Interpolation that leads to the narrowest intervals and its application to expert systems and intelligent control

In many real-life situations, we want to reconstruct the dependencyy=f(x ₁,…, x_n) from the known experimental resultsx _i ^(k) , y^(k). In other words, we want tointerpolate the functionf from its known valuesy ^(k)=f(x ₁ ^(k) ,…, x _n ^(k) ) in finitely many points $\bar x^{(k)} = (x_1^{(k)} , \ldots ,x_n^{(k)} )$ , 1≤k≤N There are many functions that go through given points. How to choose one of them? The main goal of findingf is to be able to predicty based onx _i. If we getx _i from measurements, then usually, we only getintervals that containx _i. As a result of applyingf, we get an interval y of possible values ofy. It is reasonable to choosef for which the resulting interval is the narrowest possible. In this paper, we formulate this choice problem in mathematical terms, solve the corresponding problem for several simple cases, and describe the application of these solutions to intelligent control.     

Average time behavior of distributive sorting algorithms

In this paper we investigate the expected complexityE(C) of distributive (“bucket”) sorting algorithms on a sampleX ₁, ...,X _n drawn from a densityf onR ¹. Assuming constant time bucket membership determination and assuming the use of an average timeg(n) algorithm for subsequent sorting within each bucket (whereg is convex,g(n)/n↑∞,g(n)/n ² is nonincreasing andg is independent off), the following is shown:

1. Iff has compact support, then ∫g(f(x))dx<∞ if and only ifE(C)=0(n).
2. Iff does not have compact support, then \(E(C)/n\xrightarrow{n}\infty \) .

No additional restrictions are put onf.     

Interval methods that are guaranteed to underestimate (and the resulting new justification of Kaucher arithmetic)

One of the main objectives of interval computations is, given the functionf(x ₁, ...,x _n ), andn intervals $\bar x_1 ,...,\bar x_n$ , to compute the range $\bar y = f(\bar x_1 ,...,\bar x_n )$ . Traditional methods of interval arithmetic compute anenclosure $Y \supseteq \bar y$ for the desired interval $\bar y$ , an enclosure that is often an overestimation. It is desirable to know how close this enclosure is to the desired range interval. For that purpose, we develop a new interval formalism that produces not only the enclosure, but also theinner estimate for the desired range $\bar y$ , i.e., an interval y such that $y \subseteq \bar y$ . The formulas for this new method turn out to be similar to the formulas of Kaucher arithmetic. Thus, we get a new justification for Kaucher arithmetic.     

Le cylindre des langages linéaires

Define a cylinder to be a family of languages which is closed under inverse homomorphisms and intersection with regular sets. A number of well-known families of languages are cylinders:

—CFL, the family of context-free languages, is a principal cylinder, i.e. the smallest cylinder containing a languageL _O described in [6].

—the family of deterministic context-free languages is proved to be a nonprincipal cylinder in [7].

—the family of unambiguous context-free languages is a cylinder: to prove that it is not principal seems to be a very hard problem.

In this paper we prove that Lin, the family of linear context-free languages, is a nonprincipal cylinder. This is achieved in the standard way by exhibiting a sequence of languages S_n, n∈N, such that Lin is the union of all the principal cylinders generated by these languages and is not the union of any finite number of these cylinders. This leaves open the problem raised by Sheila Greibach of whether there exists a languageL such that every linear context-free language is the image ofL in some inverse gsm mapping.     

On the degree of boolean functions as real polynomials

Every Boolean function may be represented as a real polynomial. In this paper, we characterize the degree of this polynomial in terms of certain combinatorial properties of the Boolean function. Our first result is a tight lower bound of Ω(logn) on the degree needed to represent any Boolean function that depends onn variables. Our second result states that for every Boolean functionf, the following measures are all polynomially related:

o The decision tree complexity off.

o The degree of the polynomial representingf.

o The smallest degree of a polynomialapproximating f in theL _max norm.

