arXiv subjects
Publications and source records attributed to Martin Ziegler.
We prove that every many-sorted $ω$-categorical theory is completely interpretable in a one-sorted $ω$-categorical theory. As an application, we give a short proof of the existence of non $G$--compact $ω$-categorical theories.
We introduce a new notion of computable function on $\R^N$ and prove some basic properties. We give two applications, first a short proof of Yoshinaga's theorem that periods are \el (they are actually low). We also show that the low complex numbers form a algebraically closed field closed under exponentiation and some other special functions.
It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f:A->B, additional structural information about the input x in A (that is any kind of promise that x belongs to a certain subset A' of A) should be taken advantage of. Some examples from real number computation show that such discrete advice can even make the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problems how much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation A*x=0 for a given singular real NxN-matrix A is possible when knowing rank(A)=0,1,...,N-1; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a BASIS of eigenvectors of) a given real symmetric NxN-matrix is possible when knowing the number of distinct eigenvalues: an integer between 1 and N (the latter corresponding to the nondegenerate case). And again we show that N-fold (i.e. roughly log N bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas for finding SOME SINGLE eigenvector of A, providing the truncated binary logarithm of the least-dimensional eigenspace of A--i.e. Theta(log N)-fold advice--is sufficient and optimal.
Inspired by Hofstadter's Coffee-House Conversation (1982) and by the science fiction short story SAM by Schattschneider (1988), we propose and discuss criteria for non-mechanical intelligence. Firstly, we emphasize the practical need for such tests in view of massively multiuser online role-playing games (MMORPGs) and virtual reality systems like Second Life. Secondly, we demonstrate Second Life as a useful framework for implementing (some iterations of) that test.
We propose a concise approximate description, and a method for efficiently obtaining this description, via adaptive random sampling of the performance (running time, memory consumption, or any other profileable numerical quantity) of a given algorithm on some low-dimensional rectangular grid of inputs. The formal correctness is proven under reasonable assumptions on the algorithm under consideration; and the approach's practical benefit is demonstrated by predicting for which observer positions and viewing directions an occlusion culling algorithm yields a net performance benefit or loss compared to a simple brute force renderer.
For a fixed virtual scene (=collection of simplices) S and given observer position p, how many elements of S are weakly visible (i.e. not fully occluded by others) from p? The present work explores the trade-off between query time and preprocessing space for these quantities in 2D: exactly, in the approximate deterministic, and in the probabilistic sense. We deduce the EXISTENCE of an O(m^2/n^2) space data structure for S that, given p and time O(log n), allows to approximate the ratio of occluded segments up to arbitrary constant absolute error; here m denotes the size of the Visibility Graph--which may be quadratic, but typically is just linear in the size n of the scene S. On the other hand, we present a data structure CONSTRUCTIBLE in O(n*log(n)+m^2*polylog(n)/k) preprocessing time and space with similar approximation properties and query time O(k*polylog n), where k<n is an arbitrary parameter. We describe an implementation of this approach and demonstrate the practical benefit of the parameter k to trade memory for query time in an empirical evaluation on three classes of benchmark scenes.
We turn `the' Church-Turing Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and well-defined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the Church-Turing Hypothesis (CTH), suggests to study the CTH relative to an arbitrary but specific physical theory--rather than vaguely referring to ``nature'' in general. To this end we combine (and compare) physical structuralism with (models of computation in) complexity theory. The benefit of this formal framework is illustrated by reporting on some previous, and giving one new, example result(s) of computability and complexity in computational physics.
Kolmogorov Complexity constitutes an integral part of computability theory, information theory, and computational complexity theory -- in the discrete setting of bits and Turing machines. Over real numbers, on the other hand, the BSS-machine (aka real-RAM) has been established as a major model of computation. This real realm has turned out to exhibit natural counterparts to many notions and results in classical complexity and recursion theory; although usually with considerably different proofs. The present work investigates similarities and differences between discrete and real Kolmogorov Complexity as introduced by Montana and Pardo (1998).
The unit cost model is both convenient and largely realistic for describing integer decision algorithms over (+,*). Additional operations like division with remainder or bitwise conjunction, although equally supported by computing hardware, may lead to a considerable drop in complexity. We show a variety of concrete problems to benefit from such NON-arithmetic primitives by presenting and analyzing corresponding fast algorithms.
The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskii, Tseitin, Kreisel, and Lacombe assert the existence of NON-empty co-r.e. closed sets devoid of computable points: sets which are `large' in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every non-empty co-r.e. closed real set without computable points has continuum cardinality. This leads us to investigate for various classes of computable real subsets whether they necessarily contain a (not necessarily effectively findable) computable point.
By the sometimes so-called MAIN THEOREM of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the JUMP of a representation: both a TTE-counterpart to the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of hypercomputation: and a formalization of revising computation in the sense of Shoenfield. We also consider Markov and Banach/Mazur oracle-computation of discontinuous fu nctions and characterize the computational power of Type-2 nondeterminism to coincide with the first level of the Analytical Hierarchy.
In a stable abelian group, we characterize generic types of cosets of type-definable subgroups.
The word problem for discrete groups is well-known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real extension of the word problem for a certain class of groups which are presented as quotient groups of a free group and a normal subgroup. Most important, the free group will be generated by an uncountable set of generators with index running over certain sets of real numbers. This allows to include many mathematically important groups which are not captured in the framework of the classical word problem. Our contribution extends computational group theory from the discrete to the Blum-Shub-Smale (BSS) model of real number computation. We believe this to be an interesting step towards applying BSS theory, in addition to semi-algebraic geometry, also to further areas of mathematics. The main result establishes the word problem for such groups to be not only semi-decidable (and thus reducible FROM) but also reducible TO the Halting Problem for such machines. It thus provides the first non-trivial example of a problem COMPLETE, that is, computationally universal for this model.
We exhibit a simplified version of the construction of a field of Morley rank p with a predicate of rank p-1, extracting the main ideas for the construction from previous papers and refining the arguments. Moreover, an explicit axiomatization is given, and ranks are computed.
In the BCSS model of real number computations we prove a concrete and explicit semi-decidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post's Problem over the reals significantly differs from its classical, discrete variant where advanced diagonalization techniques are only known to yield the existence of such intermediate Turing degrees. Strengthening the above result, we construct (that is, obtain again explicitly) as well an uncountable number of incomparable semi-decidable Turing degrees below the real Halting problem in the BCSS model. Finally we show the same to hold for the linear BCSS model, that is over (R,+,-,<) rather than (R,+,-,*,/,<).
By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability: * relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0''; * encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy; * non-deterministic computation. It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.
For a linearly recurrent vector sequence P[n+1] = A(n) * P[n], consider the problem of calculating either the n-th term P[n] or L<=n arbitrary terms P[n_1],...,P[n_L], both for the case of constant coefficients A(n)=A and for a matrix A(n) with entries polynomial in n. We improve and extend known algorithms for this problem and present new applications for it. Specifically it turns out that for instance * any family (p_n) of classical orthogonal polynomials admits evaluation at given x within O(n^{1/2} log n) operations INDEPENDENT of the family (p_n) under consideration. * For any L indices n_1,...,n_L <= n, the values p_{n_i}(x) can be calculated simultaneously using O(n^{1/2} log n + L log(n/L)) arithmetic operations; again this running time bound holds uniformly. * Every hypergeometric (or, more generally, holonomic) function admits approximate evaluation up to absolute error e>0 within O((log(1/e)^{1/2} loglog(1/e)) -- as opposed to O(log(1/e)) -- arithmetic steps. * Given m and a polynomial p of degree d over a field of characteristic zero, the coefficient of p^m to term X^n can be computed within O(d^2 M(n^{1/2})) steps where M(n) denotes the cost of multiplying two degree-n polynomials. * The same time bound holds for the joint calculation of any L<=n^{1/2} desired coefficients of p^m to terms X^{n_i}, n_1,...,n_L <= n.
A function f is continuous iff the PRE-image f^{-1}[V] of any open set V is open again. Dual to this topological property, f is called OPEN iff the IMAGE f[U] of any open set U is open again. Several classical Open Mapping Theorems in Analysis provide a variety of sufficient conditions for openness. By the Main Theorem of Recursive Analysis, computable real functions are necessarily continuous. In fact they admit a well-known characterization in terms of the mapping V+->f^{-1}[V] being EFFECTIVE: Given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f^{-1}[V]. Analogously, EFFECTIVE OPENNESS requires the mapping U+->f[U] on open real subsets to be effective. By effectivizing classical Open Mapping Theorems as well as from application of Tarski's Quantifier Elimination, the present work reveals several rich classes of functions to be effectively open.