automata boolean matrices and ultimate periodicity abstract this paper presents a boolean matrix based method to automata theory with an application to the study of regularity preserving functions a new characterization of such functions is derived in terms of the property of ultimate periodicity with respect to powers of boolean matrices this characterization reveals the intrinsic algebraic nature of regularity preserving functions it facilitates a concise proof of known as well as previously unknown properties of regularity preserving functions leading to the solution of the subtraction problem left open by kosaraju

on the depth of the tangent cone and the growth of the hilbert function for a d dimensional cohen macaulay local ring r mn we study the depth of the associated graded ring of r with respect to an rm primary ideal i in terms of the vallabrega valla conditions and the length of it jit where j is a j minimal reduction of i and t 1 as a corollary we generalize sally s conjecture on the depth of the associated graded ring with respect to a maximal ideal to rm primary ideals we also study the growth of the hilbert function

mean variance optimization of discrete time discounted markov decision processes abstract in this paper we study a mean variance optimization problem in an infinite horizon discrete time discounted markov decision process mdp the objective is to minimize the variance of system rewards with the constraint of mean performance different from most of works in the literature which require the mean performance already achieve optimum we can let the discounted performance equal any constant the difficulty of this problem is caused by the quadratic form of the variance function which makes the variance minimization problem not a standard mdp by proving the decomposable structure of the feasible policy space we transform this constrained variance minimization problem to an equivalent unconstrained mdp under a new discounted criterion and a new reward function the difference of the variances of markov chains under any two feasible policies is quantified by a difference formula based on the variance difference formula a policy iteration algorithm is developed to find the optimal policy we also prove the optimality of deterministic policy over the randomized policy generated in the mean constrained policy space numerical experiments demonstrate the effectiveness of our approach

axiomatizing weighted synchronization trees and weighted bisimilarity abstract we consider regular synchronization trees weighted over a semiring and provide sound and complete axiomatizations of these trees and their weighted bisimulation equivalence classes we prove that they can be both axiomatized by a finite number of identities relatively to the general axioms of the fixed point operation captured by the notion of iteration theories we present infinite equational and finite quasi equational axiomatizations

optimization of markov decision processes under the variance criterion in this paper we study a variance minimization problem in an infinite stage discrete time markov decision process mdp regardless of the mean performance for the markov chain under the variance criterion since the value of the cost function at the current stage will be affected by future actions this problem is not a standard mdp and the traditional mdp theory is not applicable in this paper we convert the variance minimization problem into a standard mdp by introducing a concept called pseudo variance then we derive a variance difference formula that quantifies the difference of variances of markov systems under any two policies with the difference formula the correlation of the variance cost function at different stages can be decoupled through a nonnegative term a necessary condition of the optimal policy is obtained it is also proved that the optimal policy with the minimal variance can be found in the deterministic policy space furthermore we propose an efficient iterative algorithm to reduce the variance of markov systems we prove that this algorithm can converge to a local optimum finally a numerical experiment is conducted to demonstrate the efficiency of our algorithm compared with the gradient based method widely adopted in the literature

the characterizations of h hemiregular hemirings and h intra hemiregular hemirings in this paper the concepts of fuzzy h bi ideals and fuzzy h quasi ideals of a hemiring are introduced and related properties are investigated the notion of h intra hemiregularity of a hemiring which is a generalization of the notion of intra regularity of a ring is provided some characterizations of h hemiregular hemirings h intra hemiregular hemirings and hemirings that are both h hemiregular and h intra hemiregular are derived in terms of fuzzy left fuzzy right h ideals fuzzy h bi ideals and fuzzy h quasi ideals

fundamental properties of infinite trees infinite trees naturally arise in the formalization and the study of the semantics of programming languages this paper investigates some of their combinatorial and algebraic properties that are especially relevant to semantics

coefficient ideals in and blowups of a commutative noetherian domain abstract the ratliff rush ideal associated to a nonzero ideal i in a commutative noetherian domain r with unity is n 1 in 1 rin is r s b i where b i r i a p a i 0 p spec r i a is the blowup of i we observe that certain ideals are minimal or even unique in the class of ideals having the same associated ratliff rush ideal if r m is local quasi unmixed and analytically unramified and if i is m primary then we show that the coefficient ideal i k of i i e the largest ideal containing i whose hilbert polynomial agrees with that of i in the highest k terms is also contracted from a blowup b i k which is obtained from b i by a process similar to s2 ification this allows us to generalise the notion of coefficient ideas we investigate these ideas in the specific context of a two dimentional regular local ring observing the interaction of these notions with the zariski theory of complete ideals

