average case intractability vs worst case intractability we show that not all sets in np or other levels of the polynomial time hierarchy have efficient average case algorithms unless the arthur merlin classes ma and am can be derandomized to np and various subclasses of p poly collapse to p furthermore other complexity classes like p pp and pspace are shown to be intractable on average unless they are easy in the worst case

leaf languages and string compression tight connections between leaf languages and strings compressed by straight line programs slps are established it is shown that the compressed membership problem for a language l is complete for the leaf language class defined by l via logspace machines a more difficult variant of the compressed membership problem for l is shown to be complete for the leaf language class defined by l via polynomial time machines as a corollary it is shown that there exists a fixed linear visibly pushdown language for which the compressed membership problem is pspace complete for xml languages it is shown that the compressed membership problem is conp complete furthermore it is shown that the embedding problem for slp compressed strings is hard for pp probabilistic polynomial time

gap definability as a closure property gap definability and the gap closure operator were defined by s fenner l fortnow and s kurth j comput system sci 48 116 148 1994 few complexity classes were known at that time to be gap definable in this paper we give simple characterizations of both gap definability and the gap closure operator and we show that many complexity classes are gap definable includingp p p p1 pspace exp nexp mp middle bitp andbp p if a class is closed under union and intersection and contains and then it is gap definable if and only if it containsspp its gap closure is the closure of this class together withsppunder union and intersection on the other hand we give some examples of classes which are reasonable and gap definable but not closed under union resp intersection complement finally we show that a complexity class such aspspaceorpp if it is not equal tospp contains a maximal gap definable many one reduction closed subclass which is properly betweensppand the class of allpspace incomplete pp incomplete sets with respect to containment the gap closure of the class of all incomplete sets inpspace resp pp ispspace resp pp

projective representations of k galilei group the projective representations of the k galilei group are found by contracting the relevant representations of the poincare group the projective multiplier is found it is shown that it is not possible to replace the projective representations of by vector representations of some of its extensions

dirac operator bicovariant differential calculus and gauge theory on minkowski space connections between the poincare covariant space of differential 1 forms on minkowski space dirac operator and alain connes formula are studied the equations and lagrangian of gauge theory are constructed the appearance of an additional spin 0 gauge field according to the non trivial structure of is studied

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

list decoding of algebraic geometric codes we generalize sudan s see j compl vol 13 p 180 93 1997 results for reed solomon codes to the class of algebraic geometric codes designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional error correction bound d 1 2 d being the minimum distance of the code our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields for the latter problem we design a polynomial time algorithm and show that the resulting overall list decoding algorithm runs in polynomial time under some mild conditions several examples are included

a root finding algorithm for list decoding of reed muller codes let f sub q x sub 1 x sub m denote the set of polynomials over f sub q in m variables and f sub q x sub 1 x sub m sub spl les u denote the subset that consists of the polynomials of total degree at most u let h t be a nontrivial polynomial in t with coefficients in f sub q x sub 1 x sub m a crucial step in interpolation based list decoding of q ary reed muller rm codes is finding the roots of h t in f sub q x sub 1 x sub m sub spl les u in this correspondence we present an efficient root finding algorithm which finds all the roots of h t in f sub q x sub 1 x sub m sub spl les u the algorithm can be used to speed up the list decoding of rm codes

enumerative counting is hard abstract an n variable boolean formula may have anywhere from 0 to 2n satisfying assignments can a polynomial time machine given such a formula reduce this exponential number of possibilities to a small number of possibilities we call such a machine an enumerator and prove that if there is a good polynomial time enumerator for p i e one where for every boolean formula f the small set has at most o f 1 e numbers then p np p p and probabilistic polynomial time equals polynomial time furthermore we show that p polynomial time turing reduces to enumerating p

complexity classes of optimization functions in this paper complexity classes of functions defined via taking maxima or minima cf the work of krentel or taking middle elements cf the work of toda are examined a number of axioms for a class to be a so called p founded class of optimization functions are given it is shown that many natural function classes fulfill these axioms then these classes are examined concerning their relationship to complexity classes of sets to this end complexity preserving operators s and f for encoding function classes into set classes and vice versa are introduced it is shown how these operators translate closure properties from one class to another how they relate operators on classes of functions and classes of sets and how they encode classes of maximum minimum or median functions into well studied classes of sets the fixpoints of the compositional operator f s are examined and shown to be exactly those function classes closed under binary search let f1 and f2 be two such fixpoints and k1 and k2 be their corresponding classes of sets i e their images under the operator s then f1 f2 if and only if k1 k2 and f1 f2 if and only if k1 k2 a number of natural classes of functions are shown to be such fixpoints thus we build hierachies of function classes with the same inclusional relationships as the polynomial time hierarchy of sets and the counting hierarchy of sets and we prove many interesting structural properties of these hierarchies

block interpolation a framework for tight exponential time counting complexity we devise a framework for proving tight lower bounds under the counting exponential time hypothesis mathsf eth introduced by dell et al our framework allows to convert many known mathsf p hardness results for counting problems into results of the following type if the given problem admits an algorithm with running time 2 o n on graphs with n vertices and mathcal o n edges then mathsf eth fails as exemplary applications of this framework we obtain such tight lower bounds for the evaluation of the zero one permanent the matching polynomial and the tutte polynomial on all non easy points except for two lines

