On equations over sets of integers

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in T} and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.Comment: 12 apges, 0 figure

On the nature of continuous physical quantities in classical and quantum mechanics

Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller (Journal of Philosophy, 1979), have thought that the answer to this question is No -- that the status of individual continuous quantities is very different in quantum mechanics than in classical mechanics. On the contrary, I shall show that the same subtle issues arise with respect to continuous quantities in classical and quantum mechanics; and that it is, after all, possible to describe a particle as possessing a sharp position value without altering the standard formalism of quantum mechanics.Comment: 26 pages, LaTe

Algorithmic iteration for computational intelligence

Machine awareness is a disputed research topic, in some circles considered a crucial step in realising Artificial General Intelligence. Understanding what that is, under which conditions such feature could arise and how it can be controlled is still a matter of speculation. A more concrete object of theoretical analysis is algorithmic iteration for computational intelligence, intended as the theoretical and practical ability of algorithms to design other algorithms for actions aimed at solving well-specified tasks. We know this ability is already shown by current AIs, and understanding its limits is an essential step in qualifying claims about machine awareness and Super-AI. We propose a formal translation of algorithmic iteration in a fragment of modal logic, formulate principles of transparency and faithfulness across human and machine intelligence, and consider the relevance to theoretical research on (Super)-AI as well as the practical import of our results

Distributed representations of the "preparatory set" in the frontal oculomotor system: a TMS study

<p>Abstract</p> <p>Background</p> <p>The generation of saccades is influenced by the level of "preparatory set activity" in cortical oculomotor areas. This preparatory activity can be examined using the gap-paradigm in which a temporal gap is introduced between the disappearance of a central fixation target and the appearance of an eccentric target.</p> <p>Methods</p> <p>Ten healthy subjects made horizontal pro- or antisaccades in response to lateralized cues after a gap period of 200 ms. Single-pulse transcranial magnetic stimulation (TMS) was applied to the dorsolateral prefrontal cortex (DLPFC), frontal eye field (FEF), or supplementary eye field (SEF) of the right hemisphere 100 or 200 ms after the disappearance of the fixation point. Saccade latencies were measured to probe the disruptive effect of TMS on saccade preparation. In six individuals, we gave realistic sham TMS during the gap period to mimic auditory and somatosensory stimulation without stimulating the cortex.</p> <p>Results</p> <p>TMS to DLPFC, FEF, or SEF increased the latencies of contraversive pro- and antisaccades. This TMS-induced delay of saccade initiation was particularly evident in conditions with a relatively high level of preparatory set activity: The increase in saccade latency was more pronounced at the end of the gap period and when participants prepared for prosaccades rather than antisaccades. Although the "lesion effect" of TMS was stronger with prefrontal TMS, TMS to FEF or SEF also interfered with the initiation of saccades. The delay in saccade onset induced by real TMS was not caused by non-specific effects because sham stimulation shortened the latencies of contra- and ipsiversive anti-saccades, presumably due to intersensory facilitation.</p> <p>Conclusion</p> <p>Our results are compatible with the view that the "preparatory set" for contraversive saccades is represented in a distributed cortical network, including the contralateral DLPFC, FEF and SEF.</p

Connectivity of the Primate Superior Colliculus Mapped by Concurrent Microstimulation and Event-Related fMRI

Background: Neuroanatomical studies investigating the connectivity of brain areas have heretofore employed procedures in which chemical or viral tracers are injected into an area of interest, and connected areas are subsequently identified using histological techniques. Such experiments require the sacrifice of the animals and do not allow for subsequent electrophysiological studies in the same subjects, rendering a direct investigation of the functional properties of anatomically identified areas impossible. Methodology/Principal Findings: Here, we used a combination of microstimulation and fMRI in an anesthetized monkey preparation to study the connectivity of the superior colliculus (SC). Microstimulation of the SC resulted in changes in the blood oxygenation level-dependent (BOLD) signals in the SC and in several cortical and subcortical areas consistent with the known connectivity of the SC in primates. Conclusions/Significance: These findings demonstrates that the concurrent use of microstimulation and fMRI can be used to identify brain networks for further electrophysiological or fMRI investigation

Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations

The new digital economy has renewed interest in how digital agents can innovate. This follows the legacy of John von Neumann dynamical systems theory on complex biological systems as computation. The Gödel-Turing-Post (GTP) logic is shown to be necessary to generate innovation based structure changing Type 4 dynamics of the Wolfram-Chomsky schema. Two syntactic procedures of GTP logic permit digital agents to exit from listable sets of digital technologies to produce novelty and surprises. The first is meta-analyses or offline simulations. The second is a fixed point with a two place encoding of negation or opposition, referred to as the Gödel sentence. It is postulated that in phenomena ranging from the genome to human proteanism, the Gödel sentence is a ubiquitous syntactic construction without which escape from hostile agents qua the Liar is impossible and digital agents become entrained within fixed repertoires. The only recursive best response function of a 2-person adversarial game that can implement strategic innovation in lock-step formation of an arms race is the productive function of the Emil Post [58] set theoretic proof of the Gödel incompleteness result. This overturns the view of game theorists that surprise and innovation cannot be a Nash equilibrium of a game