Problems of complex system modeling and applied mathematical statistics

Basic problems in the use of applied mathematical statistics for the modeling of complex systems are considered; the possibility of establishing the uniqueness of a mathematical model of optimal complexity by the group method of data handling (GMDH) is demonstrated. The basic shortcoming of contemporary mathematical statistics is that the models used are too simple because until now in regression analysis only one mean-squared error criterion has been used. To define a mathematical model of optimal complexity GMDH uses not one but two criteria and these two criteria assure a unique solution. The resulting equations are so complex that only the multilayered structure of GMDH allows us to write them down. The method works not only whenK N but also whenK >N(Kis the number of coefficients of the regression equation,N is the number of interpolation points). Increasing the area of optimization raises the accuracy of the model. The second criterion should be heuristic. Mean-squared error defined on a test sequence is used. The division of data into training and test sequences is the basic object of so-called goal-directed regularization. A second shortcoming of contemporary applied mathematical statistics is the absence of freedom of decision in the terminology of D. Gabor. The GMDH selection-type algorithm realizes both the self-organization and freedom of decision criteria. GMDH is a nonparametric procedure and does not require many of the concepts of mathematical statistics.     

Separating complexity classes related to bounded alternating ω-branching programs

We develop a theory of communication within branching programs that provides exponential lower bounds on the size of branching programs that are bounded alternating. Our theory is based on the algebraic concept of -branching programs, : , a semiring homomorphism, that generalizes ordinary branching programs, -branching programs [M2] andMOD _p-branching programs [DKMW].Due to certain exponential lower and polynomial upper bounds on the size of bounded alternating -branching programs we are able to separate the corresponding complexity classesN _ba ,co-N _{ba ba} , andMOD _p - _ba ,p prime, from each other, and from that classes corresponding to oblivious linear length-bounded branching programs investigated in the past.     

The dynamic clusters method in nonhierarchical clustering

Given a finite setE R ⁿ, the problem is to find clusters (or subsets of similar points inE) and at the same time to find the most typical elements of this set. An original mathematical formulation is given to the problem. The proposed algorithm operates on groups of points, called samplings (samplings may be called multiple centers or cores); these samplings adapt and evolve into interesting clusters. Compared with other clustering algorithms, this algorithm requires less machine time and storage. We provide some propositions about nonprobabilistic convergence and a sufficient condition which ensures the decrease of the criterion. Some computational experiments are presented.     

Specification and verification of database dynamics

Summary A framework is proposed for the structured specification and verification of database dynamics. In this framework, the conceptual model of a database is a many sorted first order linear tense theory whose proper axioms specify the update and the triggering behaviour of the database. The use of conceptual modelling approaches for structuring such a theory is analysed. Semantic primitives based on the notions of event and process are adopted for modelling the dynamic aspects. Events are used to model both atomic database operations and communication actions (input/output). Nonatomic operations to be performed on the database (transactions) are modelled by processes in terms of trigger/reaction patterns of behaviour. The correctness of the specification is verified by proving that the desired requirements on the evolution of the database are theorems of the conceptual model. Besides the traditional data integrity constraints, requirements of the form Under condition W, it is guaranteed that the database operation Z will be successfully performed are also considered. Such liveness requirements have been ignored in the database literature, although they are essential to a complete definition of the database dynamics.

Notation

Classical Logic Symbols (Appendix 1) for all (universal quantifier) - exists at least once (existential quantifier) - ¬ no (negation) - implies (implication) - is equivalent to (equivalence) - and (conjunction) - or (disjunction) Tense Logic Symbols (Appendix 1) G always in the future - G ₀ always in the future and now - F sometime in the future - F ₀ sometime in the future or now - H always in the past - H ₀ always in the past and now - P sometime in the past - P ₀ sometime in the past or now - X in the next moment - Y in the previous moment - L always - M sometime Event Specification Symbols (Sects. 3 and 4.1) (x) means immediately after the occurrence of x - (x) means immediately before the occurrence of x - (x) means x is enabled, i.e., x may occur next - { } ({w ₁} x{w ₂}) states that if w ₁ holds before the occurrence of x, then w ₂ will hold after the occurrence of x (change rule) - [ ] ([oa₁, ..., oa_n]x) states that only the object attributes oa₁, ..., oa _n are modifiable by x (scope rule) - {{ }} ({{w}}x) states that if x may occur next, then w holds (enabling rule) Process Specification Symbols (Sects. 5.3 and 5.4) :: for causal rules - for behavioural rules Transition-Pattern Composition Symbols (Sects. 5.2 and 5.3) ; sequential composition - ¦ choice composition - parallel composition - :| guarded alternative composition Location Predicates (Sect. 5.2) (z) means immediately after the occurrence of the last event of z (after) - (z) means immediately before the occurrence of the first event of z (before) - (z) means after the beginning of z and before the end of z (during) - ( z) means before the occurrence of an event of z (at)     

Mesh Selection for a Nearly Singular Boundary Value Problem

In this paper, we investigate the numerical solution of a model equation u _xx = exp(– ) (and several slightly more general problems) when 1 using the standard central difference scheme on nonuniform grids. In particular, we are interested in the error behaviour in two limiting cases: (i) the total mesh point number N is fixed when the regularization parameter 0, and (ii) is fixed when N. Using a formal analysis, we show that a generalized version of a special piecewise uniform mesh 12 and an adaptive grid based on the equidistribution principle share some common features. And the optimal meshes give rates of convergence bounded by |log()| as 0 and N is given, which are shown to be sharp by numerical tests.     

Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

We investigate three-dimensional visibility problems for scenes that consist ofn non-intersecting spheres. The viewing point moves on a flightpath that is part of a circle at infinity given by a planeP and a range of angles {(t)¦t[01]} [02]. At timet, the lines of sight are parallel to the ray inP, which starts in the origin ofP and represents the angle(t) (orthographic views of the scene). We give an algorithm that computes the visibility graph at the start of the flight, all time parameters at which the topology of the scene changes, and the corresponding topology changes. The algorithm has running time0(n + k + p) logn), wheren is the number of spheres in the scene;p is the number of transparent topology changes (the number of different scene topologies visible along the flight path, assuming that all spheres are transparent); andk denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.The second author was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).     

The mixing properties of certain processes related to Markov chains

Summary Let {Y _n ,– < n < } be either a function of a stationary Markov chain with countable state space, or a finitary process in the sense of Heller [3]. The purpose of this note is to prove that if {Y _n ,– < n < } is mixing, then it is aK-shift. (Definitions will be given below.)IfT is a measure-preserving transformation on a probability space, then the following implications relevant to the present paper are known: (1)T is aK-shift T is (r + 1)-mixing T isr-mixing T is totally ergodic T is ergodic, and (2)T is ergodic T is totally ergodic T isr-mixing T is aK-shift.It is not known if the classes ofr-mixing and (r + 1)-mixing transformations are distinct. (1-mixing is also called mixing.) The results of this note then imply that for the classes of transformation that we are consideringr-mixing and (r + 1)-mixing are equivalent.This research was partially supported by NSF Grant GP-19660.     

On parsing two-level grammars

Summary Making use of the fact that two-level grammars (TLGs) may be thought of as finite specification of context-free grammars (CFGs) with infinite sets of productions, known techniques for parsing CFGs are applied to TLGs by first specifying a canonical CFG G — called skeleton grammar — obtained from the cross-reference of the TLG G. Under very natural restrictions it can be shown that for these grammar pairs (G, G) there exists a 1 — 1 correspondence between leftmost derivations in G and leftmost derivations in G. With these results a straightforward parsing algorithm for restricted TLGs is given.     

Sufficiency conditions for existence of an optimal feedback control in stochastic mechanics

In the framework of stochastic mechanics, the following problem is considered: in a set of admissible feedback controls v, with range inE ⁿ , find one minimizing the expectationE _sx { _s ^T L(t, (t), (t, (t)))dt + W _T ((T))} for all (s, x) [0,T) E ⁿ , whereL(t, x, ) = (/12)m ² – U(t, x) is the classical action integrand and is an-dimensional diffusion process in the weak sense, (see Bensoussan, 1982) with drift and diffusion coefficientD constant > 0.W _T andU are given real functions. Sufficiency conditions for the existence of such an optimal feedback control are given. Dedicated to George Leitmann Recommended by G.J. Olsder Presented at the Third Workshop on Control Mechanics in honor of George Leitmann, January 22–24, 1990, University of Southern California, Los Angeles, California (USA).     

A forbidden subposet characterization of an order — dimension inequality

Dushnik and Miller defined the dimension of a partially ordered setX, DimX, as the minimum number of linear extensions ofX whose intersection is the partial ordering onX. The concept of dimension for a partially ordered set has applications to preference structures and the theory of measurement. Hiraguchi proved that DimX [|X|/2] when |X| 4. Bogart, Trotter, and Kimble gave a forbidden subposet characterization of Hiraguchi's inequality by constructing for eachn 2 the minimum collection _n of posets such that if [|X|/2] =n 2, then DimX < n unlessX contains one of the posets in _n . Recently Trotter gave a simple proof of Hiraguchi's inequality based on the following theorem. IfA is an antichain ofX and |X – A| =n 2, then DimX n. In this paper we give a forbidden subposet characterization of this last inequality.     

On characterizations of recursively enumerable languages

Summary Geffert has shown that earch recursively enumerable languageL over can be expressed in the formL{h(x) ^–1 g(x)x in ⁺} ^* where is an alphabet andg, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations ofL. For instance,L =(L ₀) ^* whereL ₀ is a minimal linear language and is the Dyck reductiona, A.     

Inference rules using local contexts

We construct equivalent localized versions of a formula, adding assumptions simultaneously to various locations, where the particular location determines what is added. Inference rules that take advantage of localized formulas are presented for sequent calculi in which the left hand side of sequents can be used to accumulate the background assumptions (or contexts) of assertions. The intended application is to the automatic generation of tractable justifying lemmas for substitution operations for interactive proof development systems, especially those concerned with mathematical topics where manipulation of deeply embedded terms is desirable.     

Extended topology and systems

In this paper is indicated the possible utility of isotonic spaces as a background language for discussing systems. In isotonic spaces the basic duality between neighborhood and convergent first achieves a proper background permitting applications beyond the scope of topological spaces. A generalization of continuity of mappings based on ancestral relations is presented and this definition is applied to establish a necessary and sufficient condition that mappings preserve connectedness. Fortunately for systems theory, it is not necessary to have infinite sets or infinitary operators to apply definitions of neighborhood, convergents, continuity and connectedness.This work was supported in part by a grant from the National Science Foundation.     

Context-free text grammars

A text is a triple=(, ₁, ₂) such that is a labeling function, and ₁ and ₂ are linear orders on the domain of ; hence may be seen as a word (, ₁) together with an additional linear order ₂ on the domain of . The order ₂ is used to give to the word (, ₁) itsindividual hierarchical representation (syntactic structure) which may be a tree but it may be also more general than a tree. In this paper we introducecontext-free grammars for texts and investigate their basic properties. Since each text has its own individual structure, the role of such a grammar should be that of a definition of a pattern common to all individual texts. This leads to the notion of ashapely context-free text grammar also investigated in this paper.     

Thek-server dual and loose competitiveness for paging

Weighted caching is a generalization ofpaging in which the cost to evict an item depends on the item. We study both of these problems as restrictions of the well-knownk-server problem, which involves moving servers in a graph in response to requests so as to minimize the distance traveled.We give a deterministic on-line strategy for weighted caching that, on any sequence of requests, given a cache holdingk items, incurs a cost within a factor ofk/(k–h+1) of the minimum cost possible given a cache holdingh items. The strategy generalizes least recently used, one of the best paging strategies in practice. The analysis is a primal-dual analysis, the first for an on-line problem, exploiting the linear programming structure of thek-server problem.We introduceloose competitiveness, motivated by Sleator and Tarjan's complaint [ST] that the standard competitive ratios for paging strategies are too high. Ak-server strategy isloosely c(k)-competitive if, for any sequence, foralmost all k, the cost incurred by the strategy withk serverseither is no more thanc(k) times the minimum costor is insignificant.We show that certain paging strategies (including least recently used, and first in first out) that arek-competitive in the standard model are looselyc(k)-competitive providedc(k)/Ink and bothk/c(k) andc(k) are nondecreasing. We show that the marking algorithm, a randomized paging strategy that is (Ink)-competitive in the standard model, is looselyc(k)-competitive providedk–2 In Ink and both 2 Ink–c(k) andc(k) are nondecreasing.This paper is the journal version of On-line Caching as Cache Size Varies, which appeared in theProceedings of the 2nd Annual ACM-SIAM Symposium on Discrete Algorithms (1991). Details beyond those in this paper may be found in Competitive Paging and Dual-Guided Algorithms for Weighted Caching and Matching, which is the author's thesis and is available as Technical Report CS-TR-348-91 from the Computer Science Department at Princeton University.This research was performed while the author was at the Computer Science Department, Princeton University, Princeton, NJ 08544, USA, and was supported by the Hertz Foundation.     

Modelling default and likelihood reasoning as probabilistic reasoning

This paper presents a probabilistic analysis of plausible reasoning about defaults and about likelihood. Likely and by default are in fact treated as duals in the same sense as possibility and necessity. To model these four forms probabilistically, a logicQDP and its quantitative counterpartDP are derived that allow qualitative and corresponding quantitative reasoning. Consistency and consequence results for subsets of the logics are given that require at most a quadratic number of satisfiability tests in the underlying prepositional logic. The quantitative logic shows how to track the propagation error inherent in these reasoning forms. The methodology and sound framework of the system highlights their approximate nature, the dualities, and the need for complementary reasoning about relevance.Much of this research was done while at the University of Technology, Sydney, Broadway, NSW, Australia, and some at the Turing Institute, 36 Nth. Hanover Str., Glasgow, Scotland.     

Type inference with non-structural subtyping

We present an O(n³) time type inference algorithm for a type system with a largest type, a smallest type , and the usual ordering between function types. The algorithm infers type annotations of least shape, and it works equally well for recursive types. For the problem of typability, our algorithm is simpler than the one of Kozen, Palsberg, and Schwartzbach for type inferencewithout . This may be surprising, especially because the system with is strictly more powerful.     

Unextendible Product Bases and Locally Unconvertible Bound Entangled States

Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs S and T the associated bound entangled states _S and _T cannot be converted to each other by LOCC, unless S and T coincide up to local unitaries. More specifically, there exists a finite precision (S,T) > 0 such that for any LOCC protocol mapping _S into a probabilistic ensemble (p, ), the fidelity between _T and any possible final state satisfies F(_T, ) = 1 - (S,T).PACS: 03.65.Bz; 03.67.-a; 89.70+c.     

On the accuracy of musical data, with examples from Gregorian chant and German folksong

Attention is drawn to the need for controlling (during encoding) and checking (after encoding) the quality or accuracy of musical data. Some large databases of melodies are now becoming available, and methods of control and checking are presented which are specially suited to these. Two applications are discussed in detail: to Gregorian Chant and to German folksong. An effective method in tonal and modal music is found to be the investigation of melodic progressions which remain unusual even after amalgamation by transposition to a central register. Dr. Nigel Nettheim is Senior Research Associate in the Centre for Liberal and General Studies, University of New South Wales, Australia. His research combines Mathematical Statistics and Analytical Musicology. Publications include On the Spectral Analysis of Melody, Interface,21 (1992), and The Pulse in German Folksong: A Statistical Investigation, Musikometrika5, (to appear). The author thanks the anonymous reviewers for helpful remarks.     

Software engineering education: issues and alternatives

Postgraduate degree programs in software engineering have been in existence for some time and now undergraduate degree programs with this title are beginning to appear. A number of questions and issues of only moderate importance with respect to the postgraduate programs now become critical and of overriding importance. While one could (and did) by and large gloss over these issues earlier, this will be much more difficult, and probably impossible, in the future. These issues must be resolved in a wider context than that in which they have been dealt with before, involving engineering in the universities and professional engineering societies in the larger social context. Much of the disagreement regarding these issues can, ultimately, be traced back to differing fundamental views and definitions of the term engineering and whether software engineering should be treated as just another engineering discipline or in a significantly and fundamentally different way. After examining two different definitions and views of engineering, this paper states and discusses a number of the questions and issues which planners of new undergraduate software engineering degree programs must deal with. Some pros and cons of various alternative answers are presented and a few answers suggested. The issues discussed here begin with questions of general policy and concept (e.g., which culture – that of the traditional engineering disciplines or that of computer science – should be instilled in the students in these new degree programs), include organizational matters (e.g., the position of the new degree program in the university hierarchy) and end with selected, more detailed questions regarding the curriculum (e.g., how to deal with programming, design and management topics). It is conjectured that, although the scientific foundation for the undergraduate software engineering degree programs must and will come from computer science, their culture and orientation must come from engineering if they and their graduates are to be successful in satisfying society's needs in the long term.