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pith:XUUK4HNO

pith:2026:XUUK4HNOI77GCX52SJSPFQUD5H

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Bellman Value Decomposition for Task Logic in Safe Optimal Control

Chuchu Fan, Dylan Hirsch, Oswin So, Sylvia Herbert, William Sharpless

The Bellman value for temporal logic tasks decomposes into a graph of simpler values connected by reach-avoid, avoid, and reach-avoid-loop equations.

arxiv:2602.19532 v2 · 2026-02-23 · cs.RO · cs.SY · eess.SY

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Claims

C1strongest claim

We prove the Bellman Value for a complex task defined in temporal logic can be decomposed into a graph of Bellman Values, connected by a set of well-known Bellman equations (BEs): the Reach-Avoid BE, the Avoid BE, and a novel type, the Reach-Avoid-Loop BE.

C2weakest assumption

The innate structure of the Bellman value naturally organizes temporal logic tasks so that the decomposed graph can be embedded in a two-layer neural net that bootstraps implicit dependencies without additional manual tuning or post-hoc adjustments.

C3one line summary

Bellman values for temporal logic tasks decompose into a graph of reach-avoid, avoid, and reach-avoid-loop equations solved by embedding the graph in a two-layer neural net (VDPPO) for safe high-dimensional control.

References

97 extracted · 97 resolved · 2 Pith anchors

[1] R. S. Sutton and A. G. Barto,Reinforcement Learning: An Introduction. Cambridge, MA, USA: A Bradford Book, 2018 2018

[2] LTL and beyond: Formal languages for reward function specification in reinforcement learning, 2019

[3] A time-dependent hamilton-jacobi formulation of reachable sets for continuous dynamic games 2005

[4] Reach-avoid problems with time-varying dynamics, targets and constraints, 2015

[5] Dual-objective reinforcement learning with novel hamilton-jacobi-bellman formulations, 2025

Formal links

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Cited by

1 paper in Pith

Value Functions for Temporal Logic: Optimal Policies and Safety Filters

Receipt and verification

First computed	2026-05-17T23:39:16.014036Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

bd28ae1dae47fe615fba9264f2c283e9fffb760ae7ffd66cbbf6eaeed9cc33e2

Aliases

arxiv: 2602.19532 · arxiv_version: 2602.19532v2 · doi: 10.48550/arxiv.2602.19532 · pith_short_12: XUUK4HNOI77G · pith_short_16: XUUK4HNOI77GCX52 · pith_short_8: XUUK4HNO

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/XUUK4HNOI77GCX52SJSPFQUD5H \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: bd28ae1dae47fe615fba9264f2c283e9fffb760ae7ffd66cbbf6eaeed9cc33e2

Canonical record JSON

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      "eess.SY"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "primary_cat": "cs.RO",
    "submitted_at": "2026-02-23T05:48:58Z",
    "title_canon_sha256": "33efe4b725c0d887804372495e33b7214738738197a66c3dbdc1c1ad8c4a8b0f"
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}