the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

expectation consistent approximate inference we propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation the approximation can be understood as replacing an average over the original intractable distribution with a tractable one it requires two tractable probability distributions which are made consistent on a set of moments and encode different features of the original intractable distribution in this way we are able to use gaussian approximations for models with discrete or bounded variables which allow us to include non trivial correlations these are neglected in many other methods we test the framework on toy benchmark problems for binary variables on fully connected graphs and 2d grids and compare with other methods such as loopy belief propagation good performance is already achieved by using single nodes as tractable substructures significant improvements are obtained when a spanning tree is used instead

succinct representation of regular languages by boolean automata ii abstract boolean automata are a generalization of finite automata in the sense that the next state the result of the transition function given a state and a letter is not just a single state deterministic automata or a set of states nondeterministic automata but a boolean function of the states boolean automata accept precisely the regular languages also they correspond in a natural way to certain language equation involving complementation as well as to sequential networks in a previous note we showed that for every n 1 there exists a boolean automaton b n with n states such that the smallest deterministic automaton for the same language has 2 2 n states in the present note we will show a precisely attainable lower bound on the succinctness of representing regular languages by boolean automata namely we will show that for every n 1 there exists a reduced automaton d n with n states such that the smallest boolean automaton accepting the same language has also n states

alternating pushdown and stack automata the classes of languages accepted by alternating pushdown automata alternating stack automata and alternating nonerasing stack automata both with and without an auxiliary space bounded worktape are characterized in terms of complexity classes defined by time bounded deterministic turing machines it is also shown that alternating 2 way finite state machines accept only regular languages

concatenation free languages the expressive capacity of three different types of regular expressions without concatenation is studied in particular we consider alphabetic concatenation free expressions which are ordinary regular expressions without concatenation simple concatenation free expressions where the set of literals is a finite set of words instead of letters and concatenation free expressions where additionally complementation operations are possible characterizations of the corresponding language classes are obtained in particular a characterization of unary concatenation free languages by the boolean closure of certain sets of languages is shown the characterizations are then used to derive a strict hierarchy that is in turn strictly included in the family of regular languages the closure properties of the families are investigated furthermore the position of the family of concatenation free languages in the subregular hierarchy is considered and settled for the unary case in particular there are concatenation free languages that do not belong to any of the families in the hierarchy moreover except for comets all the families considered in the subregular hierarchy are strictly included in the family of concatenation free languages

equilibria of runge kutta methods it is known that certain runge kutta methods share the property that in a constant step implementation if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system such methods are calledregular in the present paper we provide a recursive test to check whether given method is regular moreover by examining solution trajectories of linear equations we prove that the order of ans stage regular method may not exceed 2 s 2 2 and that the maximal order of regular runge kutta method with an irreducible stability function is 4

on classes of tractable unrestricted regular expressions abstract it is known that the equivalence problem for the class ure of unrestricted regular expressions is not elementary recursive this implies that constructing the corresponding reduced automaton is not tractable for some expressions in ure we say these expressions are of intractable complexity however the closure of boolean automata under efficient complementation suggests that there are large nontrivial sub classes of ure with arbitrary level of complementation and tractable complexity in this paper we introduce the class c l corresponding in a natural way to boolean automata we also consider other related subclasses of ure all these classes have tractable complexity these results suggest a revised explanation why certain unrestricted regular expressions are so hard

stability and b convergence properties of multistep runge kutta methods this paper continues earlier work by the same author concerning the stability and b convergence properties of multistep runge kutta methods for the numerical solution of nonlinear stiff initial value problems in a hilbert space a series of sufficient conditions and necessary conditions for a multistep runge kutta method to be algebraically stable diagonally stable b or optimally b convergent are established by means of which six classes of high order algebraically stable and b convergent multistep runge kutta methods are constructed in a unified pattern these methods include the class constructed by burrage in 1987 as special case and most of them can be regarded as extension of the gauss radauia radauiia and lobattoiiic runge kutta methods we find that the classes of multistep runge kutta methods constructed in the present paper are superior in many respects to the corresponding existing one step runge kutta schemes

