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| '''Generic-case complexity''' is a subfield of [[computational complexity theory]] that studies the complexity of computational problems on "most inputs".
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| Generic-case complexity is a way of measuring the complexity of a [[computational problem]] by neglecting a small set of
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| unrepresentative inputs and considering [[worst-case complexity]] on the rest.
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| Small is defined in terms of asymptotic density.
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| The apparent efficacy of generic case complexity is because for a wide variety of concrete computational problems, the most difficult instances seem to be rare. Typical instances are relatively easy.
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| This approach to complexity originated in [[combinatorial group theory]], which has a computational tradition going back to the beginning of the last century.
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| The notion of generic complexity was introduced in | |
| <ref name="KMSS">I. Kapovich, A. Myasnikov, P. Schupp and V. Shpilrain, ''[http://arxiv.org/abs/math/0203239 Generic case complexity, decision problems in group theory and random walks]'', J. Algebra, vol 264 (2003), 665–694.
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| </ref> where authors showed that for a large class of [[finitely generated group]]s the generic time complexity of some classical [[decision problems]] from combinatorial group theory, namely the [[word problem for groups|word problem]], [[conjugacy problem]] and [[membership problem]], are linear.
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| A detailed introduction of generic case complexity can be found in the surveys
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| ,<ref name="GMMU">
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| R. Gilman, A. G. Miasnikov, A. D. Myasnikov, and A. Ushakov, ''Generic Case Complexity'', unpublished first draft of a book, 143 pages.
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| </ref>
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| <ref>
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| R. Gilman, A. G. Miasnikov, A. D. Myasnikov, and A. Ushakov, ''[http://www.stevens.edu/algebraic/Files/GC/gc_survey.pdf Report on generic case complexity]'', Herald of Omsk University, Special Issue, 2007, 103–110.
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| </ref>
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| == Basic definitions ==
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| === Asymptotic density ===
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| Let ''I'' be an [[infinite set]] of inputs for a computational problem.
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| '''Definition 1.''' A size function on ''I'' is a map <math>\sigma:I\to \mathbb{N}</math> with infinite range.
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| The ball of radius ''n'' is <math>B_n=\{x\in I \mid \sigma(x)\le n \}</math>.
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| If the inputs are coded as strings over a finite alphabet, size might be the string length.
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| Let <math>\{\mu_n\}</math> be an ensemble of [[probability distributions]] where <math>\mu_n</math>
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| is a [[probability distribution]] on <math>B_n</math>.
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| If the balls <math>B_n</math> are finite, then each <math>\mu_n</math> can be taken to
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| be the equiprobable distribution which is the most common case. Notice that only finitely many
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| <math>B_n</math>'s can be empty or have <math>\mu_n(B_n) = 0</math>; we ignore them.
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| '''Definition 2.''' The asymptotic density of a subset <math>X \subset I</math> is
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| <math>\rho(X) = \lim_{n \to \infty}\mu_n(X \cap B_n)</math> when this limit exists.
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| When the balls <math>B_n</math> are finite and <math>\mu_n</math> is the equiprobable measure,
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| : <math>\rho(X)=\lim \frac{|X\cap B_n|}{|B_n|}.</math>
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| In this case it is often convenient to use spheres <math>I_n=\{x\in I\mid \sigma(x)=n\}</math> instead of balls and
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| define <math>\rho'(X)=\lim \frac{|X\cap I_n|}{|I_n|}</math>. An argument using [[Stolz theorem]] shows that <math>\rho(X)</math>
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| exists if <math>\rho'(X)</math> does, and in that case they are equal.
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| '''Definition 3''' <math>X\subseteq I</math> is generic if <math>\rho(X)=1</math> and negligible if <math>\rho(X)=0</math>.
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| ''X'' is exponentially (superpolynomially) generic if the convergence to the limit in Definition 2 is exponentially (superpolynomially) fast, etc.
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| A generic subset ''X'' is asymptotically large. Whether ''X'' appears large in practice depends
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| on how fast <math>\mu_n(X\cap B_n)</math> converges to 1. Superpolynomial convergence seems to be fast enough.
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| === Generic complexity classes ===
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| '''Definition 4''' An [[algorithm]] is in ''GenP'' (generically polynomial time) if it never gives incorrect answers and if it
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| gives correct answers in [[polynomial time]] on a generic set of inputs. A problem is in ''GenP'' if it
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| admits an algorithm in ''GenP''. Likewise for ''GenL'' (generically [[linear time]]), ''GenE'' (generically
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| [[exponential time]] with a linear exponent) ''GenExp'' (generically exponential time), etc.
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| ''ExpGenP'' is the subclass of ''GenP'' for which the relevant generic set is exponentially generic.