relations among mod classes abstract mod classes are those subclasses of p p 1 where the deterministic polynomial time computation consists only of checking whether the oracle answer for the input lies in a certain residue class modulo a fixed natural number k for the so defined classes modk p we prove that modk p modm p if k and m have the same prime divisors and as a consequence of this modk p is closed under union for all k

the complexity of bayesian networks specified by propositional and relational languages abstract we examine the complexity of inference in bayesian networks specified by logical languages we consider representations that range from fragments of propositional logic to function free first order logic with equality in doing so we cover a variety of plate models and of probabilistic relational models we study the complexity of inferences when network query and domain are the input the inferential and the combined complexity when the network is fixed and query and domain are the input the query data complexity and when the network and query are fixed and the domain is the input the domain complexity we draw connections with probabilistic databases and liftability results and obtain complexity classes that range from polynomial to exponential levels we identify new languages with tractable inference and we relate our results to languages based on plates and probabilistic relational models

computing functions with parallel queries to np the class 2p of languages polynomial time truth table reducible to sets in np has a wide range of different characterizations we consider several functional versions of 2p based on these characterizations we show that in this way the three function classes fllognp fplognp and fp np are obtained in contrast to the language case the function classes seem to all be different we give evidence in support of this fact by showing that fllognp coincides with any of the other classes then l p and that the equality of the classes fplognp and fp np would imply that the number of nondeterministic bits needed for the computation of any problem in np can be reduced by a polylogarithmic factor and that the problem can be computed deterministically with a subexponential time bound of order 2no 1 log log n

parameterized circuit complexity and the w hierarchy abstract a parameterized problem l k belongs to w t if there exists k computed from k such that l k reduces to the weight k satisfiability problem for weft t circuits we relate the fundamental question of whether the w t hierarchy is proper to parameterized problems for constant depth circuits we define classes g t as the analogues of ac 0 depth t for parameterized problems and n t by weight k existential quantification on g t by analogy with np p we prove that for each t w t equals the closure under fixed parameter reductions of n t then we prove using sipser s results on the ac 0 depth t hierarchy that both the g t and the n t hierarchies are proper if this separation holds up under parameterized reductions then the w t hierarchy is proper we also investigate the hierarchy h t defined by alternating quantification over g t by trading weft for quantifiers we show that h t coincides with h 1 we also consider the complexity of unique solutions and show a randomized reduction from w t to unique w t

approximating the permanent via nonabelian determinants since the celebrated work of jerrum sinclair and vigoda j acm 51 2004 pp 671 697 we have known that the permanent of a matrix with entries in 0 1 can be approximated in randomized polynomial time by using a rapidly mixing markov chain to sample perfect matchings of a bipartite graph a separate strand of the literature has pursued the possibility of an alternate algebraic polynomial time approximation scheme these schemes work by replacing each 1 with a random element of an algebra mathcal a and considering the determinant of the resulting matrix in the case where mathcal a is noncommutative this determinant can be defined in several ways we show that for some estimators based on the conventional determinant the critical ratio of the second moment to the square of the first and therefore the number of trials we need to obtain a good estimate of the permanent is 1 o 1 d n when mathcal a is the algebra of d times d matrices these results can be extended to group algebras and semisimple algebras in general we also study the symmetrized determinant of barvinok showing that the resulting estimator has small variance when d is large enough however if d is constant the only case in which an efficient algorithm is known we show that the critical ratio exceeds 2 n n o d thus our results do not provide a new polynomial time approximation scheme for the permanent indeed they suggest that the algebraic approach to approximating the permanent faces significant obstacles we obtain these results using diagrammatic techniques in which we express matrix products as contractions of tensor products when these matrices are chosen randomly according to the gaussian distribution we can evaluate the trace of these products in terms of the cycle structure of a suitably random permutation in the symmetrized case our estimates are then derived by a connection with the character theory of the symmetric group

complexity measures for public key cryptosystems a general theory of public key cryptography is developed that is based on the mathematical framework of complexity theory two related approaches are taken to the development of this theory and th

quantum and classical complexity classes separations collapses and closure properties we study the complexity of quantum complexity classes such as eqp bqp and nqp quantum analogs of p bpp and np respectively using classical complexity classes such as zpp wpp and c p the contributions of this paper are threefold first via oracle constructions we show that no relativizable proof technique can improve the best known classical upper bound for bqp bqp awpp journal of computer and system sciences 59 2 1999 240 to bqp wpp and the best known classical lower bound for eqp p eqp to zpp eqp second we prove that there are oracles a and b such that relative to a corp is immune to nqp and relative to b bqp is immune to pc p extending a result of de graaf and valiant technical report quant ph 0211179 quantum physics 2002 we construct a relativized world where eqp is immune to modpkp third motivated by the fact that counting classes e g lwpp awpp etc are the best known classical upper bounds on quantum complexity classes we study properties of these counting classes we prove that wpp is closed under polynomial time truth table reductions while we construct an oracle relative to which wpp is not closed under polynomial time turing reductions the latter result implies that proving the equality of the similar appearing classes lwpp and wpp would require nonrelativizable proof techniques we also prove that both awpp and app are closed under tup reductions we use closure properties of wpp and awpp to prove interesting consequences in terms of the complexity of the polynomial hierarchy of the following hypotheses nqp bqp and eqp nqp

