Checking Computations in Polylogarithmic Time
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Checking Computations in Polylogarithmic Time Leonid A Levin Lance Fortnow Laszlo Babai Dept Comp Sci Dept Comp Sci Univ of Chicago and Boston University Univ of Chicago Eotvos Univ Budap est Mario Szegedy Dept Comp Sci Univ of Chicago Abstract Motivated by Manuel Blums concept of instance checking we consider new very fast and generic mechanisms of checking computations Our results exploit recent advances in interactive pro of proto cols LFKN Sha and esp ecially the MIP NEXP proto col from BFL We show that every nondeterministic computational task S x y dened as a p olynomial time relation b etween the instance x representing the input and output combined and the witness y can b e mo died to a task S such that i the same instances remain accepted ii each instancewitness pair b ecomes checkable in polylogarithmic Monte Carlo time and iii a witness satisfying S can b e computed in p olynomial time from a witness satisfying S Here the instance and the description of S have to b e provided in errorcorrecting co de since the checker will not notice slight changes A mo dication of the MIP pro of was required to achieve p olynomial time O log log N in iii the earlier technique yields N time only This result b ecomes signicant if software and hardware reliability are regarded as a considerable cost factor The p olylogarithmic checker is the only part of the system that needs to b e trusted it can b e hard wired We use just one Checker for all problems The checker is tiny and so presumably can b e optimized and checked oline at a mo dest cost In this setup a single reliable PC can monitor the op eration of a herd of sup ercomputers working with p ossibly extremely p owerful but unreliable software and untested hardware In another interpretation we show that in p olynomial time every formal mathematical pro of can b e transformed into a transparent proof ie a pro of veriable in p olylogarithmic Monte Carlo time assuming the theoremcandidate is given in errorcorrecting co de In fact for any we can transform any O pro of P in time kP k into a transparent pro of veriable in Monte Carlo time log kP k As a bypro duct we obtain a binary error correcting co de with very ecient errorcorrection The co de transforms messages of length N into co dewords of length N and for strings within of a valid co deword it allows to recover any bit of the unique co deword within that distance in p olylogarithmic O log N time 1 Research partially supp orted by NSF Grant CCR Email lacicsuchicagoedu 2 Research partially supp orted by NSF Grant CCR Email fortnowcsuchicagoedu 3 Supp orted by NSF grant CCR Email Lndcsbuedu 4 Cummington St Boston MA 5 Current address ATT Bell Lab oratories Mountain Avenue PO Box Murray Hill NJ Email msresearchattcom 6 E th St Chicago IL Intro duction Very long mathematical pro ofs An interesting foundational problem is p osed by some mathematical pro ofs which are to o large to b e checked by a single human The pro of of the Four Color Theorem AHK considered controversial at the time started with a Lemma that the Theorem follows if certain computation terminates It was completed with the exp erimental fact that the computation did indeed terminate within two weeks on contemp orary computers The Enormous Theorem Gor provides the classication of all nite simple groups Its pro of spread over pages in Gorensteins estimate consists of a large numb er of dicult lemmas Each lemma was proven by a memb er of a large team but it seems doubtful that any one p erson was able to check all parts An even more dicult example is a statement that a given large tap e contains the correct output of a huge computation with program and input on the same tap e One might attempt to verify the claim by rep eating the computation but what if there is a systematic error in the implementation The rst two examples are sp ecial cases of the last one The requirements for mathematical pro ofs can b e completely formalized The statement of the theorem would have to incorp orate the denitions basic concepts notation and assumptions of the given mathematical area They should also include the general axiom schemes of mathematics used say Set Theory furthermore the logical axioms and inference rules parsing pro cedures needed to implement the logic etc The theorem will then b ecome very large and ugly but still easily manageable by computers The computation would then just check that the pro of adheres to the sp ecications Transparent pro ofs Randomness oers a surprisingly ecient way out of the foundational problem As we shall see all formal pro ofs can b e transformed into pro ofs that are checkable in polylogarithmic Monte Carlo time Note that no