Ray shooting on triangles in 3-space

We present a uniform approach to problems involving lines in 3-space. This approach is based on mapping lines inR ³ into points and hyperplanes in five-dimensional projective space (Plücker space). We obtain new results on the following problems:

1. Preprocessn triangles so as to answer efficiently the query: “Given a ray, which is the first triangle hit?” (Ray- shooting problem). We discuss the ray-shooting problem for both disjoint and nondisjoint triangles.
2. Construct the intersection of two nonconvex polyhedra in an output sensitive way with asubquadratic overhead term.
3. Construct the arrangement ofn intersecting triangles in 3-space in an output-sensitive way, with asubquadratic overhead term.
4. Efficiently detect the first face hit by any ray in a set of axis-oriented polyhedra.
5. Preprocessn lines (segments) so as to answer efficiently the query “Given two lines, is it possible to move one into the other without crossing any of the initial lines (segments)?” (Isotopy problem). If the movement is possible produce an explicit representation of it.

     

Drawing planar graphs using the canonical ordering

We introduce a new method to optimize the required area, minimum angle, and number of bends of planar graph drawings on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear-time-and-space algorithms can be designed for many graph-drawing problems. Our main results are as follows:

Every triconnected planar graphG admits a planar convex grid drawing with straight lines on a (2n?4)×(n?2) grid, wheren is the number of vertices.

Every triconnected planar graph with maximum degree 4 admits a planar orthogonal grid drawing on ann×n grid with at most [3n/2]+4 bends, and ifn>6, then every edge has at most two bends.

Every planar graph with maximum degree 3 admits a planar orthogonal grid drawing with at most [n/2]+1 bends on an [n/2]×[n/2] grid.

Every triconnected planar graphG admits a planar polyline grid drawing on a (2n?6)×(3n?9) grid with minimum angle larger than 2/d radians and at most 5n?15 bends, withd the maximum degree.

These results give in some cases considerable improvements over previous results, and give new bounds in other cases. Several other results, e.g., concerning visibility representations, are included.     

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.     

Stable semantics for disjunctive programs

We introduce the stable model semantics fordisjunctive logic programs and deductive databases, which generalizes the stable model semantics, defined earlier for normal (i.e., non-disjunctive) programs. Depending on whether only total (2-valued) or all partial (3-valued) models are used we obtain thedisjunctive stable semantics or thepartial disjunctive stable semantics, respectively. The proposed semantics are shown to have the following properties:

? For normal programs, the disjunctive (respectively, partial disjunctive) stable semantics coincides with thestable (respectively,partial stable) semantics.

? For normal programs, the partial disjunctive stable semantics also coincides with thewell-founded semantics.

? For locally stratified disjunctive programs both (total and partial) disjunctive stable semantics coincide with theperfect model semantics.

? The partial disjunctive stable semantics can be generalized to the class ofall disjunctive logic programs.

? Both (total and partial) disjunctive stable semantics can be naturally extended to a broader class of disjunctive programs that permit the use ofclassical negation.

? After translation of the programP into a suitable autoepistemic theory \( \hat P \) the disjunctive (respectively, partial disjunctive) stable semantics ofP coincides with the autoepistemic (respectively, 3-valued autoepistemic) semantics of \( \hat P \) .

     

Risoluzione di un sistema di equazioni non lineari con metodo numerico ai minimi quadrati

We introduce a numerical method for solving nonlinear algebraic equations related to linear least squares fitting with moving weights. The method considered introduces little overhead in the process. The approach is based on updating two matrices so that the product is an approximation of the inverse Jacobian. We may expect:

1. The tecnique performs good for stochastic systems
2. We never may yield an unbounded correction
3. We can use the algorithm for ill-conditioned, also completely singular problems.

Numerical results are given. It seems that experimental efficiency of the method compares well with mostly used iterative algorithms.     

Separability and one-way functions

We settle all relativized questions of the relationships between the following five propositions:

• P = NP.
• P = UP.
• P = NP $\cap$ coNP.
• All disjoint pairs of NP sets are P-separable.
• All disjoint pairs of coNP sets are P-separable.

We make the first widespread use of variations of generic oracles to achieve the necessary relativized worlds.     

On Bounded Rational Trace Languages

In this paper, for a finitely generated monoid M, we tackle the following three questions:

1. Is it possible to give a characterization of rational subsets of M which have polynomial growth?
2. What is the structure of the counting function of rational sets which have polynomial growth?
3. Is it true that every rational subset of M has either exponential growth or it has polynomial growth? Can one decide for a given rational set which of the two options holds?

We give a positive answer to all the previous questions in the case that M is a direct product of free monoids. Some of the proved results also extend to trace monoid.     

