t his field. it ha5 pract ical aspects. directly impact ing how real computations are done

by real cornputers. and yet seemingly fundament al. easily esplained problenis remain

iinsolred despite a yood deal of effort. -A particularly glaring esample is the famous P

v s NP problem. which asks if t hose two classes of problerns are equal. Starting froni the

XP-cornpleteness results of Cook [12] the pressure mounted with no relief. leading even

to detailed. forma1 analysis of known proof techniques and why the- are al1 ineffectual at

tackling siich problems [El. .\Inn' cornplesity classes are stiidied and conjectures about

separations and hierarchies abound. yet resiilts are elusive.

--1 different may of studying computational cornplesity is inciirectly through logic.

Nany connections betiveen the fields are known: cornplesity classes can be characterized

as those sets or fiinctions definable in certain theories: sets of nioclels of forniulas cari bc

2.-2 2 Lnrrti-ne< or c!25ses of !i~::go2zps: prF&?wtes or Li.-tiî::i Froxi ce.rt.lin cÿE:oI+--;~~- OLCCCaL- L y".""

classes cari be ~isecl to ilefine nerv logics. .-\ relevant esample comprises the hierarchies

of throrier of boiindrd arit hmetic Tf, and 3 of Biiss [-LI. As shorrri in [ 6 ] . [?SI ancl [22].

t h bouncled arit hnietiç hierarchy collapses i f and only i f proves that t h e polyrioniial

tiierarchy collapses.

pal>-rioniial-sixcd families of proposit ionnl f'orniiilas. Fiirt hrrriiorc. if t lie I)ociriilt.tl ari t h-

metic forniiila has a proof in C'ook'i systeni PI ' (correspondirig to pol>.nornial-timr rra-

soningl. t hen its translations have polynornial-sized estencled Frege proofs which can be

fourid in polynomial tinir. \.\ can replace P l ' by Ci i r i the previoiis staternent due

to hot h t heories robiist ly tlcfining polynoniial-t inie rrasonirig. t tiougti i n difkrent tvays.

Other translations are known ancl in partictilar there is a siniilar connection betwen

T: ancl C;,. and anot hcr bctween 5 and C;;. bot h due to [Z 11. There is another corre-

spondençe [ 2 3 ] betiveen ( a seconcl order systern of Buss') ancl G. aithoiigh only for

f i rs t -der . 1; formulas. In al1 of these latter correspondences. i t is also the case that

t h e boiinded arit hrnet ic dystem can prove reflection pririciples for. and t hiis sirnulate. the

proposit ional systeni.

The full circle back to computat ional cornplesity is completed wi th the work of Cook

and Reckhow in [LOI and [Hl. The? show that P=co-NP if and only if t here esists a

polynomially bounded proof system. and additionall- introdoce many of the important

definitions in the area such as those of proof systerns. polynomial simulations. and so on.

These results drive the study of propositional proof complexity and the search for lower

bounds on propositional proof systems. Fine eramples are the superpolyriomial lower

boiinds for resolotion. due to Haken [16i and bounded dept h Frege systems. due to Ajtai

[ I I . For many seeniiriglj- st ronger systerns. however. no such results are known.

1.2 Overview of Thesis

The ii1t.a stiggest ing t lie resiilt s i ii t his t liesi is yet mot lier conrier t iori bzt ween corn pu-

tat ional coniplesity and proposit ional prool coniplesity. Khen forniulated in n Gentzen

sequent style. rnariy known propositional proof systenis c m be seen to be very siniilnr.

wit 11 the only ditference betwern theni being the computational poiwr of what can bc.

writtrti at zach iine of the proof (or alternativcly what is allo~vecl in t h e cut rule). Esani-

plrs are Booleari formulas in Frege systcms. single litrrals in resoliitiori. Boolean circiiits

i r i rsteiided Frege systzriis. ;\riut lier rsariiplr is t tir systerii G. ivliicli is a sequerit-lmed

s!.steni where formulas in the sequents art. qiiantified boolean formulas ( Q B Fs). These

formiilas have proposi t ional mriables and also proposit ional quant ifiers. In t his casc.

t lien. si rice e~~nluat ing Q BFs is PSP:\C E-complete. t h e computat ional power which c m

h r harnrssed i r i sequerits is PSP.-\CE. \\> cari rest rict Ci to C;, by rest rict ing the nurnber

of alternations of quantifiers allowecl in the formulas. and the reasoning pon-er is then

that of 1: predicates.

Boolean progranis were introducecl by Cook and Solt-s in [1 II. -1 Boolean program

defines a sequence of Boolean function syrnbols. ivhere each function symbol is ciefined

using a boolean formula which can include. in addition to the arguments to the function.

in\-ocations of the previously defined syrnbols. The aut hors of t hat paper showed t hat the

problem of ei-aluating an invocation of a function symbol defined in this way. given the

inputs and the Boolean program. is PSPACE complete. The question that then arises

is whether a proof system formulated around Boolean prograrns ivould be equivalent to

G'. For this to occur. not only would Boolean programs and quantified Boolean formulas

need to charactrrize the same cornplexity class. but t here would need to be an effective

mqv ol translating between the two.

This thesis answers that question in the affirmative. After revietving basic termi-

nolog- and notation in chapter 2. in chapter 3 we define oiir new system BPLIi in a

st raightfortvard way to take advantage of the expressive potver of Boolean programs. In

that chapter ive also prove some basic results abolit the two systems in consideration.

Chapter -4 contains the first of the main resiilts. which is a polynornial simulation

of C by BPLK. \.\ tirst shoiv how to translate sequents [rom the langiiage of C; into

eqili~.alerit seqiients i r i t kir langiiage of Boolran programs. As ive discuss. the t ranslat iori is

not nierely the Skoleniization one rnight espect but rat hcr somet hing niore sophisticated

and reminiscent of Hilbert's c-calculus. Following t hat ive show Iiow to sirniilatt. G b ~ -

t ra~islat irig a proof in t hat systeni. line-by-line. intu t he langage of Boolean progrmis

arid tlicn filliny in t h e gaps to niake the resiilt a proof in B P L L

Chapter 5 presrnts the converse simulation. The trarislation usctl here first tnkes a

Boolran prograni to n single forniula which ni-. be iised to siniiiltaneoiisly eultiate al1

fiirictions tlefined bu that progrnrn. This formula is iised to evaluate Fiinction symbols

occiirring in the original BPLIi-proof and yielcls a translation of seqiieiits. As in the last

chapter. a line-by-line translation followetl by sorne filling in of gaps giws the clesirecl

resul t .

Cunclticling. in chapter 6 we disciiss some open problems and other issues raised by

these results.

Chapter 2

Preliminaries

111 ttiis cliapter ive prrsent somc lorrrinl background about proof sj-steriis. which we first

forrnally cletinc. \\é present Gentzen's popiilar seqiient- based susteni L l i . which is the

foiiritlat ion for the ttvo proof' systenis compareci in t his t hesis. a ~ i d also tlisc~iss quant ifiecl

propositional logic and the system C; and its siibsystenis. Finally. we conmient on sortie

notational con\-ent ions ivhiçh we shall use.

2.1 Propositional Proof Systems

K e shall corisider a langiiage consisting of the cornplete basis . . A } . parent heses.

constants O (for false) and 1. and an infinite stipply of a tom symbols which ive shall

represent [vit h n variety of loiver-case let ters. In the standard lia!: well-formecl formiilas

in this language define truth functions or eqiiivalently. Boolean functions. of the truth

values of the atoms. TACT is the set of proposit ional tautologies. formulas which evaluate

to true on every assignment.

Definition 2.1.1. -4 proof system P for n .set 5' is n surjective polynomial-time com-

putnble function P : S' i 5 for some alphabrt 5.

