Quantum Non-Linearity: Generating Randomness

Wednesday, November 14, 2007

Generating Randomness

I have been struggling to relate algorithmics to the notion of disorder. In my last post on this subject I suggested that an algorithm that cuts a highly ordered path through a network of incremental configuration changes in order to eventually generate randomness would require a ‘very large’ number of steps in order to get through the network. This result still stands, and it probably applies to a simple algorithm like the counting algorithm, which can be modeled using simple recursive searching of a binary tree network.

But, and here’s the big but, a simple counting algorithm does not respond to the information in the current results string (the results string is the equivalent of the tape in a Turing machine). When an algorithm starts receiving feedback from its ‘tape’, then things change considerably – we then move into the realm of non-linearity.

Given the ‘feedback’ effect and using some assumptions about just what a Turing machine could at best achieve I have obtained a crude formula giving an upper limit on the speed with which a Turing machine could development a random sequence. A random sequence would require at least ‘t’ steps to develop where:

t ~ kDn/log(m)
and where ‘k’ is a constant whose factors I am still pondering, ‘D’ is the length of the sequence, ‘n’ is an arbitrarily chosen segment length whose value depends on the quality of randomness required, and ‘m’ is the number of Turing machine states. This formula is very rough, but it makes use of a frequency profile notion of randomness I have developed elsewhere. The thing to note is that the non-linearity of algorithmics (a consequence of using conditionals) means that the high complexity of randomness could conceivably be developed in linear time. This result was a bit of a surprise as I have never been able to disprove the notion that the high complexity of randomness requires exponential time to develop it – a notion not supported by the speed with which ‘random’ sequences can be generated in computers. Paradoxically, the high complexity of randomness (measured in terms of its frequency profile) may not be computationally complex. On the other hand it may be that what truly makes randomness computationally complex is that for high a quality randomness D and n are required to become ‘exponentially’ large in order to obtain a random sequence with a ‘good’ frequency profile.

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THE IDEAS-EXPERIENCE CONTENTION

“Ideas Versus Experience!" is a slogan expressing the uneasy relation between what we think the world to be and what our actual experience suggests it is. Experience makes or breaks ideas. But the reverse is also true: Well established ideas can influence the recognition, acceptance and even the perception of experience. In short there is a two way dialogue between ideas and experience. Sometimes that dialogue can turn into an argument, even a row.

The success of Science is based on a formalisation of this potentially contentious relationship as it seeks to support or refute theoretical notions by systematically comparing them with experience. In the far less formalized contexts of daily living we probably are using a similar heuristic when we display a tendency to drop ideas that lose the confrontation with experience, and retain those that win. Winning ideas, like the gladiators of the Roman arenas, live to fight another day. This Darwinian slant on the struggle for ideological survival is itself ideology that must, for consistency sake, submit itself to the very process it purports to describe. For example, it should be able to give account of the resilience of traditional theories in the face of contra indicators. And in the extreme, explanation also needs to be given of how conspiracy theorists continue to hold on to a ramifying structure of untested elaborations.

Human theoretical visions are often grand and sweeping, confidently affirming states of affairs that are far beyond immediate sensation. The vast unseen domains covered by our best theories contrasts with the limitations on human perceptual resources, resources that only allow a very sparse experimental sampling of our most ambitious ideas. Even a professional scientist only ever tests a small portion of any theoretical structure. Moreover, our theoretical concepts are ambitious enough to cover unreachable areas like the center of distant stars, sub-atomic dimensions, events long past into history, and very complex inscrutable objects like the human mind and its societies. Thus, science is metaphysical in as much as it has to admit that it covers vast domains inaccessible to testing in practice if not in principle. For the intelligent layman even the keyhole view of the experimental scientist may not be available. So whence comes the authority to affirm ideas that are often so grandiose in their sweep that they make claim to impinge upon the very meaning of life the universe and everything?

The fact is that for most of us the really big ideas diffuse through to us from our societal context, and at most these big ideas are only illuminated here and there, at a very few places, with actual direct experience of the phenomenon they conceptualise. These big ideas float around in the ether-like-media of society in the form of books, lectures, programmes, and the Internet etc, objects that the postmodernist philosophers call "texts".

For someone such as myself the battle to make sense of life was joined as soon as I was aware of mystery. But looking back it is clear to me that I haven't done an honest scientific experiment in the whole of my life. For me the effort to bring sense and integration to the complexity of life boils down to grappling with the many texts of society; These texts carry both big theoretical ideas and raw data from which ideas crystallize. These texts are compared with other texts in the search for coherence and consistency. These texts take the place and role of experiments as one tests theories using texts. These texts are in effect the experience of life. My own modest attempt at grandiose theorising has already been presented to the world. (See this blog "Physics and the Wild Web") and my excuse is that I have simply inherited the theoretical hubris of the human race.

The postmodern philosopher Jacques Derrida said, "There is nothing outside the text". I think he was at least right about the primary role texts play in the life of those like myself whose instinct has been to grapple relentlessly with the riddles of life, and to seek out meaning and go where no man has gone before. But one could equally claim, "There is nothing outside the experience". For one might read the "text" of the clouds in the sky as a predictor of impending weather just as one reads socially generated symbolic configurations as the predictor of certain kinds of thought or potential experience. For the text is just as much an experience as more primeval phenomenon like thunder and lightning. Texts are part of our cosmic experience and they imperceptibly grade into the more conventional notion of experience. Where the crossover point is, is difficult to say.

Postmodernism may have got it right about the primary role of the text, but it has made one very serious mistake. Where I would radically differ from the postmodern view is that I believe the mass of social texts, if we allow them, naturally converge upon grand consistent narratives and those narratives are here to stay. And that is because these texts, I submit, are part of a world with an underlying rational integratedness and that integratedness is being revealed to us bit by bit. For revelation it is: Revelation is what we cannot discover unless God chooses that we discover it. My perceptions and reason, limited though they may be, were not self-made, they had to be discovered as gifts. These gifts are given to all of us; one can but trust and use them. Reason is a grace of Revelation. For this reason the hubris of theory making is justified.

Quantum Non-Linearity

Wednesday, November 14, 2007

Generating Randomness

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THE IDEAS-EXPERIENCE CONTENTION