Integrated systems I: Design principles

The following design principles are being used in an ongoing project to realize an integrated family of rule based systems that can be easily used separately or together in different combinations to solve problems common to many different disciplines. Some essential features of this family are:

1. Individual members can be used in the normal way as user-friendly rule based systems or they can be transparently invoked by other user-friendly rule based systems without interrogating users.
2. The knowledge (or rule) bases of key members do not mimic the perceived mode of human thought; therefore, they can predict events that cannot be predicted by the state-of-the-art alone.
3. The Law of Conservation of Mass/Energy is used to detect and correct computational errors.

     

Contrary to time conditionals in Talmudic logic

We consider conditionals of the form A ? B where A depends on the future and B on the present and past. We examine models for such conditional arising in Talmudic legal cases. We call such conditionals contrary to time conditionals. Three main aspects will be investigated:

1. Inverse causality from future to past, where a future condition can influence a legal event in the past (this is a man made causality).
2. Comparison with similar features in modern law.
3. New types of temporal logics arising from modelling the Talmudic examples.

We shall see that we need a new temporal logic,which we call Talmudic temporal logic with linear open advancing future and parallel changing past, based on two parameters for time.     

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 globally convergent algorithm based on imbedding and parametric optimization

The continuation method, well-established for the solution of nonlinear equations is extended to restricted optimization problems. Only the locally active restrictions are considered along the homotopy path. It is assumed that there are only finitely many critical points, i. e. that there are only finitely many changes of the index set of active restrictions. The globally convergent algorithm which we present proceeds in three stages:

1. Within each stability region, the solution is computed by the classical continuation method.
2. On the boundary of a stability region, a critical point \(\bar t\) is determined.
3. A new active index set is determined when \(\bar t\) is passed.

For the class of convex problems, the hypotheses for the convergence of the algorithm may be secured. The algorithm is applied to several examples.     

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.

     

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.

     

Kernel language KLND

This paper presents a kernel language KLND on the basis of analysing the kernel languagerequirements of new generation computer systems. These requirements are: the ability ofknow-ledge processing, the parallelism, the elegant mathematical properties of the comput-ation model which is appropriate for working as the basis of the novel architecture design, andthe suitability for writing large scale softwares. The main features of KLND are as follows: 1. several new language concepts. 2. the modularity, 3. the unification of logical and functional programming styles, 4. the exploitation of the parallelism. 5. the introduction of the type concept, 6. the introduction of the storage concept.     

String Indexing for Patterns with Wildcards

We consider the problem of indexing a string t of length n to report the occurrences of a query pattern p containing m characters and j wildcards. Let occ be the number of occurrences of p in t, and σ the size of the alphabet. We obtain the following results.

• A linear space index with query time O(m+σ ^j loglogn+occ). This significantly improves the previously best known linear space index by Lam et al. (in Proc. 18th ISAAC, pp. 846–857, [2007]), which requires query time Θ(jn) in the worst case.
• An index with query time O(m+j+occ) using space \(O(\sigma^{k^{2}} n \log^{k} \log n)\) , where k is the maximum number of wildcards allowed in the pattern. This is the first non-trivial bound with this query time.
• A time-space trade-off, generalizing the index by Cole et al. (in Proc. 36th STOC, pp. 91–100, [2004]).

We also show that these indexes can be generalized to allow variable length gaps in the pattern. Our results are obtained using a novel combination of well-known and new techniques, which could be of independent interest.     

RaumComputer

The RoomComputer is an embedded system and as such offers unprecedented chances to manage buildings. Several RoomComputers can be networked via the Intra-/Internet, which makes it possible to monitor, control, and manage rooms and buildings on a unified worldwide accessible platform, irrespective of any particular local technology. It can be easily installed in any building and gives access to a full set of services. It implements a distributed system, which provides secure and controlled access to services like

1. control of light, heating, ventilation, air and climate
2. communication facilities like unified messaging, telephone, fax, etc.
3. reservation of rooms and required resources
4. localization of persons and equipment within rooms and buildings
5. entrance control (i.e. locking/unlocking doors)
6. organization of maintenance and house keeping, and
7. charging and billing.

     

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.     

Trial and error

A pac-learning algorithm isd-space bounded, if it stores at mostd examples from the sample at any time. We characterize thed-space learnable concept classes. For this purpose we introduce the compression parameter of a concept classb and design our Trial and Error Learning Algorithm. We show: b isd-space learnable if and only if the compression parameter ofb is at mostd. This learning algorithm does not produce a hypothesis consistent with the whole sample as previous approaches e.g. by Floyd, who presents consistent space bounded learning algorithms, but has to restrict herself to very special concept classes. On the other hand our algorithm needs large samples; the compression parameter appears as exponent in the sample size. We present several examples of polynomial time space bounded learnable concept classes:

- all intersection closed concept classes with finite VC-dimension.

- convexn-gons in ?².

- halfspaces in ?ⁿ.

- unions of triangles in ?².

We further relate the compression parameter to the VC-dimension, and discuss variants of this parameter.