Barbosa, Uniform Polynomial Time Bounds, and Promises

Barb osa, Uniform P olynomia l Time Bounds, and Promises ∗ Lane A. Hemaspaandra, † Kyle Murra y , and Xiao qing T ang Departmen t of Computer Science Univ ersit y of Ro c hester Ro che ster, NY 146 27, USA June 6, 2011 Abstract This note is a commentary on, and critique of, Andr´ e Luiz Barb osa ’s pap er entitled “P != NP Pr o of.” Des pite its prov o cative title, what the paper is seeking to do is not to pr ov e P 6 = NP in the standar d sense in which that notatio n is used in the litera ture. Rather, Barb os a is (and is a ware that he is) arg uing th at a different meaning should be asso cia ted with the notation P 6 = NP , and he claims to prov e the truth of the statement P 6 = NP in his q uite different sense o f that s ta tement . How ever, we note that (1) the pap er fails even on its own terms, as due to a uniformit y pro blem, the pa p er’s pro of do es not esta blish, e ven in its un usual sense of the nota tio n, that P 6 = NP; and (2) what the pa p e r means by the claim P 6 = NP in fact implies that P 6 = NP holds even under the standard meaning that that notation has in the literature (and so it is exceedingly unlikely that Barb o sa’s pr o of can b e fixed a ny time so on). 1 In tro duc tion This note is a b r ief comment ary on an arXiv.or g article by Barb osa that has the p r o v o cativ e title “P != NP Pr o of.” Despite the title, the pap er is not really pu rp ortin g to resolv e the P versus NP problem in the standard sense. Rather, the author redefi n es the notat ion P 6 = NP, casting it in effect to b e a statemen t that there exists at least one (p oten tially restricted, p romise-lik e) domain on which P and NP differ. Barb osa is a w are of this distinction and op enly d iscusses the fact, sa ying “I hav e restated the P v ersus NP question on my o wn terms.” What w e observ e in this note is that the pap er fails ev en on its own terms: Due to a uniformity pr ob lem, it do es not prov e what it claims to. But can the pr o of someho w b e fixed? W e note that what Barb osa means by P 6 = NP in fact (if true) itself implies (in the standard sense) P 6 = NP. And so, despite the fact that Barb osa’s pap er is at the momen t ∗ Also app ears as U RCS-TR-2011-969 . † URL: www.cs.rocheste r.edu/u/lane . Supported in part by gran t N SF-CCF-0915792 and a F riedric h Wilhelm Bessel Research Aw ard. 1 w e are w r iting this on its 38th public v ersion in less than t w o ye ars, w e feel it is un lik ely that Barb osa’s altered notion of what “P 6 = NP” means can b e established an y time so on. 2 Con text-Setting This en tire note is b ased on the most recen t v ersion of Ba rb osa’s pap er a v ailable as this n ote w as b eing drafted, namely , V ersion 38 (the v ersion of Ma y 21, 2011) of arXiv.org rep ort 0907.3 965 [Bar11]. W e write this simply as a v ery brief note, so we assume that the reader has Barb osa’s pap er in hand, and w e don’t even set the stage with his definitions (except in S ection 3 we briefly deal with s ome clarificatio ns that are n eeded). Readers ma y w onder whether it make s sense to r esp ond to pap ers that ha ve P = NP or P 6 = NP in their title. In the particular case of the present critique, the critique grew out of a teac hing exercise in an undergradu ate course, in particular, a pr oblem-solving course that has a tradition of, as an exercise (and arguably as a small service to the literature regarding clarifying the meaning and correctness of claims that hav e b een publicly made), lo oking at pap ers that m igh t seem to claim to resolv e P-versus-NP or the complexit y of graph isomorphism, in ord er to u n derstand w h at th e pap ers are claiming, and wh ether they ac h iev e their claims, and to share what is learned from that. The pr esent pap er is authored, jointl y with the course’s instructor, by the t wo stud en ts w ho lo oked at Barb osa’s pap er. (Among critiques to come out of earlier y ears’ instances of the same course are the pap ers of Sab o, Sc hmitt, and Silverman [SSS 07], Clingerman, Hemphill, and Pr oscia [CHP08], Christopher, Huo, and J acobs [CHJ 08], and F erraro, Hall, and W o o d [FHW09], wh ic h are resp ectiv ely resp ond ing to extremely strong P -versus-NP claims made by F einstein, Y atsenk o, Gu b in, and Aslam. W e commend to the reader’s atten tion W oeginger’s v aluable, excellen t, quite remark ably web page on attempted resolutions/refutations r egarding P ve rsu s NP [W o e ]. 3 Critique Before b eginning th e critique, we must cov er some houseke eping items. W e w on’t rep eat Barb osa’s defi n itions here, bu t w e do n eed a notation that will let us distinguish b et wee n his notions and the standard n otions, and we also need to smo oth out some of the arguable confusions in his pap er, so we can meaningfully critique what he is seemingly trying to express. Briefly , he is prop osing a promise-lik e notion in which instead of having Σ ∗ as the domain, th e domain is instead (somewhat magically) restricted to some set L z ( z is not an y particular item here; th e notation L z is simply a n otation from Barb osa), L z ⊆ Σ ∗ . Th is o ccurs in Definition 3.5 of [Bar11]. Barb osa d o es not require that L z b e, for example, a recursiv e set. It can b e an y set. T o a void an y confu sion, let us ca ll the v ersion of NP creat ed (for the restriction L z ) b y Barb osa’s defin ition NP[ L z ]. And Barb osa’s pap er’s notion of what h e prop oses as a goo d, new, c h an ged semantic s for the notation P 6 = NP is in fact 2 what one wo uld more naturally denote by the claim ( ∃ L z ⊆ Σ ∗ )[ P [ L z ] 6 = N P [ L z ]] . In a more common wa y of sp eaking of suc h things, one migh t expr ess this as follo ws (and Barb osa as his S ection 3.3.2 do es mention that his notion is essentiall y the same as the notion of promise problems [ESY84,S el88]): There exists a pr omise problem that has a solution in NP but not in P. 1 (This already should mak e clear to the reader the seemingly insurmountable problems one would f ace to prov e Barb osa’s claim, b ut let us fir s t clear up the definition b efore returning to that later.) As to Barb osa’s definition (still Definition 3.5 of his pap er), w e ment ion in passing th at (a) the big-O isn ’t used pr op erly (rather, he needs to b efore the universal quantificati on on x fix a p olynomial b ounding the length of the certificates; w e from h ere on assume that his definition is viewed as b eing mo dified to do that); and (b) he defines NP[ L z ]—whic h he extremely u nfortunately denotes simp ly NP—with L z simply app earing in the definition without an y clear quantificat ion, and so formally it isn’t clear th at his notion of P 6 = NP eve n manages to link the P and the NP to b e b ased on the same domain L z (but ha ving them b e th e same L z is the clear in tent of Barb osa’s pap er, and we fr om here on assume that that is th e case, i.e., w e treat Barb osa as if he is trying to pr o v e ( ∃ L z ⊆ Σ ∗ )[ P [ L z ] 6 = N P [ L z ]] , since that is indeed w h at his pap er is (seeking to) pr ov e). And with our ap ologies for ha ving tak en so long to get to suc h brief, straigh tforw ard p oints, we can n o w mak e our t w o comments ab out the cont ent of Barb osa’s pap er. Our fi rst p oin t is that regarding the test set (“X G-SA T”) that he claims to p ro v e, for a particular an d rather complex and clev er set L z that he defines, is in NP[ L z ] but is n ot in P[ L z ], h is pr o of that his test set b elongs to NP[ L z ] seems fla wed. I n particular, although it is called “X G-SA T ,” the set is actually ab out taking co des of programs and r unnin g them. The pro of attempts to (in the complicated and arguably ambiguous or ill-defined “Definition 2.1”) as a promise require eac h mac hin e to (mostly) ru n in p olynomial time. Ho wev er, there is no single, shared p olynomial, and Barb osa go es out of his w a y to s tate that he is not padding things do wn (in the w a y that routinely is done when making univ ersal complete sets, f or example) with a p adding string whose length is the allo we d ru nning time. The p r oblem with all this is that some mac h ines will r un in linear time, some will run in quadratic time, some in cubic time, and so on, and so the set X G-SA T has no ob vious single 1 One migh t w orry that Barbosa’s definition (see his p ap er) is not really clear about what it means when it says th at t he P-time predicate defining membership in NP[ L z ] is P-t ime—whether it means that the underlying machine is in