Relating the PSPACE Reasoning Power of Boolean Programs And

Home , AC (complexity)

Relating the PSPACE reasoning power of Boolean

Programs and Quantified Boolean Formulas

Alan Skelley

A thesis submitted in conformity with the requirements

for the degree of Master of Science

Graduate Department of Computer Science

UniversityofToronto

Abstract

Relating the PSPACE reasoning p ower of Bo olean Programs and Quantied Bo olean

Formulas

Alan Skelley

Master of Science

Graduate Department of Computer Science

UniversityofToronto

We present a new prop ositional pro of system based on a recentnewcharacterization of

p olynomial space PSPACE called Bo olean Programs due to Co ok and Soltys Weshow

that this new system BPLK is p olynomially equivalent to the system G which is based

on the familiar and very dierent quantied Bo olean formula QBF characterization of

PSPACE due to Sto ckmeyer and Meyer We conclude with a discussion of some closely

related op en problems and their implications ii

Acknowledgements

Thanks to my parents for b eing not so bad after all AnodtoNSERC for greasing

the wheels with PGSA My ocemates Steve Stevenator Myers Iannis

Axiom Tourlakis John Mo onman Watkinson Jonathan Animal Shekter Natasa

Stash Przulj and Eric Do J Joanis for many helpful discussions and pro ductive

distractions Kleoni Ioannidou for moral supp ort

Tsuyoshi Morioka for helping with some pro duction details Michael Soltys for the

topic My second reader Toniann Pitassi for reading under duress

Esp ecially thanks to my sup ervisor Stephen Co ok for countless helpful discussions

and crucial ideas not to mention a lot of reading and correcting iii

Contents

Intro duction

Background and Motivation

Overview of Thesis

Preliminaries

Prop ositional Pro of Systems

LK and Quantied Prop ositional Logic

Bo olean Programs

Notational Conventions

BPLK and G

BPLK

Basic Results on BPLK and G

BPLK PSimulates G

Sp ecial Notation

ATranslation from the Language of G to that of BPLK

A Simulation of G byBPLK

G PSimulates BPLK

ATranslation from the Language of BPLK to that of G

A Simulation of BPLK by G iv

Future Work and Conclusions

ATechnical Improvement

Witnessing and Search Problems

Subsystems of BPLK

Miscellaneous

Bibliography v

Chapter

Intro duction

Background and Motivation

We often argue that a particular mathematical concept is imp ortant if it is natural which

means that it surfaces in many places with dierent origins and denitions and robust

suchthatavariety of disparate formulations of it end up b eing equivalent or at least

closely related Likewise the applicability maturity and imp ortance of a b o dy of results

are greater when that eld is found to have a strong connection to another Three areas

of study intricately connected in such a useful way are computational complexitythe

pro of theory of arithmetic and prop ositional pro of complexity

Computational complexity is the study of computation and the resources required to

p erform it A staggering numb er of dierent kinds of computation all fall into the domain

of this eld It has practical asp ects directly impacting how real computations are done

by real computers and yet seemingly fundamental easily explained problems remain

unsolved despite a go o d deal of eort A particularly glaring example is the famous P

vs NP problem which asks if those two classes of problems are equal Starting from the

NPcompleteness results of Co ok the pressure mounted with no relief leading even

to detailed formal analysis of known pro of techniques and why they are all ineectual at

Chapter Introduction

tackling such problems Many complexity classes are studied and conjectures ab out

separations and hierarchies ab ound yet results are elusive

A dierentway of studying computational complexity is indirectly through logic

Many connections b etween the elds are known complexity classes can b e characterized

as those sets or functions denable in certain theories sets of mo dels of formulas can b e

seen as languages or classes of languages predicates or functions from certain complexity

classes can b e used to dene new logics A relevant example comprises the hierarchies

i i

of theories of b ounded arithmetic T and S of Buss As shown in and

this b ounded arithmetic hierarchy collapses if and only if S proves that the p olynomial

hierarchy collapses

Now due to Co ok there is a translation from formulas of b ounded arithmetic to

p olynomialsized families of prop ositional formulas Furthermore if the b ounded arith

metic formula has a pro of in Co oks system PV corresp onding to p olynomialtime rea

soning then its translations have p olynomialsized extended Frege pro ofs which can b e

in the previous statementdue found in p olynomial time We can replace PV by S

to b oth theories robustly dening p olynomialtime reasoning though in dierentways

Other translations are known and in particular there is a similar connection b etween

i i

T and G and another b etween S and G b oth due to There is another corre

sp ondence b etween U a second order system of Buss and G although only for

formulas In all of these latter corresp ondences it is also the case that rstorder

the b ounded arithmetic system can prove reection principles for and thus simulate the

prop ositional system

The full circle back to computational complexity is completed with the work of Co ok

and Reckhow in and They show that PcoNP if and only if there exists a

p olynomially b ounded pro of system and additionally intro duce many of the imp ortant

denitions in the area such as those of pro of systems p olynomial simulations and so on

These results drive the study of prop ositional pro of complexity and the search for lower

Chapter Introduction

b ounds on prop ositional pro of systems Fine examples are the sup erp olynomial lower

b ounds for resolution due to Haken and b ounded depth Frege systems due to Ajtai

For many seemingly stronger systems however no such results are known

Overview of Thesis

The idea suggesting the results in this thesis is yet another connection b etween compu

tational complexity and prop ositional pro of complexity When formulated in a Gentzen

sequentstyle many known prop ositional pro of systems can b e seen to b e very similar

with the only dierence b etween them b eing the computational p ower of what can b e

written at each line of the pro of or alternatively what is allowed in the cut rule Exam

ples are Bo olean formulas in Frege systems single literals in resolution Bo olean circuits

in extended Frege systems Another example is the system G whichisa sequentbased

system where formulas in the sequents are quantied b o olean formulas QBFs These

formulas have prop ositional variables and also prop ositional quantiers In this case

then since evaluating QBFs is PSPACEcomplete the computational p ower whichcan

b e harnessed in sequents is PSPACE We can restrict G to G by restricting the number

of alternations of quantiers allowed in the formulas and the reasoning p ower is then

that of predicates

Bo olean programs were intro duced byCookandSoltys in A Bo olean program

denes a sequence of Bo olean function symb ols where each function symb ol is dened

using a b o olean formula which can include in addition to the arguments to the function

invo cations of the previously dened symb ols The authors of that pap er showed that the

problem of evaluating an invo cation of a function symboldenedinthisway given the

inputs and the Bo olean program is PSPACE complete The question that then arises

is whether a pro of system formulated around Bo olean programs would b e equivalentto

