metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

sub riemannian heat kernels and mean curvature flow of graphs abstract we introduce a sub riemannian analogue of the bence merriman osher algorithm merriman et al 1992 42 and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub riemannian carnot groups the proof follows the nonlinear semi group theory approach originally introduced by l c evans 1993 27 in the euclidean setting and is based on new results on the relation between sub riemannian heat flows of characteristic functions of subgraphs and the horizontal mean curvature of the corresponding graphs

capturing complexity classes by fragments of second order logic abstract we investigate the expressive power of certain fragments of second order logic on finite structures the fragments are second order horn logic second order krom logic as well as a symmetric and a deterministic version of the latter it is shown that all these logics collapse to their existential fragments in the presence of successor relation they provide characterizations of polynomial time deterministic and nondeterministic logspace and of the complement of symmetric logspace without successor relation these logics still can express certain problems that are complete in the corresponding complexity classes but on the other hand they are strictly weaker than previously known logics for these classes and fail to express some very simple properties

hereditarily finite sets data bases and polynomial time computability abstract the first informal result and the aim of this paper which is the extended version of sazonov 1988 is a step towards a somewhat more practically oriented version of set theoretic programming language sazonov 1985 1987 which may be considered as a resource bounded language for specifying data bases and corresponding queries in fact instead of the ordinary universe hf u of hereditarily finite sets over any given class u of urelements we consider following the ideas of red ko and basarab 1987 a more general universe hfa u of hereditarily finite sets of named elements i e sets of data qualified by some attributes the second result is more mathematical it generalizes a theorem of sazonov 1985 1987 to the case of hfa u makes it more precise and implies that language is sufficiently complete theorem cf 42 43 if language involves a linear order on u then its terms define exactly all operations hfa u hfa u which 1 preserve supports of their arguments 2 transform isomorphic arguments to isomorphic ones respecting their supports and 3 are computable in polynomial time under graphical representations of sets in hfa u

hessian measures on the heisenberg group abstract in this paper we study the properties of k convex functions on the heisenberg group h n for 1 k 2 n and prove the weak continuity of k hessian measures with respect to local uniform convergence in the heisenberg setting our approach through monotonicity formulae makes use of previous research in the corresponding euclidean case of trudinger and wang the case k 2 n provides an analogue of the monge ampere measure of aleksandrov for h n we also answer a conjecture of garofalo and tournier on monotonicity for the cases n 2

asymptotic conditional probabilities the unary case motivated by problems that arise in computing degrees of belief we consider the problem of computing asymptotic conditional probabilities for first order sentences given first order sentences phi and theta we consider the structures with domain 1 ldots n that satisfy theta and compute the fraction of them in which phi is true we then consider what happens to this fraction as n gets large this extends the work on 0 1 laws that considers the limiting probability of first order sentences by considering asymptotic conditional probabilities as shown by liogon kii math notes acad ussr 6 1969 pp 856 861 and by grove halpern and koller res rep rj 9564 ibm almaden research center san jose ca 1993 in the general case asymptotic conditional probabilities do not always exist and most questions relating to this issue are highly undecidable these results however all depend on the assumption that theta can use a nonunary predicate symbol liogonkii math notes acad ussr 6 1969 pp 856 861 shows that if we condition on formulas theta involving unary predicate symbols only but no equality or constant symbols then the asymptotic conditional probability does exist and can be effectively computed this is the case even if we place no corresponding restrictions on phi we extend this result here to the case where theta involves equality and constants we show that the complexity of computing the limit depends on various factors such as the depth of quantifier nesting or whether the vocabulary is finite or infinite we completely characterize the complexity of the problem in the different cases and show related results for the associated approximation problem

rotation and entropy for a given map f x x and an observable o x llkd rotation vectors are the limits of ergodic averages of w we study which part of the topological entropy of f is associated to a given rotation vector and which part is associated with many rotation vectors according to this distinction we introduce directional and lost entropies we discuss their properties in the general case and analyze them more closely for subshifts of finite type and circle maps

