DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Standard LSTM cell

Definition

Fix positive finite dimensions $d_{x}, d_{h}$ , inputs $x_{t} \in R^{d_{x}}$ , hidden states $h_{t} \in R^{d_{h}}$ , cell states $c_{t} \in R^{d_{h}}$ , and initial values $h_{0}, c_{0}$ . A standard long short-term memory (LSTM) cell without peephole connections or projection has shared matrices $W_{i}, W_{f}, W_{o}, W_{c} \in R^{d_{h} \times d_{x}}$ , $U_{i}, U_{f}, U_{o}, U_{c} \in R^{d_{h} \times d_{h}}$ , and biases $b_{i}, b_{f}, b_{o}, b_{c} \in R^{d_{h}}$ . Apply $σ (s) = (1 + e^{- s})^{- 1}$ and $\tanh (s) = (e^{s} - e^{- s}) / (e^{s} + e^{- s})$ coordinatewise. At each time $t \geq 1$ , set

$i_{t} = σ (W_{i} x_{t} + U_{i} h_{t - 1} + b_{i}), f_{t} = σ (W_{f} x_{t} + U_{f} h_{t - 1} + b_{f}), o_{t} = σ (W_{o} x_{t} + U_{o} h_{t - 1} + b_{o}),$

$g_{t} = \tanh (W_{c} x_{t} + U_{c} h_{t - 1} + b_{c}), c_{t} = f_{t} ⊙ c_{t - 1} + i_{t} ⊙ g_{t}, h_{t} = o_{t} ⊙ \tanh (c_{t}),$

where $⊙$ is coordinatewise multiplication. The same matrices and biases are reused at every time step. The vectors $i_{t}, f_{t}, o_{t}$ are the input, forget, and output gates, and $g_{t}$ is the candidate state update.

Definition references

Goodfellow, Bengio, and Courville, Deep Learning, Chapter 10, §10.10.1, Eqs. 10.40–10.44

Depends on

Shared-parameter recurrent neural network

Used by

A scalar LSTM cell with constant gates Example

Dependency tree · one level

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Sources

Goodfellow, Bengio, and Courville, Deep Learning, Chapter 10, §10.10.1, Eqs. 10.40–10.44 (standard reference, not scraped)