We are interested in proof systems for T-UX'. A P-proof of a tautology r is a string

a ruch that P(J) = Ï. We denote by la1 the number of symbols in 7. We have the

following important notion which allows iis to compare the power of proof systems:

Definition 2.1.2. If P and () are proof s y t e r n s . irc suy that P pal!-noniiaily simiilates

(p - s i rn i~ ln t~s ) Q and u ~ r i t ~ P S p Q if ' thtre is u polynonriaf-tirne cornputuble function g

such t h d for E C E ~ ~ t r i n g S. P ( g ( + ) ) = Q(+

2.2 LK and Quantified Propositional Logic

.-\ popiilnr proof system is Gentzen's sequent system LI?. LI\: is actually a proof s p t e n i

for pretlicate logic biit ire shall consider only the propositional fragment. Each line of a n

L Ii-proof is a sequerit. a string of t he forni r - 1. tvhere r aricl 1 are possibly eriipty

firiitr seqiirnces of propositional formiilas. .-\ secpient is satisfiecl i f and only if eit tier one

of the forrntilas on t h e left ( t he anf~crr lerz t ) is falsifiecl. or m e of the fornitilas on the.

right ( t h e succçderlt) is satisfietl. Each seqiient in a proof is either a n initial seqiient of

the forrn O -. i 1 or cr - ci for an atom ( 1 . or i t is clerivecl lrom previoiis ones ( i t s

hypot tieses ) 1-ia one of the following infwerice riiles ( t his set is: t h e same as iii [y]. ivliirli

is a slight niodificat iori of the ones in [ZO] ):

weakening:

exchange:

rl. -4. B. r2 * 1 left rl. B. -4. r2 -

contraction:

r + &A. B.& and right + Al. B..4.A2

CHXPTER 2. PRELIMIXARIES

1 : introduction:

r + 1. -4 -4.r - A left and right

TA. r - 1 r + A. TA

: int rochict ion:

'\/ : int rodiict ion:

- r - 1 B. r - 1 r - 1. A. B left and right

A v E r . + A r 4 1. -4 v B

cut:

Quanti fitd proposi t iorial logic is what resiilts when ive adcl propositional qiiarit i fiers to

our language. The semant ics of Vso(r. p) is char t his Formula is sat isfirtl by n part iciilar

assignrnent if and only if O( O. B) A O( 1. p ) is. Li kewise r lie t rut h value of 3+o( +. P) is the

same as t hat of 0 ( 0 . F ) v O( 1. p l . -4s in [TI. when in the contest of cpntified propositiorial

logic ive shall divitle variables irito boirnil ritriable..; and Jmt r n r i n b l ~ s . Frer mriablrs nia!

oçcur free in fornitilas and semiformulas. but r n q never he cluantifiecl. Bouncl variables

nia! occLir Freely in seniiformulas. and ni- be cpantified in formulas and semiformiilas.

Secluents are constriicted from Formulas.

-4dditionally. we can define a hierarchy of quantified Boolean semiformulas. The

folloming is a slight adaptation of the definition in [-O]:

Definition 2.2.1. The classes FIf and Xy are d e j n e d as follows:

1 . X: = ng are the quantifier-free propositional serni)ormulas.

2- Ifo is Xf or II: then it is abo 5; and rI; for ail j > i.

'7 ' I fi. dl

7. Zf

Sow.

with the

r i r d ff: (LE closed under. v {mi ,\.

(Il:) is closed under eristentinl (uniceianl) quantification.

the proof systeni Ci' is obtaineil by atigmenting the set of iriference riiles of LI\:

following:

wlirre B is an' formula and the atom p replaced does riot occiir in the conclusiori of the

corresponding inference. C;, is G with the restriction that al1 formulas appearing in a

proof mtist be XI or nl. [t shoiild be noted that although C; arid its siibsystems derive

tatitological stûtements of quantifiecl propositional logic. in this thesis we will consider

t hem only as proof systems for proposi tional tautologies.

2.3 Boolean Prograrns

Boolean proyrams were introdoced in [l l j and are a wal- of specil~-ing Boolean fiinc-

tions. I t seems that perhaps representations can be reniarkably (even esponentially)

much shorter than with Boolean formulas or circuits. although a forma1 proof would be a

breakthrough. Boolean programs are somet hing Iike a generalization of the technique of

using new atoms to replace part of a Boolean formula. which idea is the basis of extendeci

Frege systems. The following definition is froni that papcr:

Definition 2.3.1 (Cook-Soltys). -4 Boolean Program P is specified by n finite a q i c e n c e

{fi. .... f,,,} o f function syrnbols. where ench qnlbol fi has R I Z associat~d nr-ity ki . r i r d (cri

«s.wci«frd dcjning equution

~rhcr+ jï, is ( 1 list p i . . ... p k , OJ iwrinb1e.z ( m l -4, is u /orni itln al/ of irhost rnritrblea (ire

«mong nnd d l of irhosc Junction s yn~bob (ir+ unlong fi . .... f , - l . Iri this corltcrt the

dch'nitiori of « formula is:

2. I J / i i a k-riry Jiiriction syrnbol in P nnd Bi. .... Bk air for~iirla.~. t h t n j ( B i . .... Bk)

The scrtiantics are as for propositional formulas. escept that rvhen rvaluating an

application fi(;) of a fiiriction syrnbol. the value is clelined. iising the defining eqiiatioii.

to br .-l,(o). There is no free/boiind distinction between variables in the langiiage of

Boolean prograrns.

An interesting property of Boolean programs which demonstrates their comparability

to qiiantified Boolean formiilas is the following t heorem from [L 11:

Theorem 2.3.2 (Cook-Soltys). d Language L is in PSPACE if l L is cornpated by

some crniform polporninl s i x family O/ Boofmn programs.

2.4 Notational Conventions

iSé shall use the follorving conventions of notation: Lorver case English letters will repre-

sent atoms. wit h s. y. z . .. . reserved for bound variables. and wit h the further exception

o f f . y. h. ... to be iised for fiinction symbols. Capital letterç a n d lower case C k e k letters

will be rised for formulas. An overline indicates a list: ?i is a list of variables (u 1. .. .) and - - - -4 is a list o l lists of formulas (.-Il = ...}.K. ...). A forniula .-l rnay haïe free

variables p. and when we wish to emphasize t hat fact we shall write .î(p). althoiigh ive

r n q - not esplicitly displ- al1 free variables of -4. .4( J) denotes the result of substituting

the list of formulas o For the free variables of -4. Since we haïe separated boiind arid free

1-ariables. in ttir qiiaritified case we are ailtoniatically assiirecl tha t ; is free for p iii .4(p).

~rliich is to s q v that rio free variables of o ivill end up boiincl by an>- of .-Las qiiantifiers iri

the substitution.

\\é shall use the following symbols:

Finail>-. consider t hat althoiigh in general not al1 proof systems neecl be of ttiis forni.

tre s hall consitler only systerns where a proof consists of a seqiience of lines. each derivecl

from previoiis ones. In t hese cases. we iisiially have two forrns of a s-stem P: The dag-like

form P. wherein lines may be re-used arbitrarily often as hypot heses of inferences. and

the tree-like form Pr. wherein a line can be iised only once.

Chapter 3

BPLK and G

In t h is ciiapter ive int roduce t lie secllient systeni B P L I<. \vtiich is basicaliy t h e proposi-

tional f r a p e n t of LI; enhanced with t tie reasoning p o w r of Boolean programs. \ L é t h c n

present several lenimas about BPLII; ancl Ci whicli show that certain classes of pruofs are

easy to tint1 i n the two systems. and ~ v h i c h will siniplif' arguments later on.