improved model checking of hierarchical systems we present a unified game based approach for branching time model checking of hierarchical systems such systems are exponentially more succinct than standard state transition graphs as repeated sub systems are described only once early work on model checking of hierarchical systems shows that one can do better than a naive algorithm that flattens the system and removes the hierarchy given a hierarchical system s and a branching time specification j for it we reduce the model checking problem does s satisfy j to the problem of solving a hierarchical game obtained by taking the product of s with an alternating tree automaton a j for j our approach leads to clean uniform and improved model checking algorithms for a variety of branching time temporal logics in particular by improving the algorithm for solving hierarchical parity games we are able to solve the model checking problem for the m calculus in pspace and time complexity that is only polynomial in the depth of the hierarchy our approach also leads to an abstraction refinement paradigm for hierarchical systems the abstraction maintains the hierarchy and is obtained by merging both states and sub systems into abstract states

b convergence properties of multistep runge kutta methods by using the theory of b convergence for general linear methods to the special case of multistep runge kutta methods a series of b convergence results for multistep runge kutta methods is obtained and it is proved that the family of algebraically stable r step s stage multistep runge kutta methods with parameters al a2 ar presented by burrage in 1987 is optimally bconvergent of order at least s and b convergent of order s 1 provided that r s and aj o j 1 2 r furthermore this family of methods is optimally b convergent of order s 1 if some other additional conditions are satisfied

prefix free languages we investigate the left and right quotient and the reversal operation on the class of prefix free regular languages we get the tight upper bounds 2 n 1 n 1 and 2 n 2 1 on the state complexity of these three operations respectively to prove the tightness of these bounds we use an n 1 letter alphabet for left quotient a binary alphabet for right quotient and a ternary alphabet for reversal we also prove that these bounds cannot be met using languages defined over any smaller alphabet for left quotient we prove that the tight bound for an n 2 letter alphabet is 2 n 1 1 and we provide exponential lower bounds for every smaller alphabet except for the unary case for the reversal operation on binary prefix free languages we get 2 n 2 7 lower bound in the case of n mod 3 2 and 2 n 2 15 lower bound in the remaining cases we conjecture that our lower bounds on the state complexity of reversal on binary prefix free languages are tight if n 12 our experimental results support this conjecture

on generalized language equations a system of generalized language equations over an alphabet a is a set of n equations in n variables xi gi x1 xn i 1 n where the gi are functions from p a n into p a i 1 n p a denoting the set of all languages over a furthermore the gi are expressible in terms of set operations concatenations and stars which involve the variable xi as well as certain mixed languages in this note we investigate existence and uniqueness of solutions of a certain subclass of generalized language equations furthermore we show that a solution is regular if all fixed languages are regular

on the power of bounded concurrency i finite automata we investigate the descriptive succinctness of three fundamental notions for modeling concurrency nondeterminism and pure parallelism the two facets of alternation and bounded cooperative concurrency whereby a system configuration consists of a bounded number of cooperating states our results are couched in the general framework of finite state automata but hold for appropriate versions of most concurrent models of computation such as petri nets statecharts or finite state versions of concurrent programming languages we exhibit exhaustive sets of upper and lower bounds on the relative succinctness of these features over s and s establishing that 1 each of the three features represents an exponential saving in succinctness of the representation in a manner that is independent of the other two and additive with respect to them 2 of the three bounded concurrency is the strongest representing a similar exponential saving even when substituted for each of the others for example we prove exponential upper and lower bounds on the simulation of deterministic concurrent automata by afas and triple exponential bounds on the simulation of alternating concurrent automata by dfas

unrestricted complementation in language equations over a one letter alphabet recently the complete solution of systems of language equations over a one letter alphabet has been derived where the operators are union concatenation and star this paper addresses the question of adding the complementation operator it is known that if no stars are present and if the concatenation operator is restricted such equations need not have solutions but if they do they can be solved explicitly if the constants are regular in which case the solutions are also regular in this paper we study language equations over a one letter alphabet where unrestricted concatenation interacts with complementation union and star we define a procedure for determining a regular solution of the general type of equation assuming that all constants are regular discuss its properties and show examples we then prove that in general equations with complementation and complementation and concatenation in which all constants are regular may have nonregular solutions this shows that language equations can be used to obtain non context free languages from regular languages using only operators under which the regular languages are closed

state complexity of combined operations for suffix free regular languages we investigate the state complexity of combined operations for suffix free regular languages suffix free deterministic finite state automata have a unique structural property that is crucial for obtaining the precise state complexity of basic operations based on the same property we establish the state complexity of four combined operations star of union star of intersection star of reversal and star of catenation in the case of star of intersection we only have an upper bound and the lower bound is open

