formally reasoning about quality in recent years there has been a growing need and interest in formally reasoning about the quality of software and hardware systems as opposed to traditional verification in which one considers the question of whether a system satisfies a given specification or not reasoning about quality addresses the question of how well the system satisfies the specification we distinguish between two approaches to specifying quality the first propositional quality extends the specification formalism with propositional quality operators which prioritize and weight different satisfaction possibilities the second temporal quality refines the eventually operators of the specification formalism with discounting operators whose semantics takes into an account the delay incurred in their satisfaction in this article we introduce two quantitative extensions of linear temporal logic ltl one by propositional quality operators and one by discounting operators in both logics the satisfaction value of a specification is a number in 0 1 which describes the quality of the satisfaction we demonstrate the usefulness of both extensions and study the decidability and complexity of the decision and search problems for them as well as for extensions of ltl that combine both types of operators

on computing structural changes in evolving surfaces and their appearance as a surface undergoes a one parameter family of deformations its shape and its appearance change smoothly except at certain critical parameter values where abrupt structural changes occur this paper considers the case of surfaces defined as the zero set of smooth density functions undergoing a gaussian diffusion process and addresses the problem of computing the critical parameter values corresponding to structural changes in the parabolic curves of a surface and in its aspect graph an algorithm based on homotopy continuation and curve tracing is proposed in the case of polynomial density functions whose zero set is an algebraic surface it has been implemented and examples are presented

an analysis of completely positive trace preserving maps on m2 abstract we give a useful new characterization of the set of all completely positive trace preserving maps m 2 m 2 from which one can easily check any trace preserving map for complete positivity we also determine explicitly all extreme points of this set and give a useful parameterization after reduction to a certain canonical form this allows a detailed examination of an important class of non unital extreme points that can be characterized as having exactly two images on the bloch sphere we also discuss a number of related issues about the images and the geometry of the set of stochastic maps and show that any stochastic map on m 2 can be written as a convex combination of two generalized extreme points

results on the propositional μ calculus abstract in this paper we define and study a propositional calculus l which consists essentially of propositional modal logic with a least fixpoint operator l is syntactically simpler yet strictly more expressive than propositional dynamic logic pdl for a restricted version we give an exponential time decision procedure small model property and complete deductive system theory subsuming the corresponding results for pdl

metrics for labelled markov processes the notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions thus a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent this instability is due to the quantitative nature of probabilistic processes in a situation where the process behavior has a quantitative aspect there should be a more robust approach to process equivalence this paper studies a metric between labelled markov processes this metric has the property that processes are at zero distance if and only if they are bisimilar the metric is inspired by earlier work on logics for characterizing bisimulation and is related in spirit to the hutchinson metric

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

transience and recurrence of quantum markov semigroups this article introduces a concept of transience and recurrence for a quantum markov semigroup and explores its main properties via the associated potential we show that an irreducible semigroup is either recurrent or transient and characterize transient semigroups by means of the existence of non trivial superharmonic operators

weighted automata and weighted logics with discounting we introduce a weighted logic with discounting and we establish the buchi elgot theorem for weighted automata over finite words and arbitrary commutative semirings then we investigate buchi and muller automata with discounting over the max plus and the min plus semiring we show their expressive equivalence with weighted mso sentences with discounting in this case our logic has a purely syntactic definition for the finite case we obtain a purely syntactically defined weighted logic if the underlying semiring is additively locally finite

on the mixing property and the ergodic principle for nonhomogeneous markov chains abstract we study different types of limit behavior of infinite dimension discrete time nonhomogeneous markov chains we show that the geometric structure of the set of those markov chains which have asymptotically stationary density depends on the considered topologies we generalize and correct some results from ganikhodjaev et al 2006 3