3.1 BPLK

Definition 3.1.1 ( B P L K ) . The systtrn BPLf ï k like the pr*t>positiorid ~ p t t r n L K. bu!

trith t h e fol10 [ring chariges:

1. l r i dd i t i on to ntqirerlt~. (1 proof nlso iricliides n Boolerin progrnrn irhich de)rie,s

funclions. Il?tt-newr we re fer to n BPL lt-proof, ict- shall nlu*« ys erplicitly «*rite; it

as the pair < a. P > O/' the proof (sequents) and the h o f f a n progrnnt dcfir~liing the

lunction syrribob occurrlng in the scquents.

2. Formulas in s~querit.s urc /onn uln-s in the contert of Bookan progsama. u~ defincd

radier.

3. lf the Boolean program contains a definition of the fonn

f ip) := 4 p ) .

0 If o iç a irariable then the resuit is immediate.

0 If o is TL*. then to the sequent

apply : right and : left.

0 I f O is L- v 0 tlien appl?. weakening then \ : right to zach of the sequerits

L' -+ t'

the11 appi>- \V : left.

The case fOr ,\ is symmetric.

0 I f o is (;)IL-( 1.) for sume quantifier () t hen to the sqiierit

apply both introcluction riiles for the quantifier. left first for uniwrsal . right h s t

for esistent i d .

0 If O is j (C) . then by a separate induction o n i. derive

~ J P ) - L ( P ~

using subst and the other cases of the current lemma to derive

-41(p) + - A i ( p ) -

and then applying f*-introduction. Finall. apply subst to f (p) t f (p).

We apply at most 5 in ferences for each connect ive in (3 (and P). and the proof can easily

be cornputed using a reciirsion on the structure of o. so r ( x ) = x' is sufficient. a

Lemrna 3.2.2 (Simple Manipulations). T ' h m is a polynorrlial I - .wch &il if Z i l i r

- cur-iiibles i r r l d .-I iirr foi-nitilïis. und if

Pr-"of. Sinipl>- siibstit lite T for ii in t h e original proof. The only case to consider is if a

sequent in the original proof is an initial sequent. in which case we can apply lemnia :J.2.1.

The rtibstittition espantls t h e original proof by at most a factor of 1x1 and ndcling the

subproofs obtained from lemrna 3.2.1 ntlds n factor of at most 51x/' so r ( k . 1x1) = b(lX1)"

is sufficieut, Ei

Lemma 3.2.3 (Build Lemma for G and B P L K ) . T h e m [ire potgnonriai-six proofs

of

- -4 = B i c(X) = c(B).

-- /or ang formulas -4.B.C. in either languag~/systern. In the case of BPLIi. the s i x of' P .

thc Boolcan program dejïning the junction syrnbois occurring in -4. B. C . is an argument

to the poiynornial.

CHXPTER 3. BPLK .GD CI'

ProoJ The proof is a siniple ind~ ic t ion on the structure of C.

If C is a variable. then ive need to prove 3 = --+ -4; = B: for sonie i. which is

trivial.

- - II C' is Q.r D ( r ) for sonie quantifier Q. Ive Arst prove 7 = B - D(.-\. p ) =

D(B. p ) . Sortit. siniple nianipulatioris yielcl X = B. D ( X . pl - ~ $ 7 . pi arid t hm - -

tivo q~iantifier introductions $ive .-L = B. ~ . r ~ ( 3 . .r 1 - ~ . r D(B. r 1. Likewise. ive -

obtairi .-\ = B.QZD(B.X) i ~ s ~ ( x . . r ) and theri sonie siniple nianipiilations

'-ield t tie resiilt.

If C' is f, ( D i . .... D k ) tiien tirst pro\.e

t hen apply subst to the sequent

P = 7 --+ fJP) = fi(?)

(rvhich is proved by a simple induction on j ) to siibstitute ~ ( 3 ) for p a n d D ( B )

for ij and t hen cut to obtain the result.

\\.\é end with a final lemma showing that substitution is like a derived rule in C.:

Lemma 3.2.4. Thcre k n polynornial r such that a G-proof of the sequent s (Z ) /rom

the sequent S ( p ) (60th in the langvage of G) can be found in tirne ~ ( r ( l ~ ( o ) l ) ) .

Proof. The proof is as follov-s: First apply : right then V : right to obtain an etluiva-

lent sequent with a single formula.

Apply Y : right to obtairi

By lemmn 32.1. the secpient

~ j ( 3 ) i ~ ~ ( 0 )

Ilas a short proof. and ttitis V : left. weakening and cut prodtice

- F ~ ( G ) .

The clesirecf secltient follows after sortir simple manipulations.

Chapter 4

BPLK P-Simulates G

[rl t [lis chapter ive jhow t hnt t hr systeni BP LI\: pal>-riotiiiall>- sitiiiilates G'. \\'e do so by

.Itwri birig a procrcliirr for t ranilat irig a srqiient in t lie language of C; i rito a itmant iixll!-

r i t w Boulean prograni delines fiinct ion syriibuls IV hich urctir iri the t ranslated sequerit. aiicl

which replace t h r quanti fiers and bound variables from the orijinal one. This translatiori

is usecl to translate a Ci-proof line-by-line into the langiiage of BPLti. and ive show how

to f i l [ in t hr gnps to forni a correct proof in the latter systeni. Inçiclrntally. t his terhriiqiir

coiilcl easily be adapted to pro\-ide an alternate proof of the PSP:\ÇE-cornpleteness of

Boolean prograrns. as the proof in [ l LI dicf not in\-olve a retltiction from QBFs.

Perhaps the tirsr kind of translation one ivould think of woiiltl be to lise Baolean

programs to calculate Skolem functions. and use these to replace variables boiintl by

esistent ial c p n t ifiers. Variables bound hy universal qiiant ifiers would t hen becorne free

variables. Horvever. t here are many problems \rit h t his approach. First of all. t his type of

translation would add free variabLes. so in sorne sense it is a less fait hful translation t han

we wvould like. Second. formulas in the antecedent of a sequent occur negatively in the

contest of the eotire sequent. so we would have to translate t hose quantifiers differentl~

t han t hose in t h e succedent. This means t hat the sarne formula would have two different

translations depending on which side of the sequent it occiirred in. and this would greatly

complicate simulating the cut rule.

Instead. the translation we adopt is ver? much reminiscent of the work of Hilbert ancl

Bernays ( (1 71 and [ 1 d l ) . Tliose aiit hors iised a different language: instead of quanti fiers

they used the so-callrcl e-calcul~is. wherein 3 x . - l ( x ) ivoiilcl be rvritten as . - \ ( é , ( . - \ ( r i ) )

a r - \ as - - 1 . Ttir r .;ynihol rzi, -\!.r!! r ~ p r w ~ n r . : a n ohjert w h i r h i v i l l

satisfy .-I. if such an ubject esists. Those authors clic1 rioc ascribe an- kind of fiinctional

interpretation to the c syrnbols. but since mir iiniverse is of only two elenients. it tiirns

out that Boolean programs can easil- coiripute the values of such synbols. aricl t h is

ivhere we base our t rarislat ion.

;\ri alternat it-e translation siiggestetl by Toniann Pitassi mirrors the definit ion of

proposit ional quantifiers. and woulcl replace Y s . - l ( r ) by . - \ (O) /\ A( L ancl 3s.-\(s) by

O ) v ( 1) . Ti> avoid esponential increase in forniula size. each of tliese ivoiilrl ttieri

be replacecl by a fiinct ion synibol detined by a Boolean program. [..ri fort m a t ely. t tie

principal ~lifficiilty with the translation we do adopt. namely that the narrirs of t h r fiinc-

t ion syrnbols in translations change after int rocliict ion of quant ifiers. is also present wit li

t his translation. The cornpiesi ty of the arguments woiilcl t hus be siniilar.

