on certain polynomial time truth table reducibilities of complete sets to sparse sets let sigma be a finite alphabet a set s subset sigma is called sparse if the number of members of s having length at most n is bounded above by a polynomial in n let leqq m p denote polynomial time many one reducibility and let leqq bptt p denote the more general polynomial time bounded positive truth table reducibility we prove 1 leqq bptt p reducibility of a conp hard set to a sparse set implies text np text p and 2 leqq bptt p reducibility of an np hard set to a sparse set which is itself in np implies text np text p 1 generalizes fortune s result f he proved it for the case of leqq m p reducibility 2 for the case of leqq m p reducibility was proved by mahaney m even without assuming that the sparse set itself is in np our results imply that if a conp hard set is a finite union of sets which are leqq m p reducible to sparse sets then text np text p we then investigate a certain non

on sparse hard sets for counting classes abstract in this paper we study one word decreasing self reducible sets which are introduced by lozano and toran 1991 these are the usual self reducible sets with the peculiarity that the self reducibility machine makes at most one query and this is lexicographically smaller than the input we show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one word decreasing self reducible using this fact we can prove that for any class k chosen from pp np c p mod2p mod3 p it holds that 1 if there is a sparse pbtt hard set for k then k p and 2 if there is a sparse snbtt hard set for k then k np co np this generalizes the result of ogiwara and watanabe 1991 to the mentioned complexity classes

strong and robustly strong polynomial time reducibilities to sparse sets reducibility defined by oracle strong nondeterministic machines is studied two definitions of relativized strength are presented and separated the corresponding reduction classes of the sparse sets are characterized in terms of nonuniform complexity classes an oracle restricted positive relativization of the probabilistic class zpp is developed

on some natural complete operators abstract an operator is a mapping from integer functions to integer functions it is known from a result by baker gill and solovay 1975 that there exists an operator computable in polynomial time by a nondeterministic oracle turing machine called an np operator but not computable in polynomial time by any deterministic oracle turing machine we investigate several natural operators which share similar properties we use the concept of completeness to give a precise classification of the complexity of these operators for example the question of finding maximum values of polynomial time computable functions can be formulated as an operator complete for the class of np operators

nonuniform proof systems a new framework to describe nonuniform and probabilistic complexity classes the concept of non uniform proof systems is introduced this notion allows a uniform description of non uniform complexity classes 10 probabilistic classes e g bpp 8 15 27 32 am 2 3 and language classes defined by simultaneous non uniform and nondeterministic time bounds non uniform proof systems provide a better understanding of many results concerning these classes particularly their connections to uniform complexity measures we give an uniform approach to lowness results 19 20 for various complexity classes for instance we show that co np poly np is contained in the third level of the low hierarchy and that np c np co np poly implies that the polynomial time hierarchy collapses to its second level see also 1 finally some evidence is given that the low hierarchy cannot be extended beyond its third level by the current techniques

satisfiability allows no nontrivial sparsification unless the polynomial time hierarchy collapses consider the following two player communication process to decide a language l the first player holds the entire input x but is polynomially bounded the second player is computationally unbounded but does not know any part of x their goal is to decide cooperatively whether x belongs to l at small cost where the cost measure is the number of bits of communication from the first player to the second player for any integer d 3 and positive real e we show that if satisfiability for n variable d cnf formulas has a protocol of cost o nd e then conp is in np poly which implies that the polynomial time hierarchy collapses to its third level the result even holds when the first player is conondeterministic and is tight as there exists a trivial protocol for e 0 under the hypothesis that conp is not in np poly our result implies tight lower bounds for parameters of interest in several areas namely sparsification kernelization in parameterized complexity lossy compression and probabilistically checkable proofs by reduction similar results hold for other np complete problems for the vertex cover problem on n vertex d uniform hypergraphs this statement holds for any integer d 2 the case d 2 implies that no np hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of o k2 e edges unless conp is in np poly where k denotes the size of the deletion set kernels consisting of o k2 edges are known for several problems in the class including vertex cover feedback vertex set and bounded degree deletion

orbit equivalence rigidity and bounded cohomology we establish new results and introduce new methods in the theory of measurable orbit equivalence using bounded cohomology of group representations our rigidity statements hold for a wide uncountable class of groups arising from negative curvature geometry amongst our applications are a measurable mostow type rigidity theorems for products of negatively curved groups b prime factorization results for measure equivalence c superrigidity for orbit equivalence d the first examples of continua of type ii1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic

two results on polynomial time truth table reductions to sparse sets inspired by the recent solution of the berman hartmanis conjecture siam j comp 6 1977 pp 305 322 that np cannot have a sparse complete set for many one reductions unless text p text np we analyze the implications of np and pspace having sparse hard or complete sets for reduction types more general than the many one reductions three special cases of truth table reductions are considered first we show that if np pspace has a tally leqq k t p hard set then text p text np text p text pspace here leqq k t p denotes a subcase of polynomial time turing reducibility in which for some constant k the reducing turing machine is allowed to make at most k queries to the oracle we also show that if co np pspace has a sparse hard set for conjunctive polynomial time reductions then text p text np text p text pspace

logarithmic advice classes abstract karp and lipton 1980 introduced the notion of nonuniform complexity classes where a certain amount of additional information the advice is given for free the advice only depends on the length of the input karp and lipton initiated the study of classes with either logarithmic or polynomial advice however later yap 1983 schoning 1984 balcazar 1987 and ko and schoning 1985 concentrated on the study of classes of the form c poly where c is p np or pspace and poly denotes a polynomial size advice this paper considers classes of the form c log as a main result it is shown that in the context of an np log computation log bounded advice is equivalent to a sparse oracle in np in contrast it has been shown that a poly bounded advice to an arbitrary sparse oracle set furthermore a general theorem is presented that generalizes karp and lipton s round robin tournament method

on similarity and duality of computation i this paper describes what is the parallel time for sequential models what is the sequential time for parallel models and proves that the well known computational models ram vector machines turing machines uniform circuits uniform aggregates storage modification machines hardware modification machines are all similar in the sense that their parallel time their sequential time and their space complexities are polynomially related simultaneously

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

differentially amplitude and phase encoded qam for the correlated rayleigh fading channel with diversity reception this paper studies the differentially amplitude and phase encoded dape quadrature amplitude modulation qam transmission over correlated rayleigh channels with diversity reception operating over two successive received symbols the optimum and an asymptotic maximum likelihood aml differentially coherent receiver are developed and compared with a conventional switched diversity combining sdc grid receiver it is shown that the aml and sdc grid receivers are much simpler in complexity than the optimum receiver in that no channel side information is required in their realization an exact expression of the bit error probability bep is obtained for the sdc grid receiver based on a union bound argument a bep upper bound for the aml receiver is also derived and verified by simulation numerical results on 16 and 64 point constellations show that the aml receiver exhibits an almost optimum performance and the sdc grid receiver with a small level of diversity is nearly optimum it is also shown by simulation that the conventional equal gain diversity receiver is almost optimal for demodulating a 16 point dape qam signal

geometry of the mapping class groups i boundary amenability we construct a geometric model for the mapping class group mathcal m mathcal c mathcal g of a non exceptional oriented surface s of genus g with k punctures and use it to show that the action of mathcal m mathcal c mathcal g on the compact metrizable hausdorff space of complete geodesic laminations for s is topologically amenable as a consequence the novikov higher signature conjecture holds for every subgroup of mathcal m mathcal c mathcal g

data reduction for graph coloring problems this paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances we are interested in how well polynomial time data reduction can provably shrink instances of coloring problems in terms of the chosen parameter it is well known that deciding 3 colorability is already np complete hence parameterizing by the requested number of colors is not fruitful instead we pick up on a research thread initiated by cai dam 2003 who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial time solvable for example parameterizing by the number k of vertex deletions needed to make the graph chordal we obtain various upper and lower bounds for kernels of such parameterizations of q coloring complementing email protected s study of the time complexity with respect to these parameters our results show that the existence of polynomial kernels for q coloring parameterized by the vertex deletion distance to a graph class f is strongly related to the existence of a function f q which bounds the number of vertices which are needed to preserve the no answer to an instance of q list coloring on f

the existence and density of generalized complexity cores if c is a class of sets and a is not in c then an infinite set h is a proper hard core for a with respect to c if h a and for every c e c such that c a c c h is finite it is shown that if c is a countable class of sets of strings that is closed under finite union and finite variation then every infinite set not in c has a proper hard core with respect to c in addition the density of such generalized complexity cores is studied

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

the boolean hierarchy of np partitions we introduce the boolean hierarchy of k partitions over np for k 3 as a generalization of the boolean hierarchy of sets i e 2 partitions over np whereas the structure of the latter hierarchy is rather simple the structure of the boolean hierarchy of k partitions over np for k 3 turns out to be much more complicated we formulate the embedding conjecture which enables us to get a complete idea of this structure this conjecture is supported by several partial results

relativizing complexity classes with sparse oracles baker gill and solovay constructed sparse sets a and b such that p a np a and np b co np b in contrast to their results we prove that p np if and only if for every tally language t p t np t and that np co np if and only if for every tally language t np t co np t we show that the polynomial hierarchy collapses if and only if there is a sparse set s such that the polynomial hierarchy relative to s collapses similar results are obtained for several other complexity classes

the size of a revised knowledge base abstract in this paper we address a specific computational aspect of belief revision the size of the propositional formula obtained by means of the revision of a formula with a new one in particular we focus on the size of the smallest formula which is logically equivalent to the revised knowledge base the main result of this paper is that not all formalizations of belief revision are equal from this point of view for some of them we show that the revised knowledge base can be represented by a polynomial size formula we call these results compactability results on the other hand for other ones the revised knowledge base does not always admit a polynomial space representation unless the polynomial hierarchy collapses at a sufficiently low level non compactability results we also show that the time complexity of query answering for the revised knowledge base has definitely an impact on being able to represent the result of the revision compactly nevertheless formalisms with the same complexity may have different compactability properties we also study compactability properties for a weaker form of equivalence called query equivalence which allows to introduce new propositional symbols moreover we extend our analysis to the special case in which the new formula has constant size and to the case of sequences of revisions i e iterated belief revision a complete analysis along these four coordinates is shown

preprocessing of intractable problems some computationally hard problems e g deduction in logical knowledge bases are such that part of an instance is known well before the rest of it and remains the same for several subsequent instances of the problem in these cases it is useful to preprocess off line this known part so as to simplify the remaining on line problem in this paper we investigate such a technique in the context of intractable i e np hard problems recent results in the literature show that not all np hard problems behave in the same way for some of them preprocessing yields polynomial time on line simplified problems we call them compilable while for other ones their compilability implies some consequences that are considered unlikely our primary goal is to provide a sound methodology that can be used to either prove or disprove that a problem is compilable to this end we define new models of computation complexity classes and reductions we find complete problems for such classes completeness meaning they are the less likely to be compilable we also investigate preprocessing that does not yield polynomial time on line algorithms but generically decreases complexity this leads us to define hierarchies of compilability that are the analog of the polynomial hierarchy a detailed comparison of our framework to the idea of parameterized tractability shows the differences between the two approaches

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

algebraic properties for selector functions the nondeterministic advice complexity of the p selective sets is known to be exactly linear regarding the deterministic advice complexity of the p selective sets i e the amount of karp lipton advice needed for polynomial time machines to recognize them in general the best current upper bound is quadratic k ko j comput system sci 26 1983 pp 209 221 and the best current lower bound is linear l hemaspaandra and l torenvliet theoret comput sci 154 1996 pp 367 377 we prove that every associatively p selective set is commutatively associatively p selective using this we establish an algebraic sufficient condition for the p selective sets to have a linear upper bound which thus would match the existing lower bound on their deterministic advice complexity if all p selective sets are associatively p selective then the deterministic advice complexity of the p selective sets is linear the weakest previously known sufficient condition was p np we also establish related results for algebraic properties of and advice complexity of the nondeterministically selective sets