Parallelly Tempered Generative Adversarial Nets: Toward Stabilized Gradients

Jinwon Sohn; Qifan Song

REVIEW 3 major objections 5 minor 65 references

Parallelly Tempered Generative Adversarial Nets: Toward Stabilized Gradients

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Sohn and Song claim that optimizing a GAN against convex mixtures of real samples, across all mixture weights at once, reduces the variance of the critic's gradients and thereby stabilizes training on multimodal targets.

desk verdict A genuinely new tempered-GAN scheme with strong empirical results, but the central variance-reduction claim rests on an unproven inequality and needs a serious referee to pin it down. read the letter →

arxiv 2411.11786 v2 pith:LSYQNGZI submitted 2024-11-18 stat.ML cs.LG

Jinwon Sohn , Qifan Song This is my paper · ORCID

classification stat.MLcs.LG MSC 68T07

keywords generativeadversarialnetworksparalleltemperinggradientvariancemodecollapseconvexinterpolationtempereddistributionsfairdatagenerationWassersteindistance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that GAN training instability on multimodal targets is, at root, a variance problem: when the generator collapses onto a subset of modes, the critic's gradient estimates become noisy because the remaining, unlearned modes are far away. To fix this, it proposes training the generator against a continuum of tempered distributions $Q_\alpha = \alpha X_1 + (1-\alpha) X_2$ built by convexly interpolating pairs of real data points, with $\alpha$ sampled to favor the true distribution at $\alpha = 1$. The central claim is that this 'parallel tempering' reduces the variance of the critic's gradients because the tempered distribution is closer to the current, incomplete generator than the original target is, and the generator learns all temperature levels simultaneously. Experiments on image and tabular benchmarks, and a variant for fair synthetic data, show gains over standard stabilizers.

What carries the argument

The load-bearing object is the tempered variable $Q_\alpha = \alpha X_1 + (1-\alpha) X_2$, where $X_1, X_2$ are random draws from the target and $\alpha \sim r\delta_1 + (1-r)\,\mathrm{Unif}(0,1)$; its distribution $p_{Q_\alpha,\alpha}$ is the training target, and the objective (6) approximates the scaled 1-Wasserstein neural distance $d_D(p_{Q_\alpha,\alpha}, p_{G(Z,\alpha)},\alpha)$ with a critic that sees $\alpha$ as an extra input. Proposition 3 is the key identity driving the method: for a two-mode target with means $\mu_1, \mu_2$ and variance $\sigma^2$, the tempered components have mean separation $(3/4)|\mu_1-\mu_2|$ and variance $3\sigma^2/4 + 5|\mu_1-\mu_2|^2/192$, so the between-mode distance shrinks. The coherency penalty (8) regularizes the critic so that $D$ values are coherent across $\alpha$, preventing separate per-temperature GAN training and also controlling weight-matrix norms, which connects back to the gradient-variance bounds.

What would settle it

Run a mode-collapsed GAN on a two- or eight-component mixture and, during the collapse phase, estimate both $d_D(p_{Q_\alpha,\alpha}, p_{G^{(t)}(Z,\alpha)},\alpha)$ and $d_D(p_X,1, p_{G^{(t)}(Z,1)},1)$ for $\alpha \in (0,1)$; the central premise predicts the tempered distance is smaller. Recording the gradient-variance trace of the critic in the same runs provides the quantitative check: the stabilization claim fails if variance does not drop when interpolated samples are added.

Watch

Extended reading notes

Core claim

The paper's central claim is that solving $\min_G d_D(p_{Q_\alpha,\alpha}, p_{G(Z,\alpha)},\alpha)$ with $Q_\alpha = \alpha X_1 + (1-\alpha) X_2$ and $\alpha \sim r\delta_1 + (1-r)\,\mathrm{Unif}(0,1)$ stabilizes GAN training. The reason is 'mode connectivity': convex interpolation builds bridges between separated modes of $p_X$, so the tempered distribution has smaller remaining neural distance to an incomplete generator $G^{(t)}$ than the original target does; by Proposition 2 and Theorem 1, this smaller distance directly lowers the variance of the critic's gradient. The resulting generator $G(Z,\alpha)$ learns all tempered distributions at once, and at $\alpha = 1$ recovers the original target, while the proposed coherency penalty keeps learning across temperatures synchronized. The minimax analysis shows the estimator still achieves the near-optimal $O(1/\sqrt{n_e})$ rate, so the stabilization does not come at the price of asymptotic efficiency.