4.1 Special Notation

In the translation in this chapter. we shall use fuuncction syrnbols ivhose nanies are basecl

on formulas of Ci. Becaiise of the delicate nature of t hese names. we shall be especially de-

tailed and careful about sir bsti tut ions and the names of free variables. We shall t herefore.

in this chapter only. modify our noration:

LVhen a formula ;L is displa-ed as A@). we shall mean that p are al1 the free variables

of -4. LVe shall never use the notation A(q) to mean a substitution. but instead that -4

has the single free variable p. -411 substitutions will be denoted with a postfis operator

[ B / p ] ivhich means that the formula B (which ma! be a single variable) is substituted

for p in the Formula which precedes the operator. We may write several of these in a roiv.

and the nieaning is t hat t hey are performed in seqtience from left to right. CVe may also

interject lree variable lists. as in the example

4.2 A Translation from the Language of G to that of

BPLK

To aicl in t rnrislating ive introcluce the following detinit ion. wliich $\-es ils a ~vell-defi necl

nierging operator which says how to combine several Boolean programs into one. elinii-

nating ciiiplicate defini t ions:

Definition 4.2.1 (hIerging P o of Boolean Programs). If' P find (E arc Bool~nn

progrnrns. or. nt knst j'rngnicnta (not riecr.~snrily d~f ining ni! funetion .iyrnbois used in

dt-finitiortà). t h tn I h ç rnerging P o Q of P anri Q is obtairied b y jîrsl concntcn(zting P arid

Q . and then dclcting d l bu! the jîrst dcjiriition of ench fundion syrnbol.

\\,-e shall ilse t his merging operator in the following translation. and later on in the

actiial siniulation. Hoivever. whenever we rnerge two Booiean progranis which both define

a pxticular function syrnbol. it ivill aliva~-s be the case that the two definitions are

identical. Thus. w hich of t h e definitions is kept is irnmaterial.

Soit- ive can prrsent the actual translation. defined first for single Formulas:

Definition 4.2.2 (Translation < 'a'. P[o] > of O ) . CiÉ recursicely d$ne a transla-

tion of a quantijied propositional semiformula into a semanticaily equiralent quantifier-

jrce formula in the language of BPLK toge th~r with a Bookan program defining the

function symboi . in that formula.

'3 W . 5 L

II . . la,

II . . *-

la.

- la. 4 - L

B .r;

' - 3 -a

Note that if o is quantifier-free. then it is unchanged by the translation and the

Boolean program which results is ernpty.

Definition 4.2.3. t j5 is the sequent

th tn < 'P. P[S ] > is

\\B cal1 P[oj or P[S] the Boolean prograni rirkirig fruni tlie translation. A s a Hnal

leriinia in t his section. rve s h o ~ that translations are polynorninl aize:

Proof. First. note that Iro'l E O( 101 ): Induçt il-el! following the recursiw definition. al1

the Boolean connectiiee cases increase the translation size linrarly. In t h e case o l o r

3r L * ( I . P ) . tlie translation ' 07 is just f,(P). so the number of symbols in the translation is

at rnost twice the niimber in the original formula. This is becaiise to ivrite t ht. translation

ive write jiist an f . a copy of o. and t hen a list of 0 ' s free variables.

NOIF. t h e Boolean program arising froni a translation of o contains at niost tivo

function symbols for each quantifier in o. The size of the names of t h e function symbols

is linear in 101. since the name is based on a subformula of o. The defining formula for

a n e functioo symbol is linear in size from the previous paragraph. since it is rssentially

just a translation of a subformula of o. The defining formula for an f function symbol

consists of a translation of a subformula of o. of linear size. with an c function symbol

substituted for one of the variables. and therefore is at rnost quadratic in size. The size

of the entire Boolean program is thus in O(ISI3). Ci

4.3 A Simulation of G by BPLK

First. ive show that the translations defined above are semantically eqiiii-alent to tlie

original formulas.

Lemma 4.3.1. I n t he prcserlce 01 the Booleun proyrnnl P [ o ] dejîniiig the filrzctioii s y m -

bols ris nboce (nrisiriy ji-onr the t iwnslntion). n {possiblfi) quant ificd Boole<in for-rn irlir O is

sen~nnticlzlly equirti[cnt l o i f$ lrnnslntion '0'.

Prooj'. The proof is by a simple induction on t h e structure of o. The interesting cases are

when O is 3xc-(.r.p) or V x ~ - ( r . j j ) . [ r i the tirst case. if the translation holds then the value

obtainetl by r~val~iating t hc tvitnessing funct ion satisfies c*. ancl so o holds. C'onversely. i f

o holcls then eittier O or 1 satisfies c*. and i t is rasy to see that the witnessing fiinction

will evaliiate to a satisfying value.

Iri tlir aecuiid case. ul>srrw tliat ttir tvit~irssing fiiiiction rvill falsify L* i f possible. I'hiis

i f the translation holds t hrn c* is satisfird by bot11 O and 1. C'onversely. i f o liolds t hen c*

is satistied no matter what the witnessing fiinction evaluatcs to.

Sote t hat the correct ness of the cases of v and A depends on the fact that whenrver

two Boolean prograrns are merged wliile translating a QBF. they never disagree on the

detinition of an!. funct ion symbol. Thus. for example. the value of 'c* v 0' \vit h respect

to P [ c v 01 is indeed t h e Boolean v of the values of 'cnl and 'O1. each wit h respect to

t heir own Boolean programs. Cl

Xow. the operations of substitution and translation do not commute direct ly even

in the case of substituting only a single lariable. so for esample if . i ( a ) 3s(a V s).

then its translation is ' . i 7 ( a ) = f3Si,v,i(a). If ive then substitute b for a we obtain

';L1[b/a] f ) . On the other hand. if we did t hese t hings in the reverse order

ive would get ' - - l [ b / ~ ] ~ fy,(b,,)(b). mhich is difkrent. .A technical lemma is needed to

resolve t his difficult~:

CHXPTER S. BPLK P-SIMCLATES G 23

Lemma 4.3.2. Let 4 s . P. T) be a semiformula and B(p. Q) n formuln. both in t h t Inn-

guayt o j G . so B 1s autornaticallg f i e for s in .-\. and let be nll free rnriables ~ r c e p t s

which arc cornmon to -4 and B . (s rnuy be in g) Thcn prooh of

c m be /olrnd in t i m ~ polynorninl in the s i x of translations i ~ n d the .size of th€ Boolenn

piogrzrri trrisirig [rom Ihcm.

Proof. K e prow the lemma by in~ltiction on the striictiire of A. The base case is i f .-l is

- atomic. If' .-l is .S. tlien ' . -I l is .S. so '-.-\[B/s]' z ' B1 = - '.-I1['B1/s]. Otherivise .-\ is some

ut hrr variable if so '.-\[R!..;]' . ' 1 B . z r l . In rit her case. apply lernrrin : { .2 . l .

I f -4 ià not atoniic then WC have the follorvinç cases for each possible niain connecti1.e:

.-\ is C \/ D: By assuniption. ive have proofs of 'C[B / s ] ' - rC'7['B'/.s] and

'D[B/.s] ' - 'D1['B1/s]. k e weakening and \/ : right to produce new se-

cpen t s wit h siicceclent 'C"['B1/.s] v ' Dl[' B7/ s ] 5 ' ( C v DIT[' B7/.5]. and then

apply v : left. The converse sequent is proved similari>-.

The case for A is symmetric and that for is simiiarly easy.

a .-l is QxC(r . S. p. F): In this case. ive have already found a proof a, of 'C [B/s Iq i

'C'[ 'BT/s]. and a proof 71 of t h e converse sequent. \Ve must find proofs of

'.-\[B/sI1 i '.-1'['B1,/.5] and its converse. Xorv. the first step is to derive

and

CHXPTER 4. BPLK P-SIMULATES G 23

which follow directly from the defining equations for '.-1[B/sI1 f;l(Blsi(Pq. F) and

'A1 1 4 s . p. F). respectively.