rewriting extended regular expressions we concider an extened algebra of regular events languages with intersection besides the usual operations this algebra has the structure of a distributive lattice with monotonic operations the latter property is crucial for some applications we give a new complete horn equational axiomatiztion of the algebra and develop some termrewriting techniques for constructing logical inferences of valid equations

convergence of newton s method over commutative semirings we give a lower bound on the speed at which newton s method as defined in 11 converges over arbitrary continuous commutative semirings from this result we deduce that newton s method converges within a finite number of iterations over any semiring which is collapsed at some k n i e k k 1 holds in the sense of bloom and esik 2 we apply these results to 1 obtain a generalization of parikh s theorem 2 compute the provenance of datalog queries and 3 analyze weighted pushdown systems we further show how to compute newton s method over any continuous semiring by constructing a grammar unfolding w r t tree dimension we review several concepts equivalent to tree dimension and prove a new relation to pathwidth

a proof of the krohn rhodes decomposition theorem abstract we give a new proof of one part of the krohn rhodes decomposition theorem for automata

the ordinal generated by an ordinal grammar is computable abstract a prefix grammar is a context free grammar whose nonterminals generate prefix free languages a prefix grammar g is an ordinal grammar if the language l g is well ordered with respect to the lexicographic ordering it is known that from a finite system of parametric fixed point equations over ordinals one can construct an ordinal grammar g such that the lexicographic order of g is isomorphic with the least solution of the system if this solution is well ordered in this paper we show that given an ordinal grammar one can compute the cantor normal form of the order type of the lexicographic order of its language yielding that least solutions of fixed point equation systems defining algebraic ordinals are effectively computable and thus their isomorphism problem is also decidable

recursive markov chains stochastic grammars and monotone systems of nonlinear equations we define recursive markov chains rmcs a class of finitely presented denumerable markov chains and we study algorithms for their analysis informally an rmc consists of a collection of finite state markov chains with the ability to invoke each other in a potentially recursive manner rmcs offer a natural abstract model for probabilistic programs with procedures they generalize in a precise sense a number of well studied stochastic models including stochastic context free grammars scfg and multi type branching processes mt bp we focus on algorithms for reachability and termination analysis for rmcs what is the probability that an rmc started from a given state reaches another target state or that it terminatesq these probabilities are in general irrational and they arise as least fixed point solutions to certain monotone systems of nonlinear equations associated with rmcs we address both the qualitative problem of determining whether the probabilities are 0 1 or in between and the quantitative problems of comparing the probabilities with a given bound or approximating them to desired precision we show that all these problems can be solved in pspace using a decision procedure for the existential theory of reals we provide a more practical algorithm based on a decomposed version of multi variate newton s method and prove that it always converges monotonically to the desired probabilities we show this method applies more generally to any monotone polynomial system we obtain polynomial time algorithms for various special subclasses of rmcs among these for scfgs and mt bps equivalently for 1 exit rmcs the qualitative problem can be solved in p time for linearly recursive rmcs the probabilities are rational and can be computed exactly in p time we show that our pspace upper bounds cannot be substantially improved without a breakthrough on long standing open problems the square root sum problem and an arithmetic circuit decision problem that captures p time on the unit cost rational arithmetic ram model we show that these problems reduce to the qualitative problem and to the approximation problem to within any nontrivial error for termination probabilities of general rmcs and to the quantitative decision problem for termination extinction of scfgs mt bps

limiting average criteria for nonstationary markov decision processes this paper deals with the so called limiting average criteria for nonstationary markov decision processes with possibly unbounded rewards and borel state space a new set of conditions is provided under which the existence of both a solution to the optimality equations and the limiting average epsilon geq 0 optimal markov policies is derived also a rolling horizon algorithm for computing limiting average epsilon 0 optimal markov policies is developed furthermore the results in this paper are illustrated by several examples such as the water regulation problem

constrained optimization for average cost continuous time markov decision processes to the best of our knowledge this note is a first study on a multiconstrained optimization problem for average cost continuous time markov decision processes in polish spaces the optimality criterion is the long run expected average cost and the constraints are imposed on similar average costs we give suitable conditions under which the existence of a constrained optimal stationary policy is shown by introducing occupation measures and using a martingale technique also we illustrate our conditions with controlled potlach processes