ground states are generically a periodic orbit we prove that for an expanding transformation the maximizing measures of a generic lipschitz function are supported on a single periodic orbit

counting modulo quantifiers on finite structures we give a combinatorial method for proving elementary equivalence in first order logic fo with counting modulo n quantifiers dn inexpressibility results for fo dn with built in linear order are also considered for instance the class of linear orders of length divisible by n 1 cannot be expressed in fo dn using this result we prove that comparing cardinalities or connectivity of ordered graphs are not definable in fo dn we also show that the height of complete n ary trees cannot be expressed in fo dn with linear order interpreting the predicate y nx as a complete n ary tree we show that the predicate y px cannot be defined in fo dn with linear order whenever p has a prime factor that does not divide n this solves the problem raised by niwi ski and stolboushkin lics 93 we also discuss a connection between our results and the well known open problem in circuit complexity theory whether acc nc1

describing parameterized complexity classes we describe parameterized complexity classes by means of classical complexity theory and descriptive complexity theory for every classical complexity class we introduce a parameterized analogue in a natural way in particular the analogue of polynomial time is the class of all fixed parameter tractable problems we develop a basic complexity theory for the parameterized analogues of classical complexity classes and give among other things complete problems and logical descriptions we then show that most of the well known intractable parameterized complexity classes are not analogues of classical classes nevertheless for all these classes we can provide natural logical descriptions

bounded linear logic a modular approach to polynomial time computability typing is a way of describing the interactive behavior of algorithms usual typing systems are mainly concerned with input output specifications e g given terms f a b and a a the computation of f a by normalization yields a result of type b however one can dream of more refined typings that would not only ensure ethereal termination but would for instance yield feasible resource bounds it seems that time complexity does not lend itself naturally to modular manipulation we seek something more primitive

nondeterministic space is closed under complementation in this paper we show that nondeterministic space s n is closed under complementation for s n greater than or equal to log n it immediately follows that the context sensitive languages are closed under complementation thus settling a question raised by kuroda inform and control 7 1964 pp 207 233

a logical approach to asymptotic combinatorics i first order properties we shall present a general framework for dealing with an extensive set of problems from asymptotic combinatorics this framework provides methods for determining the probability that a large finite structure ran domly chosen from a given class will have a given property our main concern is the asymptotic probability the limiting value as the size of the structure increases for example a common problem in elementary probability texts is to show that the asymptotic probabity that a per mutation will have no fixed point is l e we shall say nothing about the closely related problem of determining rates of convergence although the methods presented here may extend to such problems to develop a general approach we must fix a language for specifying properties of structures thus our approach is logical logic is the branch of mathematics that deals with problems of language in this paper we con sider properties expressible in the language of first order logic and speak of probabilities of first order sentences rather than properties in the sequel to this paper we consider properties expressible in the more general language of monadic second order logic clearly we must restrict the classes of structures we consider in order for questions about asymptotic probabilities to be meaningful and significant therefore we choose to consider only classes closed under disjoint unions and components see sect 1 for definitions this condition is general enough to include the classes of graphs permutations unary functions and many other important examples the main theorems of the paper are about classes satisfying this condition theorems 5 7 and 5 8 state that the existence of asymptotic probabilities for a special set of sentences called component bounded sentences defined in sect 5 is equivalent to a con dition on the growth of such a class theorem 5 9 gives necessary and suf ficient conditions for every first order sentence to have an asymptotic probability of 0 or 1 in nonfast growing classes in section 8 we present a long list of examples from the literature

a restricted second order logic for finite structures we introduce a restricted version of second order logic so in which the second order quantifiers range over relations that are closed under the equivalence relation k of k variable equivalence for some k this restricted second order logic is an effective fragment of the infinitary logic l which differs from other such fragments in that it is not based on a fixpoint logic we explore the relationship of so with fixpoint logics showing that its inclusion relations with these logics are equivalent to problems in complexity theory we also look at the expressibility of np complete problems in this logic