Load-bearing premise

The whole variance-reduction mechanism rests on the unproved assumption that while the generator has collapsed onto some modes, the tempered distribution is closer (in the neural distance used for training) to the current generator than the original target distribution is; if that inequality fails, the promised gradient-variance reduction and stabilization do not follow.

Editorial extensions

If this is right

If the central claim holds, GAN training on severely multimodal targets becomes stable without an annealing schedule: a single run learns all tempered distributions and recovers $p_X$ at $\alpha = 1$.
The gradient-variance lens explains why existing stabilizers such as weight normalization, gradient penalties, and good initialization help: they control the ratio between the remaining neural distance and the critic's weight norm.
The method inherits the minimax sample-complexity rate of standard GANs, so the variance reduction is not bought by a slower convergence rate.
Because the framework is divergence-agnostic, the same parallel tempering applies to JSD, chi-square, or Wasserstein critics, not only the scaled 1-Wasserstein distance.
The fair-data variant gives a practical fairness knob: different $\alpha$ values yield different statistical-parity and utility trade-offs from a single trained model, whereas existing fair generators must retrain per penalty level.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the mode-connectivity mechanism predicts that any augmentation shrinking the between-mode separation of a multimodal target would reduce critic gradient variance, including multi-sample convex combinations or latent-space interpolation, not just pairwise input mixes.
The theory suggests a diagnostic: track the ratio $d_D(p_{Q_\alpha,\alpha}, p_{G^{(t)}(Z,\alpha)},\alpha) / d_D(p_X,1, p_{G^{(t)}(Z,1)},1)$ during training; when this ratio exceeds 1, tempering loses its edge and an adaptive $r$ schedule could switch to more $\alpha=1$ samples.
The continuous fairness dial of FairPTGAN could be formalized as selecting $\alpha$ on the Pareto frontier of utility versus statistical parity without retraining; the paper demonstrates the trade-off but does not develop a selection rule.
The same convex-interpolation tempering could be applied to diffusion-style samplers that use a critic or discriminator component, carrying the variance-reduction benefit beyond generative adversarial networks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, the core idea is genuinely new: instead of annealing noise, the authors temper the target by convex interpolation between data points, then train the generator on the joint distribution over all interpolation levels alpha. That is not Mixup, and it is not the old additive-noise tempering. Learning all tempered marginals at once, plus a coherency penalty to keep training pace consistent across alpha, is a real departure from prior GAN stabilization work. Second, the empirical work is unusually thorough for a theory-leaning paper: PTGAN beats strong baselines across CIFAR-10, BloodMnist, CelebA-HQ, and three tabular benchmarks, and the fairness extension gives a simple way to sweep the fairness–utility trade-off without retraining. Those results are consistent with the claim that the method stabilizes training.

The soft spots are real, and the stress-test note lands. The central theoretical justification is that tempered distributions reduce the variance of the critic's gradients, but the key condition in Remark 3—that the tempered neural distance is no larger than the untempered one during mode collapse—is asserted, not proved. Proposition 3 only handles a two-component Gaussian example, and the extension to many modes in high dimensions is explicitly left as a belief. I checked the surrounding material: Theorem 1 bounds gradient magnitude, not variance, and the variance comparison is mostly heuristic. The linear-class analysis in the supplementary material (Proposition 5) does give a verifiable variance-reduction condition, which is the strongest support the paper actually has, but it is limited to a linear critic.

Two smaller issues: the mixture weight r and the coherency penalty lambda are tuned on the test metrics, and no code or data is provided. That makes the empirical wins harder to assess independently. None of this is fatal—the method is coherent, the minimax analysis in Theorems 2 and 3 is in line with existing GAN sample-complexity results, and the citation pattern is appropriate—but the stabilization mechanism is not as established as the abstract implies.

