Downloads & Free Reading Options - Results

The parity decision tree model extends the decision tree model by allowing the computation of a parity function in one step. We prove that the deterministic parity decision tree complexity of any Boolean function is polynomially related to the non-deterministic complexity of the function or its complement. We also show that they are polynomially related to an analogue of the block sensitivity. We further study parity decision trees in their relations with an intermediate variant of the decision trees, as well as with communication complexity.

Transient algebra is a multi-valued algebra for hazard detection in gate circuits. Sequences of alternating 0's and 1's, called transients, represent signal values, and gates are modeled by extensions of boolean functions to transients. Formulas for computing the output transient of a gate from the input transients are known for NOT, AND, OR} and XOR gates and their complements, but, in general, even the problem of deciding whether the length of the output transient exceeds a given bound is NP-complete. We propose a method of evaluating extensions of general boolean functions. We introduce and study a class of functions with the following property: Instead of evaluating an extension of a boolean function on a given set of transients, it is possible to get the same value by using transients derived from the given ones, but having length at most 3. We prove that all functions of three variables, as well as certain other functions, have this property, and can be efficiently evaluated.

In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function $f$ is called the inversion complexity of $f$. In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that $\lceil \log_2(n+1) \rceil$ NOT gates are sufficient to compute any Boolean function on $n$ variables. As far as we know, no result is known for inversion complexity in Boolean formulas, i.e., the minimum number of NOT operators in a Boolean formula representing a Boolean function. The aim of this note is showing that we can determine the inversion complexity of every Boolean function in Boolean formulas by arguments based on the study of circuit complexity.

In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function $f$ is called the inversion complexity of $f$. In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that $\lceil \log_2(n+1) \rceil$ NOT gates are sufficient to compute any Boolean function on $n$ variables. As far as we know, no result is known for inversion complexity in Boolean formulas, i.e., the minimum number of NOT operators in a Boolean formula representing a Boolean function. The aim of this note is showing that we can determine the inversion complexity of every Boolean function in Boolean formulas by arguments based on the study of circuit complexity.

The minimum number of NOT gates in a Boolean circuit computing a Boolean function is called the inversion complexity of the function. In 1957, A. A. Markov determined the inversion complexity of every Boolean function and proved that $\lceil\log_{2}(d(f)+1)\rceil$ NOT gates are necessary and sufficient to compute any Boolean function $f$ (where $d(f)$ is maximum number of value changes from 1 to 0 over all increasing chains of tuples of variables values). In this paper we consider Boolean circuits over an arbitrary basis that consists of all monotone functions (with zero weight) and finite nonempty set of non-monotone functions (with unit weight). It is shown that the minimal sufficient for a realization of the Boolean function $f$ number of non-monotone gates is equal to $\lceil\log_{2}(d(f)+1)\rceil - O(1)$. Similar extends of another classical result of A. A. Markov for the inversion complexity of system of Boolean functions has been obtained.

Dies ist das erste Buch, in dem Alfred Wegener seine Theorie der Kontinentalverschiebung darlegt. Zeit seines Lebens wurde diese Theorie größtenteils abgelehnt und geriet nach seinem Tod in Vergessenheit. Erst Jahrzehnte später wurden seine Ideen als wahr erkannt und auf verschiedene Arten nachgewiesen. Alfred Wegener war ein deutscher Meteorologe, Geo- und Polarwissenschaftler. Er starb auf seiner dritten Expedition nach Grönland. (Zusammenfassung von Availle) <br><br> This is the first book - it was read from the first edition 1915 - that describes the idea and basic evidence for continental drift.