on the relative expressiveness of description logics and predicate logics abstract it is natural to view concept and role definitions in description logics as expressing monadic and dyadic predicates in predicate calculus we show that the descriptions built using the constructors usually considered in the dl literature are characterized exactly as the predicates definable by formulas in tl3 the subset of first order predicate calculus with monadic and dyadic predicates which allows only three variable symbols in order to handle number bounds we allow numeric quantifiers and for transitive closure of roles we use infinitary disjunction using previous results in the literature concerning languages with limited numbers of variables we get as corollaries the existence of formulas of fopc which cannot be expressed as descriptions we also show that by omitting role composition descriptions express exactly the formulas in tl2 which is known to be decidable

locality from circuit lower bounds we study the locality of an extension of first order logic that captures graph queries computable in ac 0 i e by families of polynomial size constant depth circuits the extension considers first order formulas over relational structures which may use arbitrary numerical predicates in such a way that their truth value is independent of the particular interpretation of the numerical predicates we refer to such formulas as arb invariant first order we consider the two standard notions of locality gaifman and hanf locality our main result gives a gaifman locality theorem an arb invariant first order formula cannot distinguish between two tuples that have the same neighborhood up to distance log n c where n represents the number of elements in the structure and c is a constant depending on the formula when restricting attention to string structures we achieve the same quantitative strength for hanf locality in both cases we show that our bounds are tight we also present an applicati

inductive definitions over finite structures inductive definitions have played a central role in the foundations of mathematics for over a century they were used in the 1970s as the backbone of one major generalization of recursive function theory moschovakis 1974 aczel 1977 in recent years the relevance of inductive definitions in particular over finite structures to database theory to descriptive computational complexity and to logics of programs has been recognized a seminal paper on inductive definitions in database theory is chandra and hare1 1982 where they define a hierarchy of queries over finite structures within which minor steps successor ordinals correspond to first order quantifier alternations and major steps limit ordinals correspond to uses of lixpoints they left open the question of whether the hierarchy remains strict above the first major step level w this problem was answered in the negative by immerman 1986 since the collection of first order lixpoint queries over finite structures is closed under composition and first order operations other than negation moschovakis 1974 the main component of immerman s solution was

lipschitz continuity aleksandrov theorem and characterizations for h convex functions in the geometries of stratified groups we show that h convex functions locally bounded from above are locally lipschitz continuous and that the class of v convex functions exactly corresponds to the class of upper semicontinuous h convex functions as a consequence v convex functions are locally lipschitz continuous in every stratified group in the class of step 2 groups we characterize locally lipschitz h convex functions as measures whose distributional horizontal hessian is positive semidefinite in euclidean space the same results were obtained by dudley and reshetnyak we prove that a continuous h convex function is a e twice differentiable whenever its second order horizontal derivatives are radon measures

sharp comparison and maximum principles via horizontal normal mapping in the heisenberg group in this paper we solve a problem raised by gutierrez and montanari about comparison principles for h convex functions on subdomains of heisenberg groups our approach is based on the notion of the sub riemannian horizontal normal mapping and uses degree theory for set valued maps the statement of the comparison principle combined with a harnack inequality is applied to prove the aleksandrov type maximum principle describing the correct boundary behavior of continuous h convex functions vanishing at the boundary of horizontally bounded subdomains of heisenberg groups this result answers a question by garofalo and tournier the sharpness of our results are illustrated by examples

the expressiveness of a family of finite set languages abstract in this paper we characterize exactly the complexity of a set based database language called srl which presents a unified framework for queries and updates by imposing simple syntactic restrictions on it we are able to express exactly the classes p and logspace we also discuss the role of ordering in database query languages and show that the hom operator of machiavelli language in ohori et al 1989 does not capture all the order independent properties