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| More generally for any <math>f : \mathbb{N} \to \mathbb{N}</math> we can define the class ''Gen(f)'' corresponding to
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| [[time complexity]] ''O''(''f'') on a generic set of input.
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| '''Definition 5.''' An algorithm solves a problem generically if it never gives incorrect answers and if it gives correct answers on a generic set of inputs. A problem is generically solvable if it is solved generically by some algorithm.
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| == Theory and applications ==
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| ===Combinatorial group theory problems ===
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| * The famous [[undecidable problem]]s: the word, conjugacy and membership decision problems are in generically polynomial.<ref name="KMSS" />
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| * The word and conjugacy [[search problem]]s are in ''GenP'' for all fixed finitely presented groups.<ref>
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| A. Ushakov, ''Dissertation'', City University of New York, 2005.</ref>
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| <!---------(Here is a description of these
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| problems: given a word w representing the identity in a finitely presented group, find a
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| product of conjugates of relators which is freely equal to w; given two words representing
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| conjugate group elements, find a conjugating element.)
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| ----> | |
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| * The well known [[coset enumeration]] procedure admits a computable upper bound on a generic set of inputs.<ref>
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| R. Gilman, ''Hard problems in group theory'', talk given at the International Conference
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| on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory,
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| May 18, 2009.</ref>
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| * The Whitehead algorithm for testing whether or not one element of a free group is mapped to another by an automorphism has an exponential worst case upper bound but runs well in practice. The algorithm is shown to be in ''GenL''.<ref>
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| I. Kapovich, P. Schupp, V. Shpilrain, ''[http://arxiv.org/abs/math.GR/0303386 Generic properties of Whiteheads algorithm and isomorphism rigidity of random one-relator groups]'', Pacific J. Math. 223 (2006)
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| </ref>
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| * The conjugacy problem in [[HNN extension]]s can be unsolvable even for [[free group]]s. However, it is generically cubic time<ref>
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| A.V. Borovik, A.G. Myasnikov, V.N. Remeslennikov, ''[http://arxiv.org/abs/0903.3754 Generic complexity of the conjugacy problem in HNN-extensions and algorithmic stratification of Miller's groups]'',
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| Internat. J. Algebra Comput. 17 (2007), 963–997.
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| </ref>
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| === The halting problem and the Post correspondence problem ===
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| * The [[halting problem]] for [[Turing machine]] with one-sided tape is easily decidable most of the time; it is in ''GenP''<ref>
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| [[Joel David Hamkins|J. D. Hamkins]] and A. Miasnikov, ''[http://arxiv.org/abs/math/0504351 The halting problem is decidable on a set of asymptotic probability one]'', Notre Dame J. Formal Logic 47 (2006), 515–524.
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| </ref>
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| The situation for two-sided tape is unknown. However, there is a kind of lower bound for machines of both types.
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| The halting problem is not in ''ExpGenP'' for any model of Turing machine,<ref>
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| A. Miasnikov and A. Rybalov, ''[http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jsl/1208359065 Generic complexity of undecidable problems]'', J. Symbolic Logic 73 (2008), 656–673.
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| </ref><ref>
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| A. Rybalov, ''[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1G-4N2TS51-3&_user=605441&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1052347431&_rerunOrigin=google&_acct=C000029138&_version=1&_urlVersion=0&_userid=605441&md5=8bee4d70a78ff9d14450f6bc7a993984 On the strongly generic undecidability of the halting problem]'', Theoret. Comput. Sci. 377 (2007), 268–270.
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| </ref>
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| * The [[Post correspondence problem]] is in ''ExpGenP''.<ref name="GMMU" />
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| === Presburger arithmetic ===
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| The [[decision problem]] for [[Presburger arithmetic]] admits a double exponential
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| worst case lower bound <ref>M. J. Fischer and M. O. Rabin, ''Super-Exponential Complexity of Presburger Arithmetic'',
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| Proceedings of the SIAM-AMS Symposium in Applied Mathematics 7 (1974)
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| 2741.</ref> and a triple exponential worst case upper bound. The
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| generic complexity is not known, but it is known that the problem is not in ''ExpGenP''.<ref>A. Rybalov, ''Generic complexity of Presburger arithmetic'', 356–361 in Second International Symposium on Computer Science in Russia, CSR 2007, Lecture Notes in Computer Science 4649, Springer 2007.
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| </ref>
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| === NP complete problems ===
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| As it is well known that [[NP-complete problems]] can be easy on average, it is not a surprise that several of them are generically easy too.
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| * The three [[satisfiability problem]] is in ''ExpGenP'' <ref>
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| R. Gilman, A. G. Miasnikov, A. D. Myasnikov, and A. Ushakov, Report on generic
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| case complexity, Herald of Omsk University, Special Issue, 2007, 103–110.