state complexity of basic operations on suffix free regular languages we investigate the state complexity of basic operations for suffix free regular languages the state complexity of an operation for regular languages is the number of states that are necessary and sufficient in the worst case for the minimal deterministic finite state automaton that accepts the language obtained from the operation we establish the precise state complexity of catenation kleene star reversal and the boolean operations for suffix free regular languages

descriptional and computational complexity of finite automata a survey finite automata are probably best known for being equivalent to right linear context free grammars and thus for capturing the lowest level of the chomsky hierarchy the family of regular languages over the last half century a vast literature documenting the importance of deterministic nondeterministic and alternating finite automata as an enormously valuable concept has been developed in the present paper we tour a fragment of this literature mostly we discuss developments relevant to finite automata related problems like for example i simulation of and by several types of finite automata ii standard automata problems such as fixed and general membership emptiness universality equivalence and related problems and iii minimization and approximation we thus come across descriptional and computational complexity issues of finite automata we do not prove these results but we merely draw attention to the big picture and some of the main ideas involved

alternation in two way finite automata abstract the study of alternation in two way finite automata 2 fa s has been quite non systematic since the 1970 s various authors with a variety of motivations have studied 2 fa s with various types of alternation and a variety of names creating a fairly long list of sporadic contributions with little internal consistency this article attempts to organize the subject into a single unifying framework we start with a detailed account of all contributions to date that reveals the large variety of approaches and the lack of consistency between them we then identify and name four types of automata that these contributions have really studied over the years general two way boolean finite automata 2 b fa s monotone 2 b fa s basic 2 b fa s and monotone basic or alternating 2 b fa s 2 a fa s next we identify four different ways by which authors have described how such automata compute each offering a distinct view onto their operation the circuit view where computation is modelled by a circuit of boolean gates the formula view where computation is modelled by a boolean formula over configuration variables the run view where decisions are determined by the existence of appropriate trees of configuration goal pairs called runs and the most classic tree view where computation is modelled by a tree of configurations after carefully defining each 2 b fa type and each view we prove the following first that within each type every two of the four views are equivalent to each other in the strong sense that each of them closely mimics the other at every step of the computation second that not all types of 2 b fa s are equivalent although general 2 b fa s are as powerful as monotone 2 b fa s and basic 2 b fa s are as powerful as 2 a fa s up to polynomial differences in the number of states general 2 b fa s may need exponentially fewer states than 2 a fa s

descriptional complexity of two way pushdown automata with restricted head reversals two way nondeterministic pushdown automata 2pda are classical nondeterministic pushdown automata pda enhanced with two way motion of the input head in this paper the subclass of 2pda accepting bounded languages and making at most a constant number of input head turns is studied with respect to descriptional complexity aspects in particular the effect of reducing the number of pushdown reversals to a constant number is of interest it turns out that this reduction leads to an exponential blow up in case of nondeterministic devices and to a doubly exponential blow up in case of deterministic devices if the restriction on boundedness of the languages considered and on the finiteness of the number of head and pushdown turns is dropped the resulting trade offs are no longer bounded by recursive functions and so called non recursive trade offs are shown

an alternating hierarchy for finite automata we study the polynomial state complexity classes 2 s k and 2 p k that is the hierarchy of problems that can be solved with a polynomial number of states by two way alternating finite automata 2afas making at most k 1 alternations between existential and universal states starting in an existential or universal state respectively this hierarchy is infinite for k 2 3 4 both 2 s k 1 and 2 p k 1 are proper subsets of 2 s k and of 2 p k since the conversion of a one way s k or p k alternating automaton with n states into a two way automaton with a smaller number of alternations requires 2 n 4 o k states the same exponential blow up is required for converting a s k bounded 2afa into a p k bounded 2afa and vice versa that is 2 s k and 2 p k are incomparable in the case of s k bounded 2afas the exponential gap applies also for intersection while in the case of p k bounded 2afas for union the same results are established for one way alternating finite automata this solves several open problems raised in c kapoutsis size complexity of two way finite automata in proc develop lang theory in lect notes comput sci vol 5583 springer verlag 2009 pp 47 66

support recovery with sparsely sampled free random matrices consider a bernoulli gaussian complex n vector whose components are vi xibi with xi c n 0 px and binary bi mutually independent and iid across i this random q sparse vector is multiplied by a square random matrix u and a randomly chosen subset of average size n p p 0 1 of the resulting vector components is then observed in additive gaussian noise we extend the scope of conventional noisy compressive sampling models where u is typically a matrix with iid components to allow u satisfying a certain freeness condition this class of matrices encompasses haar matrices and other unitarily invariant matrices we use the replica method and the decoupling principle of guo and verdu as well as a number of information theoretic bounds to study the input output mutual information and the support recovery error rate in the limit of n we also extend the scope of the large deviation approach of rangan and characterize the performance of a class of estimators encompassing thresholded linear mmse and l1 relaxation