on λ semirings abstract a semiring is an ordered semiring s equipped with a star operation such that for any a b s a b is the least fixed point of the linear mapping x ax b over s the notion of semirings is a generalization of several important concepts such as continuous semirings weak inductive semirings and the kleene algebras of conway and kozen we investigate several basic properties of semirings and obtain results on semirings in relation to preorders duality and formal power series some of these results can be seen as generalizations of relevant results on inductive semirings

group axioms for iteration iteration theories provide a sound and complete axiomatization of the equational properties of the iteration or fixed point operation in many models of theoretical computer science including ordered and metric structures trees and synchronization trees all known equational axiomatizations of iteration theories consist of a small set of equational axioms for conway theories and a complicated equation scheme the commutative identity here we associate an identity with each finite semigroup we prove that the set consisting of the conway identities and the group identities associated with the finite simple groups is complete moreover we prove that the conway identities and a subcollection of the semigroup identities associated with a subclass of the finite semigroups is complete iff each finite simple group divides one of the semigroups in the subclass we also formulate a conjecture and study its consequences the results are a generalization of krob s axiomatization of the equational theory of the regular sets

the multiplicative fragment of the yanov equational theory we give a finite equational axiomatization for free identities of regular languages which contain the empty word the axioms for the whole equational theory of such languages were given by yanov

an equational axiomatization for multi exit iteration abstract this paper presents an equational axiomatization of bisimulation equivalence over the language of basic process algebra bpa with multi exit iteration multi exit iteration is a generalization of the standard binary kleene star operation that allows for the specification of agents that up to bisimulation equivalence are solutions of systems of recursion equations of the form x 1 def p 1 x 2 q 1 x n def p n x 1 q n wherenis a positive integer and thepiand theqiare process terms the addition of multi exit iteration to bpa yields a more expressive language than that obtained by augmenting bpa with the standard binary kleene star bpa as a consequence the proof of completeness of the proposed equational axiomatization for this language although standard in its general structure is much more involved than that for bpa an expressiveness hierarchy for the family ofk exit iteration operators proposed by bergstra bethke and ponse is also offered

continuous monoids and semirings different kinds of monoids and semirings have been defined in the literature all of them named continuous we show their relations the main technical tools are suitable topologies among others a variant of the well known scott topology for complete partial orders

completeness of park induction abstract the in equational properties of iteration i e least pre fixed point solutions over cpo s are captured by the axioms of iteration theories all known axiomatizations of iteration theories consist of the conway identities and a complicated equation scheme the commutative identity the results of this paper show that the commutative identity is implied by the conway identities and a weak form of the park induction principle hence we obtain a simple first order axiomatization of the in equational theory of iteration it follows that a few simple identities and a weak form of the scott induction principle formulated to involve only inequations are also complete we also show that the conway identities and the park induction principle are not complete for the universal horn theory of iteration

discrete time controlled markov processes with average cost criterion a survey this work is a survey of the average cost control problem for discrete time markov processes the authors have attempted to put together a comprehensive account of the considerable research on this problem over the past three decades the exposition ranges from finite to borel state and action spaces and includes a variety of methodologies to find and characterize optimal policies the authors have included a brief historical perspective of the research efforts in this area and have compiled a substantial yet not exhaustive bibliography the authors have also identified several important questions that are still open to investigation

optimal control of ergodic continuous time markov chains with average sample path rewards in this paper we study continuous time markov decision processes with the average sample path reward aspr criterion and possibly unbounded transition and reward rates we propose conditions on the system s primitive data for the existence of epsilon aspr optimal deterministic stationary policies in a class of randomized markov policies satisfying some additional continuity assumptions the proof of this fact is based on the time discretization technique the martingale stability theory and the concept of potential we also provide both policy and value iteration algorithms for computing or at least approximating the epsilon aspr optimal stationary policies we illustrate with examples our main results as well as the difference between the aspr and the average expected reward criteria

equational properties of kleene algebras of relations with conversion abstract we describe a set of equational axioms for the variety generated by all algebras of binary relations with operations of union composition conversion and reflexive transitive closure and neutral elements 0 the empty relation and 1 the identity relation