fixpoint logics relational machines and computational complexity we establish a general connection between fixpoint logic and complexity on one side we have fixpoint logic parameterized by the choices of 1st order operators inflationary or noninflationary and iteration constructs deterministic nondeterministic or alternating on the other side we have the complexity classes between p and exptime our parameterized fixpoint logics capture the complexity classes p np pspace and exptime but equally is achieved only over ordered structures there is however an inherent mismatch between complexity and logic while computational devices work on encodings of problems logic is applied directly to the underlying mathematical structures to overcome this mismatch we use a theory of relational complexity which bridges the gap between standard complexity and fixpoint logic on one hand we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complexity classes on the other hand the expressive power of fixpoint logic can be precisely characterized in terms of relational complexity classes this tight three way relationship among fixpoint logics relational complexity and standard complexity yields in a uniform way logical analogs to all containments among the complexity classes p np pspace and exptime the logical formulation shows that some of the most tantalizing questions in complexity theory boil down to a single question the relative power of inflationary vs noninflationary 1st order operators

complexity of the first order theory of almost all finite structures a first order sentence of a relational type l is true almost everywhere if the proportion of its models among the structures of type l and cardinality m tends to 1 when m tends to it is shown that th l the set of sentences of type l true almost everywhere is complete in pspace further various upper and lower bounds of the complexity of this theory are obtained for example if the arity of the relation symbols of l is d 2 and if pr th l is the set of prenex sentences of th l then pr th l dspace n log n d and pr th l ntime o n log n d if r is a binary relation symbol and l r th l is the theory of almost all graphs then pr th l nspace o n log n these results are optimal modulo open problems in complexity such as ntime t dspace t and nspace s dspace s2

weak differentiability of bv functions on stratified groups we prove the almost everywhere approximate differentiability of functions with bounded variation in stratified groups and we study their approximate discontinuity set we introduce functions of bounded higher order variation and obtain a weak version of alexandroff differentiability theorem in this context we present a nontrivial class of functions with second order bounded variation arising from inf convolution formula of a suitable cost function

infinitary logic and inductive definability over finite structures the extensions of first order logic with a least fixed point operator fo lfp and with a partial fixed point operator fo pfp are known to capture the complexity classes p and pspace respectively in the presence of an ordering relation over finite structures recently abiteboul and vianu in proceedings of the 23rd acm symposium on the theory of computing 1991 investigated the relationship of these two logics in the absence of an ordering using a machine model of generic computation in particular they showed that the two languages have equivalent expressive power if and only if p pspace these languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables l see for instance kolaitis and vardi in proceedings of the 5th ieee symposium on logic in computer science pp 156 167 1990 we investigate this logic on finite structures and provide a normal form for it we also present a treatment of abiteboul and vianu s results from this point of view in particular we show that we can write a formula of fo lfp that defines an ordering of the lk types uniformly over all finite structures one consequence of this is a generalization of the equivalence of fo lfp and p from ordered structures to classes of structures where every element is definable we also settle a conjecture mentioned by abiteboul and vianu by showing that fo lfp is properly contained in the polynomial time computable fragment of l raising the question of whether the latter fragment is a recursively enumerable class

rotation entropy and equilibrium states for a dynamical system x t and function f x rd we consider the corresponding generalised rotation set this is the convex subset of rd consisting of all integrals of f with respect to t invariant probability measures we study the entropy h e of rotation vectors e and relate this to the directional entropy lhl l of geller misiurewicz for x t a mixing subshift of finite type and f of summable variation we prove that if the rotation set is strictly convex then the functions h and h are in fact one and the same for those rotation sets which are not strictly convex we prove that 1h e and h e can differ only at non exposed boundary points e

existential second order logic over strings existential second order logic eso and monadic second order logic mso have attracted much interest in logic and computer science eso is a much expressive logic over successor structures than mso however little was known about the relationship between msoand syntatic fragments of eso we shed light on this issue by completely characterizing this relationship for the prefix classes of eso over strings i e finite successor structures moreover we determine the complexity of model checking over strings for all eso prefix classes let eso q denote the prefix class containing all sentences of the shape r q 4 where r is a list of predicate variables q is a first order predicate qualifier from the prefix set q and 4 is quantifier free we show that eso and eso are the maximal standard eso prefix classes contained in mso thus expressing only regular languages we further prove the following dichotomy theorem an eso prefix class either expresses only regular languages and is thus in mso or it expresses some np complete languages we also give a precise characterization of those eso prefix classes that are equivalent to mso over strings and of the eso prefix classes which are closed under complementation on strings