a generalization of the berlekamp massey algorithm for multisequence shift register synthesis with applications to decoding cyclic codes a generalization of the berlekamp massey algorithm is presented for synthesizing minimum length linear feedback shift registers for generating prescribed multiple sequences a more general problem is first considered that of finding the smallest initial set of linearly dependent columns in a matrix over an arbitrary field which includes the multisequence problem as a special case a simple iterative algorithm the fundamental iterative algorithm fia is presented for solving this problem the generalized algorithm is then derived through a refinement of the fia application of this generalized algorithm to decoding cyclic codes up to the hartmann tzeng ht bound and roos bound making use of multiple syndrome sequences is considered conditions for guaranteeing that the connection polynomial of the shortest linear feedback shift register obtained by the algorithm will be the error locator polynomial are determined with respect to decoding up to the ht bound and special cases of the roos bound

recursion theoretic characterizations of complexity classes of counting functions abstract there has been a great effort in giving machine independent algebraic characterizations of complexity classes especially of functions astonishingly no satisfactory characterization of the prominent class p is known up to now here we characterize p as the closure of a set of simple arithmetical functions under summation and weak product based on that result the hierarchy of counting functions which is the closure of p under substitution is characterized remarkably without using the operator of substitution since we can show that in the context of this hierarchy the operation of modified subtraction is as powerful as substitution this leads us to a number of consequences concerning closure of p under certain arithmetical operations analogous results are achieved for the class gap p which is the closure of p under subtraction

hyper polynomial hierarchies and the polynomial jump abstract assuming that the polynomial hierarchy ph does not collapse we show the existence of ascending sequences of ptime turing degrees of length 1 ck in pspace such that successors are polynomial jumps of their predecessors moreover these ptime degrees are all uniformly hard for ph this is analogous to the hyperarithmetic hierarchy which is defined similarly but with the computable turing degrees the lack of uniform least upper bounds for ascending sequences of ptime degrees causes the limit levels of our hyper polynomial hierarchy to be inherently non canonical this problem is investigated in depth and various possible structures for hyper polynomial hierarchies are explicated as are properties of the polynomial jump operator on the languages which are in pspace but not in ph

hardness randomness tradeoffs for bounded depth arithmetic circuits in this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits more formally if there exists an explicit polynomial f that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic black box algorithm to test whether a given depth d 5 circuit that computes a polynomial of relatively small individual degrees is identically zero or not in particular if we are guaranteed that the tested circuit computes a multilinear polynomial then we can perform the identity test efficiently to the best of our knowledge this is the first hardness randomness tradeoff for bounded depth arithmetic circuits the above results are obtained using the arithmetic nisan wigderson generator of kabanets and impagliazzo together with a new theorem on bounded depth circuits which is the main technical contribution of our work this theorem deals with polynomial equations of the form p x 1 dots x n y equiv0 and shows that if p has a circuit of depth d and size s and if the polynomial f x 1 dots x n satisfies p x 1 dots x n f equiv0 then f has a circuit of depth d 3 and size mathrm poly s m r where m is the total degree of f and r is the degree of y in p this circuit for f can be found probabilistically in time mathrm poly s m r in the other direction we observe that the methods of kabanets and impagliazzo can be used to show that derandomizing identity testing for bounded depth circuits implies lower bounds for the same class of circuits more formally if we can derandomize polynomial identity testing for bounded depth circuits then nexp does not have bounded depth arithmetic circuits that is either mathrm nexp not subseteq p mathrm poly or the permanent is not computable by polynomial size bounded depth arithmetic circuits

some observations on the connection between counting and recursion abstract based on valiant s class p of all functions counting the number of accepting computations of nondeterministic polynomial time turing machines the polynomial time hierarchy of counting functions is introduced the class phcf of all functions of this hierarchy and some of its subclasses are characterized by recursion theoretic means it turns out that from the recursion theoretic point of view phcf is an analogue to kalmar s class e of elementary functions to the class pspace of polynomial space computable functions as well as to the class p of polynomial time computable functions

on the equivalence between berlekamp s and euclid s algorithms corresp it is shown that berlekamp s iterative algorithm can be derived from a normalized version of euclid s extended algorithm simple proofs of the results given recently by cheng are also presented

the size of spp derandomization techniques are used to show that at least one of the following holds regarding the size of the counting complexity class spp 1 p spp 0 2 ph spp in other words spp is small by being a negligible subset of exponential time or large by containing the entire polynomial time hierarchy this addresses an open problem about the complexity of the graph isomorphism problem it is not weakly complete for exponential time unless ph is contained in spp it is also shown that the polynomial time hierarchy is contained in sppnp if np does not have p measure 0

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

the gamma poincare quantum group from quantum group contraction we propose a contraction of the de sitter quantum group leading to a poincare quantum group in any dimensions the method relies on the coaction of the de sitter quantum group on a non commutative space and the deformation parameter q is sent to one the bicrossproduct structure of this gamma poincare quantum group is exhibited and shown to be dual to the one of the kappa poincare hopf algebra at least in two dimensions

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

the quantum two dimensional poincare group from quantum group contraction a new derivation of the algebra of functions on the two dimensional euclidean poincare group is proposed it is based on a contraction of the hopf algebra fun soq 3 where the deformation parameter q is sent to one in this geometric construction the action of fun soq 3 on the noncommutative euclidean space is a key ingredient

a hensel lifting to replace factorization in list decoding of algebraic geometric and reed solomon codes this article presents an algorithmic improvement to sudan s see j complexity vol 13 p 180 93 1997 list decoding algorithm for reed solomon codes and its generalization to algebraic geometric codes from shokrollahi and wasserman see ibid vol 45 p 432 37 1999 instead of completely factoring the interpolation polynomial over the function field of the curve we compute sufficiently many coefficients of a hensel development to reconstruct the functions that correspond to codewords we prove that these hensel developments can be found efficiently using newton s method we also describe the algorithm in the special case of reed solomon codes

computing elementary symmetric polynomials with a subpolynomial number of multiplications elementary symmetric polynomials s n k are the building blocks of symmetric polynomials in this work we prove that for constant k s s n k modulo composite numbers m p1p2 can be computed using only no 1 multiplications if the coefficients of monomials xi1xi2 xik are allowed to be 1 either mod p1 or mod p2 but not necessarily both to the best of our knowledge no previous result yielded even a sublinear i e n varepsilon 0 varepsilon 1 number of multiplications for similar tasks moreover our algorithm fits in the model of the most restrictive depth 3 arithmetic circuits homogeneous multilinear or the graph model in contrast by a lower bound of nisan and wigderson comput complexity 6 1997 pp 217 234 any homogeneous depth 3 circuit needs size omega n 2k k 2 for computing s n k modulo primes moreover the number of multiplications in our algorithm remains sublinear while k o log log n our results generalize for other nonprime power composite moduli as well the proof uses perfect hashing functions and the famous bbr polynomial of barrington beigel and rudich

two remarks on the power of counting the relationship between the polynomial hierarchy and valiant s class p is at present unknown we show that some low portions of the polynomial hierarchy namely deterministic polynomial algorithms using an np oracle at most a logarithmic number of times can be simulated by one p computation we also show that the class of problems solvable by polynomial time nondeterministic turing machines which accept whenever there is an odd number of accepting computations is idempotent that is closed under usage of oracles from the same class

the counting complexity of group definable languages abstract a group family is a countable family b b n n 0 of finite black box groups i e the elements of each group bn are uniquely encoded as strings of uniform length polynomial in n and for each bn the group operations are computable in time polynomial in n in this paper we study the complexity of np sets a which has the following property the set of solutions for every x a is a subgroup or is the right coset of a subgroup of a group bi x from a given group family b where i is a polynomial such an np set a is said to be defined over the group family b decision problems like graph automorphism graph isomorphism group intersection coset intersection and group factorization for permutation groups give natural examples of such np sets defined over the group family of all permutation groups we show that any such np set defined over permutation groups is low for pp and c p as one of our main results we prove that np sets defined over abelian black box groups are low for pp the proof of this result is based on the decomposition theorem for finite abelian groups as an interesting consequence of this result we obtain new lowness results membership testing group intersection group factorization and some other problems for abelian black box groups are low for pp and c p as regards the corresponding counting problem for np sets over any group family of arbitrary black box groups we prove that exact counting of number of solutions is in fpam consequently none of these counting problems can be p complete unless ph collapses

the re encoding transformation in algebraic list decoding of reed solomon codes the main computational steps in algebraic soft decoding as well as sudan type list decoding of reed solomon codes are bivariate polynomial interpolation and factorization we introduce a computational technique based upon re encoding and coordinate transformation that significantly reduces the complexity of the bivariate interpolation procedure this re encoding and coordinate transformation converts the original interpolation problem into another reduced interpolation problem which is orders of magnitude smaller than the original one a formal proof is presented to show that the two interpolation problems are indeed equivalent an efficient factorization procedure that applies directly to the reduced interpolation problem is also given

on the reducibility of sets inside np to sets with low information content this paper studies for various natural problems in np whether they can be reduced to sets with low information content such as branches p selective sets and membership comparable sets the problems that are studied include the satisfiability problem the graph automorphism problem the undirected graph accessibility problem the determinant function and all logspace self reducible languages some of these are complete for complexity classes within np but for others an exact complexity theoretic characterization is not known reducibility of these problems is studied in a general framework introduced in this paper prover verifier protocols with low complexity provers it is shown that all these natural problems indeed have such protocols this fact is used to show for certain reduction types that these problems are not reducible to sets with low information content unless their complexity is much less than what it is currently believed to be the general framework is also used to obtain a new characterization of the complexity class l l is the class of all logspace self reducible sets in ll sel

the complexity of counting self avoiding walks in subgraphs of two dimensional grids and hypercubes valiant siam j comput 8 1979 410 421 showed that the problem of computing the number of simple s t paths in graphs is p complete both in the case of directed graphs and in the case of undirected graphs welsh complexity knots colourings and counting cambridge university press cambridge 1993 p 17 asked whether the problem of computing the number of self avoiding walks of a given length in the complete two dimensional grid is complete for p1 the tally version of p this paper offers a partial answer to the question of welsh it is p complete to compute the number of self avoiding walks of a given length in a subgraph of a two dimensional grid several variations of the problem are also studied and shown to be p complete this paper also studies the problem of computing the number of self avoiding walks in a subgraph of a hypercube similar completeness results are shown for the problem by scaling the computation time to exponential it is shown that computing the number of self avoiding walks in hypercubes is a complete problem for exp in the case when a subgraph of a hypercube is specified by its dimension and a boolean circuit that accepts the nodes finally this paper studies the complexity of testing whether a given word over the four letter alphabet u d l r represents a self avoiding walk in a two dimensional grid a linear space lower bound is shown for nondeterministic turing machines with a 1 way input head to make this test

non commutative arithmetic circuits depth reduction and size lower bounds abstract we investigate the phenomenon of depth reduction in commutative and non commutative arithmetic circuits we prove that in the commutative setting uniform semi unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree and unrestricted depth earlier proofs did not work in the uniform setting this also provides a unified proof of the circuit characterizations of the class logcfl and its counting variant logcfl we show that ac1 has no more power than arithmetic circuits of polynomial size and degree no log log n improving the trivial bound of no log n connections are drawn between tc1 and arithmetic circuits of polynomial size and degree then we consider non commutative computation we show that over the algebra max concat arithmetic circuits of polynomial size and polynomial degree can be reduced to o log2 n depth and even to o log n depth if unbounded fanin gates are allowed this establishes that optlogcfl is in ac1 this is the first depth reduction result for arithmetic circuits over a non commutative semiring and it complements the lower bounds of kosaraju and nisan showing that depth reduction cannot be done in the general non commutative setting we define new notions called short left paths and short right paths and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible this class also can be characterized using the auxpda model finally we characterize the languages generated by efficient circuits over the semiring 2 union concat in terms of simple one way machines and we investigate and extend earlier lower bounds on non commutative circuits

tally np sets and easy census functions abstract we study the question of whether every p set has an easy i e polynomial time computable census function we characterize this question in terms of unlikely collapses of language and function classes such as p1 fp where p1 is the class of functions that count the witnesses for tally np sets we prove that every p1ph function can be computed in fp p1 p1 consequently every p set has an easy census function if and only if every set in the polynomial hierarchy does we show that the assumption p1 fp implies p bpp and ph modk p for each k 2 we also relate a set s property of having an easy census function to other well studied properties of sets such as rankability and scalability the closure of the rankable sets under p isomorphisms finally we prove that it is no more likely that the census function of any set in p can be approximated more precisely can be n enumerated in time n for fixed and than that it can be precisely computed in polynomial time

competing provers yield improved karp lipton collapse results via competing provers we show that if a language a is self reducible and has polynomial size circuits then s2a s2 building on this we strengthen the kamper afk theorem namely we prove that if np np conp poly then the polynomial hierarchy collapses to s2np conp we also strengthen yap s theorem namely we prove that if np conp poly then the polynomial hierarchy collapses to s2np under the same assumptions the best previously known collapses were to zppnp and zppnpnp respectively siam journal on computing 28 1 1998 311 journal of computer and system sciences 52 3 1996 421 building on proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 journal of computer and system sciences 39 1989 21 theoretical computer science 85 2 1991 305 theoretical computer science 26 3 1983 287 it is known that s2 zppnp proceedings of the 42nd ieee symposium on foundations of computer science ieee computer society press silver spring md 2001 pp 620 629 that result and its relativized version show that our new collapses indeed improve the previously known results the kamper afk theorem and yap s theorem are used in the literature as bridges in a variety of results ranging from the study of unique solutions to issues of approximation and so our results implicitly strengthen those results

determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy abstract it is shown that determining whether a quantum computation has a non zero probability of accepting is at least as hard as the polynomial time hierarchy this hardness result also applies to determining in general whether a given quantum basis state appears with nonzero amplitude in a superposition or whether a given quantum bit has positive expectation value at the end of a quantum computation this result is achieved by showing that the complexity class nqp of adleman demarrais and huang a quantum analog of np is equal to the counting class co c equals p

polynomial terse sets abstract let a be a set and k n be such that we wish to know the answers to x1 a x2 a xk a for various k tuples x1 x2 xk if this problem requires k queries to a in order to be solved in polynomial time then a is called polynomial terse or pterse we show the existence of both arbitrarily complex pterse and non pterse sets and that p np iff every np complete set is pterse we also show connections with p immunity p selective p generic sets and the boolean hierarchy in our framework unique satisfiability and a variation of it called ksat is in some sense close to satisfiability

some connections between bounded query classes and non uniform complexity let a x be the characteristic function of a consider the function cka x1 xk a x1 a xk we show that if cka can be computed in polynomial time with fewer than k queries to some set x then a p poly a generalization of this result has applications to bounded query classes circuits and enumerability in particular we obtain the following 1 assuming 3p 3p there are functions computable using f n 1 queries to sat that are not computable using f n queries to sat for f n o log n 2 if cka restricted to length n inputs can be computed by an unbounded fanin oracle circuit of size s n and depth d n with k 1 queries to some set x then a can be computed with an unbounded fanin non oracle circuit of size no k s n and depth d n o 1 3 assuming that ph 4p 4p and e 1 sat is not 2ne enumerable

meet your expectations with guarantees beyond worst case synthesis in quantitative games abstract classical analysis of two player quantitative games involves an adversary modeling the environment of the system which is purely antagonistic and asks for strict guarantees while markov decision processes model systems facing a purely randomized environment the aim is then to optimize the expected payoff with no guarantee on individual outcomes we introduce the beyond worst case synthesis problem which is to construct strategies that guarantee some quantitative requirement in the worst case while providing a higher expected value against a particular stochastic model of the environment given as input we study the beyond worst case synthesis problem for two important quantitative settings the mean payoff and the shortest path in both cases we show how to decide the existence of finite memory strategies satisfying the problem and how to synthesize one if one exists we establish algorithms and we study complexity bounds and memory requirements

counting classes are at least as hard as the polynomial time hierarchy in this paper it is shown that many natural counting classes such as pp c p and text mod k text p are at least as computationally hard as ph the polynomial time hierarchy in the following sense for each bf k of the counting classes above every set in bf k ph is polynomial time randomized many one reducible to a set in bf k with two sided exponentially small error probability as a consequence of the result it is seen that all the counting classes above are computationally harder than ph unless ph collapses to a finite level some other consequences are also shown

note the complexity of power index comparison abstract we study the complexity of the following problem given two weighted voting games g and g that each contain a player p in which of these games is p s power index value higher we study this problem with respect to both the shapley shubik power index and the banzhaf power index our main result is that for both of these power indices the problem is complete for probabilistic polynomial time i e is pp complete we apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games we also study the complexity of the raw shapley shubik power index deng and papadimitriou showed that the raw shapley shubik power index is p metric complete we strengthen this by showing that the raw shapley shubik power index is many one complete for p and our strengthening cannot possibly be further improved to parsimonious completeness since we observe that in contrast with the raw banzhaf power index the raw shapley shubik power index is not p parsimonious complete

complexity of finite horizon markov decision process problems controlled stochastic systems occur in science engineering manufacturing social sciences and many other cntexts if the systems is modeled as a markov decision process mdp and will run ad infinitum the optimal control policy can be computed in polynomial time using linear programming the problems considered here assume that the time that the process will run is finite and based on the size of the input there are mny factors that compound the complexity of computing the optimal policy for instance there are many factors that compound the complexity of this computation for instance if the controller does not have complete information about the state of the system or if the system is represented in some very succint manner the optimal policy is provably not computable in time polynomial in the size of the input we analyze the computational complexity of evaluating policies and of determining whether a sufficiently good policy exists for a mdp based on a number of confounding factors including the observability of the system state the succinctness of the representation the type of policy even the number of actions relative to the number of states in almost every case we show that the decision problem is complete for some known complexity class some of these results are familiar from work by papadimitriou and tsitsiklis and others but some such as our pl completeness proofs are surprising we include proofs of completeness for natural problems in the as yet little studied classes np pp

frequency computation and bounded queries there have been several papers over the last ten years that consider the number of queries needed to compute a function as a measure of its complexity the following function has been studied extensively in that light faa x1 xa a x1 a xa we are interested in the complexity in terms of the number of queries of approximating faa let b a and let f be any function such that faa x1 xa and f x1 xa agree on at least b bits for a general set a we have matching upper and lower bounds on f that depend on coding theory these are applied to get exact bounds for the case where a is semirecursive a is superterse and assuming p np a sat we obtain exact bounds when a is the halting problem using different methods

geometric sets of low information content abstract in this paper we study language classes defined by nonuniform families of hyperplanes and halfspaces these are subclasses of p poly upto many one equivalence these classes are essentially the classes elt1 and lt1 of languages accepted by nonuniform families of polynomial size depth 1 circuits with a weighted exact threshold gate at the root respectively weighted threshold gate at the root we investigate the consequences of intractable sets being reducible to sets in elt1 and lt1 using the polynomial time algorithm for linear programming and ideas from the theory of convex polytopes we prove that every disjunctively self reducible bd cylinder that is many one reducible to a set in elt1 or lt1 is already in p as a consequence we derive that if a complete set for any class k np mod k p pp c p many one reduces to a set in elt1 or lt1 then k p for families of hyperplanes over finite fields we prove that every turing self reducible set that is many one reducible to a set defined by such a family is in p and every word decreasing self reducible set that is many one reducible to a set defined by such a family is in np co np furthermore as an interesting connection it turns out that several sparse and tally reduction classes are many one reducible to elt1 or lt1 thus our results concerning reductions to elt1 and lt1 subsume the strongest existing collapse results concerning reductions of any class k np mod k p pp c p to sparse sets for various reducibilities

classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy we consider quantum computations comprising only commuting gates known as iqp computations and provide compelling evidence that the task of sampling their output probability distributions is unli

optimal series parallel trade offs for reducing a function to its own graph consider the problem of computing a function given only an oracle for its graph for this problem we present optimal trade offs between serial and parallel queries in particular we give a fuction for which parallel access to its own graph is exponentially more expensive than sequential access

relating polynomial time to constant depth going back to the seminal paper of furst saxe and sipser 1984 analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers oracles separating polynomial time classes have been obtained by diagonalization making essential use of lower bounds for circuit classes in this note we show how separating oracles can be obtained uniformly from circuit lower bounds without the need of carrying out a particular diagonalization our technical tool is the leaf language approach to the definition of complexity classes

erratum the polynomial time hierarchy collapses if the boolean hierarchy collapses it is shown that if the boolean hierarchy bh collapses then there exists a sparse set s such that text co np subseteq text np s and therefore the polynomial time hierarchy ph col

on fixed polynomial size circuit lower bounds for uniform polynomials in the sense of valiant we consider the problem of fixed polynomial lower bounds on the size of arithmetic circuits computing uniform families of polynomials assuming the generalised riemann hypothesis grh we show that for all k there exist polynomials with coefficients in ma having no arithmetic circuits of size o n k over c allowing any complex constant we also build a family of polynomials that can be evaluated in am having no arithmetic circuits of size o n k then we investigate the link between fixed polynomial size circuit bounds in the boolean and arithmetic settings in characteristic zero it is proved that np size n k or ma size n k or np ma imply lower bounds on the circuit size of uniform polynomials in n variables from the class vnp over c assuming grh in positive characteristic p uniform polynomials in vnp have circuits of fixed polynomial size if and only if both vp vnp over f p and mod p p has circuits of fixed polynomial size

polylogarithmic round interactive proofs for conp collapse the exponential hierarchy if every language in conp has a constant round interactive proof system then the polynomial time hierarchy collapses r b boppana j hastad s zachos does co np have short interactive proofs information processing letters 25 2 1987 127 132 on the other hand the well known lfkn protocol gives o n round interactive proof systems for all languages in conp c lund l fortnow h karloff n nisan algebraic methods for interactive proof systems journal of the association for computing machinery 39 4 1992 859 868 we consider the question of whether it is possible for conp to have interactive proof systems with polylogarithmic round complexity we show that this is unlikely by proving that if a conp complete set has a polylogarithmic round interactive proof system then the exponential time hierarchy collapses we also consider exponential versions of the karp lipton theorem and yap s theorem

on approximation algorithms for p the theme of this paper is to investigate to what extent approximation possibly together with randomization can reduce the complexity of problems in valiant s class p in general any function in p can be approximated to within any constant factor by a function in the class delta 3 p of the polynomial time hierarchy relative to a particular oracle delta 3 p cannot be replaced by delta 2 p in this result another part of the paper introduces a model of random sampling where the size of a set x is estimated by checking for various sample sets s whether or not s intersects x for various classes of sample sets upper and lower bounds on the number of samples required to estimate the size of x are discussed this type of sampling is motivated by particular problems in p such as computing the size of a backtrack search tree in the case of backtrack search trees a sample amounts to checking whether a certain path exists in the tree one of the lower bounds suggests that such tests

depth reduction for circuits of unbounded fan in we prove that constant depth circuits of size nlogo 1 n over the basis and or parity are no more powerful than circuits of this size with depth four similar techniques are used to obtain several other depth reduction theorems in particular we show every set in ac0 can be recognized by a family of depth three threshold circuits of size nlogo 1 n the size bound nlogo 1 n is optimal when considering depth reduction over and or and parity most of our results hold for both the uniform and the nonuniform case

an abstract model for branching and its application to mixed integer programming the selection of branching variables is a key component of branch and bound algorithms for solving mixed integer programming mip problems since the quality of the selection procedure is likely to have a significant effect on the size of the enumeration tree state of the art procedures base the selection of variables on their lp gains which is the dual bound improvement obtained after branching on a variable there are various ways of selecting variables depending on their lp gains however all methods are evaluated empirically in this paper we present a theoretical model for the selection of branching variables it is based upon an abstraction of mips to a simpler setting in which it is possible to analytically evaluate the dual bound improvement of choosing a given variable we then discuss how the analytical results can be used to choose branching variables for mips and we give experimental results that demonstrate the effectiveness of the method on miplib 2010 tree instances where we achieve a 5 geometric average time and node improvement over the default rule of scip a state of the art mip solver

on representations of algebraic geometry codes we show that all algebraic geometric codes possess a succinct representation that allows for list decoding algorithms to run in polynomial time we do this by presenting a root finding algorithm for univariate polynomials over function fields when their coefficients lie in finite dimensional linear spaces and proving that there is a polynomial size representation given which the root finding algorithm runs in polynomial time

collapsing modular counting in bounded arithmetic and constant depth propositional proofs je abek introduced fragments of bounded arithmetic which are axiomatized with weak surjective pigeonhole principles and support a robust notion of approximate counting we extend these fragments of bounded arithmetic to accommodate modular counting quantifiers these theories can formalize and prove the relativized versions of toda s theorem on the collapse of the polynomial hierarchy with modular counting we introduce a version of the paris wilkie translation for converting formulas and proofs of bounded arithmetic with modular counting quantifiers into constant depth propositional logic with modular counting gates we also define paris wilkie translations to nullstellensatz and polynomial calculus refutations as an application we the first author was supported in part by nsf grant dms 1101228 in the preliminary stages of this work the second and third authors were supported by grant no n n201 382234 of the polish ministry of science and higher education most of this work was carried out while the second author was visiting the university of california san diego supported by polish ministry of science and higher education programme mobilno c plus with additional support from a grant from the simons foundation 208717 to sam buss

nonuniform acc circuit lower bounds the class acc consists of circuit families with constant depth over unbounded fan in and or not and modm gates where m 1 is an arbitrary constant we prove the following nexp the class of languages accepted in nondeterministic exponential time does not have nonuniform acc circuits of polynomial size the size lower bound can be slightly strengthened to quasipolynomials and other less natural functions enp the class of languages recognized in 2o n time with an np oracle doesn t have nonuniform acc circuits of 2no 1 size the lower bound gives an exponential size depth tradeoff for every d m there is a 0 such that enp doesn t have depth d acc circuits of size 2n with modm gates previously it was not known whether expnp had depth 3 polynomial size circuits made out of only mod6 gates the high level strategy is to design faster algorithms for the circuit satisfiability problem over acc circuits then prove that such algorithms entail these lower bounds the algorithms combine known properties of acc with fast rectangular matrix multiplication and dynamic programming while the second step requires a strengthening of the author s prior work

quantum circuits and low degree polynomials over in this work we explore a correspondence between quantum circuits and low degree polynomials over the finite field any quantum circuit made up of hadamard z controlled z and controlled controlled z gates gives rise to a degree 3 polynomial over such that calculating quantum circuit amplitudes is equivalent to counting zeroes of the corresponding polynomial we exploit this connection which is especially clean and simple for this particular gate set in two directions first we give proofs of classical hardness results based on quantum circuit concepts second we find efficient classical simulation algorithms for certain classes of quantum circuits based on efficient algorithms for classes of polynomials

hardness of decoding quantum stabilizer codes in this paper we address the computational hardness of optimally decoding a quantum stabilizer code much like classical linear codes errors are detected by measuring certain check operators which yield an error syndrome and the decoding problem consists of determining the most likely recovery given the syndrome the corresponding classical problem is known to be np complete and are appropriate a similar decoding problem for quantum codes is also known to be np complete however this decoding strategy is not optimal in the quantum setting as it does not consider error degeneracy which causes distinct errors to have the same effect on the code here we show that optimal decoding of stabilizer codes previously known to be np hard is in fact computationally much harder than optimal decoding of classical linear codes it is p complete

universal relations and p completeness this paper follows the methodology introduced by agrawal and biswas in manindra agrawal somenath biswas universal relations in structure in complexity theory conference 1992 pp 207 220 based on a notion of universality for the relations associated with np complete problems the purpose was to study np complete problems by examining the effects of reductions on the solution sets of the associated witnessing relations this provided a useful criterion for np completeness while suggesting structural similarities between natural np complete problems we extend these ideas to the class p the notion we find also yields a practical criterion for p completeness as illustrated by a varied set of examples and strengthens the argument for structural homogeneity of natural complete problems

permanent does not have succinct polynomial size arithmetic circuits of constant depth we show that over fields of characteristic zero there does not exist a polynomial p n and a constant free succinct arithmetic circuit family f n using division by constants where f n has size at most p n and depth o 1 such that f n computes the nxn permanent a circuit family f n is succinct if there exists a nonuniform boolean circuit family c n with o logn many inputs and size n o 1 such that c n can correctly answer direct connection language queries about f n succinctness is a relaxation of uniformity to obtain this result we develop a novel technique that further strengthens the connection between black box derandomization of polynomial identity testing and lower bounds for arithmetic circuits from this we obtain the lower bound by giving an explicit construction computable in the polynomial hierarchy of a hitting set for arithmetic circuits

on pseudorandomness and resource bounded measure abstract in this paper we extend a key result of nisan and wigderson j comput system sci 49 1994 149 167 to the nondeterministic setting for all 0 we show that if there is a language in e dtime 2 o n that is hard to approximate by nondeterministic circuits of size 2 n then there is a pseudorandom generator that can be used to derandomize bp np in symbols bp np np by applying this extension we are able to answer some open questions in lutz theory comput systems 30 1997 429 442 regarding the derandomization of the classes bp p k and bp p k under plausible measure theoretic assumptions as a consequence if p 2 does not have p measure 0 then am coam is low for p 2 thus in this case the graph isomorphism problem is low for p 2 by using the nisan wigderson design of a pseudorandom generator we unconditionally show the inclusion ma zpp np and that ma coma is low for zpp np