on equations for union free regular languages in this paper we consider the variety uf generated by all algebras of binary relations equipped with the operations of composition reflexive transitive closure and the empty set and the identity relation as constants this variety coincides with the variety generated by the union free reducts of kleene algebras of languages and its free objects are formed by union free regular languages that is regular languages represented by regular expressions having no occurrence of we show that the variety uf is not finitely based the situation does not change if we consider the variety uf generated by the above algebras of binary relations equipped with the conversion operation

integrally closed and componentwise linear ideals in a two dimensional regular local ring integrally closed ideals have a unique factorization property and their associated graded ring is cohen macaulay in higher dimension these properties do not hold and the goal of the paper is to identify a subclass of integrally closed ideals for which they do we restrict our attention to 0 dimensional homogeneous ideals in polynomial rings r of arbitrary dimension we identify a class of integrally closed ideals the goto class mathcal g which is closed under product and it has a suitable unique factorization property ideals in mathcal g have a cohen macaulay associated graded ring if either they are monomial or dim r 3 our approach is based on the study of the relationship between the notions of integrally closed contracted full and componentwise linear ideals

on dynamic algebras abstract dynamic algebras are the lindenbaum tarski algebras of dynamic logics these algebras can be considered as boolean algebras with some operators indexed by the elements of some kleene algebra in this paper we prove that there are infinitely many finitely generated varieties of dynamic algebras having undecidable equational theories all these varieties are generated by representable dynamic algebras

equational logic of circular data type specification abstract iteration theories introduced by bloom elgot and wright in 1980 formalize the equational properties of the strong behaviors of flowchart algorithms we show that the same equational properties are shared by the functors used to specify circular data types lehmann and smyth 1981 have shown how to specify circular data types such as stacks as fixed points of certain functors the called gw functors for example the set of stacks of elements in the set a is a solution to the equation in the variable x x f a x where f x x 1 a x is the functor on set taking the pair a x to the disjoint union of the singleton set 1 and the product a x such equations have initial solutions f a which in turn determine a solution functor f the equational properties of the operation f f are precisely captured by the axioms for iteration theories more precisely we show how the structure th c of all functors c n c p n p 0 on the category c forms an iteration theory and conversely any identity valid in all such structures is valid in the class of all iteration theories

a mezei wright theorem for categorical algebras the main result of this paper is a generalization of the mezei wright theorem a result on solutions of a system of fixed point equations in the typical setting one solves a system of fixed point equations in an algebra equipped with a suitable partial order there is a least element suprema of w chains exist the operations preserve the ordering and least upper bounds of w chains in this setting one solution of this kind of system is provided by least fixed points the mezei wright theorem asserts that such a solution is preserved by a continuous order preserving algebra homomorphism in several settings such as countable words or synchronization trees there is no well defined partial order but one can naturally introduce a category by considering morphisms between the elements the generalization of this paper consists in replacing ordered algebras by categorical algebras the least element is replaced by an initial element and suprema of w chains are replaced by colimits of w diagrams then the mezei wright theorem for categorical algebras is that initial solutions are preserved by continuous morphisms we establish this result for initial solutions of parametric fixed point equations one use of the theorem is to characterize an algebraic element as one that can arise as a solution of some system of fixed point equations in familiar examples an algebraic element is one that is context free regular or rational then if h a b is a continuous morphism of categorical algebras the algebraic objects in b are those isomorphic to h images of algebraic objects in a

left handed completeness abstract we give a new proof of the completeness of the left handed star rule of kleene algebra the proof is significantly shorter than previous proofs and exposes the rich interaction of algebra and coalgebra in the theory of kleene algebra

equivalences and transformations of regular systems applications to recursive program schemes and grammars abstract this work presents a unified theory of recursive program schemes context free grammars grammars on arbitrary algebraic structures and in fact recursive definitions of all kind by means of regular systems the equivalences of regular systems associated with either all their solutions or their least solutions in all domains of appropriate type satisfying a set of algebraic laws expressed by equations are systematically investigated and characterized in some cases in terms of system transformations by folding unfolding and rewriting according to the equational algebraic laws grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union and the same questions are raised for them we also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system this theory applies to grammars of many kinds which generate trees graphs etc we formulate some classical transformations of context free grammars in terms of correct transformations which only use folding unfolding and algebraic laws and we immediately obtain their correctness

the variety of kleene algebras with conversion is not finitely based given an arbitrary set a one obtains the full kleene algebra of binary relations over a by considering the operations of union composition reflexive transitive closure conversion and the empty set and the identity relation as constants such algebras generate the variety of kleene algebras with conversion as a result of a general analysis of identities satisfied by varieties having an involution operation we prove that the variety of kleene algebras with conversion has no finite equational axiomatization in our argument we make use of the fact that the variety of kleene algebras without conversion is not finitely based and that relatively to this variety the variety of kleene algebras with conversion is finitely axiomatized