an analysis of fixed point queries on binary trees lindell s an analysis of fixed point queries on binary trees theoretical computer science 85 1991 75 95 the presence of ordering appears to play an essential role in the logical expressibility of polynomialtime queries on finite structures by examining the expressibility and complexity of inductive queries on the class of complete unordered binary trees we are able to show that the ability to calculate cardinality is strictly less powerful than the assumption of order

d languages for sets and logspace computable graph transformers abstract we discuss several versions of a set theoretic language as a reasonable prototype for nested database query language where database states and queries are considered respectively as hereditarily finite sets and set theoretic operations in a previous work such a language exactly corresponding to ptime computability was introduced it is supposed that hf sets are naturally presented by vertices of acyclic graphs here we consider a number of languages for sub ptime computable set operations via corresponding graph transformers two such languages lead to a notion of nlogspace and respectively dlogspace computable queries over hf which appear the most natural at our present knowledge among others considered here unlike the flat relational databases the problem of finding sufficiently good corresponding approach for hf proves to be more intricate and furthermore gives rise to some interesting questions in finite model theory cf section 13

from almost optimal algorithms to logics for complexity classes via listings and a halting problem let c denote one of the complexity classes polynomial time logspace or nondeterministic logspace we introduce a logic l c inv and show generalizations and variants of the equivalence l c inv captures c if and only if there is an almost c optimal algorithm in c for the set taut of tautologies of propositional logic these statements are also equivalent to the existence of a listing of subsets in c of taut by corresponding turing machines and equivalent to the fact that a certain parameterized halting problem is in the parameterized complexity class xcuni

equivalence and normal forms for the restricted and bounded fixpoint in the nested algebra the nested model is an extension of the traditional flat relational model in which relations can also have relation valued entries its default query language the nested algebra is rather weak unfortunately since it is only a conservative extension of the traditional flat relational algebra and thus can express only a small fraction of the polynomial time queries therefore it was proposed to extend the nested algebra with a fixpoint construct but the resulting language turned out to be too powerful many inherently exponential queries could also be expressed two polynomial time restrictions of the fixpoint closure of the nested algebra were proposed the restricted fixpoint closure by gyssens and van gucht and the bounded fixpoint closure by suciu here we prove two results first we show that both restrictions are equivalent in expressive power the proof technique relies on known encodings of nested relations into flat ones and on a novel technique called type substitution by which we reduce the equivalence of the two restrictions to its obvious counterpart in the flat relational model second we prove that both the bounded fixpoint queries and the restricted fixpoint queries admit normal forms in which the fixpoint occurs exactly once the proof technique relies on a novel encoding method of nested relations into flat ones

the computational complexity of asymptotic problems 1 partial orders the class of partial orders is shown to have ol laws for first order logic and for inductive fixed point logic a logic which properly contains first order logic this means that for every sentence in one of these logics the proportion of labeled or unlabeled partial orders of size n satisfying the sentence has a limit of either 0 or 1 as n goes to co this limit called the asymptotic probability of the sentence is the same for labeled and unlabeled structures the computational complexity of the set of sentences with asymptotic probability 1 is determined for first order logic it is pspace complete for inductive fixed point logic it is exptime complete

random graphs and the parity quantifier the classical zero one law for first order logic on random graphs says that for every first order property p in the theory of graphs and every p 0 1 the probability that the random graph g n p satisfies p approaches either 0 or 1 as n approaches infinity it is well known that this law fails to hold for any formalism that can express the parity quantifier for certain properties the probability that g n p satisfies the property need not converge and for others the limit may be strictly between 0 and 1 in this work we capture the limiting behavior of properties definable in first order logic augmented with the parity quantifier foltr over g n p thus eluding the above hurdles specifically we establish the following modular convergence law for every foltr sentence p there are two explicitly computable rational numbers a0 a1 such that for i 0 1 as n approaches infinity the probability that the random graph g 2npi p satisfies p approaches ai our results also extend appropriately to fo equipped with modq quantifiers for prime q in the process of deriving this theorem we explore a new question that may be of interest in its own right specifically we study the joint distribution of the subgraph statistics modulo 2 of g n p namely the number of copies mod 2 of a fixed number of graphs f1 fe of bounded size in g n p we first show that every foltr property p is almost surely determined by subgraph statistics modulo 2 of the above type next we show that the limiting joint distribution of the subgraph statistics modulo 2 depends only on n mod 2 and we determine this limiting distribution completely interestingly both these steps are based on a common technique using multivariate polynomials over finite fields and in particular on a new generalization of the gowers norm the first step is analogous to the razborov smolensky method for lower bounds for ac0 with parity gates yet stronger in certain ways for instance it allows us to obtain examples of simple graph properties that are exponentially uncorrelated with every foltr sentence which is something that is not known for ac0ltr

the quaternionic monge ampere operator and plurisubharmonic functions on the heisenberg group many fundamental results of pluripotential theory on the quaternionic space mathbb h n are extended to the heisenberg group we introduce notions of a plurisubharmonic function the quaternionic monge ampere operator differential operators d 0 and d 1 and a closed positive current on the heisenberg group the quaternionic monge ampere operator is the coefficient of d 0d 1u n we establish the chern levine nirenberg type estimate the existence of quaternionic monge ampere measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic monge ampere operator unlike the tangential cauchy riemann operator overline partial b on the heisenberg group which behaves badly as partial b overline partial b ne overline partial b partial b the quaternionic counterpart d 0 and d 1 satisfy d 0d 1 d 1d 0 this is the main reason that we have a good theory for the quaternionic monge ampere operator than partial b overline partial b n

on the complexity of single rule datalog queries datalog programs containing a unique rule and possibly some facts are known as single rule programs or sirups we study the complexity of evaluating sirups over variable and fixed databases respectively as well as the descriptive complexity of sirups i e their expressive power in all cases it turns out that even very restricted classes of sirups have the same complexity and essentially the same expressive power as general datalog programs in particular the evaluation of single clause programs is exptime complete combined complexity and if restricted to linear recursive rules pspace complete moreover sirups with one recursive rule and one fact capture ptime on ordered structures if a certain data representation is assumed and certain predefined relations are provided we also prove that the datalog clause implication problem i e deciding whether a datalog clause implies another one is exptime complete our main technical tool is a product construction which maps a datalog programs to an essentially equivalent sirup

regularity of mean curvature flow of graphs on lie groups free up to step 2 abstract we consider smooth solutions of the mean curvature flow of graphs over bounded domains in a lie group free up to step two and not necessarily nilpotent endowed with a one parameter family of riemannian metrics collapsing to a subriemannian metric 0 as 0 we establish c k estimates for this flow that are uniform as 0 and as a consequence prove long time existence for the subriemannian mean curvature flow of the graph our proof extend to the setting of every step two carnot group not necessarily free and can be adapted following our previous work in capogna et al 2013 to the total variation flow

the kolmogorov expression complexity of logics the purpose of the paper is to propose a completely new notion of complexity of logics in finite model theory it is the kolmogorov variant of the vardi sexpression complexity we define it by considering the value of the kolmogorov complexityc la of the infinite stringla of all truth values of sentences oflin a the higher is this value the more expressive is the logiclin a if d is a class of finite models then the value ofc la over all a d is a measure of expressive power oflin d unboundedness ofc la c l a for a d implies nonexistence of a recursive interpretation oflinl a version of this statement with complexities modulo oracles implies the nonexistence of any interpretation oflinl thus the valuesc la modulo oracles constitute an invariant of the expressive power of logics over finite models depending on their real absolute expressive power and not on the syntax we investigate our notion for fragments of the infinitary logic l least fixed point logic lfp and partial fixed point logic pfp we prove a precise characterization of 0 1 laws for these logics in terms of a certain boundedness condition placed onc la we get an extension of the notion of a 0 1 law by imposing an upper bound on the value ofc la growing not too fast with cardinality of a which still implies inexpressibility results similar to those implied by 0 1 laws we also discuss classes d in whichc pfpka is very high it appears that then pfp or its simple extension can define all the pspace subsets of d

