polynomial time 1 turing reductions from ph to p in this paper we investigate relative complexity between p and other classes of functions our particular interest is to compare p with ph and with pfh by using polynomial time reducibility and to demonstrate that a weaker notion of polynomial time reducibility is sufficiently powerful for reducing ph functions to p functions our main result is stated as follows every function in ph is polynomial time 1 turing reducible to some function in p that is ph pf p 1 some consequences of this result are as follows every function in pfh is polynomial time 1 turing reducible to some function in p if pf p 1 ph then ph collapses to a finite level furthermore if either p pfh or pfh p then ph collapses to a finite level we also give an affirmative answer to an open question posed by valiant 1979 and we show a generalized result about p rankability by hemachandra 1987

on resource bounded instance complexity we prove the instance complexity conjecture of ko orponen schoning and watanabe for all recursive tally sets and for all recursive sets which are np hard under honest reductions in particular it holds for all natural np hard problems on the other hand the conjecture is shown to be oracle dependent additional results concern the nonrecursiveness of the instance complexity measure and a comparison of time bounded c and cd complexity

threshold circuits of small majority depth we investigate the complexity of computations with constant depth threshold circuits such circuits are composed of gates that determine if the sum of their inputs is greater than a certain threshold when restricted to polynomial size these circuits compute exactly the functions in the class tc sp0 these circuits are usually studied by measuring their efficiency in terms of their total depth using this point of view the best division and iterated multiplication circuits have depth three and four respectively in this thesis we propose a different approach since threshold gates are much more powerful than and or gates we allow the explicit use of and or gates and consider the main measure of complexity to be the majority depth of the circuit i e the maximum number of threshold gates on any path in the circuit using this approach we obtain division and iterated multiplication circuits of total depth four and five but of majority depth two and three the technique used is called chinese remaindering we present this technique as a general tool for computing functions with integer values and use it to obtain depth four threshold circuits of majority depth two for other arithmetic problems such as the logarithm and power series approximation we also consider the iterated multiplication problem for integers modulo q and for finite fields the notion of majority depth naturally leads to a hierarchy of subclasses of tc sp0 we investigate this hierarchy and show that it is closely related to the usual depth hierarchy

approximable sets much structural work on np complete sets has exploited sat s d self reducibility in this paper we exploit the additional fact that sat is a d cylinder to show that np complete sets are p superterse unless p np in fact every set that is np hard under polynomial time no 1 tt reductions is p superterse unless p np in particular no p selective set is np hard under polynomial time no 1 tt reductions unless p np in addition no easily countable set is np hard under turing reductions unless p np self reducibility does not seem to suffice far our main result in a relativized world we construct a d self reducible set in np p that is polynomial time 2 tt reducible to a p selective set

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

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

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

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

representations of error terms in jensen s and some related inequalities with applications abstract we give integral representations of error terms in the classical jensen s inequality as well as in four related inequalities starting from these results it is possible to derive various lower and upper asymptotic expansions for solutions of the functional equation f x 1 f x g x in particular we present several results concerning the gamma function

exact analysis of dodgson elections lewis carroll s 1876 voting system is complete for parallel access to np in 1876 lewis carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters preferences becomes a condorcet winner a candidate who beats all other candidates in pairwise majority rule elections bartholdi tovey and trick provided a lower bound np hardness on the computational complexity of determining the election winner in carroll s system we provide a stronger lower bound and an upper bound that matches our lower bound in particular determining the winner in carroll s system is complete for parallel access to np that is it is complete for theta 2 p for which it becomes the most natural complete problem known it follows that determining the winner in carroll s elections is not np complete unless the polynomial hierarchy collapses

the capacity of the semideterministic relay channel corresp the capacity of the class of relay channels with sender x 1 a relay sender x 2 a relay receiver y 1 f x 1 x 2 and ultimate receiver y is proved to be c max min p x 1 x 2 i x 1 x 2 y h y 1 x 2 i x 1 y x 2 y 1

the computational complexity of maximization and integration on montre que ces deux problemes de la theorie de la complexite de calcul aussi bien que les autres resolvant la maximisation et l integration sont equivalents aux problemes de complexite de calcul discret naturel dont la plupart ont deja beaucoup ete etudies