μ semirings and μ lamda semirings we introduce and study semirings and semirings which generalize inductive semirings and weak inductive semirings respectively also we discuss the semiring of formal power series with coefficients in such a semiring and prove that the semiring of formal power series with coefficients in a weak inductive semiring semiring semiring semiring is a weak inductive semirng semiring semiring semiring respectively this gives a positive answer to one of esik and kuich s open problems

depth of fiber cones of ideals with almost minimal mixed multiplicity let r m be a cohen macaulay local ring of dimension d 0 i an m primary ideal with almost minimal mixed multiplicity such that depth g i d 1 we show that fm i has almost maximal depth i e depth fm i d 1

on the last hilbert samuel coefficient of isolated singularities abstract j lipman proved that the last hilbert samuel coefficient of normal ideals of a 2 dimensional complete local ring r are bounded by the geometric genus of x spec r in this paper we extend this result to ideals i of a d dimensional cohen macaulay local ring r such that the associated graded ring of r with respect to i n is cohen macaulay for n 0 we study the ideals such that their last hilbert samuel coefficients equal the geometric genus

a unified approach to markov decision problems and performance sensitivity analysis with discounted and average criteria multichain cases we propose a unified framework to markov decision problems and performance sensitivity analysis for multichain markov processes with both discounted and average cost performance criteria with the fundamental concept of performance potentials we derive both performance gradient and performance difference formulas which play the central role in performance optimization the standard policy iteration algorithms for both discounted and average reward mdps can be established using the performance difference formulas in a simple and intuitive way and the performance gradient formulas together with stochastic approximation may lead to new optimization schemes this sensitivity based point of view of performance optimization provides some insights that link perturbation analysis markov decision processes and reinforcement learning together the research is an extension of the previous work on ergodic markov chains cao automatica 36 2000 771

generalized hilbert coefficients and northcott s inequality abstract let r be a cohen macaulay local ring of dimension d with infinite residue field let i be an r ideal that has analytic spread l i d satisfies the g d condition and the weak artin nagata property a n d 2 we provide a formula relating the length i n 1 j i n to the difference p i n h i n where j is a general minimal reduction of i p i n and h i n are respectively the generalized hilbert samuel polynomial and the generalized hilbert samuel function we then use it to establish formulas to compute the generalized hilbert coefficients of i as an application we extend northcott s inequality to non m primary ideals furthermore when equality holds we prove that the ideal i enjoys nice properties indeed if this is the case then the reduction number of i is at most one and the associated graded ring of i is cohen macaulay we also recover results of g colome nin c polini b ulrich and y xie on the positivity of the generalized first hilbert coefficient j 1 i our work extends that of s huckaba c huneke and a ooishi to ideals that are not necessarily m primary

on a question of a salomaa the equational theory of regular expression salomaa 1969 p 143 asked whether the equational theory of regular expressions over a singleton alphabet has a finite equational base in this paper we provide a negative answer to this long standing question the proof of our main result rests upon a model theoretic argument for every finite collection of equations that are sound in the algebra of regular expressions over a singleton alphabet we build a model in which some valid regular equation fails the construction of the model mimics the one used by conway 1971 p 105 in his proof of a result originally due to redko to the effect that infinitely many equations are needed to axiomatize equality of regular expressions our analysis of the model however needs to be more refined than the one provided by conway 1971

a mean variance optimization problem for discounted markov decision processes in this paper we consider a mean variance optimization problem for markov decision processes mdps over the set of deterministic stationary policies different from the usual formulation in mdps we aim to obtain the mean variance optimal policy that minimizes the variance over a set of all policies with a given expected reward for continuous time mdps with the discounted criterion and finite state and action spaces we prove that the mean variance optimization problem can be transformed to an equivalent discounted optimization problem using the conditional expectation and markov properties then we show that a mean variance optimal policy and the efficient frontier can be obtained by policy iteration methods with a finite number of iterations we also address related issues such as a mutual fund theorem and illustrate our results with an example

sample path optimality and variance minimization of average cost markov control processes this paper studies several average cost criteria for markov control processes on borel spaces with possibly unbounded costs under suitable hypotheses we show i the existence of a sample path average cost spac optimal stationary policy ii a stationary policy is spac optimal if and only if it is expected average cost eac optimal and iii within the class of stationary spac optimal equivalently eac optimal policies there exists one with a minimal limiting average variance

observer design for max linear systems this technical note deals with the state estimation for max plus linear systems this estimation is carried out following the ideas of the observer method for classical linear systems the system matrices are assumed to be known and the observation of the input and of the output is used to compute the estimated state the observer design is based on the residuation theory which is suitable to deal with linear mapping inversion in idempotent semiring

axiomatizing rational power series over natural numbers iteration semi rings are conway semi rings satisfying conway s group identities we show that the semi rings n r a t of rational power series with coefficients in the semi ring n of natural numbers are the free partial iteration semi rings moreover we characterize the semi rings n r a t as the free semi rings in the variety of iteration semi rings defined by three additional simple identities where n is the completion of n obtained by adding a point of infinity we also show that this latter variety coincides with the variety generated by the complete or continuous semirings as a consequence of these results we obtain that the semi rings n r a t equipped with the sum order are free in the class of symmetric inductive semi rings this characterization corresponds to kozen s axiomatization of regular languages

characterizations of h intra and h quasi hemiregular hemirings the concepts of c email protected q d fuzzy h bi h quasi ideals of hemirings are introduced some new characterization theorems of these kinds of fuzzy h ideals are also given in particular some characterizations of the h intra hemiregular and h quasi hemiregular hemirings are investigated by these kinds of fuzzy h ideals

mean variance portfolio selection under partial information this paper is concerned with a continuous time mean variance portfolio selection problem in a possibly incomplete market with multiple stocks and a bond only the past price movements of the stocks and the bond are the information available to the investors a separation principle is shown to hold in this setting efficient strategies based on the aforementioned partial information are derived which involve the optimal filter of the stock appreciation rate processes the main methodological contribution of the paper is to employ the particle system representation to develop analytical and numerical approaches in obtaining the filter as well as solving the related backward stochastic differential equation

on the depth of the associated graded ring of an m primary ideal of a cohen macaulay local ring abstract let r m be a cohen macaulay local ring with maximal ideal m and positive dimension d let us assume r has infinite residue field and let i be an m primary ideal of r by gr l r we denote the associated graded ring of i and by depth gr l r we mean depth gr l r m where m is the maximal homogeneous ideal of gr l r in this paper we individuate some conditions on i that allow us to determine the value of depth gr l r it is proved that if j i is a minimal reduction of i such that formula then depth gr l r d 1 let us remark that lambda denotes the length function

j multiplicity and depth of associated graded modules abstract let r be a noetherian local ring we define the minimal j multiplicity and almost minimal j multiplicity of an arbitrary r ideal on any finite r module for any ideal i with minimal j multiplicity or almost minimal j multiplicity on a cohen macaulay module m we prove that under some residual conditions the associated graded module gr i m is cohen macaulay or almost cohen macaulay respectively our work generalizes the results for minimal multiplicity and almost minimal multiplicity obtained by sally rossi valla wang huckaba elias corso polini and vaz pinto

drift and monotonicity conditions for continuous time controlled markov chains with an average criterion we give conditions for the existence of average optimal policies for continuous time controlled markov chains with a denumerable state space and borel action sets the transition rates are allowed to be unbounded and the reward cost rates may have neither upper nor lower bounds in the spirit of the drift and monotonicity conditions for continuous time markov processes we propose a new set of conditions on the controlled process primitive data under which the existence of optimal deterministic stationary policies in the class of randomized markov policies is proved using the extended generator approach instead of kolmogorov s forward equation used in the previous literature and under which the convergence of a policy iteration method is also shown moreover we use a controlled queueing system to show that all of our conditions are satisfied whereas those in the previous literature fail to hold

iteration theories of synchronization trees synchronization trees are shown to form an iteration theory in a natural way the class of grove iteration theories is introduced the regular synchronization trees are shown to be the free theories in the subclass of synchronization theories moreover the bisimulation equivalence classes of regular synchronization trees are shown to be the free synchronization theories satisfying an infinite idempotency law

a probabilistic analysis of bias optimality in unichain markov decision processes focuses on bias optimality in unichain finite state and action space markov decision processes using relative value functions we present methods for evaluating optimal bias this leads to a probabilistic analysis which transforms the original reward problem into a minimum average cost problem the result is an explanation of how and why bias implicitly discounts future rewards

bisimulation through probabilistic testing abstract we propose a language for testing concurrent processes and examine its strength in terms of the processes that are distinguished by a test by using probabilistic transition systems as the underlying semantic model we show how a testing algorithm can distinguish with a probability arbitrarily close to one between processes that are not bisimulation equivalent we also show a similar result in a slightly stronger form for a new process relation called 2 3 bisimulation which lies strictly between that of simulation and bisimulation finally the ultimately strength of the testing language is shown to identify a new process relation called probabilistic bisimulation which is strictly stronger than bisimulation

the bisimulation problem for equational graphs of finite out degree the bisimulation problem for equational graphs of finite out degree is shown to be decidable we reduce this problem to the eta bisimulation problem for deterministic rational vectors of boolean series on the alphabet of a deterministic pushdown automaton cal m we then exhibit a complete formal system for deducing equivalent pairs of such vectors

equational theories of tropical semirings this paper studies the equational theories of various exotic semirings presented in the literature exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum and addition as product two prime examples of such structures are the max semiring and the tropical semiring it is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations and that similar results hold for the commutative idempotent weak semirings that underlie them for each of these commutative idempotent weak semirings the paper offers characterizations of the equations that hold in them decidability results for their equational theories explicit descriptions of the free algebras in the varieties they generate and relative axiomatization results

inductive semirings one of the most well known induction principles in computer science is the fixed point induction rule or least pre fixed point rule inductive semirings are partially ordered semirings equipped with a star operation satisfying the fixed point equation and the fixed point induction rule for linear terms inductive semirings are extensions of continuous semirings and the kleene algebras of conway and kozen we develop in a systematic way the rudiments of the theory of inductive semirings in relation to automata languages and power series in particular we prove that if s is an inductive semiring then so is the semiring of matrices sn n for any integer n 0 and that if s is an inductive semiring then so is any semiring of power series s a as shown by kozen the dual of an inductive semiring may not be inductive in contrast we show that the dual of an iteration semiring is an iteration semiring kuich proved a general kleene theorem for continuous semirings and bloom and esik proved a kleene theorem for all conway semirings since any inductive semiring is a conway semiring and an iteration semiring as we show there results a kleene theorem applicable to all inductive semirings we also describe the structure of the initial inductive semiring and conjecture that any free inductive semiring may be given as a semiring of rational power series with coefficients in the initial inductive semiring we relate this conjecture to recent axiomatization results on the equational theory of the regular sets

a note on optimality conditions for continuous time markov decision processes with average cost criterion this note deals with continuous time markov decision processes with a denumerable state space and the average cost criterion the transition rates are allowed to be unbounded and the action set is a borel space we give a new set of conditions under which the existence of optimal stationary policies is ensured by using the optimality inequality our results are illustrated with a controlled queueing model moreover we use an example to show that our conditions do not imply the existence of a solution to the optimality equations in the previous literature

a complete system of identities for one letter rational expressions with multiplicities in the tropical semiring abstract we construct a complete system of rational identities for the equational theory of the algebra of one letter rational expressions with multiplicities in the tropical semiring

bisimulation relations for weighted automata bisimulation is a well known equivalence relation for discrete event systems and has been extended to probabilistic and stochastic systems this paper introduces a general definition of bisimulation which can be applied to finite automata where weights and labels are assigned to transitions it is shown that this general view contains several known bisimulations as special cases and defines naturally equivalences for different classes of models apart from the well known forward bisimulation also backward bisimulation is introduced and it is shown that both types of bisimulation preserve different types of results furthermore it is shown that forward and backward bisimulation are congruences according to commonly known composition operations for automata