The next step is to derive

First. use subst on the endsequent of JI froni the induction hvpothesis to sub-

stitiite i (or O. as appropriate for 0 ) for r ro obtain rC ' [B /s ]7 [ l !~ ] (~ .< i . r ) - [ B 1 / ] . ] . 7). Theri. t,(e:,i introcliiçtiori is appliecl on the lelt and e,\

introcltiction o n the right. The process is repeated with 7r2 and the introduction

rules iisetl o n the opposite sides of the seqiient. and the equality follo~vs.

The Hnal step is as follows: First siibstitute e.4pl,j(P.?j. F) for .r in the endsecllient

of 71. We obtain a proof of

Then Lise the eqiiality (4.3.3) derived in the previous paragraph wit h the build

lemma to obtain a proof of

N w . tising (4.3.4) anci (-L.:l..j) ive cari easily obtain

The desired sequent ' A [ B Js]' + '.-L1[' BT/,s] is t hen obtained using (4.3. i ) and

(4*:3.2).

The proof of the other desired sequent is obtained symmet.rically to the final step

above. but starting with q.

By induction therefore. we can obtain ;i; and r i . proofs of the desired sequents. in

polynomial t ime. Cl

CHAPTER 4. BPLK P-SIKLATES C;

The main result of the chapter follorvs:

Theorem 4.3.3. If S is a quantifier-jree sequfnt with a prooj 71 in Ci. then 5' has u

BPLii-proof< r2. P [ T ~ ] > u-hich. giren 71. can be found iri timç polynornid in j ; i l l ( a n d

thus h m polynorniul six).

Proof. First of all. P [ q ] is formed by nierging the Boolean programs arising Croni t rans-

lating al1 the secpents in TI. 'rtore precisely. if 7 1 is T l . .... SL. then P [ T ~ ] = P[$] 3 ...

P [&].

K e r hen forni 7 2 direct ly by t ranslat irig ri seqiient-by-sequent into the langiiage of

BPLK. and ivhen necessar!. atltling some estra sequents betrveen the trnrislatetl unes. \.\é

h a v e the following cases. depericling on t lie in ference ride iisecl:

O Observe that al1 initial seqiients of C; are their own translations. antl are also initial

seqiients of BPLK.

i f 5' is non-initial and is inferrecl uriy rule rscept a qiia~itifier iritrotliiction with

hypottiesis(rs) ï' (and l - j . t hen '5'' follows froni 'TT (antl ' lm ' ) by t h sanie rule.

This is becaiise translations of ident ical formiilas are ident ical. and t ht' translation

operator cornmutes with the connectives V. A and l.

a if 5' is inferred lrom T by the introduction of a iinirersal qiiantifier on the right. or

t hat of a n esistential qiiantifier on the left. then considering the first case ive have

that T is

and S is

Their translations are thus

respectively. For convrriiencc. let -4 F C[x /a l . In this notation. ' T T is

First. '.-l[n/s]' i r.-l'[rc~Ï/s] follorvs frorn lemma 4.3.2. taking B G il. and s s 1..

' T T ancl cut t hen yield

S e s t . obseri-e that cr cannot occur in r or L (or therefore their translations) due to

t h e restriction on Y : right. So. rvlien tre apply subst t o this srqiient. siibstitiiting

cv,.l(p) for a to obtain

'T and '1' are i i n c h a n g d Finaily. '.-V[t~~,,i@)/rl - A,.., ( p) f 'o l lo~ froiti

detinition of t h e funçtion symbol far..\ iisitig the introductiori right r i ik for tliat

syrnbol. 'Y follu~vs with weakening and cut.

The interesting case is when S follows from T h ~ . t h e introduction of an rsisteritiai

quantifier on the right. or t hat of a iiniversal quantifier on the left. The two cases

are symmet rical. so consider the tirst. T is

and S is

Thus. it siiffices to prove '.-I[B/xI1 i f3,n(ji) and apply cut.

First. we derive '.-1[B/rI1 i '.4'['B1/r] using lenima 4.:1.2.

The riest task is to prove

This àeqiirrit h a a proof in\-011-irig -1 applications of the builcl lenima. I r i thiv proof

sketch w\.e omit the corner brackers and r hc subscript 011 c:,+-,. \\-e tirst show t h

eitlier 1 satisties .-I or not. Sest. i f 1 satisties .-I theri t niLisr. since in tliis case t

will evaliiate to one. \.\é then show tliat i f 1 tioes not satisb -4. but El tlors. ttien t-

niiist also. since in this case B and c will both evaltiate to O. Finally. we combine

t hese t hree pieces to get t h e resiilt.

BP Def 'ri

Edsy

1

1

4. weakening

- 1. weakening

5. 6. butlà k m m a

7. contract ion

3. 7 : left. weakening

O --t. weakening

3. 10. burld lemma

1. weakening

13. B. 1 -+ B B - B. weakening

14. B. . - I [ B / x ] i .4[l/x] 12.13. bulld lernrna

1.5. -A[l/r]. .-I[B/r]. B - O 1 4.7 : Ieft , weakening

16. -.-I[L/x]. .-1[B/x]. O + B O +. weakening

17. -.4[l/x].-I[B/xj. . -L[B/x] - ,I[O/r] 1 5 . 16. build lernrna

IS. -.4[l/r]. --l[B/r] -t .-l[t(p)/x] 17. contraction. 1 t . we,&ening. cu t

i9. .-i[B/.r] i .-\[c(p)/r] 13. Y, 2. weakeuing. eut

Tow. starting wi th ' T T and usin3 each of tliese last threc sccliients in tiirn w i t h

weakening and cut. one obtains the secpient '5''. as clesired.

Firially. in ilit. case tliat .$' is qiiaritifier-frw (for zsairiple. t h e tirial srquent in a proufi.

t hen t h e translation of C is S. niid tliis corripletes the proof of t h e t heorrrri. EI

Chapter 5

G P-Simulates BPLK

t r i ttiis chapter ive define a translation of Booleari programs into formulas in the langiiage

of (7. t hat is. usirig proposit ional qiiantifirrs. Cltirnatel'. ive mant to t ranslatr srqiirnts in

BPLIi into eqiii\*alent ones in G. analogotisly to the previoiis cliapter. and ive shall iisr t he

t ranslntrd Boolean program to ohtairi the values of fiinçt ion sj-tribol npplicatioris. Sincr a

l inr of the prograni can referençe previous lities ( in etlect iising a s horter Boolean program

to clrfine the current fiinct ion synibol). an indiict ive corist ruct ion is indicated. using the

translations of previoiis lines to obtain the translation of the current one. Howewr. if

more t han one copy of previoiis translations are ever iisecl to translate a line. t liere is a

possi bili t~ t hat the formula sizes ivill increase esponent ially. Therefore a i r translation

will protliice one formula wliich can be iised to simultaneoiisly evaliiate al1 the fiinction

symbols in the Boolean program. and one copy of this formula will be incorporatecl into

the translation of the nest line.

The met hod we shall Lise to -multiplesn the single occurrence of a translation is

inspired bx a trick iised bu Stockmeyer and l [ e o r in [26] to show that the problem of

ek-aluat ing a given quant ified Boolean formula ( QB F) is PSPACE-complete. The idea is

to abbreviate *(si. y i ) ~ o ( r 2 . y?) as V r . y[((r = riAy = yi )V(x = xzAy = y*)) 3 Q(.L y)].

CHAPTER 5 . Cr' P-SIXLATES BPLK

This is not exactlx what is needed but is quite close.

