a framework for non interactive instance dependent commitment schemes nic zero knowledge protocols are often studied through specific problems like graph isomorphism in many cases this approach prevents an important level of abstraction and leads to limited results whereas in fact the constructions apply to a wide variety of problems we propose to address this issue with a formal framework of non interactive instance dependent commitment schemes nic we define nic in the perfect statistical and computational settings and formally characterize problems admitting nic in all of these settings we also prove other useful lemmas such as closure properties consequently results that previously applied only to specific problems are now strengthened by our framework to apply to classes of problems by providing formal yet intuitive tools our framework facilitates the construction of zero knowledge protocols for a wide variety of problems in various settings without the need to refer to a specific problem our results are unconditional

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

survey of disjoint np pairs and relations to propositional proof systems we survey recent results on disjoint np pairs in particular we survey the relationship of disjoint np pairs to the theory of proof systems for propositional calculus

a novel characterization of the complexity class θkp based on counting and comparison abstract the complexity class 2 p which is the class of languages recognizable by deterministic turing machines in polynomial time with at most logarithmic many calls to an np oracle received extensive attention in the literature its complete problems can be characterized by different specific tasks such as deciding whether the optimum solution of an np problem is unique or whether it is in some sense odd e g whether its size is an odd number in this paper we introduce a new characterization of this class and its generalization k p to the k th level of the polynomial hierarchy we show that problems in k p are also those whose solution involves deciding for two given sets a and b of instances of two k 1 p complete or k 1 p complete problems whether the number of yes instances in a is greater than those in b moreover based on this new characterization we provide a novel sufficient condition for k p hardness we also define the general problem comp valid k which is proven here k 1 p complete comp valid k is the problem of deciding given two sets a and b of quantified boolean formulas with at most k alternating quantifiers whether the number of valid formulas in a is greater than those in b notably the problem comp sat of deciding whether a set contains more satisfiable boolean formulas than another set which is a particular case of comp valid 1 demonstrates itself as a very intuitive 2 p complete problem nonetheless to our knowledge it eluded its formal definition to date in fact given its strict adherence to the count and compare semantics here introduced comp valid k is among the most suitable tools to prove k p hardness of problems involving the counting and comparison of the number of yes instances in two sets we support this by showing that the 2 p hardness of the max voting scheme over mcp nets is easily obtained via the new characterization of k p introduced in this paper

locating p poly optimally in the extended low hierarchy the low hierarchy in np and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes and almost all of them could be located optimally therein however until now the exact location of p poly remained open

implication of clauses is undecidable abstract clause implication a b of two clauses a and b is shown to be undecidable using the undecidability of finitely generated stable transitive relations on free terms clause implication is undecidable even in the case where a consists of four literals the decision problem of clause implication is equivalent to the decision problem of clause sets that consist of one clause and some ground units hence the undecidability results hold also for these clause sets clause implication is an important problem in automated deduction systems as it can be used advantageously to reduce the search space

bounded query classes polynomial time machines having restricted access to an np oracle are investigated restricted access means that the number of queries to the oracle is restricted or the way in which the queries are made is restricted e g queries made during truth table reductions very different kinds of such restrictions result in the same or comparable complexity classes in particular the class p text np o log n can be characterized in very different ways furthermore the boolean hierarchy is generalized in such a way that it is possible to characterize p text np and p text np o log n in terms of the generalization

complexity results for explanations in the structural model approach we analyze the computational complexity of halpern and pearl s causal explanations in the structural model approach which are based on their notions of weak and actual cause in particular we give a precise picture of the complexity of deciding explanations partial explanations and partial explanations and of computing the explanatory power of partial explanations moreover we analyze the complexity of deciding whether all explanation or an partial explanation over certain variables exists we also analyze the complexity of deciding explanations and partial explanations in the case of succinctly represented context sets the complexity of deciding explanations in the general case of situations and the complexity of deciding subsumption and equivalence between causal models all complexity results are derived for the general case as well as for the restriction to the case of binary causal models in which all endogenous variables may take only two values to our knowledge no complexity results for explanations in the structural model approach have been derived so far our results give insight into the computational structure of halpern and pearl s explanations and pave the way for efficient algorithms and implementations

hard promise problems and nonuniform complexity for every recursive set a let pp a denote the following promise problem input x and y promise x a y a property x a we show that if l is a solution of pp a then a p l poly from this result it follows that if a is t p hard for np then all solutions of pp a are hard for np under a reduction that generalizes both t p and t sn specifically if a is np hard then all solutions of pp a are generalized high2 bbs86b the main theorem that leads to this result states that if b is a self reducible set b t p a and a p l poly then 2 p b 2 p l several interesting connections between uniform and nonuniform complexity follow directly from this theorem

langevin dynamics with constraints and computation of free energy differences in this paper we consider langevin processes with mechanical constraints the latter are a fundamental tool in molecular dynamics simulation for sampling purposes and for the computation of free energy differences the results of this paper can be divided into three parts i we propose a simple discretization of the constrained langevin process based on a standard splitting strategy we show how to correct the scheme so that it samples em exactly the canonical measure restricted on a submanifold using a metropolis rule in the spirit of the generalized hybrid monte carlo ghmc algorithm moreover we obtain in some limiting regime a consistent discretization of the overdamped langevin brownian dynamics on a submanifold also sampling exactly the correct canonical measure with constraints the corresponding numerical methods can be used to sample without any bias a probability measure supported by a submanifold ii for free energy computation using thermodynamic integration we rigorously prove that the longtime average of the lagrange multipliers of the constrained langevin dynamics yields the gradient of a rigid version of the free energy associated with the constraints a second order time discretization using the lagrange multipliers is proposed iii the jarzynski crooks fluctuation relation is proved for langevin processes with mechanical constraints evolving in time an original numerical discretization without time step error is proposed numerical illustrations are provided for ii and iii

relativized cryptography it appears to be very difficult to give a formal definition of computational security for public key cryptography a slightly different notion called transient key cryptography is defined for which a natural definition of security against chosen plaintext attacks is given the main result presented here is the existence of a relativized model of computation under which there does exist a secure transient key cryptosystem indeed there exists a computable oracle that can be used by cryptographers to efficiently encipher and decipher messages yet it is of no help to the cryptanalyst trying to decode messages not intended for him as a corollary there also exists a length preserving permutation the inverse of which is hard to compute on most elements of its domain even if arbitrary evaluations of the function itself are allowed for free

quantum finite automata we first outline main steps and achievements along bertoni s research path in quantum finite automata theory from the very basic definitions of the models of quantum finite automata throughout the investigation of their computational and descriptional power next we choose to focus on bertoni s studies on quantum finite automata descriptional complexity in particular we expand on a statistical framework for the synthesis of succinct quantum finite automata discussing its adaptation to the case of multiperiodic events and languages we then improve such a framework to obtain even more succinct quantum finite automata for some multiperiodic languages finally we introduce some promise problems for multiperiodic inputs showing that even on this class of problems the descriptional power of quantum finite automata greatly outperforms that of equivalent classical finite automata we build small size monte carlo quantum finite automata for multiperiodic languages we analyze the modular architecture of such monte carlo quantum finite automata we solve promise problems on multiperiodic inputs by small quantum finite automata

a geometrical approach to computing free energy landscapes from short ranged potentials particles interacting with short ranged potentials have attracted increasing interest partly for their ability to model mesoscale systems such as colloids interacting via dna or depletion we consider the free energy landscape of such systems as the range of the potential goes to zero in this limit the landscape is entirely defined by geometrical manifolds plus a single control parameter these manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system equilibrium or nonequilibrium can be computed for arbitrary potentials to consider dynamical quantities we compute the asymptotic limit of the fokker planck equation and show that it becomes restricted to the low dimensional manifolds connected by sticky boundary conditions to illustrate our theory we compute the low dimensional manifolds for identical particles providing a complete description of the lowest energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom the results can be directly tested on colloidal clusters this limit is a unique approach for understanding energy landscapes and our hope is that it can also provide insight into finite range potentials

explicit time stepping for stiff odes we present a new strategy for solving stiff odes with explicit methods by adaptively taking a small number of stabilizing small explicit time steps when necessary a stiff ode system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis for many stiff problems the cost of the stabilizing small time steps is small so the improvement is large we illustrate the technique on a number of well known stiff test problems

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

more complicated questions about maxima and minima and some closures of np abstract starting from np complete problems defined by questions of the kind max m and min m we consider problems defined by more complicated questions about these maxima and minima as for example max m min m and is max odd this continues a work started by papadimitriou and yannakakis 1982 it is shown that these and other problems are complete in certain subclasses of the boolean closure of np and other classes in the interesting area below the class p 2 of the polynomial time hierarchy special methods are developed to prove such completeness results for this it is necessary to establish some properties of the classes in question which might be interesting in their own right

compactly representing utility functions using weighted goals and the max aggregator weighted propositional formulas can be used to model preferences over combinatorial domains each formula represents a goal we would like to see satisfied the weight of a formula represents the importance of the goal in question and to assess the desirability of a given alternative we aggregate the weights of the goals satisfied by that alternative one of several options is to aggregate by using the maximum of the weights of the satisfied goals this approach gives rise to a family of preference representation languages one for each of a range of possible restrictions we can impose on either formulas or weights we analyze the properties of these languages and establish results regarding their expressivity and absolute and relative succinctness we also study the computational complexity of the problem of finding the best and the worst alternative for a given set of weighted goals and of finding an alternative that is optimal for a group of agents for a range of different notions of collective optimality proposed in social choice theory and welfare economics

the boolean hierarchy ii applications the boolean hierarchy i structural properties j cai et al siam j comput 17 1988 pp 1232 252 explores the structure of the boolean hierarchy the closure of np with respect to boolean

on the complexity of data disjunctions we study the complexity of data disjunctions in disjunctive deductive databases dddbs a data disjunction is a disjunctive ground clause r c 1pt1 r ck k 2 which is derived from the database such that all atoms in the clause involve the same predicate r we consider the complexity of deciding existence and uniqueness of a minimal data disjunction as well as actually computing one both for propositional data and nonground program complexity of the database our results extend and complement previous results on the complexity of disjunctive databases and provide newly developed tools for the analysis of the complexity of function computation

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

unambiguous computations and locally definable acceptance types abstract hertrampf s locally definable acceptance types show that many complexity classes can be defined in terms of polynomial time bounded ntms with simple local conditions on the nodes of its computation tree rather than global concepts like number of accepting paths etc we introduce a modification of hertrampf s locally definable acceptance types which allows to get a larger number of characterizable complexity classes among others the newly characterizable classes are up and mod z k p it is shown how different types of oracle access e g guarded access can be characterized by this model this sheds new light on the discussion on how to access unambiguous computation we present simple functions that describe precisely objects of current research as the unambiguous oracle alternation and promise hierarchies we exhibit the new class uap which seems to be an unambiguous analogue of wagner s p uap and thus p contains few and is currently the smallest class known with this property

serk2v3 solving mildly stiff nonlinear partial differential equations abstract multidimensional nonlinear parabolic partial differential equations pdes appear in a large variety of disciplines usually the scientific literature advises against the use of explicit ode solvers for the integration of stiff problems however in this manuscript we would like to show how some explicit methods can be very efficient for specific problems stabilized explicit runge kutta methods serk are a class of explicit methods with extended stability domains along the negative real axis it is necessary to evaluate the function s times but the stability region is o s 2 hence the computational cost is o s times lower than for traditional explicit algorithms in this way moderately stiff problems can be integrated by the use of simple explicit evaluations serk schemes can easily be applied to many different classes of problems with large dimensions and additionally they have low memory demand since these methods are explicit they do not require algebra routines to solve large nonlinear systems associated to odes and therefore are especially well suited for the method of lines mol discretizations of parabolic multi dimensional pdes additionally the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way that is with minimal number of additional stages hence adaptation of the length step is possible at practically no extra cost in this work we study several strategies to change the time step for non smooth functions and finally propose a new technique this procedure improves the numerical results especially when the data is non smooth we also derive two algorithms to estimate the spectral radius in the new code a nonlinear power method and another procedure based on the gershgorin theorem although serk algorithms are only second order schemes in this manuscript we propose to utilize them after higher order discretizations in space to obtain higher accurate solutions efficiently and usually faster since these higher order schemes allow the use of a smaller number of nodes some numerical experiments shown in the paper support these conclusions

the boolean hierarchy i structural properties in this paper we study the complexity of sets formed by boolean operations union intersection and complement on np sets these are the sets accepted by trees of hardware with np predicates as leaves and together these form the boolean hierarchy we present many results about the structure of the boolean hierarchy separation and immunity results natural complete languages and structural asymmetries between complementary classes we show that in some relativized worlds the boolean hierarchy is infinite and that for every k there is a relativized world in which the boolean hierarchy extends exactly k levels we prove natural languages variations of vertex cover complete for the various levels of the boolean hierarchy we show the following structural asymmetry though no set in the boolean hierarchy is text d text p immune there is a relativized world in which the boolean hierarchy contains text cod text p immune sets thus this paper explores the structural properties of the

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

hybrid monte carlo methods for sampling probability measures on submanifolds probability measures supported on submanifolds can be sampled by adding an extra momentum variable to the state of the system and discretizing the associated hamiltonian dynamics with some stochastic perturbation in the extra variable in order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed hamiltonian dynamics a metropolis rejection procedure can be considered the so obtained scheme belongs to the class of generalized hybrid monte carlo ghmc algorithms we show here how to generalize to ghmc a procedure suggested by goodman holmes cerfon and zappa for metropolis random walks on submanifolds where a reverse projection check is performed to enforce the reversibility of the algorithm for any timesteps and hence avoid biases in the invariant measure we also provide a full mathematical analysis of such procedures as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples

hard core theorems for complexity classes nancy lynch proved that if a decision problem a is not solvable in polynomial time then there exists an infinite recursive subset x of its domain on which the decision is almost everywhere complex in this paper general theorems of this kind that can be applied to several well known automata based complexity classes including a common class of randomized algorithms are proved

graph isomorphism is in spp we show that graph isomorphism is in the complexity class spp and hence it is in p in fact in modkp for each k 2 these inclusions for graph isomorphism were not known prior to membership in spp we derive this result as a corollary of a more general result we show that a generic problem find group has an fpspp algorithm this general result has other consequences for example it follows that the hidden subgroup problem for permutation groups studied in the context of quantum algorithms has an fpspp algorithm also some other algorithmic problems over permutation groups known to be at least as hard as graph isomorphism e g coset intersection are in spp and thus in modkp for each k 2

model checking for fragments of the interval temporal logic hs at the low levels of the polynomial time hierarchy abstract some temporal properties of reactive systems such as actions with duration and temporal aggregations which are inherently interval based can not be properly expressed by the standard point based temporal logics ltl ctl and ctl as they give a state by state account of system evolution conversely interval temporal logics which feature intervals instead of points as their primitive entities naturally express them we study the model checking mc problem for halpern and shoham s modal logic of time intervals hs interpreted on kripke structures under the homogeneity assumption hs is the best known interval based temporal logic which has one modality for each of the 13 ordering relations between pairs of intervals allen s relations apart from equality we focus on mc for some hs fragments featuring modalities for a subset of allen s relations meet met by started by and finished by showing that it is in p np additionally we provide some complexity lower bounds to the problem

exponential fitting bdf algorithms explicit and implicit 0 stable methods we present bdf type formulas capable of the exact integration with only round off errors of differential equations the solutions of which belong to the space generated by the linear combinations of 1 eax xeax plots of their 0 stability regions in terms of are provided we will see that the explicit method is 0 stable with many parameters h plots of their regions absolute stability that include all the negative real axis are provided numerical examples show the efficiency of the proposed codes specially when we are integrating stiff problems with the explicit method

default reasoning from conditional knowledge bases complexity and tractable cases abstract conditional knowledge bases have been proposed as belief bases that include defeasible rules also called defaults of the form which informally read as generally if then such rules may have exceptions which can be handled in different ways a number of entailment semantics for conditional knowledge bases have been proposed in the literature however while the semantic properties and interrelationships of these formalisms are quite well understood about their computational properties only partial results are known so far in this paper we fill these gaps and first draw a precise picture of the complexity of default reasoning from conditional knowledge bases given a conditional knowledge base kb and a default does kb entail we classify the complexity of this problem for a number of well known approaches including goldszmidt et al s maximum entropy approach and geffner s conditional entailment where we consider the general propositional case as well as natural syntactic restrictions in particular to horn and literal horn conditional knowledge bases as we show the more sophisticated semantics for conditional knowledge bases are plagued with intractability in all these fragments we thus explore cases in which these semantics are tractable and find that most of them enjoy this property on feedback free horn conditional knowledge bases which constitute a new meaningful class of conditional knowledge bases furthermore we generalize previous tractability results from horn to q horn conditional knowledge bases which allow for a limited use of disjunction our results complement and extend previous results and contribute in refining the tractability intractability frontier of default reasoning from conditional knowledge bases they provide useful insight for developing efficient implementations

note on the computational complexity of j radii of polytopes in r n we show that for fixed dimensionn the approximation of inner and outer j radii of polytopes in n endowed with the euclidean norm is in our method is based on the standard polynomial time algorithms for solving a system of polynomial inequalities over the reals in fixed dimension

the logarithmic alternation hierarchy collapses aσ 2l aπ 2l abstract we show that a 2 l a k l k 2 by proving that a 2 l coincides with a 2 l essentially this is done by reducing the a 2 l complete set gap cogap to the question whether of two vectors a and b of n components a contains more solvable components i e components which are contained in gap than b moreover using a similar technique we show a 2 l l hd nl finally we consider the relevance of our proof technique for polynomial time classes e g the boolean np hierarchy

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

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

smoothing schemes for reaction diffusion systems with nonsmooth data cox and matthews s m cox p c matthews exponential time differencing for stiff systems j comput phys 176 2002 430 455 developed a class of exponential time differencing runge kutta schemes etdrk for nonlinear parabolic equations kassam and trefethen a k kassam ll n trefethen fourth order time stepping for stiff pdes siam j sci comput 26 2005 1214 1233 have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth order etdrk4 scheme they use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials but this approach creates new difficulties in choosing and evaluating the contour for larger problems neither treatment addresses problems with nonsmooth data where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications including computational chemistry and financial engineering we introduce a new version of the fourth order cox matthews kassam trefethen etdrk4 scheme designed to eliminate the remaining computational difficulties this new scheme utilizes an exponential time differencing runge kutta etdrk scheme using a diagonal pade approximation of matrix exponential functions while to deal with the problem of nonsmooth data we use several steps of an etdrk scheme using a sub diagonal pade formula the new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel in this approach it is only required that several backward euler linear problems be solved in serial or parallel numerical experiments are described to support the new scheme

extrapolated stabilized explicit rungekutta methods extrapolated stabilized explicit rungekutta methods eserk are proposed to solve multi dimensional nonlinear partial differential equations pdes in such methods it is necessary to evaluate the function nt times per step but the stability region is o nt2 hence the computational cost is o nt times lower than for a traditional explicit algorithm in that way stiff problems can be integrated by the use of simple explicit evaluations in which case implicit methods usually had to be used therefore they are especially well suited for the method of lines mol discretizations of parabolic nonlinear multi dimensional pdes in this work first s stages first order methods with extended stability along the negative real axis are obtained they have slightly shorter stability regions than other traditional first order stabilized explicit rungekutta algorithms also called rungekuttachebyshev codes later they are used to derive nt stages second and fourth order schemes using richardson extrapolation the stability regions of these fourth order codes include the interval 0 01nt2 0 nt being the number of total functions evaluations which are shorter than stability regions of rock4 methods for example however the new algorithms neither suffer from propagation of errors as other rungekuttachebyshev codes as rock4 or dumka nor internal instabilities additionally many other types of higher order and also lower order methods can be obtained easily in a similar way these methods also allow adaptation of the length step with no extra cost hence the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way i e with minimal number of additional stages we compare the new techniques with other well known algorithms with good results in very stiff diffusion or reactiondiffusion multi dimensional nonlinear equations

promise problems complete for complexity classes abstract a general framework is given to obtain hardness results for promise problems that derive from self reducible decision problems the principal theorem is that if a set a is d p equivalent to a disjunctive self reducible set in np then the natural promise problem associated with a is as hard to solve as it is to recognize a np hardness of the satisfiability promise problem follows and graph isomorphism hardness of a promise problem that derives from the graph isomorphism problem is proved

piecewise linearized methods for initial value problems with oscillating solutions two piecewise linearized methods for the numerical solution of nonlinear second order differential equations which contain the first order derivative are presented these methods provide piecewise analytical solutions in open intervals result in explicit finite difference equations for both the displacement and the velocity or only the displacement at nodal points and continuous solutions everywhere one of the piecewise linearized methods is a two level method that provides smooth solutions and is self starting the other is a non self starting three level technique the paper also presents several iterative schemes a time linearized three point technique and a time linearized numerov method an extensive assessment of the piecewise linearized methods is carried out in order to assess their accuracy in nine examples and it is shown that in some cases these techniques provide as accurate results as exponentially or trigonometrically fitting methods that do require an estimate of the dominant frequency of the solution the piecewise linearized methods presented here do not require such an estimate at all but may exhibit spurious fixed points attractors in nonlinear problems and their accuracy is a strong function of both the nonlinearities and the time step however they are applicable to nonlinear second order differential equations which contain the first order derivative

resource bounded kolmogorov complexity revisited we take a fresh look at cd complexity where cdt x is the size of the smallest program that distinguishes x from all other strings in time t x we also look at cnd complexity a new nondeterministic variant of cd complexity and time bounded kolmogorov complexity denoted by c complexity we show several results relating time bounded c cd and cnd complexity and their applications to a variety of questions in computational complexity theory including the following showing how to approximate the size of a set using cd complexity without using the random string as needed in sipser s earlier proof of a similar result also we give a new simpler proof of this result of sipser s improving these bounds for almost all strings using extractors a proof of the valiant vazirani lemma directly from sipser s earlier cd lemma a relativized lower bound for cnd complexity exact characterizations of equivalences between c cd and cnd complexity showing that satisfying assignments of a satisfiable boolean formula can be enumerated in time polynomial in the size of the output if and only if a unique assignment can be found quickly this answers an open question of papadimitriou a new kolmogorov complexity based proof that bpp subseteq sigma 2 p new kolmogorov complexity based constructions of the following relativized worlds there exists an infinite set in p with no sparse infinite np subsets exp nexp but there exists a nexp machine whose accepting paths cannot be found in exponential time satisfying assignments cannot be found with nonadaptive queries to sat

on truth table reducibility to sat we show that polynomial time truth table reducibility via boolean circuits to sat is the same as logspace truth table reducibility via boolean formulas to sat and the same as logspace turing reducibility to sat in addition we prove that a constant number of rounds of parallel queries to sat is equivalent to one round of parallel queries we give an oracle relative to which 2p is not equal to the class of predicates polynomial time truth table reducible to sat

exponential fitting bdf algorithms and their properties we present two families of explicit and implicit bdf formulas capable of the exact integration with only round off errors of differential equations whose solutions belong to the space generated by the linear combinations of exponential of matrices products of the exponentials by polynomials and products of those matrices by ordinary polynomials those methods are suitable for stiff and highly oscillatory problems then we will study their properties of local truncation error and stability plots of their 0 stability regions in terms of ah are provided plots of their regions of absolute stability that include all the negative real axis in the implicit case are provided exponential fitting algorithms depend on a parameter ah how can we find this parameter is a big question here we show different ways to find a good parameter finally numerical examples show the efficiency of the proposed codes specially when we are integrating stiff problems where the jacobian of the function has complex eigenvalues or problems where the jacobian has positive eigenvalues but the solutions of the problems have not positive exponentials

on the complexity of entailment in existential conjunctive first order logic with atomic negation we consider the entailment problem in the fragment of first order logic fol composed of existentially closed conjunctions of literals without functions denoted by fol a this problem can be recast as several fundamental problems in artificial intelligence and databases namely query containment for conjunctive queries with negation clause entailment for clauses without functions and query answering with incomplete information for boolean conjunctive queries with negation over a fact base entailment in fol a is p 2 p complete whereas it is only np complete when the formulas contain no negation we investigate the role of specific literals in this complexity increase these literals have the property of being exchangeable with this notion taking the structure of the formulas into account to focus on the structure of formulas we shall see them as labeled graphs graph homomorphism which provides a sound and complete proof procedure for positive formulas is at the core of this study let entailment k be the following family of problems given two formulas g and h in fol a such that g has at most k pairs of exchangeable literals is g entailed by h the main results are that entailment k is np complete if k is less or equal to 1 and p n p complete for any value of k greater or equal to 3 as a corollary of our proofs we are able to classify exactly entailment k for any value of k 2 when g is decomposable into a tree

eserk5 a fifth order extrapolated stabilized explicit runge kutta method abstract a new algorithm is developed and analyzed for multi dimensional non linear parabolic partial differential equations pdes which are semi discretized in the spatial variables leading to a system of ordinary differential equations odes it is based on fifth order extrapolated stabilized explicit runge kutta schemes eserk they are explicit methods and therefore it is not necessary to employ complicated software for linear or non linear system of equations additionally they have extended stability regions along the negative real semi axis hence they can be considered to solve stiff problems coming from very common diffusion or reaction diffusion problems previously only lower order codes up to fourth order have been constructed and made available in the scientific literature however at the same time higher order codes were demonstrated to be very efficient to solve equations where it is necessary to have a high precision or they have transient zones that are very severe and where functions change very fast the new schemes allow changing the step length very easily and with a very small computational cost thus a variable step length with variable number of stages algorithm is constructed and compared with good numerical results in relation to other well known ode solvers

reductions between disjoint np pairs disjoint np pairs are pairs a b of nonempty disjoint sets in np we prove that all of the following assertions are equivalent there is a many one complete disjoint np pair there is a strongly many one complete disjoint np pair there is a turing complete disjoint np pair such that all reductions are smart reductions there is a complete disjoint np pair for one to one invertible reductions the class of all disjoint np pairs is uniformly enumerable let a b c and d be nonempty sets belonging to np a smart reduction between the disjoint np pairs a b and c d is a turing reduction with the additional property that if the input belongs to a b then all queries belong to c d we prove under the reasonable assumption that up co up has a p bi immune set that there exist disjoint np pairs a b and c d such that a b is truth table reducible to c d but there is no smart reduction between them this paper contains several additional separations of reductions between disjoint np pairs we exhibit an oracle relative to which distnp has a truth table complete disjoint np pair but has no many one complete disjoint np pair

on the query complexity of selecting minimal sets for monotone predicates propositional satisfiability sat solvers are routinely used for solving many function problems a natural question that has seldom been addressed is what is the number of calls to a sat solver for solving some target function problem this article improves upper bounds on the query complexity of solving several function problems defined on propositional formulas these include computing the backbone of a formula and computing the set of independent variables of a formula for the general case of monotone predicates the article improves upper bounds on the query complexity of computing a minimal set when the number of minimal sets is constant this applies for example to the computation of a minimal unsatisfiable subset mus for cnf formulas but also to the computation of prime implicants and implicates with immediate application in a number of ai settings

a hierarchy based on output multiplicity abstract the class np k v consists of those partial multivalued functions that can be computed by a nondeterministic polynomial time bounded transducer that has at most k distinct values on any input we define the output multiplicity hierarchy to consist of the collection of classes np k v for all positive integers k 1 in this paper we investigate the strictness of the output multiplicity hierarchy and establish three main results pertaining to this 1 1 if for any k 1 the class np k v collapses into the class np k 1 v then the polynomial hierarchy collapses to 2 p 2 2 if the converse of the above result is true then any proof of this converse cannot relativize we exhibit an oracle relative to which the polynomial hierarchy collapses to p np but the output multiplicity hierarchy is strict 3 3 relative to a random oracle the output multiplicity hierarchy is strict this result is in contrast to the still open problem of the strictness of the polynomial hierarchy relative to a random oracle in introducing the technique for the third result we prove a related result of interest relative to a random oracle up np

circuit principles and weak pigeonhole variants this paper considers the relational versions of the surjective partial surjective and multifunction weak pigeonhole principles for pv 1 b 1 b and b 1 b formulas as well as relativizations of these formulas to higher levels of the bounded arithmetic hierarchy we show that the partial surjective weak pigeonhole principle for 1 b formulas implies that for each k there is a string of length 2 2 n k which is hard to block recognize by circuits of size n k these principles in turn imply the partial surjective principle for 1 b formulas we show that the surjective weak pigeonhole principle for b 1 b formulas in s 2 1 implies our hard string principle which in turn implies the surjective weak pigeonhole principle for 1 b formulas we introduce a class of predicates corresponding to poly log length iterates of polynomial time computable predicates and show that over s 2 1 the multifunction weak pigeonhole principle for such predicates is equivalent to an iterative circuit block recognition principle a consequence of this is that if s 2 1 proves this principle then rsa is vulnerable to polynomial time attacks

on the isomorphism problem for decision trees and decision lists we study the complexity of isomorphism testing for boolean functions that are represented by decision trees or decision lists our results are the following isomorphism testing of rank 1 decision trees is complete for logspace for any constant r 2 isomorphism testing for rank r decision trees is polynomial time equivalent to graph isomorphism as a consequence of our reduction we obtain our main result for decision trees a 2 n log s o 1 time algorithm for isomorphism testing of decision trees of size s over n variables the isomorphism problem for decision lists admits a schaefer type trichotomy depending on the class of base functions the isomorphism problem is either in l or polynomial time equivalent to graph isomorphism or conp hard

algorithmic design of self folding polyhedra self assembly has emerged as a paradigm for highly parallel fabrication of complex three dimensional structures however there are few principles that guide a priori design yield and defect tolerance of self assembling structures we examine with experiment and theory the geometric principles that underlie self folding of submillimeter scale higher polyhedra from two dimensional nets in particular we computationally search for nets within a large set of possibilities and then test these nets experimentally our main findings are that i compactness is a simple and effective design principle for maximizing the yield of self folding polyhedra and ii shortest paths from 2d nets to 3d polyhedra in the configuration space are important for rationalizing experimentally observed folding pathways our work provides a model problem amenable to experimental and theoretical analysis of design principles and pathways in self assembly

the shrinking property for np and conp we study the shrinking and separation properties two notions well known in descriptive set theory for np and conp and show that under reasonable complexity theoretic assumptions both properties do not hold for np and the shrinking property does not hold for conp in particular we obtain the following results 1 np and conp do not have the shrinking property unless ph is finite in general s n p and p n p do not have the shrinking property unless ph is finite this solves an open question posed by selivanov 1994 33 2 the separation property does not hold for np unless email protected conp 3 the shrinking property does not hold for np unless there exist np hard disjoint np pairs existence of such pairs would contradict a conjecture of even et al 1984 6 4 the shrinking property does not hold for np unless there exist complete disjoint np pairs moreover we prove that the assumption np conp is too weak to refute the shrinking property for np in a relativizable way for this we construct an oracle relative to which email protected conp np conp and np has the shrinking property this solves an open question posed by blass and gurevich 1984 3 who explicitly ask for such an oracle

locally testable codes and pcps of almost linear length we initiate a systematic study of locally testable codes that is error correcting codes that admit very efficient membership tests specifically these are codes accompanied with tests that make a constant number of random queries into any given word and reject non codewords with probability proportional to their distance from the code locally testable codes are believed to be the combinatorial core of pcps however the relation is less immediate than commonly believed nevertheless we show that certain pcp systems can be modified to yield locally testable codes on the other hand we adapt techniques that we develop for the construction of the latter to yield new pcps our main results are locally testable codes and pcps of almost linear length specifically we prove the existence of the following constructs locally testable binary linear codes in which k information bits are encoded by a codeword of length k exp o log k this improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet pcp systems of almost linear length for sat the length of the proof is n exp o log n and verification in performed by a constant number i e 19 of queries as opposed to previous results that used proof length n 1 p o 1 q for verification by q queries the novel techniques in use include a random projection of certain codewords and pcp oracles that preserves local testability an adaptation of pcp constructions to obtain linear pcp oracles for proving conjunctions of linear conditions and design of pcps with some new soundness properties a direct construction of locally testable linear codes of subexponential length

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

random vectors in the isotropic position abstract letybe a random vector in r n satisfying e y y id letmbe a natural number and lety1 ymbe independent copies ofy we study the question of approximation of the identity operator by finite sums of the tensorsyi yi we prove that for some absolute constantc e 1 m i 1 m y i y i id c log n m e y log m 1 log m provided that the last expression is smaller than 1 we apply this estimate to improve a result of bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position

free energy of singular sticky sphere clusters networks of particles connected by springs model many condensed matter systems from colloids interacting with a short range potential and complex fluids near jamming to self assembled lattices and various metamaterials under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero frequency vibrational modes yet such singular modes are at the epicenter of many interesting behaviors in the systems above we consider a system of n spherical particles and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged although the contribution to the partition function from singular clusters diverges in the limit its asymptotic value can be calculated and depends on only two parameters characterizing the depth and range of the potential the result holds for systems that are second order rigid a geometric characterization that describes all known ground state rigid sticky clusters to illustrate the applications of our theory we address the question of emergence how does crystalline order arise in large systems when it is strongly disfavored in small ones we calculate the partition functions of all known rigid clusters up to n 21 and show the cluster landscape is dominated by hyperstatic clusters those with more than 3n 6 contacts singular and isostatic clusters are far less frequent despite their extra vibrational and configurational entropies since the most hyperstatic clusters are close to fragments of a close packed lattice this underlies the emergence of order in sticky sphere systems even those as small as n 10

the complexity of promise problems with applications to public key cryptography a promise problem is a formulation of partial decision problem complexity issues about promise problems arise from considerations about cracking problems for public key cryptosystems using a notion of turing reducibility between promise problems this paper disproves a conjecture made by even and yacobi 1980 that would imply nonexistence of public key cryptosystems with np hard cracking problems in its place a new conjecture is raised having the same consequence in addition the new conjecture implies that np complete sets cannot be accepted by turing machines that have at most one accepting computation for each input word

computing answers with model elimination abstract we demonstrate that theorem provers using model elimination me can be used as answer complete interpreters for disjunctive logic programming more specifically we introduce a family of restart variants of model elimination and we introduce a mechanism for computing answers building on this we develop a new calculus called ancestry restart me this variant admits a more restrictive regularity restriction than restart me and as a side effect it is in particular attractive for computing definite answers the presented calculi can also be used successfully in the context of automated theorem proving we demonstrate experimentally that it is more difficult to compute nontrivial answers to goals than to prove the existence of answers

separating complexity classes with tally oracles abstract long and selman 1986 proved that for most familiar pairs of complexity classes separating the classes with a tally oracle is no easier than truly separating the classes refuting a claim in the literature we prove for the first time that for many familiar pairs of complexity classes separating the classes with a tally oracle is easier than truly separating the classes

revisiting deutsch jozsa algorithm abstract the deutsch jozsa algorithm is essentially faster than any possible deterministic classical algorithm for solving a promise problem that is in fact a symmetric partial boolean function named as the deutsch jozsa problem the deutsch jozsa problem can be equivalently described as a partial function d j n 0 0 1 n 0 1 defined as d j n 0 x 1 for x n 2 d j n 0 x 0 for x 0 n and it is undefined for the remaining cases where n is even and x is the hamming weight of x the deutsch jozsa algorithm needs only one query to compute d j n 0 but the classical deterministic algorithm requires n 2 1 queries to compute it in the worse case we present all symmetric partial boolean functions with degree 1 and 2 we prove the exact quantum query complexity of all symmetric partial boolean functions with degree 1 and 2 we prove deutsch jozsa algorithm can compute any symmetric partial boolean function f with exact quantum 1 query complexity

projection of diffusions on submanifolds application to mean force computation we consider the problem of sampling a boltzmann gibbs probability distribution when this distribution is restricted in some suitable sense on a manifold of r n implicitly defined by n constraints q1 x qn x 0 this problem arises for example in systems subject to hard constraints or in the context of free energy calculations we prove that the constrained stochastic di erential equations i e di usions proposed in w e and e vanden eijnden in multiscale modelling and simulation eds s attinger and p koumoutsakos lncse 39 springer 2004 and g ciccotti r kapral and e vanden eijnden chemphyschem 6 1809 2005 are ergodic with respect to this restricted distribution we also construct numerical schemes for the integration of the constrained di usions finally we show how these schemes can be used to compute the gradient of the free energy associated with the constraints

an oracle separating conjectures about incompleteness in the finite domain abstract pudlak 19 lists several major complexity theoretic conjectures relevant to proof complexity and asks for oracles that separate pairs of corresponding relativized conjectures among these conjectures are disjnp the class of all disjoint np pairs does not have many one complete problems sat np does not contain many one complete sets that have p optimal proof systems up up does not have many one complete problems np conp np conp does not have many one complete problems as one answer to this question we construct an oracle relative to which among others disjnp sat up and np conp hold hence there is no relativizable proof for the implication disjnp up np conp sat in particular regarding the conjectures by pudlak this extends a result by khaniki 11

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

frequency evaluation in exponential fitting multistep algorithms for odes we consider the linear multistep algorithms for first order odes and examine the problem of how the frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms we find out that the key of the answer consists in analysing the behaviour of the error on further investigating the simple case of two step bdf algorithms we produce formulae for the optimal s and show that if the optimal s are used the order of the method is increased by one unit the reported numerical illustrations suggest that further investigations along these lines deserve a real attention

removing redundancy from a clause abstract this paper deals with the problem of removing redundant literals from a given clause we first consider condensing a weak type of redundancy elimination a clause is condensed if it does not subsume any proper subset of itself it is often useful and sometimes necessary to replace a non condensed clause c by a condensation i e by a condensed subset of c which is subsumed by c after studying the complexity of an existing clause condensing algorithm we present a more efficient algorithm and provide arguments for the optimality of the new method we prove that testing whether a given clause is condensed is co np complete and show that several problems related to clause condensing belong to complexity classes that are probably slightly harder than np we also consider a stronger version of redundancy elimination a clause c is strongly condensed iff it does not contain any proper subset c such that c logically implies c we show that the problem of testing whether a clause is strongly condensed is undecidable

bounded queries to sat and the boolean hierarchy we study the complexity of decision problems that can be solved by a polynomial time turing machine that makes a bounded number of queries to an np oracle depending on whether we allow some queries to depend on the results of other queries we obtain two probably different hierarchies we present several results relating the bounded np query hierarchies to each other and to the boolean hierarchy we also consider the similarly defined hierarchies of functions that can be computed by a polynomial time turing machine that makes a bounded number of queries to an np oracle we present relations among these two hierarchies and the boolean hierarchy in particular we show for all k that there are functions computable with 2k parallel queries to an np set that are not computable in polynomial time with k serial queries to any oracle unless p np as a corollary k 1 parallel queries to an np set allow us to compute more functions than are computable with only k parallel queries to an np set unless p np the same is true of serial queries similar results hold for all tt self reducible sets using a mind change technique we show that 2k 1 parallel queries to an np set allow us to accept in polynomial time exactly the same sets as can be accepted in polynomial time with k serial queries to an np set in fact the same is true for any class in place of np that is closed under polynomial time positive bounded truth table reductions this contrasts with the expected result for function computations with an np oracle beigel 1988 in addition we show that the boolean hierarchy and the bounded query hierarchies of languages either stand or collapse together finally we show that if the boolean hierarchy collapses to any level but the zeroth deterministic polynomial time then for all k there are functions computable in polynomial time with k parallel queries to an np set that are not computable in polynomial time with k 1 serial queries to any set np complete sets are p superterse

oracles that compute values this paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in npmv the class of all multivalued partial functions that are computable nondeterministically in polynomial time concerning deterministic polynomial time reducibilities it is shown that a multivalued partial function is polynomial time computable with k adaptive queries to npmv if and only if it is polynomial time computable via 2k 1 nonadaptive queries to npmv a characteristic function is polynomial time computable with k adaptive queries to npmv if and only if it is polynomial time computable with k adaptive queries to np unless the boolean hierarchy collapses for every k k adaptive nonadaptive queries to npmv are different than k 1 adaptive nonadaptive queries to npmv nondeterministic reducibilities lowness and the difference hierarchy over npmv are also studied the difference hierarchy for partial functions does not collapse unless the boolean hierarchy collapses but surprisingly the levels of the difference and bounded query hierarchies do not interleave as is the case for sets unless the polynomial hierarchy collapses

on the stability of exponential fitting bdf algorithms we present bdf type formulas capable of the exact integration with only round off errors of differential equations whose solutions are linear combinations of an exponential with parameter l and ordinary polynomials for l 0 new formulas reduces to the classical bdf formulas plots of their 0 stability regions in terms of l are provided plots of their regions absolute stability that include all the negative real axis are provided numerical examples shows the efficiency of the proposed codes specially when we are integrating stiff oscillatory problems

computational complexity studies of synchronous boolean finite dynamical systems on directed graphs abstract a finite dynamical system is a system consisting of some finite number of objects that take upon a value from some domain as a state in which after initialization the states of the objects are updated based upon the states of the other objects and themselves according to a certain update schedule this paper studies a subclass of finite dynamical systems the synchronous boolean finite dynamical system synchronous bfds for short where the states are boolean and the state update takes place in discrete time and at the same on all objects the paper is concerned with three problems convergence path intersection and cycle length of the synchronous bfds in which the state update functions or the local state transition functions are chosen from a predetermined finite basis of boolean functions b the paper results characterize their computational complexity