circumscribing datalog expressive power and complexity abstract in this paper we study a generalization of datalog the language of function free definite clauses it is known that standard datalog semantics i e least herbrand model semantics can be obtained by regarding programs as theories to be circumscribed with all predicates to be minimized the extension proposed here called datalog circ consists in considering the general form of circumscription where some predicates are minimized some predicates are fixed and some vary we study the complexity and the expressive power of the language thus obtained we show that this language and actually its non recursive fragment is capable of expressing all the queries in db co np and as such is much more powerful than standard datalog whose expressive power is limited to a strict subset of ptime queries both data and combined complexities of answering datalog circ queries are studied data complexity is proved to be co np complete combined complexity is shown to be in general hard for co ne and complete for co ne in the case of herbrand bases containing k distinct constant symbols where k is bounded

a comparison between algebraic query languages for flat and nested databases abstract recently much attention has been paid to query languages for nested relations in the present paper we consider the nested algebra and the powerset algebra and compare them both mutually as well as to the traditional flat algebra we show that either nest or difference can be removed as a primitive operator in the powerset algebra while the redundancy of the nest operator might have been expected the same cannot be said of the difference basically this result shows that the presence of one nonmonotonic operator suffices in the powerset algebra as an interesting consequence of this result the nested algebra without the difference remains complete in the sense of bancilhon and paredaens finally we show there are both similarities and fundamental differences between the expressiveness of query languages for nested relations and that of their counterparts for flat relations

a zero one law for logic with a fixed point operator the logic obtained by adding the least fixed point operator to first order logic was proposed as a query language by aho and ullman in proc 6th acm sympos on principles of programming languages 1979 pp 110 120 and has been studied particularly in connection with finite models by numerous authors we extend to this logic and to the logic containing the more powerful iterative fixed point operator the zero one law proved for first order logic in glebskii kogan liogonki and talanov 1969 kibernetika 2 31 42 fagin 1976 j symbolic logic 41 50 58 for any sentence of the extend logic the proportion of models of among all structures with universe 1 2 n approaches 0 or 1 as n tends to infinity we also show that the problem of deciding for any whether this proportion approaches 1 is complete for exponential time if we consider only s with a fixed finite vocabulary or vocabularies of bounded arity and complete for double exponential time if is unrestricted in addition we establish some related results

linear time and the power of one first order universal quantifier the aim of this paper is to give two new logical characterizations of nlin nondeterministic linear time improving significantly a previous characterization of e grandjean 1994 siam j comput 23 573 597 and f olive 1998 comput complexity 7 54 97 it is known that nlin coincides with the class of problems definable by formulas of the prenex form f1 fk where the fi are unary function symbols and is first order prenex with only one universal quantifier we show that the characterization remains true in the two following cases a unary functions are replaced by a single binary relation whose outdegree is bounded by some fixed constant h b when only ordered structures are considered unary functions are restricted to be bijective

lower bounds for invariant queries in logics with counting we study the expressive power of counting logics in the presence of auxiliary relations such as orders and preorders the simplest such logic is the first order logic with counting this logic captures the complexity class tc0 over ordered structures we also consider first order logic with arbitrary unary quantifiers and with infinitary extensions we start by giving a simple direct proof that first order logic with counting in the presence of pre orders that are almost everywhere linear orders cannot express the transitive closure of a binary relation the proof is based on locality of formulae we then show that the technique cannot be extended to linear orders we further show that this result does not say anything about the power of invariant queries in first order logic with counting vs the class tc0 in the presence of these preorders in the second part of the paper we prove a separation result showing that for all the counting logics above a linear order is more powerful than a preorder that is a linear order almost everywhere in fact we prove that the expressive power of invariant queries in the presence of such preorders can be characterized by a property normally associated with first order definability over unordered structures we do this by using locality techniques from finite model theory however as some standard notions of locality fail in this setting we have to modify them to prove the main result