Who is this for? Anyone working on GAN training dynamics, tempered generative models, or interpolation-based regularizers. It deserves a serious referee: a good reviewer could ask for a proof or substantial empirical validation of Remark 3, code release, and a cleaner tuning protocol. I would not desk-reject this.

Referee Report

3 major / 5 minor

Summary. The paper proposes parallelly tempered generative adversarial networks (PTGAN), a GAN framework that trains the critic and generator on the joint distribution of tempered interpolants Qα = αX1 + (1−α)X2, where X1, X2 are data points and α is drawn from rδ1 + (1−r)Unif(0,1) (Eq. 7). The generator is asked to learn all temperature levels jointly, with a coherency penalty (Eq. 8) that aligns the critic's behavior across levels. The central theoretical claim is that this objective reduces the variance of the critic's gradient in the presence of mode collapse, mitigating instability: gradient variance is linked to the neural distance dD(pSct, pG(t)(Z)) via a lower bound (Proposition 2) and magnitude bounds (Theorem 1), and the tempered distance is argued to be smaller (Propositions 1 and 3, Remark 3, Corollary 1). The paper also provides minimax estimation and approximation results for the joint objective (Theorems 2–3, Proposition 4) at the standard n^{−1/2} rate. Empirically, PTGAN with the coherency penalty outperforms spectral normalization and Lipschitz-GAN baselines on CIFAR-10, BloodMnist, and CelebA-HQ and on three tabular benchmarks, and the fairness variant (FairPTGAN) achieves better utility–parity trade-offs than two competitors. The main theoretical weakness is that the variance-reduction mechanism rests on an unproven inequality in Remark 3 and an explicitly conjectural extension in Section 3.2.

Significance. The paper is a serious and substantial contribution with three distinct strengths. First, it offers a systematic analysis of GAN instability through the variance of the critic's gradient, with explicit bounds (Theorem 1, Corollary 1) expressed in terms of the neural distance; this reframing is original relative to the existing tempering literature, which focuses on support smoothing. Second, the proposed tempering mechanism is genuinely different from additive-noise annealing: convex interpolation shrinks between-mode variability (Proposition 3) rather than only inflating within-mode variability, and the joint objective with the coherency penalty avoids annealing schedules. Third, the theoretical infrastructure is coherent: Theorems 2–3 and Proposition 4 reproduce the established Ji et al. (2021)/Huang et al. (2022) sample-complexity framework for the joint objective, and Proposition 5 (SM A.3) provides a fully rigorous variance-reduction comparison for a linear critic under a checkable condition, showing the mechanism is not vacuous.

major comments (3)

[§4.2, Remark 3; §3.2] The variance-reduction mechanism asserted in the abstract and Section 4.2 is conditional on the inequality dD(pQα∗,α∗, pG(t)(Z,α∗),α∗) ≤ dD(pX,1, pG(t)(Z,1),1) stated in Remark 3, but this inequality is not proven. Its only quantitative support is Proposition 3, which analyzes a two-component Gaussian target with a specific construction of pα_1 and pα_2, and its extension to general high-dimensional multimodal targets is presented as a belief in Section 3.2: 'We believe that the same argument holds even when pX has more underlying unimodal distributions in higher dimensions.' The derivation in SM B.5 expresses the joint distance only as a weighted average r·dD(pX,1,...) + (1−r)·dD(pQα*,...); this is a convex combination, not an inequality between the two terms. Corollary 1 merely transfers the Theorem 1 bound to the tempered objective, so the chain 'tempered distance smaller → upper bound smaller → variance smaller' rests on exactly the unproven inequality, and Remark 3 itself acknowledges the reverse possibility when G(t) already covers S. I recommend either proving the inequality under explicit, checkable conditions on the mode geometry and the state of G(t), or reframing the theoretical contribution as conditional, with Proposition 5 (linear class, SM A.3) and the empirical results carrying the unconditional variance-reduction claim.
[§4.2, Theorem 1/Corollary 1; Abstract] The transition from the bounds in Theorem 1 and Corollary 1 to a claim of variance reduction is an inference, not a derivation: those results bound the magnitude of individual gradient entries (Eq. 4), and the variance comparison uses Var[X] ≤ E[X²] together with a term-wise comparison of upper bounds. This legitimately bounds each variance, but it does not prove Var(PTGAN gradient) ≤ Var(vanilla gradient), since both variances are only shown to lie below comparable quantities. The abstract's sentence 'We theoretically analyze that such significant improvement can arise from reducing the variance of gradient estimates' thus overstates what the theorems establish, even though Remark 3 is appropriately cautious ('Corollary 1 suggests that the variance can decrease'). A rigorous variance comparison in the general setting would require either a direct calculation of the covariance of the tempered gradient or a matching lower-bound–upper-bound sandwich; absent that, the theoretical claims should be calibrated.
[§3.1–3.2, Proposition 3; SM B.3] Proposition 3 contains a concrete arithmetic slip and an unclear representational claim. The variance stated in Proposition 3 is σ*² = 3σ²/4 + 5(µ1−µ2)²/192, but the component-wise computation in SM B.3 (σ²E[α²] + Var(X2)E[(1−α)²] + (µ1−E[X2])²Var(α) with E[α²]=7/12, E[(1−α)²]=1/12, Var(α)=1/48, Var(X2)=σ²+(µ1−µ2)²/4) yields σ*² = 2σ²/3 + 5(µ1−µ2)²/192. In addition, the text states that the construction with α ~ Unif(0.5, 1) gives 'the same distribution' as the original Qα with α ~ Unif(0, 1); these two laws differ in general, and the moment formulas are distribution-dependent. Because Proposition 3 is the only quantitative evidence for the mode-connectivity property that underlies Remark 3, the arithmetic should be corrected and the status of the two-component representation (exact versus approximate) clarified, possibly recomputing the comparison under the α-distribution (7) actually used in the algorithm.