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

on bounded queries and approximation this paper investigates the computational complexity of approximating several np optimization problems using the number of queries to an np oracle as a complexity measure the results show a tradeoff between the closeness of the approximation and the number of queries required for an approximation factor k n log log k n n queries to an np oracle can be used to approximate the maximum clique size of a graph within a factor of k n however this approximation cannot be achieved using fewer than log log k n n c queries to any oracle unless pe np where c is a constant that does not depend on k these results hold for approximation factors k n geq 2 that belong to a class of functions which includes any integer constant function log n log a n and n 1 a similar results are obtained for graph coloring set cover and other np optimization problems

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

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

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

optimal proof systems imply complete sets for promise classes a polynomial time computable function h whose range is a set l is called a proof system for l in this setting an h proof for x l is just a string w with h w x cook and reckhow defined this concept in 13 and in order to compare the relative strength of different proof systems for the set taut of tautologies in propositional logic they considered the notion of p simulation intuitively a proof system h p simulates h if any h proof w can be translated in polynomial time into an h proof w for h w we also consider the related notion of simulation between proof systems where it is only required that for any h proof w there exists an h proof w whose size is polynomially bounded in the size of w a proof system is called p optimal for a set l if it p simulates every other proof system for l the question whether p optimal or optimal proof systems for taut exist is an important one in the field in this paper we show a close connection between the existence of p optimal proof systems and the existence of complete problems for certain promise complexity classes like up np sparse rp or bpp for this we introduce the notion of a test set for a promise class c and prove that c has a many one complete set if and only if c has a test set t with a p optimal proof system if in addition the machines defining a promise class have a certain ability to guess proofs then the existence of a p optimal proof system for t can be replaced by the presumably weaker assumption that t has an optimal proof system strengthening a result from krajicek and pudlak 20 we also give sufficient conditions for the existence of optimal and p optimal proof systems

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

alternating and empty alternating auxiliary stack automata we consider variants of alternating auxiliary stack automata and characterize their computational power when the number of alternations is bounded by a constant or unlimited in this way we get new characterizations of np the polynomial hierarchy pspace and bounded query classes like co dp nl np 1 and 2p pnp o logn in a uniform framework

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

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

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

quantifying the amount of verboseness we study the fine structure of the classification of sets of natural numbers a according to the number of queries which are needed to compute the n fold characteristic function of a a complete characterization is obtained relating the question to finite combinatorics in order to obtain an explicit description we consider several interesting combinatorial problems

time bounded frequency computations 1 we obtain two new results concerning the inclusion problem of polynomial time frequency classes with equal numbers of errors 1 m m d p m 1 m d 1 pform 2d 2 m m d p m 1 m d 1 pform c d wherec d is large enough this disproves a conjecture of kinber 2 we give a transparent proof of a generalization of kinber s result that there exist arbitrarily complex problems admitting a polynomial time frequency computation several corollaries provide more insight into the structure of the hierarchy of polynomial time frequency classes 3 the relationships between polynomial time frequency classes and selectivity classes are studied

logarithmic convexity and inequalities for the gamma function abstract we propose a method based on logarithmic convexity for producing sharp bounds for the ratio x x as an application we present an inequality that sharpens and generalizes inequalities due to gautschi chu boyd lazarevic lupas and kershaw

