Pebbles and Branching Programs for Tree Evaluation
Bibliographic record
Abstract
We introduce the tree evaluation problem , show that it is in LogDCFL (and hence in P ), and study its branching program complexity in the hope of eventually proving a superlogarithmic space lower bound. The input to the problem is a rooted, balanced d -ary tree of height h , whose internal nodes are labeled with d -ary functions on [ k ] = {1,..., k }, and whose leaves are labeled with elements of [ k ]. Each node obtains a value in [ k ] equal to its d -ary function applied to the values of its d children. The output is the value of the root. We show that the standard black pebbling algorithm applied to the binary tree of height h yields a deterministic k -way branching program with O ( k h ) states solving this problem, and we prove that this upper bound is tight for h = 2 and h = 3. We introduce a simple semantic restriction called thrifty on k -way branching programs solving tree evaluation problems and show that the same state bound of Θ ( k h ) is tight for all h ≥ 2 for deterministic thrifty programs. We introduce fractional pebbling for trees and show that this yields nondeterministic thrifty programs with Θ ( k h/2+1 ) states solving the Boolean problem “determine whether the root has value 1”, and prove that this bound is tight for h = 2,3,4. We also prove that this same bound is tight for unrestricted nondeterministic k -way branching programs solving the Boolean problem for h = 2,3.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.006 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".