minor comments (5)

[§2.3.2, Proposition 2] The assumption E[L(t)_i | L(t)_i ≤ ε] = 0 is introduced without justification or guidance on the choice of ε; since the variance decomposition in SM B.2 (Lemma 2) holds without this assumption, it should either be justified as a normalization condition or relaxed.
[§4.4, Theorems 2–3] The phrase 'achieving the same minimax convergence rate √n' is imprecise: the bounds in (9) and (11) scale as 1/√ne (i.e., O(n^{−1/2}) with ne = n/2). Please state the rate consistently.
[§5.1, Table 3] The S_T metric (average absolute difference between Strain and St over the second half of training) is interpreted as 'quick and accurate convergence,' but a low S_T could also reflect a stably poor generator; acknowledging this alternative reading would strengthen the evaluation.
[§1, Introduction] The novelty claim 'To our knowledge, this is the first work to theoretically analyze the GAN mechanism through the lens of the gradients' variance of D' is strong; given the existing literature on gradient-variance-aware analyses of nonconvex-nonconcave min-max problems (e.g., Yang et al., 2020, cited in §4.2), it would be safer to phrase the claim relative to the GAN training objective specifically.
[§2.1, (A1)] The set-builder notation for the network parameter families W and V uses 'N' in a nonstandard way (apparently as a collection or intersection symbol), which obscures the definition of the network classes; a standard set-union/product notation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation chain found: the tempered objective is an external construction, and the variance-reduction claim is conditional on an explicitly stated inequality rather than being equivalent to an input by construction.

full rationale

The paper's central object, dD(pQα,α, pG(Z,α),α), is defined externally via convex interpolation Qα = αX1 + (1−α)X2 with α ~ rδ1 + (1−r)Unif(0,1), and is not fitted to the variance-reduction outcome. Proposition 3 computes the means and variances of the constructed tempered mixture and compares the between-component separation to the original; this is a direct calculation, not a restatement of the conclusion. The gradient analysis in Theorem 1 and Corollary 1 provides upper bounds on gradient magnitude in terms of neural distances, which are genuine inequalities. Remark 3 then explicitly assumes dD(pQα*,α*, pG(t)(Z,α*),α*) ≤ dD(pX,1, pG(t)(Z,1),1) to conclude that variance can decrease; this is a stated sufficient condition, and the paper openly acknowledges it may fail ('There might be the reverse relationship...'). While the condition is asserted rather than proven for general high-dimensional multimodal targets—and Section 3.2 says 'We believe that the same argument holds even when pX has more underlying unimodal distributions in higher dimensions'—that is an unproven assumption or support gap, not circular reasoning. The linear-class result in Proposition 5 independently verifies variance reduction under an explicit condition, further separating the derivation from the claim. The minimax analysis benchmarks the estimator against the joint tempered objective and a lower-bound family over pQα,α; this is a well-defined statistical target and is not equivalent to assuming the desired conclusion. Self-citations (e.g., Kim et al. 2023, Sohn et al. 2024) are peripheral and not load-bearing for the core stabilization mechanism. No uniqueness theorem or ansatz is smuggled in via citation; the approximation results are imported from external, code-independent sources and are not used to forbid alternatives. Overall, the derivation chain is self-contained in the sense that its conclusions are not constructed to equal its inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new physical or latent entities are postulated. The tempered random variable Qalpha and the alpha input are data-space constructions, not additional model parameters or hidden variables. The main load-bearing assumptions are the unproven mode-connectivity generalization and the conditional variance-reduction inequality.

