Foliations for solving equations in groups: free, virtually free, and hyperbolic groups

We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips classification of measured band complexes. Using canonical representatives, we deduce an algorithm for solving equations and inequations in hyperbolic groups (maybe with torsion). Additionnally, we can deal with quasi-isometrically embeddable rational constraints.Comment: 70 pages, 7 figures, revised version. To appear in Journal of Topolog

The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases

We briefly survey the basic concepts and results concerning the computational power of neural net-orks which basically depends on the information content of eight parameters. In particular, recurrent neural networks with integer, rational, and arbitrary real weights are classi ed within the Chomsky and finer complexity hierarchies. Then we re ne the analysis between integer and rational weights by investigating an intermediate model of integer-weight neural networks with an extra analog rational-weight neuron (1ANN). We show a representation theorem which characterizes the classification problems solvable by 1ANNs, by using so-called cut languages. Our analysis reveals an interesting link to an active research field on non-standard positional numeral systems with non-integer bases. Within this framework, we introduce a new concept of quasi-periodic numbers which is used to classify the computational power of 1ANNs within the Chomsky hierarchy

Nonmonotonic Probabilistic Logics between Model-Theoretic Probabilistic Logic and Probabilistic Logic under Coherence

Recently, it has been shown that probabilistic entailment under coherence is weaker than model-theoretic probabilistic entailment. Moreover, probabilistic entailment under coherence is a generalization of default entailment in System P. In this paper, we continue this line of research by presenting probabilistic generalizations of more sophisticated notions of classical default entailment that lie between model-theoretic probabilistic entailment and probabilistic entailment under coherence. That is, the new formalisms properly generalize their counterparts in classical default reasoning, they are weaker than model-theoretic probabilistic entailment, and they are stronger than probabilistic entailment under coherence. The new formalisms are useful especially for handling probabilistic inconsistencies related to conditioning on zero events. They can also be applied for probabilistic belief revision. More generally, in the same spirit as a similar previous paper, this paper sheds light on exciting new formalisms for probabilistic reasoning beyond the well-known standard ones.Comment: 10 pages; in Proceedings of the 9th International Workshop on Non-Monotonic Reasoning (NMR-2002), Special Session on Uncertainty Frameworks in Nonmonotonic Reasoning, pages 265-274, Toulouse, France, April 200

Word formation in contemporary Liangmai: A morphological study

This paper attempts to discuss the different processes of word formation in contemporary Liangmai, a Tibeto-Burman (TB) language of the Kuki-Chin-Naga sub group (Bradley 1997). The language is spoken by around 50,000 speakers in the state of Manipur and Nagaland, in the northeastern part of India. This paper discusses a detailed description of the word formation processes that are relevant in Liangmai, namely affixation, compounding and reduplication. Like the other TB languages of the region, Liangmai is an agglutinative language in which almost all the syllable boundary corresponds to morpheme boundary. Most of the Liangmai words are monosyllabic. In the case of disyllabic/polysyllabic words, various morphemes which composed the word are easily segmentable

The reals as rational Cauchy filters

We present a detailed and elementary construction of the real numbers from the rational numbers a la Bourbaki. The real numbers are defined to be the set of all minimal Cauchy filters in $\mathbb{Q}$ (where the Cauchy condition is defined in terms of the absolute value function on $\mathbb{Q}$) and are proven directly, without employing any of the techniques of uniform spaces, to form a complete ordered field. The construction can be seen as a variant of Bachmann's construction by means of nested rational intervals, allowing for a canonical choice of representatives

The complexity of finite-valued CSPs

We study the computational complexity of exact minimisation of rational-valued discrete functions. Let $\Gamma$ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, $\operatorname{VCSP}(\Gamma)$, is the problem of minimising a function given as a sum of functions from $\Gamma$. We establish a dichotomy theorem with respect to exact solvability for all finite-valued constraint languages defined on domains of arbitrary finite size. We show that every constraint language $\Gamma$ either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of $\operatorname{VCSP}(\Gamma)$ exactly, or $\Gamma$ satisfies a simple hardness condition that allows for a polynomial-time reduction from Max-Cut to $\operatorname{VCSP}(\Gamma)$