on the universality of noiseless linear estimation with respect to the measurement matrix in a noiseless linear estimation problem one aims to reconstruct a vector x from the knowledge of its linear projections y x there have been many theoretical works concentrating on the case where the matrix is a random i i d one but a number of heuristic evidence suggests that many of these results are universal and extend well beyond this restricted case here we revisit this problematic through the prism of development of message passing methods and consider not only the universality of the l1 transition but also the one of the optimal bayesian reconstruction we observed that the universality extends to the bayes optimal minimum mean squared mmse error and to complicated matrices such as those appearing in randomly statistical retrieval problems

high order algebraically stable multistep runge kutta methods in this paper we examine some of the stability properties of a special class of multivalue methods called in k burrage order properties of multivalue methods ima j numer anal submitted multistep runge kutta methods and construct families of algebraically stable methods of arbitrarily high order

on the state complexity of reversals of regular languages we compare the number of states between minimal deterministic finite automata accepting a regular language and its reversal mirror image in the worst case the state complexity of the reversal is 2n for an n state language we present several classes of languages where this maximal blow up is actually achieved and study the conditions for it in the case of finite languages the maximal blow up is not possible but still a surprising variely of different growth types can be exhibited

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

quotient complexity of ideal languages a language l over an alphabet s is a right left ideal if it satisfies email protected email protected l it is a two sided ideal if email protected email protected and an all sided ideal if email protected l the shuffle of s with l ideal languages are not only of interest from the theoretical point of view but also have applications to pattern matching we study the state complexity of common operations in the class of regular ideal languages but prefer to use the equivalent term quotient complexity which is the number of distinct left quotients of a language we find tight upper bounds on the complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of the minimal generator and also on the complexity of the minimal generator in terms of the complexity of the language moreover tight upper bounds on the complexity of union intersection set difference symmetric difference concatenation star and reversal of ideal languages are derived

efficient implementation of regular languages using reversed alternating finite automata alternating finite automata afa provide a natural and succinct way to denote regular languages we introduce a bit wise representation of reversed afa r afa transition functions and describe an efficient implementation method for r afa and their operations using this representation experiments have shown that this implementation is much more efficient than for instance the grail dfa implementation in both space and time

expressiveness and complexity of graph logic we investigate the complexity and expressive power of a spatial logic for reasoning about graphs this logic was previously introduced by cardelli gardner and ghelli and provides the simplest setting in which to explore such results for spatial logics we study several forms of the logic the logic with and without recursion and with either an exponential or a linear version of the basic composition operator we study the combined complexity and the expressive power of the four combinations we prove that without recursion the linear and exponential versions of the logic correspond to significant fragments of first order fo and monadic second order mso logics the two versions are actually equivalent to fo and mso on graphs representing strings however when the two versions are enriched with m style recursion their expressive power is sharply increased both are able to express pspace complete problems although their combined complexity and data complexity still belong to pspace

alternating finite automata and star free languages for a given extended regular expression e we construct an equational representation of an alternating finite automaton accepting the language denoted by e for star free extended regular expressions the construction yields a loop free alternating finite automaton also the inclusion in the opposite direction holds and thus we obtain a new characterization for the class of star free languages

partial orders on words minimal elements of regular languages and state complexity abstract the classical partial orders on strings prefix suffix subsegment subsequence lexical and dictionary order can be generalized to the case where the alphabet itself has a partial order this was done by higman for the subsequence order and by kundu for the prefix order higman proved that for any language l the set min l of minimal elements in l with respect to the generalized subsequence order is finite kundu proved that for any regular language l the set min l of minimal elements in l with respect to the generalized prefix order is also regular here we extend his result to the other orders and give upper bounds for the number of states of the finite automata recognizing min l the main contribution of this paper however is the proof of lower bounds the upper bounds are shown to be tight in particular if l is recognized by a deterministic finite automaton with n states then any deterministic or even nondeterministic finite automaton recognizing min l needs exponentially many states in n here min is taken with respect to a generalized prefix suffix or subsegment order with a partially ordered alphabet of 4 letters whose hasse diagram contains just one edge or with respect to the ordinary subsequence order we also give a new proof of a theorem of sakoda and sipser about the complementation of nondeterministic finite automata