P-time o ver all inputs that belong to L z or whether it is P-time globally (ov er Σ ∗ ). How ever, there is no issue here, as if th e former holds, one can take a polyn omial p ′ of the form n i + i t h at ma jorizes that machine’s p olynomial run-time boun d on L z , and can then take one’s original underlying mac hine M and from it build a new mac hine, M ′ , that on eac h inpu t sim ulates M for a num b er of steps that is p ′ applied to the input length. Note that M ′ is globally P-time. F or this reason, w e from here on will assume th at languages in NP[ L z ] and P[ L z ] are defined and instan tiated by mac hines that globally ob ey nondeterministic or deterministic p olynomial time b ounds. F or completeness, we mention in passing that Barbosa actually in th e stated definition defin es the P-time predicate R ’s domain as b eing L z × Σ ∗ , and so he ma y indeed b e th in k ing of the time b ound as app lying just on that region. 3 p olynomial upp er -b oundin g the nondeterministic runnin g time of a NTM accepting it. So the pro of that XG-SA T b elongs to NP[ L z ] seems inv alid. No w we turn to our second p oin t. Can one hop e to someho w fix Barb osa’s pro of and establish his claim? Recall that h is claim is that ( ∃ L z ⊆ Σ ∗ )[ P [ L z ] 6 = N P [ L z ]] . But supp ose that that h olds. That means (k eeping in mind the comments of F o otnote 1 ab out slapp ing on a global clo ck as n eeded) that there is an NPTM, call it N ′ , wh ose language when r estricted to L z differs, for ev ery DPTM M ′ , f r om the restriction to L z of the language of M ′ . Ho wev er, note that if P = NP (in the standard sens e the literature h as for what that means), then the language of N ′ is, since P = NP, accepted by some DPT M, call it c M . And so certainly the restriction of L ( c M ) to L z is iden tical to the restriction of L ( N ′ ) to L z , th us putting this in to P[ L z ], con trary to what w e assumed. Put another wa y , clearly it follo ws easily from the defin itions that: If ( ∃ L z ⊆ Σ ∗ )[ P [ L z ] 6 = N P [ L z ]], th en P 6 = NP. So p ro ving Barb osa’s main result w ould implicitly separate NP from P in the standard sense of the literature. Thus we fin d it u nlik ely that Barb osa’s main result can b e correctly pro v en in the im m ediate futur e (or at least it cannot b e pr o v en without the pro ve r winning the Cla y I nstitute’s million-dollar p rize). Ac kno wledgmen ts W e thank the memb ers of the Sprin g 2011 C SC200/200H Under- graduate Problem Seminar for a memorable course, and we th an k course T A J ake S c heib er for his feedbac k on an earlier v ersion of th is pap er. References [Bar11] A. Barb osa. P != NP pr o of. T ec hn ical Rep ort arXiv:0907.396 5 [cs.CC], Comput- ing Researc h R ep ository , h ttp://arXiv.org/ corr/ , Ma y 2011. V ersion 38 (of Ma y 21, 2011) ; the d ate of V ersion 1 was July 23, 2009. [CHJ08] I. Chr istoph er, D. Huo, and B. Jacobs. A criti que o f a p olynomial-time SA T solver devised by Sergey Gu b in. T ec hnical Rep ort arXiv:080 4.2699 [cs.CC], Computing Researc h Rep ository , http://arXiv.o rg/corr/ , April 2008. [CHP08] C. Clingerman, Jeremiah Hemphill, and Corey Proscia. Analysis and counterex- amples regarding Y atsenk o’s p olynomial-time algorithm f or solving the tra veling salesman problem. T ec hnical Rep ort arXiv:0801.047 4 [cs.CC], Compu ting Re- searc h Rep ository , h ttp://arXiv.org/ corr/ , Jan uary 2008. [ESY84] S. Ev en, A. S elman, and Y. Y acobi. The complexit y of promise problems with applications to public-k ey cryptograph y . Information and Contr ol , 61(2):159–1 73, 1984. 4 [FHW09] F. F erraro, G. Hall, and A. W o o d . Refutatio n of Aslam’s pro of that NP = P. T ec hnical Rep ort arXiv:0904.391 2 [cs.CC], Comp uting Researc h Rep ository , h ttp://arXiv.org/co rr / , April 2009. [Sel88] A. S elman. Promise problems complete for complexit y classes. Information and Computation , 78:87–98 , 1988. [SSS07] K. S ab o, R. Schmitt, a nd M. Silverman. Critique of F einstein’s pro of th at P is n ot equal to NP. T ec h nical Report arXiv:0706.2035 , Computing Researc h Rep ository , h ttp://arXiv.org/co rr / , Jun e 2007. [W o e] G. W o eginger. T h e P-v ersus-NP page. h ttp://www.win.tue.nl/ ˜ gw o egi/P-v ersus-NP .htm, w eb page. 5