It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time … So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequerboard with all its apparent complexities.
Richard Feynman, 1964.
(T.B. and Vincenzo Ciancia)
Notions of observation play a central role in Physics, but also in Theoretical Computer Science, notably in Process Algebra. The importance in Physics of the mutual influences between observer and observed phenomena is well recognized, and yet the properties of the former are in general fuzzily specified, in spite (or because) of the fact that they may include sophisticated cognitive and operative skills.
Our purpose is to transpose the observer/observed interplay in a simple formal context that greatly facilitates their being treated on an equal footing. We plan to identify observers and observed entities in the context of discrete models of spacetime – in particular, algorithmic causal sets (causets). We shall define simple forms of observer, with associated synchronisation/observation mechanisms, and detect their emergence in dynamic causets. The search can be automated by model checkers using spatial or spatio-temporal logics.
We shall focus on the internal (frog’s) view, as opposed to the external (bird’s) view, by assimilating frogs with simple causet substructures, e.g. fattened causal chains provided with some persistent identity (akin to ‘digital particles’ in cellular automata), that synchronise and communicate with their environment. We shall investigate what ‘observation’ means to these entities and how they might ‘subjectively’ picture their neighborhood or remote environment. These partial observations may be reminiscent, in spirit, of stroboscopic sampling or Poincaré maps.
The emergent features of various models of computation have been widely investigated, notably for the spatio-temporal diagrams of cellular automata, but always under an external viewpoint. Our aim is to restart this analysis under the radically different perspective of an internal (proto-)observer, with a focus on stochastic and deterministic, labelled or unlabelled causets. Simply stated, our goal is to discover qualitative differences between the bird’s and the frog’s view. The issue becomes more challenging when considering different classes of observers of the same phenomena. Identifying their different viewpoints leads to considerations on invariance, akin to Lorentz covariance.
Suitable notions of abstraction shall be considered, since an observer may be blind to the tiniest causet details but sensitive to a coarse-grained version of it. Additionally, ‘smart’ observers may be required to tell apart regular from random-like causet regions.
In our investigation on observations within causets, we will adopt techniques that proved successful in theoretical computer science and process calculi, such as partially ordered Event Structures, and category-theoretical models. Coalgebras, in particular, provide flexibile notions of observation. Causality and locality are achieved using the emerging nominal computation model that endows observers with primitive forms of naming, whose power is balanced by a finite memory principle. We consider it an interesting additional research question to explore the impact of such basic computational constraints on algorithmic causal sets.
Notions of event and event occurrence play a central role in various areas of computer science and ICT (Information and Communication Technology).
In this proposal we are particularly interested in event concepts from process algebras such as Milner’s Calculus of Communicating Systems (CCS) and Hoare’s Communicating Sequential Processes (CSP), and related languages (e.g. LOTOS), since their high abstraction level makes it possible and attractive to investigate the impact of these event notions, and associated constructions, in the apparently remote field of causal sets, intended as discrete models of spacetime.
While, in line with General Relativity, a spacetime event in a causal set is simply a node in a directed, acyclic graph (DAG), in process algebras events represent the building blocks of more elaborate structures called processes. Events may occur ‘internally’ – inside a process – or as part of the interaction with another process; they may be atomic or involve the exchange of data items; they may or may not induce changes in the local states of the interacting parties; they may be organized in temporal/causal patterns and be encapsulated in processes to be used, in turn, as building blocks of more complex event patterns.
In spite of the richer event-related constructions offered by process algebras, the formal, ‘true concurrency semantics’ of the latter maps the syntax of an algebraic specification, no matter how complex, into a DAG, thus providing a common basis for comparing the event patterns arising in the two fields.
In particular, we are interested in exploring algorithmic causal sets – those obtained by deterministic rather than stochastic procedures – and in verifying the extent to which their emergent properties match the structured behavioral patterns typical of process algebra.