5.1 A Translation from the Language of BPLK to

that of G

- since we have an ~nhnite supply ot variables. let ils resert-e F. 7. T. (1. and Ti as neiv

1-ariables not occtirring in the BPLIC-proof being rrarislatetl. There will be a variables r..,

and t ..\ for each forniilla .-\ i n the langiiage of BP LI;. and we shall replace ocçiirrenccs of

I~inct ion synibols by t hese variables as part of our translation.

Example 5.1.2 (Hat Operator). I f ~ ( p ) is tht forfiluln

LVe shall assume tha t a Boolean program defines funct ions fl. f2. ... and t hat each

function symbol in the Boolean program is defined as f i ( p ) := . . l , (p) . ( In other rvords.

.-Li is the defining formula of fi). Now. oc(?. üj mil1 be the translation of the Boolean

- program ~ i p to and inclucling the definition of fk. The meaning of O L ( ~ . .... 21.. ii 1. .... i ~ i )

will br that this formula is satisfied if and only if fl(z) = 111. .-.. and fk(zI = U L .

\\e are now in a position to defiiie the translation of a Boolean program more fornially :

Definition 5.1.3. Consider n Boolenn program defining j! ... fi-. Define oo := 1. . V w .

sny the definition of fi includes the function symbol occurrences j,, (5). .... f,, (x) in

ordfr . aonte of u~hich nzny b~ nested in othfrs. (Here jL. j?. .... jm nr f jus! tht intitres of

Exarnple 5.1.4. Consider the foi10 sing simple contriccd Boolcan program:

fl(~1. P?) := 'Pi A p?

P 1

- -2.1

'29 9 -.-

L'fi (-,2,,.=2.1 t

- - 2.1

l Z 2 - 2

z2.1

r fl(z*.J.-:2,2) V - q I ( f ~ ( ' : 2 . 2 * = l . i ).=1.1)

CHXPTER 5. G P-SISICLATES BPLK

.Yoa. Q is constructed /rom the nboce as indicated in the dejnition

Claim 5.1.5. For fnch 1 .

ProoJ First. the tat te ment vacuously holds for i = 0.

Now suppose it holds for i - 1. IF O,(?. ii) holds. t hen t here esist i*'s satisfying the

part of O, market1 ('). which ensures chat the! have the same d u e s as the function

symbol applications t hey replace. so indeed fi(%) = u ; . The conjuncts (") ensure t hat

f,(7) = U].] < i.

Conversely. if f i(%) = (11 A ... A Ji(,) = holds. then the r's satisfying (') (which CI

esist and are unique) must have the correct d u e s and so A,(%. c) = 11;. Also. (") is

clearly sat isfied. and t hus al1 of O* is. Cf

L V e can norv define the translation of sequents. This translation is in the context of

a BPLIi-proof. so the Boolean progam and the rest of the sequents in the proof are

CHAPTER 5 . C; P-SI.LICL.-ITES BPLK 34

already fised. Exactly which proof a particular translation is relative to is not indicated

in the notation. but it r d 1 a lmqs be clear from the context.

Definition 5.1.6 (Translation 'Sq of the sequent S relative to 7) . Fir a BPLIï-

proof; and ils <issociattd Boolenn progrm dejining fi. ....fk. Let ~ J T ; ) be n list of nll

subforrnulas in a irhose main con nec tir^ is a junction symbol. (C <ire nrgunltrltd to f,,.

nnà agnrrc ji n r r srnipiy rrlderes/.

Thhen the .zequent S .

Herc the 'Ci1 und the corresponding t > arc in the correct places to br tht q u r r t c n t . ~

to. nrid the C I I ~ U E S of: t h t Junctiorz sy~r~bol fil. The t1 nrç dtrmrny ixri«bles. IIE eocild use

tj,r,(u, instmil of il: ( s i n c ~ rlf rrill be corzstrnirml to th€ m l u t fJO) ) but it tri11 be c ~ o r w ~ n i ~ n t

/«ter. on thut the d+.5 «rc distinct. I I shnll cul1 the occurrences of oc riboce the prefis o j

the tran.dation. tzncl thhe r m u l i n d ~ r th€ siiffis.

Sow. t hese translations may have free variables t hat the original ones did not ( t e s

and dos). \\é cannot. therefore. assert semantic equivaience of the two. However. we are

çoncerned rv i t h proving valid sequents. and we can s q - sornet hing nearly as good:

The idea is t hat if the translation of a sequent is satisfied by some assignment. t hen

either one of the t or d variables has an incorrect value. falsifying the corresponding

instance of oi. or else the- dl have the correct values and the remainder of the translated

sequent is sat isfied. In t hat case. the original sequent is sat isfied by the same assignment.

Conversely. if the original sequent is valid. then every assignment to the translation

will either falsify one of the ok's. or else al1 the t 's rvill have the correct value and t hus

the remainder of the sequent will be satisfied. Therefore.

Clairn 5.1.7. For {zn y sequent S frorn the lnng trng~ of BPLIC. S is ralid if anri orily '5"

Id.

The final leriima in t his section shows t hat translations are polynomial size:

h

Piaof. First note that for any BPLI i forniiila o. we Iiave (01. I'oll E O(lo1). These

operators add a constant niiniber of synibols for each replacertirrit t hey perforni. and t his

ntiniber is boiinderl b ~ - the ~ i z r of the fornilila.

Nest . consiclrr t h e coristriict ion of O, frorn o, - I . The followitig are n&lt.d:

h

'1 copies uf -4,

A

a 2 copies of B. for each B wtiich is the argument to a function symbol in -4; (in the

section ('1)

a 3 occurrences of the çorresponding r variables

section (") whose size is in O(IP1).

(in the section (') and the quantifier)

Therefore summing these al1 up for oo throiigh ok ive see that the last item dominates

the sum and that lokl E O(I PI').

Finally. ' S 7 consists of the prefis. a t most In1 occurrences of ot. each with substi-

tutions of size at most 1 ~ 1 . followed by the suffix. of size O(IS1). Therefore I'S1l E

O(1 P121~12). a

CHXPTER 5 . C; P-SIML~LATES BPLEC

5.2 A Simulation of BPLK by G

We first show trhat proofs of secpeots from two special classes are efficient to find.

Lemma 5.2.1 (Existence Sequents). T h e r ~ is a polynominl r such that for E L . E ~ ~ i.

the s~querl! E,:

has (L procf rhich cnn bc found in tinrc O ( r ( 1 E J ) ) . nrid ii*hose length ià tlius .5irriilurly

b0urlcl€rl.

ha.< a pro01 crhich cnrc bc jound in tinic O(r(11;1)). and ichose length 1s thus similarly

bu il n tir d.

ProoJ These two lernmas are proved by induction in parallel.

For i = 0. the result is trivial. - -

Xorv assume the two lemmas are proved for i - 1. Let B = B;. ... . B, be al1 formulas

appearing as arguments to function symbols in the definition of ft. 6il: as arguments to

f,,. Existence and uniqueness for oi-1 plus some simple manipulations give

h - - Sorne niore simple niariipiilations (siniply conjoining t h e taiitology .-l,(Z. ci = A,( z , . r , )

inside t hc outermost qiiantifier) give

1 A

and then i : right (on the u's and one instance of .Ai) and 'J : right (on the 1'5) ?-ield

the existence sequent for O;.

Xotv in the case of uniqueness. note that

has a short proof using existence and uniqueness for i - 1 and some siniple nianipulat ions.

lolloivs b ~ . unicpeness for i - L. Sotv ive proceetl as follow:

First by the tletinition of O;.

Theri rennming the clliant ifiecl variables a n d doirig sonie si t i i ple riiarii p d a t iotis.

Liiqueness for i - 1 and more manipulations ailotvs LIS to prove that the r's in one of

t h e conjoncts are eqiial to those in the other. and thus procliice

\\ cari similarly consolidate t h e 2's by adtling a hypothesis:

Coot racting.

