Abstract
A robot’s action are carried out over time through space. The computational mechanisms that the robot uses to choose and guide actions must therefore at some level reflect the structure of time and space. This essay discusses what ontology for space and time should be built into a robot. I argue that the designers of a robot generally need to have in mind explicit otologies of time and space, and that these will generally be the real line and three-dimensional Euclidean space, though it is often advantageous to project down to a lower-dimensional space, when possible. I then discuss which spatial or spatio-temporal regions should be considered “well-behaved” and which should be excluded as “monsters.” I propose some principles that the class of well-behaved regions should observe, and discuss some examples.
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Notes
- 1.
For purposes of both brevity and concreteness, I will largely frame the discussion in this essay in terms of robots. However, most AI systems are not connected to physical robots; and much of the discussion in this paper applies equally to AI systems of many other kinds that deal with spatial information; e.g. to natural language processing systems that have to interpret spatial terms in text.
- 2.
This data structure by definition denotes this exact algebraic number, because that is what the programmer means by the data structure. The programmer presumably provides functions that carry out exact calculations over these data structures, e.g. an addition function that takes two addends represented in this form and returns the sum, also represented in this form.
- 3.
The space-filling curve, the Banach-Tarski paradox, and some other technical terms are defined in a glossary at the end of the chapter.
- 4.
It also breaks down in the neighborhood of a black hole; but that is the least of the problems facing a robot in the neighborhood of a black hole.
- 5.
The Pythagorean proof that the ratio of the diagonal of an isosceles right triangle to its leg is irrational fundamentally depends on rotation. Comparing the lengths of two non-parallel line segments depends on rotating one of the lines so that the two are collinear. In affine geometry, where rotation is undefined, the ratio of the lengths of two non-parallel lines is also undefined.
- 6.
For instance, an ontology of space in which all points have rational coordinates would be consistent with a theory in which all rotations are rational rotations, i.e. (in 2D) rotations by angle θ such that \(\cos (\theta ) = a/c\) and \(\sin (\theta ) = b/c\) where a, b, c are a Pythagorean triple (see Glossary). But then the position at which a rotating object collides with a stationary object would in general be undefined; it would either have to overshoot (in which the two objects interpenetrate) or undershoot (in which the collision occurs while there is still a finite gap between the objects).
- 7.
Physicists and engineers rarely if ever worry about this; they use their physical intuition to avoid worrying about physically meaningless problems. But AI programs only have the physical intuitions that we program into them. The purpose of our analysis here is precisely to determine what are the physical intuitions that will enable the AI program to avoid worrying about meaningless problems.
- 8.
One semester as an undergraduate, I took both Analysis II, which was mostly the theory of distributions, and Functions of a Complex Variable. In Analysis, the ne plus ultra of well-behaved functions were C ∞ functions with compact support; distributions could be defined as linear functionals over that space. In Complex Variables, these were all monsters because they are not analytic.
- 9.
Both Dalton’s chemical theory and the Maxwell/Boltzmann kinetic theory of heat have the unsettling feature that, though they are based on particles, they give no indication of how large the particles should be.
- 10.
There is more than one measure of approximation for regions in Euclidean space; Hausdorff distance (see Glossary) will do fine here.
- 11.
The amount of change that physics predicts can certainly be very small, even for reasonable sized objects undergoing perceptible changes. For instance, if a 1 cm cubed object is moving at a speed of 1 cm per second, the relativistic contraction is 5 ⋅ 10−23 m.
- 12.
More precisely, one would want to posit that any bump or indentation contains a sphere of 0.01 mm; in the technical jargon, the region is r-regular with r = 0. 01 mm (see Glossary).
- 13.
John Stillwell’s chapter in this collection discusses in great depth the intricate relations between set theory and the theory of the real numbers and Euclidean space.
- 14.
Slight variants on standard AI reasoning techniques actually do generate a version of the Achilles paradox. Suppose that you posit there are important times, important places, and important events; when an object arrives at an important place, that is an important event; the time when an important event occurs is an important time; and the place of an object at an important time is an important place. Then, as Achilles chases the tortoise, infinitely many important events occur, by the standard argument.
References
K. Cho et al. “Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation,” Conference on Empirical Methods in Natural Language Processing (EMNLP), 2014.
P. Stelldinger Image Digitization and its Influence on Shape Properties in Finite Dimensions, IOS Press, 2008.
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Davis, E. (2015). How should robots think about space?. In: Davis, E., Davis, P. (eds) Mathematics, Substance and Surmise. Springer, Cham. https://doi.org/10.1007/978-3-319-21473-3_5
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