Perspex machine VII: The universal perspex machine

The perspex machine arose from the unification of projective geometry with the Turing machine. It uses a total arithmetic, called transreal arithmetic, that contains real arithmetic and allows division by zero. Transreal arithmetic is redefined here. The new arithmetic has both a positive and a negative infinity which lie at the extremes of the number line, and a number nullity that lies off the number line. We prove that nullity, 0/0, is a number. Hence a number may have one of four signs: negative, zero, positive, or nullity. It is, therefore, impossible to encode the sign of a number in one bit, as floating-, point arithmetic attempts to do, resulting in the difficulty of having both positive and negative zeros and NaNs. Transrational arithmetic is consistent with Cantor arithmetic. In an extension to real arithmetic, the product of zero, an infinity, or nullity with its reciprocal is nullity, not unity. This avoids the usual contradictions that follow from allowing division by zero. Transreal arithmetic has a fixed algebraic structure and does not admit options as IEEE, floating-point arithmetic does. Most significantly, nullity has a simple semantics that is related to zero. Zero means "no value" and nullity means "no information." We argue that nullity is as useful to a manufactured computer as zero is to a human computer. The perspex machine is intended to offer one solution to the mind-body problem by showing how the computable aspects of mind and. perhaps, the whole of mind relates to the geometrical aspects of body and, perhaps, the whole of body. We review some of Turing's writings and show that he held the view that his machine has spatial properties. In particular, that it has the property of being a 7D lattice of compact spaces. Thus, we read Turing as believing that his machine relates computation to geometrical bodies. We simplify the perspex machine by substituting an augmented Euclidean geometry for projective geometry. This leads to a general-linear perspex-machine which is very much easier to pro-ram than the original perspex-machine. We then show how to map the whole of perspex space into a unit cube. This allows us to construct a fractal of perspex machines with the cardinality of a real-numbered line or space. This fractal is the universal perspex machine. It can solve, in unit time, the halting problem for itself and for all perspex machines instantiated in real-numbered space, including all Turing machines. We cite an experiment that has been proposed to test the physical reality of the perspex machine's model of time, but we make no claim that the physical universe works this way or that it has the cardinality of the perspex machine. We leave it that the perspex machine provides an upper bound on the computational properties of physical things, including manufactured computers and biological organisms, that have a cardinality no greater than the real-number line.

Extreme value distribution for singular measures

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems that have a singular measure. Using the block maxima approach described in Faranda et al. [2011] we show that, numerically, the Extreme Value distribution for these maps can be associated to the Generalised Extreme Value family where the parameters scale with the information dimension. The numerical analysis are performed on a few low dimensional maps. For the middle third Cantor set and the Sierpinskij triangle obtained using Iterated Function Systems, experimental parameters show a very good agreement with the theoretical values. For strange attractors like Lozi and H\`enon maps a slower convergence to the Generalised Extreme Value distribution is observed. Even in presence of large statistics the observed convergence is slower if compared with the maps which have an absolute continuous invariant measure. Nevertheless and within the uncertainty computed range, the results are in good agreement with the theoretical estimates.

Non-Refoulement between 'Common Article 1' and 'Common Article 3'

This article considers whether, in the context of armed conflicts, certain non-refoulement obligations of non-belligerent States can be derived from the 1949 Geneva Conventions. According to Common Article 1 (CA1) thereof, all High Contracting Parties (HCPs) undertake to ‘respect and to ensure respect’ for the four conventions ‘in all circumstances’. It is contended that CA1 applies both in international armed conflicts (IACs) and in non-international armed conflicts (NIACs). In turn, it is suggested that Common Article 3 (CA3) which regulates conduct in NIACs serves as a ‘minimum yardstick’ also applicable in IACs. It is widely (though not uniformly) acknowledged that the undertaking to ‘ensure respect’ in a given armed conflict extends to HCPs that are not parties to it; nevertheless, the precise scope of this undertaking is subject to scholarly debate. This article concerns situations where, in the course of an (international or non-international) armed conflict, persons ’taking no active part in hostilities’ flee from States where violations of CA3 are (likely to be) occurring to a non-belligerent State. Based on the undertaking in CA1, the central claim of this article is that, as long as risk of exposure to these violations persists, persons should not be refouled notwithstanding possible assessment of whether they qualify as refugees based on the 1951 Refugee Convention definition, or could be eligible for complementary or subsidiary forms of protection that are regulated in regional arrangements. The analysis does not affect the explicit protection from refoulement that the Fourth Geneva Convention accords to ‘protected persons’ (as defined in Article 4 thereof). It is submitted that CA1 should be read in tandem with other obligations of non-belligerent States under the 1949 Geneva Conventions. Most pertinently, all HCPs are required to take specific measures to repress ‘grave breaches’ and to take measures necessary for the suppression of all acts contrary to the 1949 Geneva Conventions other than the grave breaches. A HCP that is capable of protecting displaced persons from exposure to risks of violations of CA3 and nonetheless refoules them to face such risks is arguably failing to take lawful measures at its disposal in order to suppress acts contrary to the conventions and, consequently, fails to ‘ensure respect’ for the conventions. KEYWORDS Non-refoulement; International Armed Conflict; Non-International Armed Conflict; Common Article 1; Common Article 3