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| </ref>
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| * The [[subset sum problem]] is in ''GenP''.<ref name="GMMU"/>
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| === One way functions ===
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| There is a generic complexity version of a [[one-way function]] <ref>
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| A. D. Myasnikov, ''[http://arxiv.org/abs/0802.3734 Generic Complexity and One-Way Functions]'', Groups, Complexity and Cryptography, 1, (2009), 13–31.
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| </ref> which yields the same class of functions but allows one to consider different security assumptions than usual.
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| === Public-key cryptography ===
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| A series of articles,<ref>
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| R. Gilman, A. G. Miasnikov, A. D. Myasnikov, and A. Ushakov, ''New developments in commutator key exchange'', Proc. First Int. Conf. on Symbolic Computation and
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| Cryptography (SCC-2008), Beijing, 2008.
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| </ref><ref>A. G. Myasnikov, V. Shpilrain, A. Ushakov, ''A practical attack on a braid group based cryptographic protocol'', in Lecture Notes in Computer Science, 3621, Springer Verlag,
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| 2005, 86–96.</ref><ref>A. D. Myasnikov, and A. Ushakov, ''Length based attack and braid groups: cryptanalysis of Anshel–Anshel–Goldfeld key exchange protocol'', in Public Key Cryptography PKC 2007, 76–88, Lecture Notes in Comput. Sci., 4450, Springer, Berlin, 2007.
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| </ref> is devoted to cryptanalysis of the [[Anshel–Anshel–Goldfeld key exchange]] protocol,
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| whose security is based on assumptions about the [[braid group]]. This series culminates in <ref>
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| A. G. Miasnikov and A. Ushakov, ''Random subgroups and analysis of the length-based and quotient attacks'', Journal of Mathematical Cryptology, 2 (2008), 29–61.</ref> which applies techniques from
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| generic case complexity to obtain a complete analysis of the [[length based attack]] and the
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| conditions under which it works. The generic point of view also suggests a kind of new
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| attack called the quotient attack, and a more secure version of the Anshel–Anshel–Goldfeld protocol.
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| === List of general theoretical results ===
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| *A famous [[Rice's theorem]] states that if ''F'' is a subset of the set of partial computable functions from <math>\mathbb{N}</math> to <math>\{0,1\}</math>, then unless ''F'' or its complement is empty, the problem of deciding whether or not a particular [[Turing machine]] computes a function in ''F'' is undecidable. The following theorem gives a generic version.
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| '''Theorem 1''' <ref>
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| A. Miasnikov and A. Rybalov, ''Generic complexity of undecidable problems'', J. Symbolic Logic 73 (2008), 656–673.
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| </ref> Let ''I'' be the set of all Turing machines. If ''F'' is a subset of the set of all
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| partial computable function from <math>\mathbb{N}</math> to itself such that ''F'' and its complement are both non-empty,
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| then the problem of deciding whether or not a given Turing machine computes a function from
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| ''F'' is not decidable on any exponentially generic subset of ''I''.
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| * The following theorems are from.<ref name="KMSS" />
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| '''Theorem 2''' The set of [[formal languages]] which are generically computable has measure zero.
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| '''Theorem 3''' There is an infinite hierarchy of generic complexity classes. More precisely
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| for a proper complexity function ''f'', <math>Gen(f) \subsetneq Gen(f^3)</math>. | |
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| * The next theorem shows that just as there are [[average case complete problems]] within distributional NP problems,
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| there are also generic case complete problems. The arguments in the generic case are similar to
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| those in the average case, and the generic case complete problem is also average case complete.
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| It is the distributional [[bounded halting problem]].
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| '''Theorem 4''' <ref name="GMMU" /> There is a notion of generic-polynomial-time reduction with respect to
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| which the distributional bounded halting problem is complete within class of distributional NP problems.
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| == Comparisons with previous work ==
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| === Almost polynomial time ===
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| Meyer and Paterson <ref>A. R. Meyer and M. S. Paterson, ''With what frequency are apparently intractable
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| problems difficult?'', M.I.T. Technical Report, MIT/LCS/TM-126, February, 1979.
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| </ref> define an algorithm to be almost polynomial time, or APT, if it halts | |
| within ''p(n)'' steps on all but ''p(n)'' inputs of size ''n''. Clearly APT algorithms are included in our
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| class ''GenP''. We have seen several [[NP complete]] problems in ''GenP'', but Meyer and Paterson
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| show that this is not the case for APT. They prove that an NP complete problem is reducible to
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| a problem in APT if and only if [[P = NP]]. Thus APT seems much more restrictive than ''GenP''.
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| === Average-case complexity ===
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| Generic case complexity is similar to [[average-case complexity]]. However there are some significant differences.
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| Generic case complexity is a direct measure of the performance of an algorithm on most inputs while average case complexity
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| gives a measure of the balance between easy and difficult instances. In addition Generic-case complexity naturally applies to [[undecidable problem]]s.