     

Online Metric Tracking and Smoothing

We consider the online smoothing problem, in which a tracker is required to maintain distance no more than Δ≥0 from a time-varying signal f while minimizing its own movement. The problem is determined by a metric space (X,d) with an associated cost function c:?→?. Given a signal f ₁,f ₂,…∈X the tracker is responsible for producing a sequence a ₁,a ₂,… of elements of X that meet the proximity constraint: d(f _i ,a _i )≤Δ. To complicate matters, the tracker is on-line—the value a _i may only depend on f ₁,…,f _i —and wishes to minimize the cost of his travels, ∑c(d(a _i ,a _i+1)). We evaluate such tracking algorithms competitively, comparing this with the cost achieved by an optimal adversary apprised of the entire signal in advance. The problem was originally proposed by Yi and Zhang (In: Proceedings of the 20th annual ACM-SIAM symposium on discrete algorithms (SODA), pp. 1098–1107. ACM Press, New York, 2009), who considered the natural circumstance where the metric spaces are taken to be ? ^k with the ? ₂ metric and the cost function is equal to 1 unless the distance is zero (thus the tracker pays a fixed cost for any nonzero motion).

We begin by studying arbitrary metric spaces with the “pay if you move” metric of Yi and Zhang (In: Proceedings of the 20th annual ACM-SIAM symposium on discrete algorithms (SODA), pp. 1098–1107. ACM Press, New York, [2009]) described above and describe a natural randomized algorithm that achieves a O(logb _Δ)-competitive ratio, where b _Δ=max _x∈X |B _Δ(x)| is the maximum number of points appearing in any ball of radius Δ. We show that this bound is tight.

We then focus on the metric space ? with natural families of monotone cost functions c(x)=x ^p for some p≥0. We consider both the expansive case (p≥1) and the contractive case (p<1), and show that the natural lazy algorithm performs well in the expansive case. In the contractive case, we introduce and analyze a novel deterministic algorithm that achieves a constant competitive ratio depending only on p. Finally, we observe that by slightly relaxing the guarantee provided by the tracker, one can obtain natural analogues of these algorithms that work in continuous metric spaces.

     

How many minimal upper bounds of minimal upper bounds

We construct a family such that the maximal elementX _n ∈ ? _n can be obtained inductively by computing minimal upper bounds over already generated setsG, involving not less than exponentially many (with respect to |X _n |) such intermediate setsG, starting with minimal ones in ? _n .     

Optimal interval enclosures for fractionally-linear functions,and their application to intelligent control

One of the main problems of interval computations is, given a functionf(x ₁, ...,x _n ) andn intervals x₁, ..., x _n , to compute the range y=f(x₁, ..., x _n ). This problem is feasible for linear functionsf, but for generic polynomials, it is known to be computationally intractable. Because of that, traditional interval techniques usually compute theenclosure of y, i.e., an interval that contains y. The closer this enclosure to y, the better. It is desirable to describe cases in which we can compute theoptimal enclosure, i.e., the range itself. In this paper, we describe a feasible algorithm for computing the optimal enclosure forfractionally linear functionsf. Applications of this result tointelligent control are described.     

Regular prefix relations

We define a class ofn-ary relations on strings called the regular prefix relations, and give four alternative characterizations of this class:

1. the relations recognized by a new type of automaton, the prefix automata,
2. the relations recognized by tree automata specialized to relations on strings,
3. the relations between strings definable in the second order theory ofk successors,
4. the smallest class containing the regular sets and the prefix relation, and closed under the Boolean operations, Cartesian product, projection, explicit transformation, and concatenation with Cartesian products of regular sets.

We give concrete examples of regular prefix relations, and a pumping argument for prefix automata. An application of these results to the study of inductive inference of regular sets is described.     

Some computational problems in linear algebra as hard as matrix multiplication

We define the complexity of a computational problem given by a relation using the model of computation trees together with the Ostrowski complexity measure. Natural examples from linear algebra are:

KER _n : Compute a basis of the kernel for a givenn×n-matrix,

OGB _n : Find an invertible matrix that transforms a given symmetricn×n-matrix (quadratic form) into diagonal form,

SPR _n : Find a sparse representation of a givenn×n-matrix.

     

Eine stets quadratisch konvergente Modifikation des Steffensen-Verfahrens

It is shown that the following modification of the Steffensen procedurex _n+1=x _n ?k _s (x _n )f(x _n ) (f[x _n ,x _n ?f(x _n )])^?1 (n=0,1,...) withk _s (x)=(1?z _s (x))^?1,z _s (x)=f(x) 2f[x?f(x),x,x+f(x)]×(f[x,x?f(x)])^?2 is quadratically convergent to the root of the equation \(f(x) = (x - \bar x)^p g(x) = 0(p > 0,g(\bar x) \ne 0)\) . Furthermore \(\mathop {\lim }\limits_{n \to \infty } k_s (x_n ) = p\) holds.     