LYe can now drop t h e quantifier:

Some simple manipulations to combine this last sequent ivith ('). and then : right

and several applications of V : right produce the uniqueness sequent for i. O

Finally. ive can state and prove the main result:

Theorem 5.2.3. I/ 5' hns <i BPLfi-praof~~. then 5 hus n G'-proof al which. gictr i 71.

cnri bc fourd in i irrle pulrrnorrzid in jrl i i«nd thus h m volyionii«l .six 1.

Prooj. K e çonat riict n- direct ly by t ranslat ing z l . seqiient-b~v-secliirnt . into t lie language

of CL re1atit.e to the Boolean program of 71. If necessary. ive insert seqilerits to prove the

translation of a secpient from the translations of its hypotheses.

First of all. i f 5' is an initial sequent of BPLIï . tlirn it is fiinction synibol fret! and so

its translation is itself. and t hiis alreacly an initial seqiient of C;.

Socv consider a non-initial sequent S inferred from previoiis ones. if t he inferencc was

weakening. contraction. or introduction of l. A or V. then the same riile yields '5'.

if 5' = T ( c * ) is inferrecl frorri T ( p ) by subst. then note that ivithoiit l o s of generality

ive ma!. assume that p dors not occiir in L*. Otherwisr ive coiild rrioclify 71 to prrforrn

subst tivice: once to siibstitute ~ * ( q ) for p ((1 is a variable which dors not occlir i r i T ) and

ttirn again to siibstitute p for q. To simiilate the substitution in G. first use leninia :).2.-l

to siibstitute 'c*' for p in T ) . obtaining 'T1('c.'. T ) . Finally. apply lemnia 5.2.-1.

which follotvs after this proof.

The last case in the roof is ivhen 5' is inferred by Ji-introduction. introdiicing

fJB(P)) . Then clearly the seqiient

together ivith some simple manipulations. mil1 produce 'ST (basically jiist by using cut).

We derive the desired sequent as follorvs: First. t h e lolloming is st raightforward:

CHXPTER 5. G P-SIMI:L.ATES BPLK

Expanding the oie

Note that the -4, occurrence above contains t variables. from the 'Bq siibstit~ited for

the 5. and also c variables. Ironi function symbols occurring in the definition of f;.

V riiqiieness for O, - ancl

- one secllient for each function synibol occurrence f , (( ' ( (z , ))) in the cletinition of jt. alloir

h

us to renarne the rf,,=, in the occurrence of .-\, abovc to t j 3 pro<liicirig '.-\,( B)'. ,( ( 1 ) '

anil tlirri we drop t tir existent in1 quaritifier and sotrir cotijtinçts ro ar t

which is the clesireci sequent.

Nearing the end of t h e proof now. i f S is t he Iast sequent of the proof. then it is

function syrnbol-free. i v e need only remove the prefis from 'S1 to obtain 5. The t

[variable corresponding to the outer-most function symbol application in 71 ( t here may

be man! outer-most applications) is defined by an occurrence of o k . but i t is not iised

in the definition of an' of the other t variables. We m q thus use 3 : left on the t

and the d ' s . follorvecl by V : left on the B's and the 0's. to change t bis occurrence into

~ ? 3 ~ i o ~ ( ? . ~ i ) . which ive can ciit atv- with the existence sequent and weakening. R é

can now do the same for the nest most outer function symbol application. and so on.

The resulting sequent at the end of this process is S. which comptetes the proof.

-411 that remains is to prove lemma 5.2.4. This lemma is analogous to lernma 4.R.2

of the previous chapter. and is needed because substitution does not commute with

t ransiat ion.

Lemma 5.2.4. 1f T ( p ) is a sequent in n BPLK-proof and L* is a BPLK formula irt

which p does not occur. then a G-pro01 o f r T ( ~ ) ' from 'T7( 'u1) can be found in time

po[ynoniid in the s i x O! its endsequent.

Proof. The first step is to use simple nianipiilations to renanie al1 the variables t in

'T1('~+'). :\ lariable tB( , ) is renaniecl to ts(,.) by an application of lemnia 3.2.4. This

rrns..mifiir O he d o ~ e In ZI?!; i~rc!er. 2nd C-!! the TCEL:!!IE~ ~ e c j ï ~ z t !'. YGY:. 1: I S cas:; :O

see that for rvery occurrence of a subforriiula of the forrn ' C ( p ) ? in 'T'. the correspotid-

ing ocçurrcnce in [ ' is 'C'( L-)': This follows because w henever t lie translation operator

replaces a siibformiila B ( p ) of C ( p ) bj- a fiinction synbol. the syrnbol's name is t e c p i . and

50 after the renaming it rvill be t s i L . ) as i t stioiild be.

Sorv. consider any variable t f t(8(p), occiirring in ' T l . This variable is driîned by an

occurrence of ol; in the prefis of ' T l :

( In façt. it is possible that this variable occurs only in the prefis. ) After the siibstitiition

of ' L * ~ into "P. the corresporiding occurrence became

o ~ ( O . .... 0. B(p)l(rt-*'). O. .... O . d l . .... c l ! - , . t,,,, . il!,

After the retiaming. in I' this occurrence beconirs

- 1 ok(O ..... O.'B(L-)'.O ..... 0.d : ..... tif-,.! ,,,,c,, .cf ,,,.

which correct ly defines t f t i g g i .

Sow before the final step. note that the suffis of l' is identical to the sufix of ' T ( r * ) ' .

ancl those occurrences of ok defining t variables in the siiffis of 'T(L*) ' also occur in

. The only ditference. then. betmeen I - and ' T (L* ) ' is that the former sequent r n -

have some prefis formulas which the latter does not. and vice versa. We can thus use

the existence sequents (or contraction. in the case of a duplicate) to cut away the

superfluous prefix formulas from I - . and weakening to add the missing ones. The result

is the desired sequent. 17

Chapter 6

Future Work and Conclusions

Ir1 t h i a t tiesis ive cirnioiist ratetl a strorig connrction between trvo proposit iorial proof

systenis bot h hased on PS P.-\C E reasoning. These rrsults raise rriany interest irig qiirstions

whiçh reniain unsolved:

6.1 A Technical Improvement

First of d l . from a technical perspective it ivould be nice to get rid of the subst riile from

BPLIi. [t is shown in Dowcl (151 that estended Frege systems p-simiilate s.iibstitittion

Frege. Boolean programs would appear to be a generalization of the extension ride. so it

seerns reasonable t hat a sirnilar resiilt to Dowcl's niight hold which woulci allow a version

of B PLI\: wit hout subst to p-simulate the subst-augmentecl version.

6.2 Witnessing and Search Problems

Buss and KrajiCek in [5] show t hat those functions which are E: definable in T-,' are

exact 1'- polynomial t ime projections of PLS funct ions. PLS is Papadimit riou's class of

polynomial local search problems and is discussed in (191. [--LI and ['XI. Because of the

correspondence between T i and Gi. it is therefore the case that the problem of finding

mit nesses for the quantifiers in a proof in G1 is also esactly as hard as PLS.

Several lines of research are suggested: First. it would be interesting to characterize

the hardness of the iritnessing problems for the other siibsystems of G. and indeed

different kinds of definability in the subsystems of T2 and S2. Part of this ivork has

recently been done by Chiari and Krajiëek in [SI For X: and 5: definability in T;? but

not hing gerieral is known yet. Secondly. t here are ot lier local search problems t han P LS.

soriie of d i i c h are disciissrtl in [LY! anci in niore cletail in ['LI. It would h e interesting to

fincl propositional proof systenis whosr ivitnessing problems were exactly projections of

t hese ot her local searc ti problern classes.

