Equation 3 · Why the Physical World Is Harder
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol p^H
is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
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Its accuracy depends on the assumptions and range of use described in the article.
What the article says around this equation
This compounds badly over long horizons. If a task requires H decisions and each succeeds independently with probability p , the task succeeds with probability . The independence assumption is generous — real failures correlate, because one slip puts the system into a region where subsequent steps are also more likely to fail — but even the generous version is brutal. For a 300-step manipulation sequence to complete nine times in ten, per-step reliability must be about 0.99965. A per-step reliability of 0.99, which would be an excellent number for almost any perception module, yields a task success rate of roughly five percent. This is straightforward arithmetic rather…
Read the full surrounding passage
This compounds badly over long horizons. If a task requires H decisions and each succeeds independently with probability p , the task succeeds with probability . The independence assumption is generous — real failures correlate, because one slip puts the system into a region where subsequent steps are also more likely to fail — but even the generous version is brutal. For a 300-step manipulation sequence to complete nine times in ten, per-step reliability must be about 0.99965. A per-step reliability of 0.99, which would be an excellent number for almost any perception module, yields a task success rate of roughly five percent. This is straightforward arithmetic rather than a finding from any cited study, but it explains an otherwise puzzling pattern: policies that look near-perfect on individual skills collapse on chained ones, and the collapse is not a bug to be found, it is the exponent.
Sources cited in the article section
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