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

np trees and carnap s modal logic in this paper we consider problems and complexity classes definable by interdependent queries to an oracle in np how the queries depend on each other is specified by a directed graph g we first study the class of problems where g is a general dag and show that this class coincides with d p 2 we then consider the class where g is a tree our main result states that this class is identical to p np o log n the class of problems solvable in polynomial time with a logarithmic number of queries to an oracle in np this result has interesting applications in the fields of modal logic and artificial intelligence in particular we show that the following problems are all p np o log n complete validity checking of formulas in carnap s modal logic checking whether a formula is almost surely valid over finite structures in modal logics k t and s4 a problem recently considered by halpern and kapron 1992 and checking whether a formula belongs to the stable set of beliefs generated by a propositional theory we generalize the case of dags to the case where g is a general possibly cyclic directed graph of np oracle queries and show that this class corresponds to p p 2 we show that such graphs are easily expressible in autoepistemic logic finally we generalize our complexity results to higher classes of the polynomial time hierarchy

the complexity of kemeny elections kemeny proposed a voting scheme which is distinguished by the fact that it is the unique voting scheme that is neutral consistent and condorcet bartholdi tovey and trick showed that determining the winner in kemeny s system is np hard we provide a stronger lower bound and an upper bound matching the lower bound namely we show that determining the winner in kemeny s system is complete for p np the class of sets solvable via parallel access to np

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

on the unique satisfiability problem unique sat is the problem of deciding whether a given boolean formula has exactly one satisfying truth assignment this problem is a typical moreover complete representative of a natural class of problems about unique solutions all these problems belong to the class dife l1 l2 l1 lz np studied by papadimitriou and yannakakis we consider the relationship between these two classes particularly whether unique sat is dife complete it is if np co np we construct an oracle relative to which unique sat is not complete for dif and another oracle relative to which unique sat is complete for dif e whereas np v co np

gurland s ratio for the gamma function we consider the ratio t x y x y 2 x y 2 and its properties related to convexity logarithmic convexity schur convexity and complete monotonicity several new bounds and asymptotic expansions for t are derived sharp bounds for the function x x 1 e x are presented as well as bounds for the trigamma function the results are applied to a problem related to the volume of the unit ball in r n and also to the problem of finding the inverse of the function x t 1 x 3 x which is of importance in applied statistics

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

bounds on capacity and minimum energy per bit for awgn relay channels upper and lower bounds on the capacity and minimum energy per bit for general additive white gaussian noise awgn and frequency division awgn fd awgn relay channel models are established first the max flow min cut bound and the generalized block markov coding scheme are used to derive upper and lower bounds on capacity these bounds are never tight for the general awgn model and are tight only under certain conditions for the fd awgn model two coding schemes that do not require the relay to decode any part of the message are then investigated first it is shown that the side information coding scheme can outperform the block markov coding scheme it is also shown that the achievable rate of the side information coding scheme can be improved via time sharing in the second scheme the relaying functions are restricted to be linear the problem is reduced to a single letter nonconvex optimization problem for the fd awgn model the paper also establishes a relationship between the minimum energy per bit and capacity of the awgn relay channel this relationship together with the lower and upper bounds on capacity are used to establish corresponding lower and upper bounds on the minimum energy per bit that do not differ by more than a factor of 1 45 for the fd awgn relay channel model and 1 7 for the general awgn model

propositional circumscription and extended closed world reasoning are p p 2 complete abstract circumscription and the closed world assumption with its variants are well known nonmonotonic techniques for reasoning with incomplete knowledge their complexity in the propositional case has been studied in detail for fragments of propositional logic one open problem is whether the deduction problem for arbitrary propositional theories under the extended closed world assumption or under circumscription is p 2 complete i e complete for a class of the second level of the polynomial hierarchy we answer this question by proving these problems p 2 complete and we show how this result applies to other variants of closed world reasoning

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

on the hardness of approximate reasoning abstract many ai problems when formalized reduce to evaluating the probability that a propositional expression is true in this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability we consider various methods used in approximate reasoning such as computing degree of belief and bayesian belief networks as well as reasoning techniques such as constraint satisfaction and knowledge compilation that use approximation to avoid computational difficulties and reduce them to model counting problems over a propositional domain we prove that counting satisfying assignments of propositional languages is intractable even for horn and monotone formulae and even when the size of clauses and number of occurrences of the variables are extremely limited this should be contrasted with the case of deductive reasoning where horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms what is even more surprising is that as we show even approximating the number of satisfying assignments i e approximating approximate reasoning is intractable for most of these restricted theories we also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given

krull s theory for the double gamma function abstract the barnes g function g x 1 2 satisfies the functional equation log g x 1 log g x log x we complement w krull s work in bemerkungen zur differenzengleichung g x 1 g x x math nachrichten 1 1948 365 376 with additional results that yield a different characterization of the function g new expansions and sharp bounds for g on x 0 in terms of gamma and digamma functions a new expansion for the gamma function and summation formulae with polygamma functions

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

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

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

polynomial size ω branching programs and their computational power abstract in the following new types of branching programs so called branching programs b 2 are introduced the complexity classes related to polynomial size branching programs will be completely classified in addition to identifying a new class p bp l poly between l poly and p poly new characterizations of such fundamental space complexity classes like nl poly co nl poly and p poly are obtained using these characterizations we relate the complexity of the mentioned classes to those of some extremely restricted problems resembling the graph accessibility problem

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

study of gaussian relay channels with correlated noises in this paper we consider the full duplex and half duplex gaussian relay channels where the noises at the relay and destination are arbitrarily correlated we first derive the capacity upper bound and the achievable rates with three existing schemes decode and forward df compress and forward cf and amplify and forward af we present two capacity results under specific noise correlation coefficients one being achieved by df and the other being achieved by direct link transmission or a special case of cf the channel for the former capacity result is equivalent to the traditional gaussian degraded relay channel and the latter corresponds to the gaussian reversely degraded relay channel for cf and af schemes we show that their achievable rates are strictly decreasing functions of the correlation coefficient when the correlation coefficient is negative moreover when the noise correlation coefficient is positive the cf achievable rate may also outperform the independent noise case if the noise correlation coefficient is within a certain range through numerical comparisons under different channel settings we observe that although df completely disregards the noise correlation while the other two can potentially exploit such extra information none of the three relay schemes always outperforms the others over different correlation coefficients moreover the exploitation of noise correlation by cf and af accrues more benefit when the source relay link is weak this paper also considers the optimal power allocation problem under the correlated noise channel setting with individual power constraints at the relay and the source it is shown that the relay should use all its available power to maximize the achievable rates under any correlation coefficient with a total power constraint across the source and the relay the achievable rates are proved to be concave functions over the power allocation factor for af and cf under full duplex mode where the closed form power allocation strategy is derived

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

analysis of the generalized df cf for gaussian relay channels decode or compress we consider a three node quasi static communication system with a full duplex relay the goal is to determine the relaying mode that enables rate efficient communication under given channel conditions to achieve this goal we consider a generalized scheme that subsumes the decode and forward df and compress and forward cf schemes as special cases the generalized scheme is considered when the source and relay signals are synthesized from commonly used gaussian codebooks which are shown to be capacity achieving in two asymptotic cases perfect relay destination link and broken source destination link studying the generalized df cf scheme it is shown that for two non asymptotic cases in which the signal to noise ratios snrs of the links satisfy certain conditions this scheme reduces to either df or cf for another set of non asymptotic snrs the generalized scheme is shown to yield strictly higher rates than both df and cf despite the complexity of the generalized scheme its rate advantage over df and cf is shown to be upper bounded by 0 5 bits per channel use this indicates that the practical benefit of the analysis of this scheme is to enable selecting the relaying mode that suits a given channel realization numerical results show that under rayleigh fading conditions this selection yields significant gains over fixed df and cf

observations on complete sets between linear time and polynomial time there is a single set that is complete for a variety of nondeterministic time complexity classes with respect to related versions of m reducibility this observation immediately leads to transfer results for determinism versus nondeterminism solutions also an upward transfer of collapses of certain oracle hierarchies built analogously to the polynomial time or the linear time hierarchies can be shown by means of uniformly constructed sets that are complete for related levels of all these hierarchies a similar result holds for difference hierarchies over nondeterministic complexity classes finally we give an oracle set relative to which the nondeterministic classes coincide with the deterministic ones for several sets of time bounds and we prove that the strictness of the tape number hierarchy for deterministic linear time turing machines does not relativize

a decisive characterization of bpp the complexity class bpp defined by gill contains problems that can be solved in polynomial time with bounded error probability a new and simple characterization of bpp is given it is shown that a language l is in bpp iff x l y zp x y z x l y z p x y z for a polynomial time predicate p and for y z poly x the formula yp y with the random quantifier means that the probability pr y p y for a fixed this characterization allows a simple proof that bpp zpp np which strengthens the result of lautemann inform process lett 17 1983 215 217 sipser in proceedings 15th annu acm sympos theory of comput 1983 pp 330 335 that bpp 2 p 2 p several other results about probabilistic classes can be proved using similar techniques e g np r zpp np and 2 p bpp 2 p

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

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

a second step towards complexity theoretic analogs of rice s theorem abstract rice s theorem states that every nontrivial language property of the recursively enumerable sets is undecidable borchert and stephan 1997 initiated the search for complexity theoretic analogs of rice s theorem in particular they proved that every nontrivial counting property of circuits is up hard and that a number of closely related problems are spp hard the present paper studies whether their up hardness result itself can be improved to spp hardness we show that their up hardness result cannot be strengthened to spp hardness unless unlikely complexity class containments hold nonetheless we prove that every p constructibly bi infinite counting property of circuits is spp hard we also raise their general lower bound from unambiguous nondeterminism to constant ambiguity nondeterminism

bi immunity results for cheatable sets abstract an oracle a is k cheatable if there is a polynomial time algorithm to determine the answers to 2k parallel queries to a from the answers to only k queries to some other oracle b it is known that 1 cheatable sets cannot be bi immune for p in contrast we construct 2 cheatable sets that are bi immune for arbitrary time complexity classes in addition for each k we construct a set that is k 1 cheatable but not k cheatable we show that this separation does not hold with bi immunity we show that if a recursive set a is bi immune for p then there exists a nontrivial 1 cheatable set that is polynomial time m reducible to a consequently if np contains a set that is bi immune for p then np contains a set that is not polynomial time turing equivalent to a self reducible set

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

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

a simple and efficient algorithm to compute tail probabilities from transforms we present an algorithm to approximately compute the tail probability that a random variable exceeds a specified number given only an expression for its transform we also show that the problem is p hard more difficult than np hard suggesting that no efficient procedure can solve it exactly our method consists essentially of summing a power series and thus is easy to perform and requires little memory furthermore its computational effort is nearly linear in the reciprocal of a prespecified worst case error bound

how many queries are needed to learn we investigate the query complexity of exact learning in the membership and proper equivalence query model we give a complete characterization of concept classes that are learnable with a polynomial number of polynomial sized queries in this model we give applications of this characterization including results on learning a natural subclass of dnf formulas and on learning with membership queries alone query complexity has previously been used to prove lower bounds on the time complexity of exact learning we show a new relationship between query complexity and time complexity in exact learning if any honest class is exactly and properly learnable with polynomial query complexity but not learnable in polynomial time then p np in particular we show that an honest class is exactly polynomial query learnable if and only if it is learnable using an oracle for p 4

measure on p we give a notion of measure at p in a paradigm that differs somewhat from the standard theory our new notion overcomes some limitations of earlier formulations specifically concerning closure of null sets under union first we analyze formally some of the difficulties in defining measure at p we then present the new definitions and determine the basic properties of the notion including the density and immunity characteristics of a random language we argue that these results are parallel to previous measure results at exponential time

lower bounds for the low hierarchy the low hierarchy in np sc 83 and the extended low hierarchy bbs 86 have been useful in characterizing the complexity of certain interesting classes of sets however until now there has been no way of judging whether or not a given lowness result is the best possible

on boolean lowness and boolean highness the concepts of lowness and highness originate from recursion theory and were introduced into the complexity theory by schoning sch85 informally a set is low high resp for a relativizable class k of languages if it does not add adds maximal resp power to k when used as an oracle in this paper we introduce the notions of boolean lowness and boolean highness informally a set is boolean low boolean high resp for a class k of languages if it does not add adds maximal resp power to k when combined with k by boolean operations we prove properties of boolean lowness and boolean highness which show a lot of similarities with the notions of lowness and highness using kadin s technique of hard strings see kad88 wag87 ck96 bco93 we show that the sets which are boolean low for the classes of the boolean hierarchy are low for the boolean closure of 2p furthermore we prove a result on boolean lowness which has as a corollary the best known result see bco93 in fact even a bit better on the connection of the collapses of the boolean hierarchy and the polynomial time hierarchy if bh np k then ph 2p k 1 np k

kolmogorov characterizations of complexity classes abstract this paper completely characterizes the k p levels of the polynomial hierarchy in terms of kolmogorov complexity from the characterization it follows that the k p and k p levels of the polynomial hierarchy are equal if and only if every k p language is accepted by some k p machine whose pronouncements query answers are kolmogorov simple analogous results are obtained for the exponential hierarchy

logspace and logtime leaf languages the computation tree of a nondeterministic machinemwith inputxgives rise to aleaf stringformed by concatenating the outcomes of all the computations in the tree in lexicographical order we may characterize problems by considering for a particular leaf language y the set of allxfor which the leaf string ofmis contained iny in this way in the context of polynomial time computation leaf languages were shown to capture many complexity classes in this paper we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation we show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages capturing also many classes withinp in contrast logtime leaf languages basically behave like logtime reducibilities both cases are more subtle to handle than the polynomial time case we also raise the issue of balanced versus nonbalanced computation trees underlying the leaf language we indicate that it is a nontrivial problem to obtain information about the leaf string of a nonbalanced computation tree and present conditions under which it does not matter whether the computation tree is balanced or not

lower bounds and the hardness of counting properties rice s theorem states that all nontrivial language properties of recursively enumerable sets are undecidable borchert and stephan math logic quart 46 4 2000 489 504 started the search for complexity theoretic analogs of rice s theorem and proved that every nontrivial counting property of boolean circuits is up hard hemaspaandra and rothe theoret comput sci 244 1 2 2000 205 217 improved the up hardness lower bound to upo 1 hardness the present paper raises the lower bound for nontrivial counting properties from upo 1 hardness to fewp hardness i e from constant ambiguity nondeterminism to polynomial ambiguity nondeterminism furthermore we prove that no relativizable technique can raise this lower bound to fewp 1 ttp hardness we also prove a rice style theorem for np namely that every nontrivial language property of np sets is np hard and for a broad class of leaf language classes we prove a sufficient condition for the natural analog of rice s theorem to hold

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 complexity of counting in the polynomial hierarchy we investigate the time complexity of the following counting problem for a given set of words a a length n and a number k check whether a contains exactly k words of length n in symbol if ca n k oracle alternating turing machines of a constant number of alternations const atms are used to solve the problem a machine queries its oracle to check in a single step if x a for any word x we prove that no const atm can solve in polynomial time even the following restricted counting problem for a fixed l u n n l u and an integer p 2 for a given set a such that l ca u and numbers n and k check whether ca n k mod p provided that between l and u there is a function f with superpolynomial min f n 2n f n in other words it shows that counting requires superpolynomial amount of time unless a or is sparse or an unbounded number of alternations can be made then we consider an approximate counting problem and prove a new lower bound on the complexity of const atms that solve that problem in polynomial time