the complexity of mean payoff pushdown games two player games on graphs are central in many problems in formal verification and program analysis such as synthesis and verification of open systems in this work we consider solving recursive game graphs or pushdown game graphs that model the control flow of sequential programs with recursion while pushdown games have been studied before with qualitative objectives such as reachability and regular objectives in this work we study for the first time such games with the most well studied quantitative objective the mean payoff objective in pushdown games two types of strategies are relevant 1 global strategies which depend on the entire global history and 2 modular strategies which have only local memory and thus do not depend on the context of invocation but rather only on the history of the current invocation of the module our main results are as follows 1 one player pushdown games with mean payoff objectives under global strategies are decidable in polynomial time 2 two player pushdown games with mean payoff objectives under global strategies are undecidable 3 one player pushdown games with mean payoff objectives under modular strategies are np hard 4 two player pushdown games with mean payoff objectives under modular strategies can be solved in np i e both one player and two player pushdown games with mean payoff objectives under modular strategies are np complete we also establish the optimal strategy complexity by showing that global strategies for mean payoff objectives require infinite memory even in one player pushdown games and memoryless modular strategies are sufficient in two player pushdown games finally we also show that all the problems have the same complexity if the stack boundedness condition is added where along with the mean payoff objective the player must also ensure that the stack height is bounded

on the existence of stationary states for quantum dynamical semigroups we provide two criteria on the existence of stationary states for quantum dynamical semigroups the first one is based on the semigroup itself while the second criterion is based on the generator which is in general unbounded and interpreted as a sesquilinear form these results are illustrated by physical examples drawn from quantum optics

the dobrushin ergodicity coefficient and the ergodicity of noncommutative markov chains abstract in this paper we study the notion of the dobrushin ergodicity coefficient for positive contractions defined on the predual m of von neumann algebra m moreover in terms of this coefficient we prove ergodic type theorems for nonhomogeneous markov chains on m

global bifurcation sets and stable projections of nonsingular algebraic surfaces the view graph of a surface is a planar graph whose nodes are the stable views projections of the surface and whose edges represent transitional views of codimension one the space of all directions of orthogonal projection can be identified with the projective plane the set of bad projection directions associated with the degenerate views of positive codimension forms a graph in the projective plane the view bifurcation set this graph is dual to the view graph and divides the projective plane into a certain number of connected regions whose representatives are the nodes of the view graph we assume that the projected surface is nonsingular and parameterized by polynomials of degree d we present an estimate for the number of nodes in the view graph in terms of d and describe symbolic algorithms for computing the bifurcation set and the view graph of a surface from a parametrization

relative entropy under mappings by stochastic matrices the relative g entropy of two finite discrete probability distributions x x1 xn and y y1 yn is defined as hg x y kxkg yk kk 1 where g 1 r is convex and g 0 0 when g t log 1 t then hg x y kxklog xk yk the usual relative entropy let pn x rn ixi 1 xi 0 i our major results is that for any m n column stochastic matrix a the contraction coefficient defined as g a sup hg ax ay hg x y x y pn x y satisfies g a 1 a where a minj k i min aij aik is dobrushin s coefficient of ergodicity consequently g a 1 if and only if a is scrambling upper and lower bounds on g a are established analogous results hold for markov chains in continuous time

weighted automata and multi valued logics over arbitrary bounded lattices we show that l weighted automata l rational series and l valued monadic second order logic have the same expressive power for any bounded lattice l and for finite and infinite words we also prove that aperiodicity star freeness and l valued first order and ltl definability coincide this extends classical results of kleene buchi elgot trakhtenbrot and others to arbitrary bounded lattices without any distributivity assumption that is fundamental in the theory of weighted automata over semirings in fact we obtain these results for large classes of strong bimonoids which properly contain all bounded lattices

alternating finite automata on ω words abstract alternating finite automata on words are introduced as an extension of nondeterministic finite automata which process infinite sequences of symbols the classes of languages defined by alternating finite automata are investigated and characterized under four types of acceptance conditions it is shown that for one type of acceptance condition alternation increases the power in comparison with nondeterminism and for other three acceptance conditions nondeterministic finite automata on words have the same power as alternating ones

an automata theoretic approach to branching time model checking translating linear temporal logic formulas to automata has proven to be an effective approach for implementing linear time model checking and for obtaining many extensions and improvements to this verification method on the other hand for branching temporal logic automata theoretic techniques have long been thought to introduce an exponential penalty making them essentially useless for model checking recently bernholtz and grumberg 1993 have shown that this exponential penalty can be avoided though they did not match the linear complexity of non automata theoretic algorithms in this paper we show that alternating tree automata are the key to a comprehensive automata theoretic framework for branching temporal logics not only can they be used to obtain optimal decision procedures as was shown by muller et al but as we show here they also make it possible to derive optimal model checking algorithms moreover the simple combinatorial structure that emerges from the automata theoretic approach opens up new possibilities for the implementation of branching time model checking and has enabled us to derive improved space complexity bounds for this long standing problem