on winning strategies in ehrenfeucht fraisse games we present a powerful and versatile new sufficient condition for the second player the duplicator to have a winning strategy in an ehrenfeucht fra isst game on graphs we accomplish two things with this technique first we give a simpler and much easier to understand proof of ajtai and fagin s result that reachability in directed finite graphs is not in monadic np monadic np otherwise known as monadic xi corresponds to existential second order logic with the restriction that the second order quantifiers range only over sets and not over relations of higher arity such as binary relations second we show that this result holds in the presence of a larger class of built in relations than was known before

hypothetical datalog complexity and expressibility abstract we present an extension of horn clause logic which can hypothetically add and delete tuples from a database such logics have been discussed in the literature but their complexities and expressibilities have remained an open question this paper examines two such logics in the function free predicate case it is shown in particular that augmenting horn clause logic with hypothetical addition increases its data complexity from ptime to pspace when deletions are added as well complexity increases again to exptime we then augment the logic with negation as failure and develop the notion of stratified hypothetical rulebases it is shown that negation does not increase complexity to establish expressibility we view the logic as a query language for relational databases it is shown that any typed generic query that is computable in pspace can be expressed as a stratified rulebase of hypothetical additions similarly any typed generic query that is computable in exptime can be expressed as a stratified rulebase of hypothetical additions and deletions neither of these results assumes the data domain is linearly ordered in this way we establish the expressive completeness of our logics for queries in pspace and exptime respectively

the kolmogorov expressive power of boolean query languages abstract we address the question how much of the information stored in a given database can be retrieved by all boolean queries in a given query language in order to answer it we develop a kolmogorov complexity based measure of expressive power of boolean query languages over finite structures this turns the above informal question into a precisely defined mathematical one this notion gives a meaningful definition of the expressive power of a boolean query language in a single finite database the notion of kolmogorov expressive power of a boolean query language l in a finite database a is defined by considering two values the kolmogorov complexity of the isomorphism type of a equal to the length of the shortest description of this type and the number of bits of this description that can be reconstructed from truth values of all queries from l in a the closer is the second value to the first the more expressive is the query language after giving the definitions and proving that they are correct we concentrate our efforts on first order logic and its powerful extensions inflationary fixpoint logic and partial fixpoint logic we explore some connections between the proposed kolmogorov expressive power of boolean queries in these languages and their standard expressive power in particular with the definability of order we show that except of being of interest on its own our notion may have important diagnostic value for database query optimization

machine independent characterizations and complete problems for deterministic linear time this article presents two algebraic characterizations and two related complete problems for the complexity class dlin that was introduced in e grandjean ann math artif intell 16 1996 pp 183 236 dlin is essentially the class of all functions that can be computed in linear time on a random access machine which uses only numbers of linear value during its computations the algebraic characterizations are in terms of recursion schemes that define unary functions one of these schemes defines several functions simultaneously while the other one defines only one function from the algebraic characterizations we derive two complete problems for dlin under new very strict and machine independent affine reductions

relational queries computable in polynomial time we characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe we also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation this proves that the fixed point query hierarchy suggested by chandra and harel collapses at the first fixed point level it is also a general result showing that in finite model theory one application of fixed point suffices

fixed point definability and polynomial time on graphs with excluded minors we give a logical characterization of the polynomial time properties of graphs embeddable in some surface for every surface s a property p of graphs embeddable in s is decidable in polynomial time if and only if it is definable in fixed point logic with counting it is a consequence of this result that for every surface s there is a k such that a simple combinatorial algorithm namely the k dimensional weisfeiler lehman algorithm decides isomorphism of graphs embeddable in s in polynomial time we also present without proof generalizations of these results to arbitrary classes of graphs with excluded minors

convexity and horizontal second fundamental forms for hypersurfaces in carnot groups we use a riemannian approximation scheme to give a characterization for smooth convex functions on a carnot group in the sense of danielli garofalo nhieu or lu manfredi stroffolini in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph