It is shown that the counting function of n Boolean variables can be implemented with the formulae of size O(n3.06) over the basis of all 2-input Boolean functions and of size O(n4.54) over the standard basis. The same bounds follow for the complexity of any threshold symmetric function of n variables and particularly for the majority function. Any bit of the product of binary numbers of length...

The UK's political constitution rests on the checking and balancing operations of a representative system in which parliament is sovereign. By contrast, referendums are often considered instances popular sovereignty. Critics condemn them as populist appeals to singular will people that risk majority tyranny, supporters believe they allow citizens check balance elitism politicians. Such argument...

The first contribution of this article is a politico-philosophical map that, drawing upon two common sets arguments against modern natural rights, might help to explain the prevailing neo-republican position on rights. Under label ‘abstraction argument’, we explore view that rights are metaphysical construct usually ends in violent application speculative principles society. ‘self-interest disc...

1. The Andre'ev (1987) bound: explicit function with size f = ˜ Ω(n 5 2) 2. The relationship between formula size and circuit depth, and how to accomplish " depth reduction " for a general circuit 3. Monotone formulas for monotone functions (example: monotone formula for the majority function MAJ)

Let X be a linearly ordered set of arbitrary size (finite or infinite). Natural functions on such a set one can define using the linear order include maximum, minimum and median functions. While it is clear what the clone generated by the maximum or the minimum looks like, this is not obvious for median functions. We show that every clone on X contains either no median function or all median fu...