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

computing exact aspect graphs of curved objects algebraic surfaces this article presents an algorithm for computing the exact aspect graph of an opaque solid bounded by a smooth algebraic surface orthographic projection is assumed the algorithm is based on a catalog of visual events available from singularity theory it uses curve tracing cell decomposition homotopy continuation and ray tracing to construct the regions of the view sphere delineated by visual event curves the algorithm has been fully implemented and examples are presented

isomorphism of k contractible graphs a generalization of bounded valence and bounded genus a polynomial time isomorphism test for graphs called k contractible graphs for fixed k is included the class of k contractible graphs includes the graphs of bounded valence and the graphs of bounded genus the algorithm uses several new ideas including 1 it removes portions of the graph and replaces them with groups which are used to keep track of the symmetries of these portions 2 it maintains with each group a tower of equivalence relations which allow a decomposition of the group these towers are called towers of k actions

ridges crests and sub parabolic lines of evolving surfaces the ridge lines on a surface can be defined either via contact of the surface with spheres or via extrema of principal curvatures along lines of curvature certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications there is a related concept of sub parabolic line on a surface also defined via extrema of principal curvatures in this paper we study in detail the structure of the ridge lines crest lines and sub parabolic lines on a generic surface and on a surface which is evolving in a generic one parameter family the mathematical details of this study are in bruce et al 1994c

the scale space aspect graph currently the aspect graph is computed from the theoretical standpoint of perfect resolution in object shape the viewpoint and the projected image this means that the aspect graph may include details that an observer could never see in practice introducing the notion of scale into the aspect graph framework provides a mechanism for selecting a level of detail that is large enough to merit explicit representation this effectively allows control over the number of nodes retained in the aspect graph this paper introduces the concept of the scale space aspect graph defines three different interpretations of the scale dimension and presents a detailed example for a simple class of objects with scale defined in terms of the spatial extent of features in the image

visibility occlusion and the aspect graph this chapter studies the ways in which the topology of the image of a polyhedron changes with changing viewpoint we catalog the ways that the topological appearance or aspect can change this enables us to find maximal regions of viewpoints of the same aspect we use these techniques to construct the viewpoint space partition vsp a partition of viewpoint space into maximal regions of constant aspect and its dual the aspect graph here we present tight bounds on the maximum size of the vsp and the aspect graph and give algorithms for their construction first in the convex case and then in the general case in particular we give bounds on the maximum size of n 2 and n 6 under an orthographic projection viewing model and of n 3 and n 9 under a perspective viewing model the algorithms make use of a new representation of the appearance of polyhedra from all viewpoints called the aspect representation or asp we believe that this representation is one of the significant contributions of this paper

the curvature primal sketch in this paper we introduce a novel representation of the significant changes in curvature along the bounding contour of planar shape we call the representation the curvature primal sketch because of the close analogy to the primal sketch representation advocated by marr for describing significant intensity changes we define a set of primitive parameterized curvature discontinuities and derive expressions for their convolutions with the first and second derivatives of a gaussian we describe an implemented algorithm that computes the curvature primal sketch by matching the multiscale convolutions of a shape and illustrate its performance on a set of tool shapes several applications of the representation are sketched

on the complexity of numerical analysis we study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis a the blum shub smale model of computation over the reals and b a problem we call the generic task of numerical computation which captures an aspect of doing numerical computation in floating point similar to the long exponent model that has been studied in the numerical computing community we show that both of these approaches hinge on the question of understanding the complexity of the following problem which we call posslp given a division free straight line program producing an integer n decide whether n 0 in the blum shub smale model polynomial time computation over the reals on discrete inputs is polynomial time equivalent to posslp when there are only algebraic constants we conjecture that using transcendental constants provides no additional power beyond nonuniform reductions to posslp and we present some preliminary results supporting this conjecture the generic task of numerical computation is also polynomial time equivalent to posslp we prove that posslp lies in the counting hierarchy combining this with work of tiwari we obtain that the euclidean traveling salesman problem lies in the counting hierarchy the previous best upper bound for this important problem in terms of classical complexity classes being pspace in the course of developing the context for our results on arithmetic circuits we present some new observations on the complexity of the arithmetic circuit identity testing acit problem in particular we show that if n is not ultimately easy then acit has subexponential complexity