On derivative-based global sensitivity criteria

A mathematical model f(x) given in the unit n-dimensional cube, where x = (x ₁, ..., x _n ), is considered. How can one estimate the global sensitivity of f(x) with respect to x _i ? If f(x) ∈ L ₂, global sensitivity indices help answer this question. Being less reliable, derivative-based sensitivity criteria are sometimes easier-to-compute. In this work, a new derivative-based global sensitivity criterion is compared to the respective global sensitivity index. These estimates are proven to coincide in the particular case when f(x) linearly depends on x _i . However, Monte Carlo approximations to the derivative-based criterion converge faster. Thus, the global derivative-based sensitivity criterion can prove useful when f(x) depends on x _i almost linearly. It can be also used to find nonessential variables x _i .     

Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2)

To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem P _curve of minimizing $\int _{0}^{\ell} \sqrt{\xi^{2} +\kappa^{2}(s)} {\rm d}s $ for a planar curve having fixed initial and final positions and directions. Here κ(s) is the curvature of the curve with free total length ?. This problem comes from a model of geometry of vision due to Petitot (in J. Physiol. Paris 97:265–309, 2003; Math. Inf. Sci. Humaines 145:5–101, 1999), and Citti & Sarti (in J. Math. Imaging Vis. 24(3):307–326, 2006). In previous work we proved that the range $\mathcal{R} \subset\mathrm{SE}(2)$ of the exponential map of the underlying geometric problem formulated on SE(2) consists of precisely those end-conditions (x _fin,y _fin,θ _fin) that can be connected by a globally minimizing geodesic starting at the origin (x _in,y _in,θ _in)=(0,0,0). From the applied imaging point of view it is relevant to analyze the sub-Riemannian geodesics and $\mathcal{R}$ in detail. In this article we

• show that $\mathcal{R}$ is contained in half space x≥0 and (0,y _fin)≠(0,0) is reached with angle π,
• show that the boundary $\partial\mathcal{R}$ consists of endpoints of minimizers either starting or ending in a cusp,
• analyze and plot the cones of reachable angles θ _fin per spatial endpoint (x _fin,y _fin),
• relate the endings of association fields to $\partial\mathcal {R}$ and compute the length towards a cusp,
• analyze the exponential map both with the common arc-length parametrization t in the sub-Riemannian manifold $(\mathrm{SE}(2),\mathrm{Ker}(-\sin\theta{\rm d}x +\cos\theta {\rm d}y), \mathcal{G}_{\xi}:=\xi^{2}(\cos\theta{\rm d}x+ \sin\theta {\rm d}y) \otimes(\cos\theta{\rm d}x+ \sin\theta{\rm d}y) + {\rm d}\theta \otimes{\rm d}\theta)$ and with spatial arc-length parametrization s in the plane $\mathbb{R}^{2}$ . Surprisingly, s-parametrization simplifies the exponential map, the curvature formulas, the cusp-surface, and the boundary value problem,
• present a novel efficient algorithm solving the boundary value problem,
• show that sub-Riemannian geodesics solve Petitot’s circle bundle model (cf. Petitot in J. Physiol. Paris 97:265–309, [2003]),
• show a clear similarity with association field lines and sub-Riemannian geodesics.

     

A note on the average depth of tries

LetA _n be the average root-to-leaf distance in a binary trie formed by the binary fractional expansions ofn independent random variablesX ₁,...,X _n with common densityf on [0, 1). We show thateither E(A _n )=∞ for alln≥2or \(\mathop {\lim }\limits_n E(A_n )/\log _2 n = 1\) depending on whether ∫f ² (x)dx = ∞ or ∫f ² (x)dx < ∞.     

Optimal parallel algorithms for rectilinear link-distance problems

We provide optimal parallel solutions to several link-distance problems set in trapezoided rectilinear polygons. All our main parallel algorithms are deterministic and designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM). LetP be a trapezoided rectilinear simple polygon withn vertices. InO(logn) time usingO(n/logn) processors we can optimally compute:

1. Minimum réctilinear link paths, or shortest paths in theL ₁ metric from any point inP to all vertices ofP.
2. Minimum rectilinear link paths from any segment insideP to all vertices ofP.
3. The rectilinear window (histogram) partition ofP.
4. Both covering radii and vertex intervals for any diagonal ofP.
5. A data structure to support rectilinear link-distance queries between any two points inP (queries can be answered optimally inO(logn) time by uniprocessor).

Our solution to 5 is based on a new linear-time sequential algorithm for this problem which is also provided here. This improves on the previously best-known sequential algorithm for this problem which usedO(n logn) time and space.5 We develop techniques for solving link-distance problems in parallel which are expected to find applications in the design of other parallel computational geometry algorithms. We employ these parallel techniques, for example, to compute (on a CREW PRAM) optimally the link diameter, the link center, and the central diagonal of a rectilinear polygon.     

Balancing Bounded Treewidth Circuits

We use algorithmic tools for graphs of small treewidth to address questions in complexity theory. For our main construction, we prove that multiplicatively disjoint arithmetic circuits of size n ^O(1) and treewidth k can be simulated by bounded fan-in arithmetic formulas of depth O(k ²logn). From this we derive an analogous statement for syntactically multilinear arithmetic circuits, which strengthens the central theorem of M. Mahajan and B.V.R. Rao (Proc. 33rd International Symposium on Mathematical Foundations of Computer Science, vol. 5162, pp. 455–466, 2008). We show our main construction has the following three applications:

• Bounded width arithmetic circuits of size n ^O(1) can be balanced to depth O(logn), provided chains of iterated multiplication in the circuit are of length O(1).
• Boolean bounded fan-in circuits of size n ^O(1) and treewidth k can be simulated by bounded fan-in formulas of depth O(k ²logn). This strengthens in the non-uniform setting the known inclusion that SC⁰?NC¹.
• We demonstrate treewidth restricted cases of Directed-Reachability and Circuit Value Problem that can be solved in LogDCFL.

We also give a construction showing, for both arithmetic and Boolean circuits, that any circuit of size n ^O(1) and treewidth O(log ⁱ n) can be simulated by a circuit of width O(log ⁱ⁺¹ n) and size n ^c , where c=O(1), if i=0, and c=O(loglogn) otherwise.     

One-Nonterminal Conjunctive Grammars over a Unary Alphabet

Systems of equations of the form X _i =φ _i (X ₁,…,X _n ) (1≤ i ≤ n) are considered, in which the unknowns are sets of natural numbers. Expressions φ _i may contain the operations of union, intersection and elementwise addition \(S+T=\{m+n\mid m\in S\), n∈T}. A system with an EXPTIME-complete least solution is constructed in the paper through a complete arithmetization of EXPTIME-completeness. At the same time, it is established that least solutions of all such systems are in EXPTIME. The general membership problem for these equations is proved to be EXPTIME-complete. Among the consequences of the result is EXPTIME-completeness of the compressed membership problem for conjunctive grammars.     

A general theory of translation

The concept of a translation is fundamental to any theory of compiling. Formally, atranslation is any set of pairs of words. Classes of finitely describable translations are considered in general, from the point of view of balloon automata [17, 18, 19]. A translation can be defined by atransducer, a device with an input tape and an output terminal. If, with inputx, the stringy appears at the output terminal, then (x, y) is in the translation defined by the transducer. One can also define a translation by a two input taperecognizer. Ifx andy are placed on the two tapes, the recognizer tells if (x, y) is in the defined translation. One can define closed classes of transducers and recognizers by:

1. restricting the way in which infinite storage may be used (pushdown structure, stack structure, etc.),
2. allowing the finite control to be nondeterministic or deterministic,
3. allowing one way or two way motion on the input tapes.

We have some results on classes of translations which can be categorized roughly into three types.

1. Translations defined by certain classes of transducers and recognizers are equivalent.
2. Translations of a given class are sometimes closed under composition and decomposition with a finite memory translation (gsm mapping).
3. A nondeterministically defined translation can be expressed as the composition of a finitely defined translation and a related deterministically defined translation in many cases.

In addition, ifC is a class of translations, then one can write a compiler-compiler to render any translationT inC and only if the following question is solvable: For any translationT inC and stringx, does there exist ay such that (x, y) is inT? We shall show that, in general, the decidability of this question is equivalent to the decidability of one or more questions from automata theory, depending upon the type of devices defining the classC.     

Nonterminals versus homomorphisms in defining languages for some classes of rewriting systems

Given a rewriting system G (its alphabet, the set of productions and the axiom) one can define the language of G by

1. taking out of all strings generated by G only those which are over a distinguished subalphabet of G, or
2. translating the set of all strings generated by G by a fixed homomorphism.

The “trade-offs” between these two mechanisms for defining languages are discussed for both “parallel” rewriting systems from the developmental systems hierarchy and “sequential” rewriting systems from the Chomsky hierarchy.