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| Suppose <math>\mathcal{A}</math> is an algorithm whose [[time complexity]], <math>T:I\to \mathbb{N}</math> is polynomial on <math>\mu</math> average.
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| What can we infer about the behavior of <math>\mathcal{A}</math> on typical inputs?
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| '''Example 1''' Let ''I'' be the set of all words over <math>\{0,1\}</math> and define the size <math>\sigma(w)</math> to be word length,
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| <math>|w|</math>. Define <math>I_n</math> to be the set of words of length ''n'', and assume that each <math>\mu_n</math> is the equiprobable measure.
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| Suppose that ''T(w)=n'' for all but one word in each <math>I_n</math>, and <math>T(w)=2^{2^n}</math> on the exceptional words.
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| In this example ''T'' is certainly polynomial on typical inputs, but ''T'' is not polynomial on average. ''T'' is in ''GenP''.
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| '''Example 2''' Keep ''I'' and <math>\sigma(w) = |w|</math> as before, but define <math>\mu(w)= 2^{-2|w|-1}</math> and
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| <math>T(w) = 2^{|w|}</math>. ''T'' is polynomial on average even though it is exponential on typical inputs. ''T'' is not in ''GenP''.
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| In these two examples the generic complexity is more closely related to behavior
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| on typical inputs than average case complexity. Average case complexity measures something
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| else: the balance between the frequency of difficult instances and the degree of difficulty,.<ref>
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| Y. Gurevich, ''The challenger-solver game: variations on the theme of P =?NP'', Logic
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| in Computer Science Column, The Bulletin of the EATCS, October 1989, p.112-121.
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| </ref><ref>R. Impagliazzo, ''A personal view of average-case complexity'', in Proceedings of the
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| 10th Annual Structure in Complexity Theory Conference - SCT 1995, IEEE Computer
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| Society, 1995, page 134.</ref>
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| Roughly speaking an algorithm which is polynomial time on average can have only a subpolynomial
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| fraction of inputs that require superpolynomial time to compute.
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| Nevertheless in some cases generic and average case complexity are quite close to each other.
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| A function <math>f:I\rightarrow\mathbb{R}^+</math> is polynomial on <math>\mu</math>-average on spheres if there
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| exists <math>k \geq 1</math> such that <math>\sum_{w\in I_n} f^{1/k}(w) \mu_n(w) = O(n)</math> where <math>\{\mu_n\}</math>
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| is the ensemble induced by <math>\mu</math>. If ''f'' is polynomial on <math>\mu</math>-average on spheres, the ''f'' is
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| polynomial on <math>\mu</math>-average, and for many distributions the converse holds
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| <ref>Y. Gurevich, ''Average case completeness'', J. of Computer and System Science, 42
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| (1991), 346–398.
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| </ref>
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| '''Theorem''' 5 <ref name="GMMU"/> If a function <math>f:I\rightarrow \mathbb{R}^+</math> is polynomial on <math>\mu</math>-average on spheres then ''f''
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| is generically polynomial relative to the spherical asymptotic density <math>\rho'</math>.
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| '''Theorem''' 6 <ref name="GMMU"/> Suppose a complete algorithm <math>\mathcal{A}</math> has subexponential time bound ''T'' and a partial algorithm <math>\mathcal{B}</math>
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| for the same problem is in ''ExpGenP'' with respect to the ensemble <math>\{\mu_n\}</math> corresponding to a probability measure <math>\mu</math>
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| on the inputs ''I'' for <math>\mathcal{A}</math>. Then there is a complete algorithm which is <math>\mu</math>-average time complexity.
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| === Errorless heuristic algorithms ===
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| In a 2006 paper, Bogdanov and Trevisan came close to defining generic case complexity.<ref>A. Bogdanov, L. Trevisan, ''Average-case Complexity'', Found. Trends Theor. Comput. Sci. '''2''', No. 1, 111 p. (2006)..</ref> Instead of partial algorithms, they consider so-called errorless heuristic algorithms. These are
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| complete algorithms which may fail by halting with output "?". The class ''AvgnegP'' is defined
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| to consist of all errorless heuristic algorithms ''A'' which run in polynomial time and for which the | |
| probability of failure on <math>I_n</math> is negligible, i.e., converges superpolynomially fast to 0.
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| ''AvgnegP'' is a subset of ''GenP''. Errorless heuristic algorithms are essentially the same as the algorithms with
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| benign faults defined by Impagliazzo where polynomial time on average algorithms are
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| characterized in terms of so-called benign algorithm schemes.
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| == References ==
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| {{Reflist}}
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| {{DEFAULTSORT:Generic-Case Complexity}}
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| [[Category:Computational complexity theory]]
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