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On topological lower bounds for algebraic computation trees

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Nicolai Vorobjov, Andrei Gabrielov

Computational Complexity · Submitted 15 Feb 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of the set, divided by m+1.

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arXiv ID1502.04341 (version 2)
Pith IDL4KJHF2I, Pith's permanent record of this paper and its review
CitationAndrei Gabrielov, Nicolai Vorobjov. "On topological lower bounds for algebraic computation trees." arXiv:1502.04341 (2015). https://pith.science/paper/L4KJHF2I
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,718 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.96 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of the set, divided by m+1. This result complements the well known lower bound by Yao for locally closed semialgebraic sets in terms of the total Borel-Moore Betti number. We also prove that the height is bounded from below by the logarithm of m-th Betti number of a projection of the set onto a coordinate subspace, divided by (m+1)^2. We illustrate these general results by examples of lower complexity bounds for some specific computational problems.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of the set, divided by m+1. result 0.94 0.97 0.93 asserted 1.00
s1 This result complements the well known lower bound by Yao for locally closed semialgebraic sets in terms of the total Borel-Moore Betti number. background 0.59 0.19 0.05 asserted 0.99
s2 We also prove that the height is bounded from below by the logarithm of m-th Betti number of a projection of the set onto a coordinate subspace, divided by (m+1)^2. result 0.98 0.96 0.11 asserted 1.00
s3 We illustrate these general results by examples of lower complexity bounds for some specific computational problems. contribution 0.80 0.89 0.03 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of the set, divided by m+1. This result complements the well known lower bound by Yao for locally closed semialgebraic sets in terms of the total Borel-Moore Betti number. We also prove that the height is bounded from below by the logarithm of m-th Betti number of a projection of the set onto a coordinate subspace, divided by (m+1)^2. We illustrate these general results by examples of lower complexity bounds for some specific computational problems.

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