G For this to o ccur not only would Bo olean programs and quantied Bo olean formulas

Chapter Introduction

need to characterize the same complexity class but there would need to b e an eective

way of translating b etween the two

This thesis answers that question in the armative After reviewing basic termi

nology and notation in chapter in chapter we dene our new system BPLK in a

straightforward waytotake advantage of the expressivepower of Bo olean programs In

that chapter we also prove some basic results ab out the two systems in consideration

Chapter contains the rst of the main results which is a p olynomial simulation

of G by BPLK We rst showhow to translate sequents from the language of G into

equivalent sequents in the language of Bo olean programs As we discuss the translation is

not merely the Skolemization one might exp ect but rather something more sophisticated

and reminiscent of Hilb erts calculus Following that we showhowtosimulate G by

translating a pro of in that system linebyline into the language of Bo olean programs

and then lling in the gaps to make the result a pro of in BPLK

Chapter presents the converse simulation The translation used here rst takes a

Bo olean program to a single formula whichmaybeusedtosimultaneously evaluate all

functions dened by that program This formula is used to evaluate function symb ols

o ccurring in the original BPLKpro of and yields a translation of sequents As in the last

chapter a linebyline translation followed by some lling in of gaps gives the desired

result

Concluding in chapter we discuss some op en problems and other issues raised by these results

Chapter

Preliminaries

In this chapter we present some formal background ab out pro of systems whichwe rst

formally dene We present Gentzens p opular sequentbased system LK whichisthe

foundation for the two pro of systems compared in this thesis and also discuss quantied

prop ositional logic and the system G and its subsystems Finallywe comment on some

notational conventions whichwe shall use

Prop ositional Pro of Systems

We shall consider a language consisting of the complete basis f g parentheses

constants for false and and an innite supply of atom symb ols whichwe shall

represent with a varietyoflowercase letters In the standard waywellformed formulas

in this language dene truth functions or equivalently Bo olean functions of the truth

values of the atoms TAUT is the set of prop ositional tautologies formulas whichevaluate

to true on every assignment

Denition A pro of system P for a set S is a surjective polynomialtime com

putable function P S for some alphabet

Weareinterested in pro of systems for TAUT A P pro of of a tautology is a string

Chapter Preliminaries

suchthatP We denote by j j the number of symb ols in Wehavethe

following imp ortant notion which allows us to compare the p ower of pro of systems

Denition If P and Q areproof systems we say that P p olynomially simulates

psimulates Q and write P Q if thereisa polynomialtime computable function g

such that for every string x P g x Qx

LK and Quantied Prop ositional Logic

A p opular pro of system is Gentzens sequent system LK LK is actually a pro of system

for predicate logic but we shall consider only the prop ositional fragment Each line of an

LKpro of is a sequent a string of the form where and are p ossibly empty

nite sequences of prop ositional formulas A sequent is satised if and only if either one

of the formulas on the left the antecedent is falsied or one of the formulas on the

righ tthesuccedent is satised Eachsequent in a pro of is either an initial sequentof

the form ora a for an atom aoritisderived from previous ones its

hyp otheses via one of the following inference rules this set is the same as in which

is a slight mo dication of the ones in

weakening

left and right

A A

exchange

AB AB

and right left

BA BA

contraction

AA AA

left and right

A A

Chapter Preliminaries

intro duction

A A

and right left

A A

intro duction

A B A B

left and right

A B A B

intro duction

A B AB

left and right

A B A B

cut

A A

Quantied prop ositional logic is what results when we add prop ositional quantiers to

our language The semantics of xx pisthatthisformula is satised by a particular

assignment if and only if p p is Likewise the truth value of xx pisthe

same as that of p p As in when in the context of quantied prop ositional

logic we shall divide variables into bound variables and free variablesFree variables may

o ccur free in formulas and semiformulas but may never b e quantied Bound variables

may o ccur freely in semiformulas and may b e quantied in formulas and semiformulas

Sequents are constructed from formulas

Additionallywe can dene a hierarchyofquantied Bo olean semiformulas The

following is a slight adaptation of the denition in

q q

Denition The classes and are dened as fol lows

i i

q q

are the quantierfreepropositional semiformulas

q q q q

for al l ji and then it is also or If is

j j i i

Chapter Preliminaries

q q

If x is then xx is

i i

q q

If x is then xx is

i i

q q q q

If is then is respectively

i i i i

q q

and areclosed under and

i i

q q

is closed under existential universal quantication

i i

Now the pro of system G is obtained by augmen ting the set of inference rules of LK

with the following

intro duction

AB Ap

left and right

xAx xAx

intro duction

Ap AB

left and right

xAx xAx

where B is anyformula and the atom p replaced do es not o ccur in the conclusion of the

corresp onding inference G is G with the restriction that all formulas app earing in a

q q

pro of must b e or It should b e noted that although G and its subsystems derive

i i

tautological statements of quantied prop ositional logic in this thesis we will consider

them only as pro of systems for prop ositional tautologies

Bo olean Programs

Bo olean programs were intro duced in and are a way of sp ecifying Bo olean func

tions It seems that p erhaps representations can b e remarkably even exp onentially

much shorter than with Bo olean formulas or circuits although a formal pro of would b e a

breakthrough Bo olean programs are something like a generalization of the technique of

Chapter Preliminaries

using new atoms to replace part of a Bo olean formula which idea is the basis of extended

Frege systems The following denition is from that pap er

Denition Co okSoltys A Bo olean Program Pisspeciedbyanitesequence

ff f g of function symbols whereeach symbol f has an associated arity k and an

m i i

associated dening equation

f p A

i i i

where p is a list p p of variables and A is a formula al l of whose variables are

i k i

among p and al l of whose function symbols are among f f In this context the

i i

denition of a formula is

and p are formulas for any variable p

If f is a k ary function symbol in P and B B are formulas then f B B

k k

is a formula

If A and B are formulas then A B A B and A are formulas

The semantics are as for prop ositional formulas except that when evaluating an

application f of a function symbol the value is dened using the dening equation

to b e A There is no freeb ound distinction b etween variables in the language of

Bo olean programs

An interesting prop erty of Bo olean programs which demonstrates their comparability

to quantied Bo olean formulas is the following theorem from

Theorem Co okSoltys ALanguage L is in PSPACE i L is computedby

some uniform polynomial size family of Boolean programs

Notational Conventions

We shall use the following conventions of notation Lower case English letters will repre

sent atoms with x y z reserved for b ound variables and with the further exception

Chapter Preliminaries

of f g h to b e used for function symb ols Capital letters and lower case Greek letters

will b e used for formulas An overline indicates a list a is a list of variables a and

A is a list of lists of formulas A fA A g A A formula A mayhave free

variables p and when we wish to emphasize that fact we shall write Ap although we

may not explicitly display all free variables of A A denotes the result of substituting

the list of formulas for the free variables of A Since wehave separated b ound and free

variables in the quantied case we are automatically assured that is free for p in Ap

whichistosay that no free variables of will end up b ound byanyof As quantiers in

the substitution

We shall use the following symbols

Denition The symbols andare not in the language of either

system we consider but we shal l use them as abbreviations A B abbreviates

A B B A A B B A

k k k k

A B abbreviates A B The symbol willbeused to denote syntactic equality

Finally consider that although in general not all pro of systems need b e of this form

we shall consider only systems where a pro of consists of a sequence of lines each derived

from previous ones In these cases we usually havetwo forms of a system P The daglike

form P wherein lines may b e reused arbitrarily often as hyp otheses of inferences and

the treelikeform P wherein a line can b e used only once

Chapter

BPLK and G

In this chapter weintro duce the sequent system BPLK which is basically the prop osi

tional fragment of LK enhanced with the reasoning p ower of Bo olean programs Wethen

present several lemmas ab out BPLK and G whichshow that certain classes of pro ofs are

easy to nd in the two systems and which will simplify arguments later on

BPLK

Denition BPLK The system BPLK is like the propositional system LK but

with the fol lowing changes

In addition to sequents a proof also includes a Boolean program which denes

functions Whenever we refer to a BPLKproof we shal l always explicitly write it

as the pair P of the proof sequents and the Boolean program dening the

function symbols occurring in the sequents

Formulas in sequents are formulas in the context of Boolean programs as dened

earlier

If the Boolean program contains a denition of the form

p Ap f

Chapter BPLK and G

the new LK rules f left

and f right

may beused where areprecisely as many formulas as p are variables

Substitution Rule The new inference rule subst

q p q p

p p

may beused wherealloccurrences of q have been substituted for

One p oint to note ab out the ab ove denition is that we allow only one substitution

at a time with the subst rule However we can substitute for p in the sequent S pby

p to variables which rst p erforming several instances of subst to change the variables

do not o ccur in or the sequent We then substitute eachof into the sequent one at a

time The size of the pro of fragment is a p olynomial in the size of the resulting sequent

Basic Results on BPLK and G

Lemma Thereisapolynomial r such that if is any formula in the language of

either BPLK or of G then the sequent

has a proof in the appropriate system which can be found in time O r jj in the case

of G and time O r jj jP j in the case of BPLK with P the Boolean program dening

the function symbols in The length of this proof is thus bounded by the same value

Proof The pro of is a simple induction on the structure of

Chapter BPLK and G

If is a variable then the result is immediate

If is then to the sequent

apply right and left

If is then apply weakening then right to each of the sequents

and

then apply left

The case for is symmetric

If is Qx x for some quantier Q then to the sequent

a a

apply b oth intro duction rules for the quantier left rst for universal right rst

for existential

If is f then by a separate induction on i derive

f p f p

i i

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

A p A p

i i

and then applying f intro duction Finallyapply subst to f p f p i

Chapter BPLK and G

We apply at most inferences for each connectivein and P and the pro of can easily

b e computed using a recursion on the structure of so r xx is sucient

Lemma Simple Manipulations Thereisapolynomial r such that if a are

variables and A are formulas and if

a a

fol lows from

a a

via a proof of size k thenaproof of the sequent S

A A

from the sequent S

A A

A jO r k jAj jP j in the case of BPLK with P dening can be found in time O r k j

al l relevant function symbols and whose length is thus similarly bounded

Proof Simply substitute A for a in the original pro of The only case to consider is if a

sequent in the original pro of is an initial sequent in whichcasewe can apply lemma

The substitution expands the original pro of by at most a factor of jAj and adding the

subpro ofs obtained from lemma adds a factor of at most jAj so r k jAj k jAj

is sucient

Lemma Build Lemma for G and BPLK Therearepolynomialsize proofs

A B C AC B

AB C in either languagesystem In the case of BPLK the size of P for any formulas

the Boolean program dening the function symbols occurring in A B C isanargument

to the polynomial

Chapter BPLK and G

Proof The pro of is a simple induction on the structure of C

If C is a variable then we need to prove A B A B for some iwhichis

i i

trivial

A B D AD B and A B If C is D E D E or D weprove

E AE B and p erform some simple prop ositional manipulations to obtain the

desired sequent

If C is QxD x for some quantier Qwe rst prove A B D A p

D B p Some simple manipulations yield A B DA p D B p and then

two quantier intro ductions give A B QxDAx QxD B x Likewise we

obtain A B QxD B x QxD A x and then some simple manipulations

yield the result

If C is f D D then rst prove

j k

A B D AD B

then apply subst to the sequent

p q f p f q

j j

whichisproved by a simple induction on j to substitute D Aforp and D B

for q and then cut to obtain the result

We end with a nal lemma showing that substitution is like a derived rule in G

from Lemma Thereisapolynomial r such that a G proof of the sequent S

the sequent S p both in the language of G can be found in time O r jS j

Chapter BPLK and G

Proof The pro of is as follows First apply right then right to obtain an equiva

lent sequent with a single formula

p F

Apply right to obtain

xF x

By lemma the sequent

F F

S S

has a short pro of and thus left weakening and cut pro duce

The desired sequent follows after some simple manipulations

Chapter

BPLK PSimulates G

In this chapter we show that the system BPLK p olynomially simulates G Wedosoby

describing a pro cedure for translating a sequent in the language of G intoasemantically

equivalent one in the language of BPLK with an accompanying Bo olean program This

new Bo olean program denes function symb ols which o ccur in the translated sequent and

which replace the quantiers and b ound variables from the original one This translation

is used to translate a G pro of linebyline into the language of BPLK and we showhow

to ll in the gaps to form a correct pro of in the latter system Incidentally this technique

could easily b e adapted to provide an alternate pro of of the PSPACEcompleteness of

Bo olean programs as the pro of in did not involve a reduction from QBFs

Perhaps the rst kind of translation one would think of would b e to use Bo olean

programs to calculate Skolem functions and use these to replace variables b ound by

existential quantiers Variables b ound by universal quantiers would then b ecome free

variables However there are many problems with this approach First of all this typ e of

translation would add free variables so in some sense it is a less faithful translation than

wewould like Second formulas in the antecedent of a sequent o ccur negatively in the

context of the entire sequent so wewould have to translate those quantiers dierently

than those in the succedent This means that the same form ula would havetwo dierent

Chapter BPLK PSimulates G

translations dep ending on which side of the sequent it o ccurred in and this would greatly

complicate simulating the cut rule

Instead the translation we adopt is very much reminiscentofthework of Hilb ert and

Bernays and Those authors used a dierent language instead of quantiers

they used the socalled calculus wherein xAxwould b e written as A Ax

and xAxas A Ax The symbol Ax represents an ob ject which will

x x

satisfy Aifsuch an ob ject exists Those authors did not ascrib e any kind of functional

interpretation to the symb ols but since our universe is of only two elements it turns

out that Bo olean programs can easily compute the values of suchsymb ols and this is

where we base our translation

An alternative translation suggested byToniann Pitassi mirrors the denition of

prop ositional quantiers and would replace xA xby A A and xAxby

A A Toavoid exp onential increase in formula size eachofthesewould then

b e replaced by a function symb ol dened by a Bo olean program Unfortunatelythe

principal diculty with the translation we do adopt namely that the names of the func

tion symb ols in translations change after intro duction of quantiers is also presentwith

this translation The complexity of the arguments would thus b e similar

Sp ecial Notation

In the translation in this chapter we shall use function symb ols whose names are based

on formulas of G Because of the delicate nature of these names we shall b e esp ecially de

tailed and careful ab out substitutions and the names of free variables We shall therefore

in this chapter only mo dify our notation

When a formula A is displayed as A p we shall mean that p are al l the free variables

of AWe shall never use the notation Aq to mean a substitution but instead that A

has the single free variable q All substitutions will b e denoted with a p ostx op erator

Chapter BPLK PSimulates G

Bp which means that the formula B whichmay b e a single variable is substituted

for p in the formula which precedes the op erator Wemay write several of these in a row

and the meaning is that they are p erformed in sequence from left to right Wemay also

interject free variable lists as in the example

Aa bB q rsbq rsaCsDr

ATranslation from the Language of G to that of

BPLK

To aid in translating weintro duce the following denition whichgivesusawelldened

merging op erator whichsays how to combine several Bo olean programs into one elimi

nating duplicate denitions

Denition Merging P Q of Bo olean Programs If P and Q areBoolean

programs or at least fragments not necessarily dening al l function symbols usedin

denitions then the merging P Q of P and Q is obtained by rst concatenating P and

Q and then deleting al l but the rst denition of each function symbol

We shall use this merging op erator in the following translation and later on in the

actual simulation However whenever we merge two Bo olean programs which b oth dene

a particular function symb ol it will always b e the case that the two denitions are

identical Thus which of the denitions is kept is immaterial

Nowwe can present the actual translation dened rst for single formulas

Denition Translation pqP of We recursively dene a transla

tion of a quantiedpropositional semiformula into a semantical ly equivalent quantier

free formula in the language of BPLK together with a Boolean program dening the

function symbols in that formula

Chapter BPLK PSimulates G

If is an atom p then pqP is

If is then pqP is p q p qP P p q

p qP P

If is then pqP is p qP

If is x x p then pqP is f p P P Here P is a

Boolean program fragment with the fol lowing two denitions

p p qxp

f p p q pxp

If is x x p then pqP is f p P P Here P is a

Boolean program fragment with the fol lowing two denitions

p p qxp

f p p q pxp

p we Thus in the case where a quantier is the toplevel connectiveof Qx x

rst pick a function symb ol whose name dep ends on the formula ie including more

deeply nested quantiers and free variables This choice enco des b oth whichvariable

is the most newly quantied one and also what kind of quantier it is We call this

function symb ol a witnessing function for b ecause its value will satisfy if p ossible

in case Q is existential or else it will falsify if p ossible

We then substitute this function symb ol applied to all the variables free in into the

translation of Finallywe dene a new function symb ol using the resulting formula

and use it instead of the formula We do so b ecause p q p xp has more o ccur

rences of free variables than p qp and so without the new function symb ol subsequent

substitutions of this form may increase the length of the formula exp onentially

Chapter BPLK PSimulates G

Note that if is quantierfree then it is unchanged by the translation and the

Bo olean program which results is empty

Denition If S is the sequent

j k

then pS qPS is

p qp q p qp q P P

j k k

We call P orP S the Bo olean program arising from the translation As a nal

lemma in this section we show that translations are p olynomial size

Lemma If S is a sequent in the language of G then j pS qPS jO jS j

Proof First note that jpqj O jj Inductively following the recursive denition all

the Bo olean connective cases increase the translation size linearly In the case of

x x p the translation pq is just f p so the number of symb ols in the translation is

at most twice the numb er in the original formula This is b ecause to write the translation

we write just an f a copyof and then a list of s free variables

Now the Bo olean program arising from a translation of contains at most two

function symb ols for eachquantier in The size of the names of the function symb ols

is linear in jj since the name is based on a subformula of The dening formula for

an function symb ol is linear in size from the previous paragraph since it is essentially

just a translation of a subformula of The dening formula for an f function symbol

consists of a translation of a subformula of of linear size with an function symbol

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

of the entire Bo olean program is thus in O jS j

Chapter BPLK PSimulates G

ASimulation of G by BPLK

First weshow that the translations dened ab ove are semantically equivalenttothe

original formulas

Lemma In the presence of the Boolean program P dening the function sym

bols as above arising from the translation a possibly quantiedBoolean formula is

semantical ly equivalent to its translation pq

Proof The pro of is by a simple induction on the structure of The interesting cases are

when is x x porx x p In the rst case if the translation holds then the value

obtained byevaluating the witnessing function satises andso holds Converselyif

holds then either or satises and it is easy to see that the witnessing function

will evaluate to a satisfying value

In the second case observe that the witnessing function will falsify if p ossible Thus

if the translation holds then is satised by b oth and Converselyif holds then

is satised no matter what the witnessing function evaluates to

Note that the correctness of the cases of and dep ends on the fact that whenever

two Bo olean programs are merged while translating a QBF they never disagree on the

denition of any function symbol Thus for example the value of p q with resp ect

to P is indeed the Bo olean of the values of p q and p q each with resp ect to

their own Bo olean programs

Now the op erations of substitution and translation do not commute directly even

in the case of substituting only a single variable so for example if Aa xa x

then its translation is pAqa f a If we then substitute b for a we obtain

xax

pAqba f b On the other hand if we did these things in the reverse order

xax

wewould get pAbaq f b which is dierent A technical lemma is needed to

xbx

resolve this diculty

Chapter BPLK PSimulates G

Lemma Let As p r be a semiformula and B p q a formula both in the lan

guage of G soB is automatical ly free for s in Aandlet p beallfree variables except s

which arecommon to A and B s may beinq Then proofs of

pABsq p q r pAqpB qsp q r

and

p q r pABsqp q r pAqpB qs

can be found in time polynomial in the size of translations and the size of the Boolean

program arising from them

Proof Weprove the lemma by induction on the structure of A The base case is if A is

atomic If A is s then pAq is ssopABsq pB q pAqpB qs Otherwise A is some

other variable u so pABsq pAqpB qs u In either case apply lemma

If A is not atomic then wehave the following cases for each p ossible main connective

A is C D By assumption weha ve pro ofs of pC Bsq pC qpB qsand

pD Bsq pD qpB qs Use weakening and right to pro duce new se

quents with succedent pC qpB qs pD qpB qs pC D qpB qs and then

apply leftTheconverse sequentisproved similarly

The case for is symmetric and that for is similarly easy

A is QxC x s p r In this case wehave already found a pro of of pC Bsq

pC qpB qs and a pro of of the converse sequent Wemust nd pro ofs of

pABsq pAqpB qs and its converse Now the rst step is to derive

p q r x pABsq pC Bsq

ABs

and

pAqpB qs pC q s p r xpB qs pC qpB qs pB q p r x

A A

Chapter BPLK PSimulates G

which follow directly from the dening equations for pABsq f p q r and

ABs

pAq f s p r resp ectively

The next step is to derive

p q r pB q p r

ABs

First use subst on the endsequentof from the induction hyp othesis to sub

stitute or as appropriate for Qfor x to obtain pC Bsqxp q r

pC qpB qsxp q r Then intro duction is applied on the left and

ABs

intro duction on the right The pro cess is rep eated with and the intro duction

rules used on the opp osite sides of the sequent and the equality follows

p q r for x in the endsequent The nal step is as follows First substitute

ABs

of We obtain a pro of of

pC Bsq p q r x pC qpB qs p q r x

ABs ABs

Then use the equality derived in the previous paragraph with the build

lemma to obtain a pro of of

p q r x pC qpB qs pB q p r x pC qpB qs

A ABs

Now using and we can easily obtain

pC Bsq p q r x pC qpB qs pB q p r x

ABs

The desired sequent pABsq pAqpB qs is then obtained using and

The pro of of the other desired sequent is obtained symmetrically to the nal step

ab ove but starting with

By induction therefore we can obtain and pro ofs of the desired sequents in

p olynomial time

Chapter BPLK PSimulates G

The main result of the chapter follows

Theorem If S is a quantierfreesequent with a proof in G then S has a

BPLKproof P which given can be found in time polynomial in j j and

thus has polynomial size

Proof First of all P is formed by merging the Bo olean programs arising from trans

lating all the sequents in More preciselyif is S S then P P S

P S

We then form directly by translating sequentb ysequentinto the language of

BPLK and when necessary adding some extra sequents b etween the translated ones We

have the following cases dep ending on the inference rule used

Observe that all initial sequents of G are their own translations and are also initial

sequents of BPLK

If S is noninitial and is inferred by any rule except a quantier intro duction with

hyp othesises T and U then pS q follows from pT q and pU qby the same rule

This is b ecause translations of identical formulas are identical and the translation

op erator commutes with the connectives and

If S is inferred from T by the intro duction of a universal quantier on the right or

that of an existential quantier on the left then considering the rst case wehave

that T is

Ca p

and S is

p xC xax

Their translations are thus

pq pq pC qa p

Chapter BPLK PSimulates G

and

pq pqf p

xC xa

resp ectivelyFor convenience let A C xa In this notation pT q is

pq pq pAaxqa p

First pAaxq pAqpaqx follows from lemma taking B a and s x

pT q and cut then yield

pq pq pAqax

Next observe that a cannot o ccur in or or therefore their translations due to

the restriction on right So when weapplysubst to this sequent substituting

pfor a to obtain

pq pq pAq px

p x f p follows from the pq and pq are unchanged Finally pAq

xA xA

denition of the function symbol f using the intro duction right rule for that

symbol pS q follows with weakening and cut

The interesting case is when S follows from T by the intro duction of an existential

quantier on the right or that of a universal quantier on the left The twocases

are symmetrical so consider the rst T is

Ax pBx

and S is

xAx p

The translations pT q and pS q are

pq pq pABxq

Chapter BPLK PSimulates G

and

pq pqf p

Thus it suces to prove pABxq f p and apply cut

First we derive pABxq pAqpB qx using lemma

The next task is to prove

pAqpB qx pAq px

This sequent has a pro of involving applications of the build lemma In this pro of

sketchwe omit the corner brackets and the subscript on We rst show that

either satises A or not Next if satises A then must since in this case

will evaluate to one We then show that if do es not satisfy AbutB do es then

must also since in this case B and will b oth evaluate to Finallywe combine

these three pieces to get the result

p Ax BP Def n

Ax Ax Easy

p Ax

Ax p

Ax p weakening

Axp weakening

AxAx Apx build lemma

Ax Apx contraction

p left weakening Ax

Ax p weakening

AxAx A px build lemma

B B weakening

Chapter BPLK PSimulates G

B B B B weakening

B ABx Ax build lemma

AxABxB left weakening

AxABx B weakening

AxABxABx Ax build lemma

AxABx Apx contraction weakening cut

ABx Apx weakening cut

Lastly pAq px f pfollows from the denition of f

xA xA xA

Now starting with pT q and using each of these last three sequents in turn with

weakening and cut one obtains the sequent pS q as desired

Finally in the case that S is quantierfree for example the nal sequent in a pro of

then the translation of S is S and this completes the pro of of the theorem

Chapter

G PSimulates BPLK

In this chapter we dene a translation of Bo olean programs into formulas in the language

of G that is using prop ositional quantiers Ultimatelywewant to translate sequents in

BPLK into equivalentonesin G analogously to the previous chapter and we shall use the

translated Bo olean program to obtain the values of function symb ol applications Since a

line of the program can reference previous lines in eect using a shorter Bo olean program

to dene the current function symb ol an inductive construction is indicated using the

translations of previous lines to obtain the translation of the currentoneHowever if

more than one copy of previous translations are ever used to translate a line there is a

p ossibility that the formula sizes will increase exp onentially Therefore our translation

will pro duce one formula which can b e used to simultaneously evaluate all the function

symb ols in the Bo olean program and one copy of this formula will b e incorp orated into

the translation of the next line

The metho d we shall use to multiplex the single o ccurrence of a translation is

inspired by a trickusedby Sto ckmeyer and Meyer in to show that the problem of

evaluating a given quantied Bo olean formulaQBFisPSPACEcomplete The idea is

to abbreviate x y x y as x y x x y y x x y y x y

Chapter G PSimulates BPLK

This is not exactly what is needed but is quite close

A Translation from the Language of BPLK to

that of G

Since wehave an innite supply of variables let us reserve v t z d andu as new

variables not o ccurring in the BPLKpro of b eing translated There will b e a variables v

and t for eachformula A in the language of BPLK and we shall replace o ccurrences of

function symb ols by these variables as part of our translation

Denition Hat and Corner Op erators If A p is a formula in the language

p are the variables freein A then let Ap v be the result of replacing in A of BPLK

every maximal subformula f B of A whose main connective is a function symbol by the

corresponding variable v This new formula wil l have some subset of the original p

f B

and also some v s as free variables The corner operator eg pAq is dened similarly

except that corresponding t variables areusedinstead of v s If either operator is applied

to a set of formulas we mean that it should be appliedtoeach formula in the set in turn

Example Hat Op erator If p is the formula

f p g f p p p

p v is then

v v p

f p g f p p

We shall assume that a Bo olean program denes functions f f and that each

p A p In other words function symb ol in the Bo olean program is dened as f

i i

A is the dening formula of f Now z u will b e the translation of the Bo olean

i i k

Chapter G PSimulates BPLK

program up to and including the denition of f The meaning of z z u u

k k k k

will b e that this formula is satised if and only if f z u and f z u

k k k

We are now in a p osition to dene the translation of a Bo olean program more formally

Denition Consider a Boolean program dening f f Dene Now

say the denition of f includes the function symbol occurrences f B f B in

i j j m

order some of which may be nested in others Here j j j are just the indexes of

the function symbols ie a sequenceofm integers not necessarily distinct each smal ler

than i Then dene

z z u u

i i i

v A z v u

i i i

f B z f B z

j j i j j i

z z u z u u

z z u v B

f B z

j j i

v z z u B

f B i z

j j j i

z z u v B

B i f z m

j j j i

m m m

z z u u

u z z u

i i

Example Consider the fol lowing simple contrivedBoolean program

f p p p p

f p p f p p f f p p p

Chapter G PSimulates BPLK

Then by our terminology A p is p p and A p is f p p f f p p p

is Now In the denition of f therearenooccurrences of function symbols so

thereareno B We thus dene

z z u z z u

has a number of degenerate portions and is therefore not so interesting In the deni

tion of f f occurs times and is the only function symbol present By our terminology

we have the fol lowing

B p p

B p f p p

B p p

z v z B

B z v z

B z v v

f z z

B z v z

z v z B

A z v v v

f z z f f z z z

Chapter G PSimulates BPLK

Now is constructedfrom the above as indicated in the denition

z z u u

v v v v u

f z z f f z z z f z z f f z z z

z u z u

z z z z u v

f z z

z v z z u v

f z z f f z z z

z z z z u v

f z z

z z z z u u

Claim For each i

z u

is semantical ly equivalent to

f z u f z u

i i i

Proof First the statementvacuously holds for i

Now supp ose it holds for i If z u holds then there exist v s satisfying the

part of marked which ensures that they have the same values as the function

z u The conjuncts ensure that symb ol applications they replace so indeed f

i i i

f z u j i

j j j

Converselyif f z u f z u holds then the v s satisfying which

i i i

z v u Also is exist and are unique must have the correct values and so A

i i i

clearly satised and thus all of is

We can now dene the translation of sequents This translation is in the context of

a BPLKpro of so the Bo olean program and the rest of the sequents in the pro of are

Chapter G PSimulates BPLK

already xed Exactly which pro of a particular translation is relative to is not indicated

in the notation but it will always b e clear from the context

Denition Translation pS q of the sequent S relativeto Fix a BPLK

proof and its associatedBoolean program dening f f Let f C be a list of al l

k j i

subformulas in whose main connective is a function symbol C arearguments to f

i j

and again j are simply indexes

Then the sequent S

is translated as the sequent pS q

pC q d d t d d

j f C j k

m m m m

t d d pC q d d

m k

j j

f k C

m m

pq pq

pC q and the corresponding ts are in the correct places to be the arguments Herethe

to and the values of the function symbol f The d are dummy variables We could use

l l

t but it wil l beconvenient instead of d since d wil l beconstrained to the value f

i i

later on that the ds are distinct We shal l cal l the occurrences of above the prex of

the translation and the remainder the sux

Now these translations mayhavefreevariables that the original ones did not ts

and ds We cannot therefore assert semantic equivalence of the two However we are

concerned with proving valid sequents and we can say something nearly as go o d

The idea is that if the translation of a sequent is satised by some assignment then

either one of the t or d variables has an incorrect value falsifying the corresp onding

instance of or else they all have the correct values and the remainder of the translated

sequent is satised In that case the original sequent is satised by the same assignment

Chapter G PSimulates BPLK

Conversely if the original sequentisvalid then every assignment to the translation

will either falsify one of the s or else all the ts will have the correct value and thus

the remainder of the sequent will b e satised Therefore

Claim For any sequent S from the language of BPLK S is valid if and only pS q

The nal lemma in this section shows that translations are p olynomial size

Lemma Let S beasequent from the BPLKproof P Then jpS qj

O jP j j j

Proof First note that for any BPLK formula wehave jj jpqj O jj These

op erators add a constan tnumber of symb ols for each replacement they p erform and this

numb er is b ounded by the size of the formula

Next consider the construction of from The following are added

i i

copies of A

copies of B for each B which is the argument to a function symbol in A in the

section

o ccurrences of the corresp onding v variables in the section and the quantier

section whose size is in O jP j

Therefore summing these all up for through we see that the last item dominates

the sum and that j j O jP j

Finally pS q consists of the prex at most j j o ccurrences of each with substi

tutions of size at most j j followed by the sux of size O jS j Therefore jpS qj

O jP j j j

Chapter G PSimulates BPLK

ASimulation of BPLK by G

We rst show that pro ofs of sequents from two sp ecial classes are ecient to nd

Lemma Existence Sequents Thereisapolynomial r such that for every i

the sequent E

zu z u

has a proof which can be found in time O r jE j and whose length is thus similarly

bounded

Lemma Uniqueness Sequents Thereisapolynomial r such that for every

i the sequent U

z z u u z u z u

i i

z z u u

i i i i

has a proof which can be found in time O r jU j and whose length is thus similarly

bounded

Proof These two lemmas are proved by induction in parallel

For i the result is trivial

B B B b e all formulas Now assume the twolemmas are proved for i Let

app earing as arguments to function symb ols in the denition of f B as arguments to

i w

Chapter G PSimulates BPLK

Existence and uniqueness for plus some simple manipulations give f

i j

v z z u u z u

i i

z B u v

j j

B f

w w

z z u u

j j

c c

Some more simple manipulations simply conjoining the tautology A z v A z v

i i i i i i

inside the outermost quantier give

c c

v A z v A z v

i i i i i i

z z u u z u

z B u v

j f B

w w

z z u u

j j

and then right on the us and one instance of A and right on the z s yield

the existence sequent for

Now in the case of uniqueness note that

z u z u

i i

Chapter G PSimulates BPLK

has a short pro of using existence and uniqueness for i and some simple manipulations

Thus

z u z u

i i

z z u u

i i i i

follows by uniqueness for i Nowwe pro ceed as follows

First by the denition of

z u z u

i i

c c

v A z v u v A z v u

i i i i i i

Then renaming the quantied variables and doing some simple manipulations

z u z u

i i

c c

v v A z v u A z v u

i i i i i i

Uniqueness for i and more manipulations allows us to provethatthe v s in one of

the conjuncts are equal to those in the other and thus pro duce

z u z u

i i

c c

v v A z v u A z v u

i i i i i i

We can similarly consolidate the z s by adding a hyp othesis

z u z u

i i

c c

z z v v A z v u A z v u

i i i i i i i i

Contracting

z u z u z z v v u u

i i i i i i

We can now drop the quantier

z u z u z z u u

i i i i i i

Chapter G PSimulates BPLK

Some simple manipulations to combine this last sequent with and then right

and several applications of right pro duce the uniqueness sequent for i

Finallywe can state and prove the main result

Theorem If S has a BPLKproof then S has a G proof which given

can be found in time polynomial in j j and thus has polynomial size

Proof We construct directly by translating sequentbysequent into the language

of G relative to the Bo olean program of If necessarywe insert sequents to provethe

translation of a sequent from the translations of its hyp otheses

First of all if S is an initial sequent of BPLK then it is function symb ol free and so

its translation is itself and thus already an initial sequentofG

Now consider a noninitial sequent S inferred from previous ones If the inference was

weakening contractionorintro duction of or then the same rule yields pS q

If S T is inferred from T pby subst then note that without loss of generality

wemay assume that p do es not o ccur in Otherwise we could mo dify to p erform

subst twice once to substitute q for p q is a variable which does not occur in T and

then again to substitute p for q Tosimulate the substitution in G rst use lemma

to substitute p q for p in pT qp t obtaining pT qp q t Finally apply lemma

which follows after this pro of

The last case in the pro of is when S is inferred by f intro duction intro ducing

f B p Then clearly the sequent

pA B qp tt

k i

B f

together with some simple manipulations will pro duce pS q basically just by using cut

We derive the desired sequent as follows First the following is straightforward

pB q u t d pB q ut d

i k

f B f B

i i

Chapter G PSimulates BPLK

Next we add the rest of the prex

pB q u t d

k i

f B

Expanding the

v A z pB q v t

k i i

f B

Note that the A o ccurrence ab ovecontains t variables from the pB q substituted for

the z and also v variables from function symb ols o ccurring in the denition of f

i i

Uniqueness for and

pC B q ut d pC B q ut d

i k

f C B f C B

j j

one sequent for each function symb ol o ccurrence f C z in the denition of f allow

j i i

in the o ccurrence of A ab ovetot us to rename the v B q pro ducing pA

i i

C z f

i j f C B

and then we drop the existential quantier and some conjuncts to get

B q t pA

k i

f B

which is the desired sequent

Nearing the end of the pro of now if S is the last sequent of the pro of then it is

function symb olfree We need only remove the prex from pS q to obtain S The t

variable corresp onding to the outermost function symb ol application in there may

b e many outermost applications is dened by an o ccurrence of but it is not used

in the denition of anyoftheothert variables Wemaythus use left on the t

and the ds followed by left on the B s and the s to change this o ccurrence into

z u z u whichwe can cut away with the existence sequentand weakeningWe

can now do the same for the next most outer function symb ol application and so on

The resulting sequent at the end of this pro cess is S which completes the pro of

All that remains is to prove lemma This lemma is analogous to lemma

of the previous chapter and is needed b ecause substitution do es not commute with translation

Chapter G PSimulates BPLK

Lemma If T p is a sequent in a BPLKproof and is a BPLK formula in

which p does not occur then a G proof of pT q from pT qp q can be found in time

polynomial in the size of its endsequent

Proof The rst step is to use simple manipulations to rename all the variables t in

pT qp q A variable t is renamed to t by an application of lemma This

B p B

renaming can b e done in any order and call the resulting sequent U Now it is easy to

see that for every o ccurrence of a subformula of the form pC pq in pT q the corresp ond

ing o ccurrence in U is pC q This follows b ecause whenever the translation op erator

replaces a subformula B pof C pby a function symb ol the symb ols name is t and

B p

so after the renaming it will b e t as it should b e

Now consider anyvariable t o ccurring in pT qThisvariable is dened byan

f B p

o ccurrence of in the prex of pT q

pB pq d d t d d

i i k

f B p

In fact it is p ossible that this variable o ccurs only in the prex After the substitution

of p q into pT q the corresp onding o ccurrence b ecame

d d pB pqp q d d t

i k i

B p f

After the renaming in U this o ccurrence b ecomes

pB q d d t d d

i i k

f B

which correctly denes t

B f

Now b efore the nal step note that the sux of U is identical to the sux of pT q

and those o ccurrences of dening t variables in the sux of pT q also o ccur in

U The only dierence then b etween U and pT q is that the former sequentmay

have some prex formulas which the latter do es not and vice versa We can thus use

the existence sequents or contraction in the case of a duplicate to cut awaythe

sup eruous prex formulas from U andweakening to add the missing ones The result

is the desired sequent

Chapter

Future Work and Conclusions

In this thesis we demonstrated a strong connection b etween two prop ositional pro of

systems b oth based on PSPACE reasoning These results raise manyinteresting questions

which remain unsolved

ATechnical Improvement

First of all from a technical p ersp ectiveitwould b e nice to get rid of the subst rule from

BPLK It is shown in Dowd that extended Frege systems psimulate substitution

Frege Bo olean programs would app ear to b e a generalization of the extension rule so it

seems reasonable that a similar result to Dowds might hold whichwould allowaversion

of BPLK without subst to psimulate the substaugmented version

Witnessing and Search Problems

Buss and Kra jcek in show that those functions which are denable in T are

exactly p olynomial time pro jections of PLS functions PLS is Papadimitrious class of

p olynomial lo cal search problems and is discussed in and Because of the

corresp ondence b etween T and G it is therefore the case that the problem of nding

Chapter Future Work and Conclusions

witnesses for the quantiers in a pro of in G is also exactly as hard as PLS

Several lines of research are suggested First it would b e interesting to characterize

the hardness of the witnessing problems for the other subsystems of G and indeed

dierent kinds of denability in the subsystems of T and S Part of this work has

b b

recently b een done by Chiari and Kra jcek in for and denabilityin T but

nothing general is known yet Secondly there are other lo cal search problems than PLS

some of which are discussed in and in more detail in It would b e interesting to

nd prop ositional pro of systems whose witnessing problems were exactly pro jections of

these other lo cal search problem classes

Subsystems of BPLK

Another set of questions which are particularly interesting concerns the p ossibilityof

nding natural subsystems of BPLK akin to the structure of G In their pap er

the authors nd a natural restriction of Bo olean programs essentially amounting to

extension axioms for witnessing pro ofs in G Itwould b e instructive to nd restrictions

of Bo olean programs whichwould naturally witness pro ofs in other subsystems of G It

would also b e interesting to nd some kind of a hierarchy within BPLK whichmayor

may not corresp ond to the hierarchyin G

Miscellaneous

Finally it is p ossible that due to the apparent ease of use of BPLK more p ositive results

may b e forthcoming than with G For example it may not b e to o dicult to pro duce

p olynomialsized pro ofs in BPLK of some of the conjectured hard examples for Frege

and extended Frege systems As another example the connection b etween G and U

which currently is restricted to only formulas might b e generalized to handle more

general theorems in particular including the secondorder features of that system

Bibliography

M Ajtai The complexity of the pigeonhole principle In th Annual Symposium on

Foundations of Computer Science pages White Plains New York

Octob er IEEE

Paul Beame Stephen Co ok Je Edmonds Russell Impagliazzo and Toniann

Pitassi The relative complexity of NP search problems Journal of Computer and

System Sciences August

Maria Luisa Bonet Samuel R Buss and Toniann Pitassi Are there hard examples

for frege systems In P Clote J Remmel eds Feasible Mathematics II pages

Birkhauser Boston

S Buss BoundedArithmetic Bibliop olis Naples

Samuel Buss and Jan Kra jcek An application of Bo olean complexity to separation

problems in b ounded arithmetic Proceedings of the London Mathematical Society

Samuel R Buss Relating the b ounded arithmetic and p olynomial time hierarchies

Annals of Pure and AppliedLogic Septemb er

Samuel R Buss editor Handbook of Proof Theory Elsevier Science B V Amster

dam

Bibliography

Mario Chiari and Jan Kra jcek Witnessing functions in b ounded arithmetic and

search problems The Journal of Symbolic Logic Septemb er

S A Co ok CSC S Pro of Complexity and Bounded Arithmetic Course notes

URL httpwwwcstorontoedusaco okcsch Spring

Stephen Co ok and Rob ert Reckhow On the lengths of pro ofs in the prop ositional

calculus preliminary version In ConferenceRecordofSixthAnnual ACM Sympo

sium on Theory of Computing pages Seattle Washington April May

Stephen Co ok and Michael Soltys Bo olean programs and quantied prop ositional

pro of systems Bul letin of the Section of Logic

Stephen A Co ok The complexity of theoremproving pro cedures In Conference

Record of ThirdAnnual ACM Symposium on Theory of Computing pages

Shaker Heights Ohio

Stephen A Co ok Feasibly constructive pro ofs and the prop ositional calculus pre

liminary version In ConferenceRecord of Seventh Annual ACM Symposium on

Theory of Computation pages Albuquerque New Mexico May

Stephen A Co ok and Rob ert A Reckhow The relative eciency of prop ositional

pro of systems Journal of Symbolic Lo gic

Martin Dowd Mo del theoretic asp ects of P NP Typ ewritten manuscript

Armin Haken The intractability of resolution Theoretical Computer Science

August

D Hilb ert and PBernays Grund lagen der Mathematik I Springer Berlin

D Hilb ert and PBernays Grund lagen der Mathematik II Springer Berlin

Bibliography

David S Johnson Christos H Papadimitriou and Mihalis Yannakakis How easy

is lo cal search Journal of Computer and System Sciences August

Jan Kra jcek BoundedArithmetic Propositional Logic and Complexity Theory

Cambridge University Press

Jan Kra jcek and Pavel Pudlak Prop ositional pro of systems the consistency of rst

order theories and the complexity of computations The Journal of Symbolic Logic

Jan Kra jcek Pavel Pudlak and Gaisi Takeuti Bounded arithmetic and the p oly

nomial hierarchy Annals of Pure and AppliedLogic

p olynomial induction In S R Jan Kra jcek and Gaisi Takeuti On b ounded

Buss and P J Scott editors FEASMATH Feasible Mathematics A Mathematical

Sciences Institute Workshop pages Birkhauser

Christos H Papadimitriou On the complexity of the parity argument and other

inecient pro ofs of existence Journal of Computer and System Sciences

June

Alexander A Razb orov and Steven Rudich Natural pro ofs Journal of Computer

and System Sciences August

L J Sto ckmeyer and A R Meyer Word problems requiring exp onential time

Preliminary rep ort In ConferenceRecord of Fifth Annual ACM Symposium on

Theory of Computing pages Austin Texas April May

Mihalis Yannakakis Computational complexity In Emile Aarts and Jan Karel

Lenstra editors Local Search in Combinatorial Optimization pages John

Wiley and Sons Chichester UK

Bibliography

D Zamb ella Notes on p olynomially b ounded arithmetic The Journal of Symbolic

Logic