One may argue that an event happens when a trace of its occurrence gets recorded somewhere. In process algebra, event occurrences may indeed affect the local states of the interacting processes. Can we meaningfully separate event threads in algorithmic causal sets? Can we detect the emergence of process-like substructures in them – maybe a subgraph with some kind of boundary? Can such ‘processes’ be stateful too? Can we distinguish between their internal and external events? More generally, which subset of the compact set of behavioural operators of process algebras (sequential and parallel composition, choice, etc.) can find a counterpart in the emergent structures of algorithmic causal sets?
“Confronted with a pythagorean jingle derived from simple ratios, a sequence of 23 moves from knot theory, and the interaction between a billiard-ball and a zero-gravity field, a young detective soon realizes that three crimes could have been avoided if math were not so unreasonably effective in describing our physical world. Why is this so? Asimov’s fictional character Prof. Priss confirms to the detective that there is some truth in Tegmark’s Mathematical Universe Hypothesis, and reveals him that all mathematical structures entailing self-aware substructures (SAS) are computable and isomorphic. The boss at the investigation agency is not convinced and proposes his own views on the question.”
This is the abstract of ‘Let’s consider two spherical chickens‘, my contribution to the FQXi 2015 Essay Contest ‘Trick or Truth? The Mysterious Connection Between Physics and Mathematics’, which obtained a Third Prize and the mention for ‘Most Creative Presentation’, out of 203 submissions.
I wish to dedicate this essay and these results to the memory of my father Giampaolo, who was very amused by the spherical chickens concept, referring to the habit of some theoretical physicists to make over-simplifications when developing models of real systems. As a chemist he must have considered himself immune from this attitude!
Excerpt from Gerard ’t Hooft: ‘The Cellular Automaton Interpretation of Quantum Mechanics’ – June 10, 2014
We set up a systematic study of the Cellular Automaton Interpretation of quantum mechanics. We hope to inspire more physicists to do so, to consider seriously the possibility that quantum mechanics as we know it is not a fundamental, mysterious, impenetrable feature of our physical world, but rather an instrument to statistically describe a world where the physical laws, at their most basic roots, are not quantum mechanical at all. Sure, we do not know how to formulate the most basic laws at present, but we are collecting indications that a classical world underlying quantum mechanics does exist. Our models show how to put quantum mechanics on hold when we are constructing models such as string theory and “quantum” gravity, and this may lead to much improved understanding of our world at the Planck scale.
Noi percepiamo chiaramente che soltanto ora incominciamo a raccogliere materiale attendibile per saldare insieme, in un unico complesso, la somma di tutte le nostre conoscenze; ma, d’altro lato, è diventato quasi impossibile per una sola mente il dominare più di un piccolo settore specializzato in tutto ciò. Io non vedo altra via di uscita da questo dilemma (a meno di non rinunciare per sempre al nostro scopo) all’infuori di quella che qualcuno di noi si avventuri a tentare una sintesi di fatti e teorie, pur con una conoscenza di seconda mano e incompleta di alcune di esse, e correre il rischio di farsi ridere dietro. (citato in E. Klein, Sette volte la rivoluzione, 2006)
A demonstration illustrating the exponential growth of space in De Sitter spacetime was added to the Wolfram Demostration site. Find it here.
Quotes from George Ellis’ FQXi 2012 Contest Essay (2nd prize winner)
The degree of complexity that can arise by bottom-up causation alone is strictly limited. Sand piles, the game of life, bird flocks, or any dynamics governed by a local rule  do not compare in complexity with a single cell or an animal body. The same is true in physics: spontaneously broken symmetry is powerful , but not as powerful as symmetry breaking that is guided top-down to create ordered structures (such as brains and computers). Some kind of coordination of effects is needed for such complexity to emerge.
I suggest top-down effects from these [upper] levels is the key to the rise of genuine complexity (such as computers and human beings)
Hypothesis: bottom up emergence by itself is strictly limited in terms of the complexity it can give rise to. Emergence of genuine complexity is characterised by a reversal of information flow from bottom up to top down .
But can we really rule out the possibility for this ‘kind of coordination of effects’ itself to spontaneously emerge in an artificial system such as a cellular automaton? Would it not be possible to observe this type of high, ‘biological’ complexity to emerge in a simulation of an artificial system like Wolfram’s automaton n. 110, provided we are willing to wait for a sufficiently long (likely astronomic) time?