305 research outputs found
Lorenz, G\"{o}del and Penrose: New perspectives on determinism and causality in fundamental physics
Despite being known for his pioneering work on chaotic unpredictability, the
key discovery at the core of meteorologist Ed Lorenz's work is the link between
space-time calculus and state-space fractal geometry. Indeed, properties of
Lorenz's fractal invariant set relate space-time calculus to deep areas of
mathematics such as G\"{o}del's Incompleteness Theorem. These properties,
combined with some recent developments in theoretical and observational
cosmology, motivate what is referred to as the `cosmological invariant set
postulate': that the universe can be considered a deterministic dynamical
system evolving on a causal measure-zero fractal invariant set in its
state space. Symbolic representations of are constructed explicitly based
on permutation representations of quaternions. The resulting `invariant set
theory' provides some new perspectives on determinism and causality in
fundamental physics. For example, whilst the cosmological invariant set appears
to have a rich enough structure to allow a description of quantum probability,
its measure-zero character ensures it is sparse enough to prevent invariant set
theory being constrained by the Bell inequality (consistent with a partial
violation of the so-called measurement independence postulate). The primacy of
geometry as embodied in the proposed theory extends the principles underpinning
general relativity. As a result, the physical basis for contemporary programmes
which apply standard field quantisation to some putative gravitational
lagrangian is questioned. Consistent with Penrose's suggestion of a
deterministic but non-computable theory of fundamental physics, a
`gravitational theory of the quantum' is proposed based on the geometry of
, with potential observational consequences for the dark universe.Comment: This manuscript has been accepted for publication in Contemporary
Physics and is based on the author's 9th Dennis Sciama Lecture, given in
Oxford and Triest
Temporal Stream Logic: Synthesis beyond the Bools
Reactive systems that operate in environments with complex data, such as
mobile apps or embedded controllers with many sensors, are difficult to
synthesize. Synthesis tools usually fail for such systems because the state
space resulting from the discretization of the data is too large. We introduce
TSL, a new temporal logic that separates control and data. We provide a
CEGAR-based synthesis approach for the construction of implementations that are
guaranteed to satisfy a TSL specification for all possible instantiations of
the data processing functions. TSL provides an attractive trade-off for
synthesis. On the one hand, synthesis from TSL, unlike synthesis from standard
temporal logics, is undecidable in general. On the other hand, however,
synthesis from TSL is scalable, because it is independent of the complexity of
the handled data. Among other benchmarks, we have successfully synthesized a
music player Android app and a controller for an autonomous vehicle in the Open
Race Car Simulator (TORCS.
Towards "dynamic domains": totally continuous cocomplete Q-categories
It is common practice in both theoretical computer science and theoretical
physics to describe the (static) logic of a system by means of a complete
lattice. When formalizing the dynamics of such a system, the updates of that
system organize themselves quite naturally in a quantale, or more generally, a
quantaloid. In fact, we are lead to consider cocomplete quantaloid-enriched
categories as fundamental mathematical structure for a dynamic logic common to
both computer science and physics. Here we explain the theory of totally
continuous cocomplete categories as generalization of the well-known theory of
totally continuous suplattices. That is to say, we undertake some first steps
towards a theory of "dynamic domains''.Comment: 29 pages; contains a more elaborate introduction, corrects some
typos, and has a sexier title than the previously posted version, but the
mathematics are essentially the sam
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