Discontinuities in recurrent neural networks

This paper studies the computational power of various discontinuous real computational models that are based on the classical analog recurrent neural network (ARNN). This ARNN consists of finite number of neurons; each neuron computes a polynomial net-function and a sigmoid-like continuous activation-function. The authors introducePostprint (published version

Quantum Kolmogorov Complexity Based on Classical Descriptions

We develop a theory of the algorithmic information in bits contained in an individual pure quantum state. This extends classical Kolmogorov complexity to the quantum domain retaining classical descriptions. Quantum Kolmogorov complexity coincides with the classical Kolmogorov complexity on the classical domain. Quantum Kolmogorov complexity is upper bounded and can be effectively approximated from above under certain conditions. With high probability a quantum object is incompressible. Upper- and lower bounds of the quantum complexity of multiple copies of individual pure quantum states are derived and may shed some light on the no-cloning properties of quantum states. In the quantum situation complexity is not sub-additive. We discuss some relations with ``no-cloning'' and ``approximate cloning'' properties.Comment: 17 pages, LaTeX, final and extended version of quant-ph/9907035, with corrections to the published journal version (the two displayed equations in the right-hand column on page 2466 had the left-hand sides of the displayed formulas erroneously interchanged

Path Checking for MTL and TPTL over Data Words

Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are quantitative extensions of linear temporal logic, which are prominent and widely used in the verification of real-timed systems. It was recently shown that the path checking problem for MTL, when evaluated over finite timed words, is in the parallel complexity class NC. In this paper, we derive precise complexity results for the path-checking problem for MTL and TPTL when evaluated over infinite data words over the non-negative integers. Such words may be seen as the behaviours of one-counter machines. For this setting, we give a complete analysis of the complexity of the path-checking problem depending on the number of register variables and the encoding of constraint numbers (unary or binary). As the two main results, we prove that the path-checking problem for MTL is P-complete, whereas the path-checking problem for TPTL is PSPACE-complete. The results yield the precise complexity of model checking deterministic one-counter machines against formulae of MTL and TPTL