a conjectured r matrix a new spectral parameter independent r matrix that depends on all of the dynamical variables is proposed for the elliptic calogero moser models the necessary and sufficient conditions for the existence this r matrix reduces to a determinantal equality involving elliptic functions the required identity appears new and is still unproven in full generality we present it as conjecture

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

on contracted semigroup rings sufficient conditions are given for a contracted semigroup ring in which the two sided ideals have a certain property to be a semiprime b semiprimitive c prime d primitive the results are applied to the contracted semigroup rings of inverse semigroups where they provide new proofs of theorems of domanov

an algorithm for probabilistic planning abstract we define the probabilistic planning problem in terms of a probability distribution over initial world states a boolean combination of propositions representing the goal a probability threshold and actions whose effects depend on the execution time state of the world and on random chance adopting a probabilistic model complicates the definition of plan success instead of demanding a plan that provably achieves the goal we seek plans whose probability of success exceeds the threshold in this paper we present buridan an implemented least commitment planner that solves problems of this form we prove that the algorithm is both sound and complete we then explore buridan s efficiency by contrasting four algorithms for plan evaluation using a combination of analytic methods and empirical experiments we also describe the interplay between generating plans and evaluating them and discuss the role of search control in probabilistic planning

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

probabilistic supervisory control of probabilistic discrete event systems this paper considers supervisory control of probabilistic discrete event systems pdess pdess are modeled as generators of probabilistic languages the supervisory control problem considered is to find if possible a supervisor under whose control the behavior of a plant is identical to a given probabilistic specification the probabilistic supervisors we employ are a generalization of the deterministic ones previously employed in the literature at any state the supervisor enables disables events with certain probabilities necessary and sufficient conditions for the existence of such a supervisor and an algorithm for its computation are presented

a groupoid approach to discrete inverse semigroup algebras abstract let k be a commutative ring with unit and s an inverse semigroup we show that the semigroup algebra ks can be described as a convolution algebra of functions on the universal etale groupoid associated to s by paterson this result is a simultaneous generalization of the author s earlier work on finite inverse semigroups and paterson s theorem for the universal c algebra it provides a convenient topological framework for understanding the structure of ks including the center and when it has a unit in this theory the role of gelfand duality is replaced by stone duality using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups generalizing the case of an inverse semigroup with finitely many idempotents more generally we describe the irreducible representations of an inverse semigroup s that can be induced from associated groups as precisely those satisfying a certain finiteness condition this finiteness condition is satisfied for instance by all representations of an inverse semigroup whose image contains a primitive idempotent

noncrystallographic coxeter group h4 in e8 the e8 lattice is constructed in terms of icosians by matching two sets of f4 lattices described by quaternions embedding the noncrystallographic group h4 into the weyl group w e8 has been described using matrix generators with an emphasis on the relevant coxeter elements the conjugacy classes of h4 in terms of quaternions and the characters of the two four dimensional irreducible representations are explicitly calculated

shortest paths without a map we study several versions of the shortest path problem when the map is not known in advanced but is specified dynamically we are seeking dynamic decision rules that optimize the worst case ratio of the distance covered to the length of the statically optimal path we describe optimal decision rules for two cases layered graphs of bounded width and two dimensional scenes with unit square obstacles the optimal rules turn out to be intuitive common sense heuristics for slightly more general graphs and scenes we show that no bounded ratio is possible we also show that the computational problem of devising a strategy that achieves a given worst case ratio to the optimum path in a graph is a universal two person game and thus pspace complete whereas optimizing the expected ratio is p hard

quasicrystals and icosians a family of quasicrystals of dimensions 1 2 3 4 governed by the root lattice e8 is constructed the use of the icosian ring found in the quaternions with coefficients in q square root 5 allows simultaneous interpretation of the construction both in physical space and as a result of the standard cut and projection method in double dimension icosians are seen to provide a natural co ordination scheme for these quasicrystals nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians the use of coxeter diagrams clarifies the amazing relationship of e8 and quasicrystal symmetries and leads to the fundamental chain e8 contains implies d6 contains implies a4 contains implies a1 a1 that underlies five fold symmetry in quasicrystals decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed

trace functions on the algebras of certain e unitary inverse semigroups a construction is given for a trace function on the semigroup algebra of a certain type of e unitary inverse semigroup over any subfield of the complex field that is closed under complex conjugation in particular the method applies to the semigroup algebras of free inverse semigroups of arbitrary rank

generalised projections in finite state automata and decidability of state determinacy loss of sensors and communication links may lead to incomplete observation at the supervisory level of discrete event systems des under these circumstances an event may conceivably be observable at one state and unobservable at another state and the observability may become dependent on the history of event occurrences this paper presents a framework for analysis of generalised projection maps in des including the maps that introduce possibly unbounded memory

structural transformations of probabilistic finite state machines probabilistic finite state machines have recently emerged as a viable tool for modelling and analysis of complex non linear dynamical systems this paper rigorously establishes such models as finite encodings of probability measure spaces defined over symbol strings the well known nerode equivalence relation is generalized in the probabilistic setting and pertinent results on existence and uniqueness of minimal representations of probabilistic finite state machines are presented the binary operations of probabilistic synchronous composition and projective composition which have applications in symbolic model based supervisory control and in symbolic pattern recognition problems are introduced the results are elucidated with numerical examples and are validated on experimental data for statistical pattern classification in a laboratory environment

non crystallographic reduction of generalized calogero moser models we apply a recently introduced reduction procedure based on the embedding of non crystallographic coxeter groups into crystallographic ones to calogero moser systems for rational potentials the familiar generalized calogero hamiltonian is recovered for the hamiltonians of trigonometric hyperbolic and elliptic types we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio we explicitly show for the simplest of these non crystallographic models how the corresponding classical equations of motion can be derived from a lie algebraic lax pair based on the larger crystallographic coxeter group

complexity of finite horizon markov decision process problems controlled stochastic systems occur in science engineering manufacturing social sciences and many other cntexts if the systems is modeled as a markov decision process mdp and will run ad infinitum the optimal control policy can be computed in polynomial time using linear programming the problems considered here assume that the time that the process will run is finite and based on the size of the input there are mny factors that compound the complexity of computing the optimal policy for instance there are many factors that compound the complexity of this computation for instance if the controller does not have complete information about the state of the system or if the system is represented in some very succint manner the optimal policy is provably not computable in time polynomial in the size of the input we analyze the computational complexity of evaluating policies and of determining whether a sufficiently good policy exists for a mdp based on a number of confounding factors including the observability of the system state the succinctness of the representation the type of policy even the number of actions relative to the number of states in almost every case we show that the decision problem is complete for some known complexity class some of these results are familiar from work by papadimitriou and tsitsiklis and others but some such as our pl completeness proofs are surprising we include proofs of completeness for natural problems in the as yet little studied classes np pp

on the complexity of partially observed markov decision processes abstract in the paper we consider the complexity of constructing optimal policies strategies for some type of partially observed markov decision processes this particular case of the classical problem deals with finite stationary processes and can be represented as constructing optimal strategies to reach target vertices from a starting vertex in a graph with colored vertices and probabilistic deviations from an edge chosen to follow the colors of the visited vertices is the only information available to a strategy the complexity of markov decision in the case of perfect information bijective coloring of vertices is known and briefly surveyed at the beginning of the paper for the unobservable case all the colors are equal we give an improvement of the result of papadimitriou and tsitsiklis namely we show that the problem of constructing even a very weak approximation to an optimal strategy is np hard our main results concern the case of a fixed bound on the multiplicity of coloring that is a case of partially observed processes where some upper bound on the unobservability is supposed we show that the problem of finding an optimal strategy is still np hard but polytime approximations are possible some relations of our results to the max word problem are also indicated

optimal control of infinite horizon partially observable decision processes modelled as generators of probabilistic regular languages decision processes with incomplete state feedback have been traditionally modelled as partially observable markov decision processes in this article we present an alternative formulation based on probabilistic regular languages the proposed approach generalises the recently reported work on language measure theoretic optimal control for perfectly observable situations and shows that such a framework is far more computationally tractable to the classical alternative in particular we show that the infinite horizon decision problem under partial observation modelled in the proposed framework is approximable and in general is not harder to solve compared to the fully observable case the approach is illustrated via two simple examples

signed real measure of regular languages for discrete event supervisory control this paper reviews expands and clarifies the underlying concepts of a signed real measure of regular languages which has been used as a novel tool for synthesis of discrete event supervisory control systems the language measure is constructed upon the principles of automata theory and real analysis it allows total ordering of a set of partially ordered sublanguages of a regular language for quantitative evaluation of the supervised behaviour of deterministic finite state automata dfsa under different supervisors in the setting of the language measure a supervisor s performance is superior if the supervised plant is more likely to terminate at a good marked state and or less likely to terminate at a bad marked state the computational complexity of the language measure algorithm is polynomial in the number of dfsa states

the complexity of markov decision processes we investigate the complexity of the classical problem of optimal policy computation in markov decision processes all three variants of the problem finite horizon infinite horizon discounted and infinite horizon average cost were known to be solvable in polynomial time by dynamic programming finite horizon problems linear programming or successive approximation techniques infinite horizon we show that they are complete for p and therefore most likely cannot be solved by highly parallel algorithms we also show that in contrast the deterministic cases of all three problems can be solved very fast in parallel the version with partially observed states is shown to be pspace complete and thus even less likely to be solved in polynomial time than the np complete problems in fact we show that most likely it is not possible to have an efficient on line implementation involving polynomial time on line computations and memory of an optimal policy even if an arbitrary amount of precomputation is allowed finally the variant of the problem in which there are no observations is shown to be np complete

a language measure for partially observed discrete event systems recent literature has introduced and validated a signed real measure of regular languages for quantitative analysis and synthesis of discrete event supervisory des control systems where all events are assumed to be observable this paper presents a modification of the language measure for supervisory control under partial observation and shows how to generalize the analysis when some of the events may not be observable at the supervisory level in the context of des control synthesis the language measure of partially observable discrete event processes is expressed in a closed form which is structurally similar to that of completely observable discrete event processes examples are provided to elucidate the concept of des control under partial observation

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

antilinear deformations of coxeter groups an application to calogero models we construct complex root spaces remaining invariant under antilinear involutions related to all coxeter groups we provide two alternative constructions one is based on deformations of factors of the coxeter element and the other based on the deformation of the longest element of the coxeter group motivated by the fact that non hermitian hamiltonians admitting an antilinear symmetry may be used to define consistent quantum mechanical systems with real discrete energy spectra we subsequently employ our constructions to formulate deformations of coxeter models remaining invariant under these extended coxeter groups we provide explicit and generic solutions to the schrodinger equation of these models for the eigenenergies and corresponding wavefunctions a new feature of these novel models is that when compared with the undeformed case their solutions are usually no longer singular for an exchange of an amount of particles less than the dimension of the representation space of the roots the simultaneous scattering of all particles in the model leads to anyonic exchange factors for processes which have no analogue in the undeformed case

on the undecidability of probabilistic planning and related stochastic optimization problems automated planning the problem of how an agent achieves a goal given a repertoire of actions is one of the foundational and most widely studied problems in the ai literature the original formulation of the problem makes strong assumptions regarding the agent s knowledge and control over the world namely that its information is complete and correct and that the results of its actions are deterministic and known recent research in planning under uncertainty has endeavored te relax these assumptions providing formal and computation models wherein the agent has incomplete or noisy information about the world and has noisy sensors and effectors this research has mainly taken one of two approaches extend the classical planning paradigm to a semantics that admits uncertainty or adopt another framework for approaching the problem most commonly the markov decision process mdp model this paper presents a complexity analysis of planning under uncertainty it begins with the probabilistic classical planning problem showing that problem to be formally undecidable this fundamental result is then applied to a broad class of stochastic optimization problems in brief any problem statement where the agent a operates over an infinite or indefinite time horizon and b has available only probabilistic information about the system s state undecidability is established for policy existence problems for partially observable infinite horizon markov decision processes under discounted and undiscounted total reward models average reward models and state avoidance models the results also apply to corresponding approximation problems with undiscounted objective functions the paper answers a significant open question raised by papadimitriou and tsitsiklis math oper res 12 3 1987 441 450 about the complexity of infinite horizon pomdps

renormalized measure of regular languages this paper modifies the signed real measure of regular languages which has been reported in recent literature for analysis and synthesis of discrete event supervisory control laws a new concept of renormalized measure is introduced to eliminate a user selectable parameter in the present version of the language measure the concept of measure renormalization is illustrated by an example