parabolic curves of evolving surfaces in this article we show how certain geometric structures which are also associated with a smooth surface evolve as the shape of the surface changes in a 1 parameter family we concentrate on the parabolic set and its image under the gauss map but the same techniques also classify the changes in the dual of the surface all these have significance for computer vision for example through their connection with specularities and apparent contours with the aid of our complete classification which includes all the phenomena associated with multi contact tangent planes as well as those associated with parabolic sets we re examine examples given by j koenderink in his book 1990 under the title of morphological scripts we also explain some of the connections between parabolic sets and ridges of a surface where principal curvatures achieve turning values along lines of curvature the point of view taken is the analysis of the contact between surfaces and their tangent planes a systematic investigation of this yields the results using singularity theory the mathematical details are suppressed here and appear in bruce et al 1993

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

efficiently computing and representing aspect graphs of polyhedral objects an efficient algorithm and a data structure for computing and representing the aspect graph of polyhedral objects under orthographic projection are presented the aspect graph is an approach to representing 3 d objects by a set of 2 d views for the purpose of object recognition in this approach the viewpoint space is partitioned into regions such that in each region the qualitative structure of the line drawing does not change the viewing data of an object is the partition of the viewpoint space together with a representative view in each region the algorithm computes the viewing data for line drawings of polyhedral objects under orthographic projection

ergodicity of nonlinear markov operators on the finite dimensional space abstract a nonlinear markov chain is a discrete time stochastic process whose transition matrices may depend not only on the current state of the process but also on the current distribution of the process in this paper we study strong and uniform ergodicity of nonlinear markov operators defined by stochastic hypermatrices higher order matrix we introduce dobrushin s ergodicity coefficient for a stochastic hypermatrix which enables to study ergodicity of nonlinear markov operators by introducing a notion of scrambling stochastic hypermatrix we study the strong ergodicity of scrambling nonlinear markov operators

computing exact aspect graphs of curved objects solid of revolution this paper introduces a new approach to computing the exact orthographic aspect graph of curved objects curves corresponding to various visual events partition the view sphere into regions where the image structure is stable a catalog of these events for piecewise smooth objects is available from singularity theory for a solid of revolution whose generator is an algebraic curve each visual event is characterized by a system of polynomial equations whose roots can be computed by continuation methods within each region the stable image structure is characterized by a variation of cylindrical algebraic decomposition and ray tracing this approach has been implemented and several examples are presented

asymptotic properties of quadratic stochastic operators acting on the l1 space quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours andthese have been introduced and studied recently in the l1 space it turns out that inprinciple most of the resu

generalising automaticity to modal properties of finite structures we introduce a complexity measure of modal properties of finite structures which generalises the automaticity of languages it is based on graph automata like devices called labelling systems we define a measure of the size of a structure that we call rank and show that any modal property of structures can be approximated up to any fixed rank n by a labelling system the function that takes n to the size of the smallest labelling system doing this is called the labelling index of the property we demonstrate that this is a useful and fine grained measure of complexity and show that it is especially well suited to characterise the expressive power of modal fixed point logics from this we derive several separation results of modal and non modal fixed point logics some of which are already known whereas others are new

non homogeneous quantum markov states and quantum markov fields abstract the program relative to the investigation of quantum markov states for general one dimensional spin models is carried on following a strategy developed in the last years in such a way the emerging structure is fully clarified this analysis is a starting point for the solution of the basic still open problem concerning the construction of a satisfactory theory of quantum markov fields i e quantum markov processes with multi dimensional indices

reasoning about infinite computations we investigate extensions of temporal logic by connectives defined by finite automata on infinite words we consider three different logics corresponding to three different types of acceptance conditions finite looping and repeating for the automata it turns out however that these logics all have the same expressive power and that their decision problems are all pspace complete we also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem