completely degenerate lower dimensional invariant tori for hamiltonian system abstract we study the persistence of lower dimensional invariant tori for a nearly integrable completely degenerate hamiltonian system it is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness

degenerate lower dimensional tori in reversible systems abstract in this paper we prove the persistence of lower dimensional invariant tori with prescribed frequencies and singular normal matrices in reversible systems the normal variable is two dimensional and the unperturbed nonlinear terms in the differential equation for this variable have a special structure

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

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

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

degenerate lower dimensional tori in hamiltonian systems abstract we study the persistence of lower dimensional tori in hamiltonian systems of the form h x y z y 1 2 z m z e p x y z where x y z t n r n r 2 m e is a small parameter and m can be singular we show under a weak melnikov nonresonant condition and certain singularity removing conditions on the perturbation that the majority of unperturbed n tori can still survive from the small perturbation as an application we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable properly degenerate hamiltonian system for which neither the kolmogorov nor the g nondegenerate condition is satisfied

the local multiplier algebra of a c algebra ii abstract let a be a separable c algebra and let m loc a be the local multiplier algebra of a it is shown that every minimal closed prime ideal of m loc a is primitive if prim a has a dense g consisting of closed points for instance if prim a is a t 1 space and a is unital then m loc a is its own local multiplier algebra and has only inner derivations the same is true for m loc m loc a if a is non unital if a is postliminal then m loc m loc a is the regular completion of a which is an aw algebra

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

perturbations of lower dimensional tori for hamiltonian systems abstract this paper provides a generalized kolmogorov arnold moser theorem for lower dimensional tori in hamiltonian systems which applies to multiple normal frequency case the proof is based on newton s iteration method and a generalized version of small divisor conditions

persistence of lower dimensional hyperbolic tori for reversible system abstract in this paper we consider the persistence of hyperbolic lower dimensional invariant tori with prescribed frequency for reversible system and prove that if the frequency mapping has nonzero topological degree at some diophantine frequency then the hyperbolic lower invariant dimensional torus with this frequency persists under small perturbations

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

abelian strict approximation in aw algebras and weyl von neumann type theorems in this paper for a c algebra a with m m a an aw algebra or equivalently for an essential norm closed two sided ideal a of an aw algebra m we investigate the strict approximability of the elements of m from commutative c subalgebras of a in the relevant case of the norm closed linear span a of all finite projections in a semi finite aw algebra m we shall give a complete description of the strict closure in m of any maximal abelian self adjoint subalgebra masa of a we shall see that the situation is completely different for discrete respectively continuous m in the discrete case for any masa c of a the strict closure of c is equal to the relative commutant c m while in the continuous case under certain conditions concerning the center valued quasitrace of the finite reduced algebras of m satisfied by all von neumann algebras c is already strictly closed thus in the continu ous case no elements of m which are not already belonging to a can be strictly approximated from commutative c subalgebras of a in spite of this pathology of the strict topology in the case of the norm closed linear span of all finite projections of a continuous semi finite aw algebra we shall prove that in general situations including also this case any normal y 2 m is equal modulo a to some x 2 m which belongs to an order theoretical closure of an appropriate commutative c subalgebra of a in other words if we replace the strict topology with order theoretical approximation weyl von neumann berg sikonia type theo rems will hold in substantially greater generality

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

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

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

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

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

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

on lower dimensional invariant tori in reversible systems abstract in this paper a result on the persistence of lower dimensional invariant tori in reversible systems is obtained under some weak non degenerate conditions such a result can be applied to the case where normal frequencies are not simple and some of them may be zero we also give an answer of a conjecture of m b sevryuk 1986 reversible systems lecture notes in math vol 1211 springer verlag new york berlin about the existence of lower dimensional invariant tori in resonant zones

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

semantical and computational aspects of horn approximations abstract selman and kautz proposed a method called horn approximation for speeding up inference in propositional knowledge bases their technique is based on the compilation of a propositional formula into a pair of horn formulae a horn greatest lower bound glb and a horn least upper bound lub in this paper we focus on glbs and address two questions that have been only marginally addressed so far 1 what is the semantics of the horn glbs 2 what is the exact complexity of finding them we obtain semantical as well as computational results the major semantical result is the set of minimal models of a propositional formula and the set of minimum models of its horn glbs are the same the major computational result is finding a horn glb of a propositional formula in cnf is np equivalent

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

birkhoff kolmogorov arnold moser tori in convex hamiltonian systems for the hamiltonian systems of kam type it is proved that some lower dimensional invariant tori always exist in the resonance gaps although those maximum tori can not survive small perturbations in the generic case

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

lower dimensional invariant tori with prescribed frequency for the nonlinear schrodinger equation abstract in this paper one dimensional 1d nonlinear schrodinger equation i u t u x x m u u 2 u f u 2 u 0 subject to dirichlet boundary conditions is considered where the nonlinearity f is a real analytic function near u 0 with f 0 f 0 0 it is proved that for each given constant potential m and each prescribed integer b 1 the above equation admits a whitney smooth family of small amplitude time quasi periodic solutions whose b dimensional frequencies are just small dilation of a prescribed diophantine vector accordingly we obtain the existence of lower dimensional invariant kam tori with tangential frequencies constrained to a given diophantine direction in an infinite dimensional phase space setting

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