matter how huge a pro of we have the logarithm of its size is very small The logarithm of the numb er of particles in the visible Universe is under O A probabilistic proof system is a kT P k time algorithm AT P It has random access to the source of coin ips and two input arrays T theorem candidate and P pro of candidate T is given in an error correcting co de If T is not a valid co de word but is within say Hamming distance of one this valid co deword T is uniquely dened and recoverable in nearly linear time Each pair T P is either correct accepted for all or wrong rejected for most or imperfect can b e easily transformed into correct T P with unique T close to T In the last case the checker is free either to reject there are errors or to accept errors are inessential Using a sp ecial error correcting co de we can guarantee the acceptance with high probability if there are of errors and make easily transformed to mean p olylogarithmic time p er digit P is a proof of T if T P is correct T is a theorem if it has a pro of A deterministic pro of system is the sp ecial case with no use of Extension is a system with more pro ofs but not more theorems O A pro of P of T is transparent if it is accepted in a sp ecic p olylogarithmic time log kT P k The O system is friend ly if every pro of P can b e transformed in kT P k time into a transparent pro of of the same theorem Theorem Each deterministic proof system has a friend ly probabilistic extension It will b e sucient to consider just one pro of system which is NP complete with resp ect to p olylog arithmic time reductions see denition in Section In fact using a RAM mo del rather than Turing machines we construct a pro of system complete for nearlylinear nondeterministic time with resp ect to such reductions This enables us in time kT P k for any to put the pro ofs into transparent form O veriable in time log kT P k It is an interesting problem to eliminate from this exp onent Comments O O Polynomial and polylog ab ove refer to kT P k and log kT P k resp where kxk denotes the length of the string x Nearly linear means linear times p olylog Theorem asserts that given T as a valid co deword and a correct pro of P one can compute in e p olynomial time another pro of P of T which is accepted in p olylog time by the extended system Note that acceptance of T P do es not guarantee the correctness of T P But if acceptance o ccurs with a noticeable probability then b oth T and P can easily b e corrected Correcting the Theorem Errorcorrecting format enco ding of the input refers to any sp ecic p olynomial time computable enco ding with the prop erty that every co deword can b e recovered in p olynomial time from any distortion in a small constant fraction of the digits Such co des can b e computed very eciently by logdepth nearly linear size networks eg variants of FFT The errorcorrecting enco ding of the theoremcandidate is crucial otherwise P could b e a correct pro of of a slightly mo died and therefore irrelevant theorem and it would not b e p ossible to detect in p olylog time the slight alteration In case T fails to b e in valid errorcorrecting form acceptance of the pro of means correctness of the unique T to which T is close One would not need errorcorrecting enco ding if we were only concerned ab out short theorems with long pro ofs This is rare in computing where inputsoutputs tend to b e long But even in mathematics there are go o d reasons to assume that truly selfcontained theorem statements will b e very long Indeed as discussed in Section the statement of a theorem in top ology for instance would include lots of relevant textb o ok material on logic algebra analysis geometry top ology Correcting the Pro of If T P has a noticeable chance of acceptance in the extended system then T after errorcorrection if not a valid co deword is guaranteed to b e a theorem ie to have a correct pro of We do not guarantee that P itself is a correct pro of but a simple MonteCarlo pro cedure with random access to P will correct it revising each digit indep endently in p olylog time This is a selfcorrection feature related to a selfcorrection concept intro duced by BlumLubyRubinfeld and Lipton The pro of uses the interp olation trick of BeaverFeigenbaumLipton Checking computations We shall now interpret Theorem in the context of checking of computations There will b e two parties involved the Solver comp eting in the op en market to serve the user and the Checker a tiny but highly reliable device A universal Checker A nondeterministic programming task is sp ecied by a deterministic p olynomialtime predicate S x y W meaning W is a witness that x y is an acceptable inputoutput pair in errorcorrecting form A Solver may present a program which is claimed to pro duce a p ossibly long witness