on semigroup rings with decreasing hilbert function abstract given a one dimensional semigroup ring r k s in this article we study the behaviour of the hilbert function h r by means of the notion of support of the elements in s for some classes of semigroup rings we give conditions on the generators of s in order to have decreasing h r when the embedding dimension v and the multiplicity e verify v 3 e v 4 the decrease of h r gives an explicit description of the apery set of s in particular for e v 3 we prove that h r is non decreasing if e 12 and we classify the semigroups with e 13 and h r decreasing finally we deduce that h r is non decreasing for every gorenstein semigroup ring with e v 4 this fact is not true in general through numerical duplication and some of the above results another recent paper shows the existence of infinitely many one dimensional gorenstein rings with decreasing hilbert function

mean variance criteria for finite continuous time markov decision processes this technical note deals with the mean variance problem known as the average variance av minimization problem for finite continuous time markov decision processes we first introduce a so called g condition which is weaker than the well known ergodicity and unichain conditions and sufficient for the finiteness of the av of a policy also we present an example of a policy having infinite av when the g condition is not satisfied under the g condition we prove that the av criterion can be transformed into an equivalent mean or expected average criterion by using a martingale technique and an observation from the canonical form of a transition rate matrix and thus the existence and calculation of an av minimal policy over a class of mean optimal policies are obtained by a policy iteration algorithm in an finite number of iterations as byproduct we obtain some interesting new results about the mean average optimality

artin nagata properties minimal multiplicities and depth of fiber cones abstract minimal values of multiplicities of ideals have a strong relation with the depth of blowup algebras in this paper we introduce the notion of goto minimal j multiplicity for ideals of maximal analytic spread in a cohen macaulay ring inspired by the work of s goto a jayanthan t puthenpurakal and j verma we study the interplay among this new notion the notion of minimal j multiplicity introduced by c polini and y xie and the cohen macaulayness of the fiber cone of ideals satisfying certain residual assumptions we are also able to provide a bound on the reduction number of ideals of goto minimal j multiplicity having either cohen macaulay associated graded algebra or linear decay in the depth of their powers

hilbert coefficients of hilbert filtrations adic filtrations see 7 11 18 21 wx in the first section of this paper we extend most of the techniques of 4 w5 to the case of hilbert filtrations and by doing so we are also able toxoffer alternative simple proofs of some results of 7 wxin the second section thanks to the setting just described if

the semiring of matrices over a finite chain let l m denote the chain 0 1 2 m 1 with the usual ordering and m n l m the matrix semiring of all nxn matrices with elements in l m we firstly introduce some order preserving semiring homomorphisms from m n l m to m l k by using these homomorphisms we show that a matrix over the finite chain l m can be decomposed into the sum of some matrices over the finite chain l k where k

the equational theory of regular words courcelle introduced the study of regular words i e words isomorphic to frontiers of regular trees heilbrunner showed that a nonempty word is regular iff it can be generated from the singletons by the operations of concatenation omega power omega op power and the infinite family of shuffle operations we prove that the algebra of nonempty regular words on the set a equipped with these operations is freely generated by a in a variety which is axiomatizable by an infinite collection of some natural equations we also show that this variety has no finite equational basis and that its equational theory is decidable in polynomial time

axiomatizing the identities of binoid languages we present a nontrivial axiomatization for the equational theory of binoid languages the subsets of a free binoid in doing so we prove that a conjecture given in our previous paper was true the identical laws of ordinary string languages written separately using horizontal and vertical operation symbols form a required complete system of axioms

graded rings associated with contracted ideals the study of the ideals in a regular local ring r m of dimension 2 has a long and important tradition dating back to the fundamental work of zariski zs more recent contributions are due to several authors including cutkosky huneke lipman rees sally and teissier among others see c1 c2 h hs l lt r one of the main result in this setting is the unique factorization theorem for complete i e integrally closed ideals proved originally by zariski zs theorem 3 appendix 5 it asserts that any complete ideal can be factorized as a product of simple complete ideals in a unique way up to the order of the factors by definition an ideal is simple if it cannot be written as a product of two proper ideals another important property of a complete ideal i is that its reduction number is 1 which in turns implies that the associated graded ring gri r is cohen macaulay and its hilbert series is well understood this is due to lipman and teissier lt see also hs the class of contracted ideals plays an important role in the original work of zariski as well as in the work of huneke h an ideal i of r is contracted if i r ir m x for some x m m2 any complete ideal is contracted but not the other way round the associated graded ring gri r of a contracted ideal i need not be cohen macaulay and its hilbert series can be very complicated