6. Discussion
This paper developed techniques for analyzing the internal structure of distributed measurements. We introduced entanglement, which quantifies the extent to which a measurement is indecomposable. Entanglement can be shown to quantify context-dependence. Moreover, positive entanglement is necessary for a system to generate more information than the sum of its subsystems. Along the way, we constructed the quale, which geometrically represents the compositional structure of a distributed measurement. The information-theoretic approach developed here is dual, in a precise sense, to the algorithmic perspective on computation. Studying duals ${\mathrm{\u0111\u0165\u201d\u015e}}^{\mathrm{\xe2\u2122\circledR}}$ instead of mechanisms $\mathrm{\u0111\u0165\u201d\u015e}$ shifts the focus from what the algorithm does to how it does it: instead of analyzing rules we analyze functional dependencies.
The intuition driving the paper is that the structure presheaf^{} $\mathrm{\xe2\u201e\pm}$ is an information-theoretic analogue of a tangent space. A particle moving in a manifold $X$ defines a vector field â€“ a section^{} of the tangent space to $X$, which is a sheaf. The tangent vector at a point depends on the particleâ€™s location at â€śnearby time-pointsâ€ť: it is computed by taking the limit of difference in positions at $t$ and $t+h$ as $h\xe2\u2020\u20190$. Similarly, a system performing a measurement generates a quale, a section of the structure presheaf consisting of â€śnearby counterfactualsâ€ť. The quale is computed by applying Bayesâ€™ rule to determine which inputs could have led to the output.^{1}^{1}A counterfactual input is â€śnearbyâ€ť to an output if it causes (leads to) that output. How far this analogy can be developed remains to be seen.
Entanglement can be loosely considered as an information-theoretic analogue of curvature: the extent to which interactions within a system â€śwarpâ€ť sections of $\mathrm{\xe2\u201e\pm}$ away from a product structure. A related approach to geometrically analyzing the complexity of interactions was proposed in [1]. In fact, this project began as an attempt to reformulate [2] in terms of sheaf cohomology using ideas from [1]. We failed at the first step since the structure presheaf is not a sheaf. However, the failure was instructive since it is precisely the obstruction to forming a sheaf that is of interest since it is the obstruction (entanglement) that quantifies indecomposability and context-dependence, and only systems whose measurements are entangled are able to generate more information than the sum of their subsystems.
References
- 1 NÂ Ay, EÂ Olbrich, NÂ Bertschinger & JÂ Jost (2006): A unifying framework for complexity measures of finite systems. In: Proceedings of ECCS06, European Complex Systems Society, Oxford, UK, pp. ECCS06â€“174.
- 2 David Balduzzi & Giulio Tononi (2009): Qualia: the geometry of integrated information. PLoS Comput Biol 5(8), p. e1000462, doi:10.1371/journal.pcbi.1000462.
Title | 6. Discussion |
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Canonical name | 6Discussion |
Date of creation | 2014-04-22 15:13:32 |
Last modified on | 2014-04-22 15:13:32 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 2 |
Author | rspuzio (6075) |
Entry type | Feature |