Mathematics

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(Original Image by everyone’s idle.)

This post was a originally published on Luma Labs, now dead.

As old as stimulus-response techniques are, they still form an important part of many AI systems, even if it is a thin layer underneath a sophisticated decision, planning, or learning system. In this tutorial I give some advice to their design and implementation, mostly out of experience gained from implementing the AI for some racing games.

A stimulus response agent (or a reactive agent) is an agent that takes inputs from its world through sensors, and then takes action based on those inputs through actuators. Between the stimulus and response, there is a processing unit that can be arbitrarily complex. An example of such an agent is one that controls a vehicle in a racing game: the agent “looks” at the road and nearby vehicles, and then decides how much to turn and break.

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1052727062_0ec2c67ea4_smallI have not posted in a while; one reason is that I got sucked into some interesting mathematics; the work-in-progress Reference for Functional Equations is the result. If you are interested in such things – have a look.

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26 February 2009 | No comments

1052727062_0ec2c67ea4

Original image by openDemocracy.

The document below contains tables and formulas useful for working with functional equations, especially di fference equations, and to a lesser extent, quotient equations (where differences are replaced by quotients).

The reference contains tables for differences, sums, quotients, and products. There is also a table of z-transforms, binomial transofroms, formulas for converting certain kinds of functional equations to difference equations and some discrete Taylor series. There are more than 300 formulas in its 30 pages.

This is a work in progress, so be sure to read the preface (which highlights some of the issues with this document). If you find any errors, please comment below.

Download

FunctionalEquationReference_v1_1.pdf (175 KB).

reference_example

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For many applications, detailed statistical models are overkill. Instead, we can get away with a rough description of the distribution – not in mathematical formula form, but just as a graph with a few sample points.

For example, when trying to model the traffic around a school, you might know that the graph looks something like this:

school

The input is the number of minutes before the first bell rings, and the output the number of children dropped off at that time. You know that most kids are brought before the bell rings, and that the closer to the bell, the more kids are being brought every minute. Only a few kids are late.

This tutorial describes how to generate random numbers that can generate a distribution described by an arbitrary (piece-wise linear) curve, as the one above.

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