complexity and succinctness issues for linear time hybrid logics full linear time hybrid logic hl is a non elementary and equally expressive extension of standard ltl past obtained by adding the well known binder operators 7 and we investigate complexity and succinctness issues for hl in terms of the number of variables and nesting depth of binder modalities first we present direct automata theoretic decision procedures for satisfiability and model checking of hl which require space of exponential height equal to the nesting depth of the binder modalities the proposed algorithms are proved to be asymptotically optimal by providing matching lower bounds second we show that for the one variable fragment of hl the considered problems are elementary and precisely expspace complete finally we show that for all 0 h

a rice style theorem for parallel automata we present a general result similar to rice s theorem concerning the complexity of detecting properties on finite automata enriched by bounded cooperative concurrency such as statecharts and abstract parallel automata which we denote by cfas concurrent finite automata on one extreme the complexity of detecting non trivial properties that preserve equivalence of machines i e properties of the accepted language on finite automata can be as little as o 1 on the other extreme rice s theorem states that all such properties on turing machines are undecidable we state that all the non trivial properties of the regular or w regular languages are pspace hard on cfas with e moves and on cfas without e moves accepting infinite words we also extend this result to cfas without e moves accepting finite words that satisfy a condition that holds for many properties

state complexity of inversion operations the reversal operation is well studied in the literature and the deterministic respectively nondeterministic state complexity of reversal is known to be 2 n respectively n we consider the inversion operation where some substring of the given string is reversed formally the inversion respectively prefix inversion of a language l consists of all strings u x r v such that u x v l respectively all strings u r x where u x l we show that the nondeterministic state complexity of prefix inversion is n 2 and that of inversion is n 3 we show that the deterministic state complexity of prefix inversion is at most 2 n log n n and has lower bound 2 n log n the same lower bound holds for the state complexity of inversion but for inversion we do not have a matching upper bound we also study the state complexity of other variants of the inversion operation

equilibrium attractive properties of a class of multistep runge kutta methods the main purpose of this paper is to discuss the equilibrium attractive properties of a class of multistep runge kutta methods for initial value problems of ordinary differential equations some algebraic conditions insuring the equilibrium attractivity are given and some methods satisfying these algebraic conditions are constructed some numerical examples confirm our results

on the numerical solution of stiff ivps by lobatto iiia runge kutta methods in this paper the numerical solution of stiff initial value problems by means of lobatto iiia runge kutta methods is considered new iterative schemes for an efficient solution of the implicit equations associated to the fourth and sixth order runge kutta lobatto iiia formulae are proposed finally the results of some numerical experiments are presented to show that when implemented with the new iterative schemes lobatto iiia formulae can be competitive with standard codes used in solving stiff systems

propositional dynamic logic of looping and converse is elementarily decidable propositional dynamic logic is a formal system for reasoning about the before after behavior of regular program schemes an extension of propositional dynamic logic which includes both an infinite looping construct and a converse or backtracking construct is considered and it is proved that the satisfiability problem for this logic is elementarily decidable in order to establish this result deterministic two way automata on infinite trees are defined and it is shown how they can be simulated by nondeterministic one way automata the satisfiability problem for propositional dynamic logic of looping and converse is then reduced to the emptiness problem for these two way automata

alternation and bounded concurrency are reverse equivalent numerous models of concurrency have been considered in the framework of automata among the more interesting concurrency models are classical nondeterminism and pure concurrency the two facets of alternation and the bounded concurrency model bounded concurrency was previously considered to be similar to nondeterminism and pure concurrency in the sense of the succinctness of automata augmented with these features in this paper we show that when viewed more broadly the power of succinctness of bounded concurrency is in fact most similar to the power of alternation our contribution is that just like nondeterminism and pure concurrency are complement equivalent bounded concurrency and alternation are reverse equivalent over finite automata the reverse equivalence is expressed by the existence of polynomial transformations in both directions between bounded concurrency and alternation for the reverse of the language accepted by the other it follows that bounded concurrency is double exponentially more succinct than dfas with respect to reverse while alternation only saves one exponent this is as opposed to the direct case where alternation saves two exponents and bounded concurrency saves only one an immediate corollary is that for languages over a one letter alphabet bounded concurrency and alternation are equivalent we complete the picture of succinctness results for these languages by considering the different combinations of the concurrency models using additional lower bounds

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