Relating the PSPACE reasoning power of Boolean Programs and Quantified Boolean Formulas by Alan Skelley A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science UniversityofToronto c Copyright by Alan Skelley 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 i i sp ondence b etween U a second order system of Buss and G although only for b 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 i of alternations of quantiers allowed in the formulas and the reasoning p ower is then p that of predicates i 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