6.3 Subsystems of BPLK

Anot her set of questions which are part iciilarly iriterest ing concrrris t tie possi bility of

finding natural siibsystenis of BPLI i . akiii to the structure of C;. In th& papcr [ I I ] .

t lie aiit hors f nd a natiiral restriction of Boolean programs. essentially arnount ing to

estension asiorns. for witnessirig proofs in C;;. I t rvould be instriictivc to îind restrictions

of Boolean prograrns which would naturally ivitness proofs in other siibsystems uf C;. It

trotilcl also be interesting to fincl some kind of a liierarchy within BPLK whicli may or

n i q not correspond to the hierarchy in G.

6.4 Miscellaneous

Finally. it is possible t hat due to the apparent ease of use of BP Lf i . more positive results

may be forthcoming than with G. For esarnple. it may not be too difficult t o produce

polynornial-sized proofs in BPLK of some of the conjectured hard examples for Frege [ 3 ]

a n d evtended Frege systems. .As another example. the connection between G and I Z .

rvhich currentiy is restricted to only 5: formulas. rnight be generalized to handie more

generai t heorems. in part icular including the second-order features of t hat systern.

Bibliography

[LI 11. Ajtai. The complesity of t h e pigeonhole principle. In 29th .-lrinirnl Syiripo.:' .IL u rri on

Fo urirlntiona of C'ornputcr Scie rice. pages 3-16-:155. K h i t e Plains. New York. 2-1-26

Octobcr 193s. IEEE.

[2] Paul Beatne. Stephen Cook. .leR Edmoncls. Russell Inipagliazzo. and Toriiarin

Pitassi. The relati\-e cornplesity of SP search problems. . J o i r r m d of C'vrrlputer und

.Sy.slslc ni Science.5. .77( 1 )::3- l!). Aiiçiist l!M.

[:II Maria L u k a Bonet. Samuel R. Bus. and Toniann Pitassi. Arc th( iples

for frege systerns? in P. Clot E. J. Remniel ( tx1 .s . ) : Fensiblc .\Inthe rntitic.5 Il. pages

30-56. Birkhiit~ser. Boston. 19'35.

[-Il S. Biiss. Bourlded .-\rithmetic. Bibliopolis. 'laples. 1936.

[5] Samuel Buss and .Jan Iirajitek. An application of Boolean coniplexity to separation

problems in bounded arithmetic. Proceedings of the London .IIn~hematical Society.

69:L-21. 199-4.

[6] Samuel R. Btiss. Relat ing the bounded arit hmet ic and polynomial t ime hierarchies.

.-lnnnk of Pure and -4pplied Logic. Z(1-2):6ï-77. 12 Septernber 199.5.

(71 Samuel R. Buss. editor. Handbook of Proof Theory. Elsevier Science B. V.. Amster-

dam. 1998.

[SI Mario C'hiari and Jan lirajii-ek. Witnessing functions in bounded arithmetic and

search problems. The .Journal of Synzbolic Logic. 63(3):1095-LLIS. September 199%

[9] S. -4. Cook. CSC 2429s: Proof C'omplexity and Bounded .-\rithmetic. C'ourse notes.

I'RL: " ht t p://~vw~v.cs.toronto.ec~u~--sacook/csc2429 h l . Spring k W S .

[LOI Stephen Cook ancl Robert Reckhow. On the lengt Lis of proofs in the propositional

calciilus i preliniinar- version). In C'Ortj~i+iice Rtcord of Sixth .-lnriud -4 C'.Il Sqrnpo-

siam o n Thto iy of C'ornpatirig. pages 13.7- 1-1s. Scat t le. Kashiiigton. 30 :\pril-2 .\Lay

197-4.

[ L l ] Steptien C'ook and Michad Soltys. Boolean programs anci quantifiecl propositional

proof systems. Bulletin of the Sectiori 01 Logic. 2S(:3). 1999.

[12] Stephen A. C'ook. The cornplesity of theoreni-proving procediirt~s. In C'orlfe~rncc

Record O! Third .-lrtrluiil -4C.11 Symposium on Thcor9 of C'on~putirig. pages 141-155.

Shaker Heights. Ohio. 3-5 19; 1 197 1.

[KI) Strphen ;\. C'ook. Feasibly constructive proofs ancl the propositioriril calciilus ( p r c

lirninary version). In C o n J ~ r m c c Record of S'ermth . - l n r d .AC '.\l Syrnyosiurn on

Thcory of Computation. pages $3-97. .-\lbiicliierqiie. Sew Mesico. 3-7 Slay 1075.

[LI] Stephen A. Cook and Robert A. Reckhow. The relative efficiency of propositional

proof systems. *JO unla1 of Sy m bolir Logic. -I4:36-30. 1970.

[l5] lh r t in Dowd. .\Iode1 theoretic aspects of P f NP. Typewritten maniiscript.

[16] Armin Haken. The int ractability of resolut ion. Theoretical Cornputer Science. :KI(?-

:I):'L97-308. August 1985.

[KI D. Hilbert and P. Bernays. Gmndlagen der .llathematit I. Springer. Berlin. 1934.

[Ml D. Hilbert and P. Bernays. Grundlagen der .Chthematil; II. Springer. Berlin. 1939.

David S. Johnson. Christos H. Papadimitriou. and hIihalis Iannakakis. How e a q

is local search'.' J O ui-na1 of Cornputt r uritl Systenl Sciences. d7( 1 ):;y- LOO. Aiigust

19SS.

Jan Iiraj ièek. Bo unded -4rith rrietic. Propositional Logic. and COrnplerit y Theoi-y.

Cambridge Cniversity Press. 1995.

.Jan IirajiCek and Pavel Ptlcilak. Propositional proof systems. the consistency of first

ortler t heories and the complesity of compiitat ions. The Jo t in id o j Sy rn bolic Logic.

~5-C(:3):106:3-10';!1. L9S9.

.Jan IirajiCek. Pavel Piicllik. arid Gaisi Takeuti. Boundecl arithnietic and thc poly-

nuniial tiierarcli!*. .-Lrinds of Ptirc [rnd .-Ipplitd Logic. 52( L -2 ): 1-KI- 133. 199 1.

.Jan Iirajièek and Gaisi Taketiti. Or1 boiiritlc.cl -1 polynomial induction. [ r i S. R.

Btiss ancl P. .J. Scott . d i tors. FE:\S.\I.-\ Tf l : Fmsibl~ .\lnthr rrl nt ics: .-I .\lathe rn(~ticcr1

Sciçricçs [nsti! u t c Ilork.sliop. pages 259-SO. Bi rkhaiiser. L !)!IO.

Çtiristos CI. Papadimit rioii. On the complesity of the parity argument ancl other

inefficient proofs of existence. Journnl of C'ornputtr and Systtrn Sci~riccs. -IS(:j):-IYY-

532 . .J une 199-4.

Alexander A. Razborov and Steven Rudich. Natural proofs. .Journal of Computer

nrid Sgsk rn Scit n c m 55( 1 ):%-35. Augiist 1997.

L. .J. Stockrneyer and A. R. 1Ieyer. Word probtems reqiiiring exponential time:

Prelirninary report. in Confcrcnc~ Record of FiRh .-Lnnunl .4 C.11 Spposiurn on

Theorg of Cornputing. pages 1-9. Austin. Texas. ;30 .-\pril-Z hIay 1973.

Mihalis Yannakakis. Comptitational coniplexity. In Emile -\arts and Jan Iiarel

Lenstra. editors. Local Search in Combinatorial Optimization. pages 19-55. .John

Kiley and Sons. Chichester. CI<. L99ï.

[-Y] D. Zarnbella. Xotes on polynomially bounded arit hmet ic. The JO urnai 01 Sy nlbolic

Logic. 61(:3):942-966. 1996.