free parameters (2)

r (mixture proportion in alpha distribution, Eq. 7) = 0.9/0.99 (CIFAR10, BloodMnist), 0.98 (CelebA-HQ), 0.5 (tabular), 0.2 (FairPTGAN)
Controls the bias-variance trade-off between original and tempered samples; for images, selected as the value that maximizes IS/FID on the evaluation sets.
lambda (coherency penalty coefficient, Eq. 8) = 100
Default penalty strength used in all simulations; no sensitivity analysis is reported.

assumptions (5)

domain assumption (A1)-(A3): bounded weight matrices, Lipschitz activations, bounded support of X and Z
Standard regularity conditions used throughout Section 2 and 4 for all gradient and sample-complexity bounds.
domain assumption (A4): positive homogeneous activations (e.g., ReLU, lReLU)
Needed for the Rademacher complexity bound in Lemma 10; relaxed in Remark 12 to activations with kappa(0)=0 under a bounded 1-norm condition.
ad hoc to paper Existence of epsilon > 0 with E[L_i | L_i <= epsilon] = 0 in Proposition 2
This condition is introduced inside the proof of the lower variance bound and is not guaranteed by the standing assumptions.
ad hoc to paper Mode connectivity extends from the 2-component toy example to general high-dimensional multimodal pX
Section 3.2 states this as 'We believe that the same argument holds even when pX has more underlying unimodal distributions in higher dimensions.'
domain assumption Divergence correlation: if dD decreases, d_kappa (W1 or TV) also decreases
Remark 3 relies on a positive relationship between neural distance and Wasserstein/total variation, citing Chae and Walker (2020).

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Cite this review

Pith. "Pith review of Parallelly Tempered Generative Adversarial Nets: Toward Stabilized Gradients." pith.science (2026). https://pith.science/paper/LSYQNGZI

@misc{pith2026241111786,
  author       = {Pith},
  title        = {Pith review of: Parallelly Tempered Generative Adversarial Nets: Toward Stabilized Gradients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSYQNGZI}},
  note         = {Machine review of arXiv:2411.11786}
}

read the original abstract

A generative adversarial network (GAN) has been a representative backbone model in generative artificial intelligence (AI) because of its powerful performance in capturing intricate data-generating processes. However, the GAN training is well-known for its notorious training instability, usually characterized by the occurrence of mode collapse. Through the lens of gradients' variance, this work particularly analyzes the training instability and inefficiency in the presence of mode collapse by linking it to multimodality in the target distribution. To ease the raised training issues from severe multimodality, we introduce a novel GAN training framework that leverages a series of tempered distributions produced via convex interpolation. With our newly developed GAN objective function, the generator can learn all the tempered distributions simultaneously, conceptually resonating with the parallel tempering in statistics. Our simulation studies demonstrate the superiority of our approach over existing popular training strategies in both image and tabular data synthesis. We theoretically analyze that such significant improvement can arise from reducing the variance of gradient estimates by using the tempered distributions. Finally, we further develop a variant of the proposed framework aimed at generating fair synthetic data which is one of the growing interests in the field of trustworthy AI.

Figures

Figures reproduced from arXiv: 2411.11786 by the authors.

**Figure 2.** Mode collapse induces multimodality. The leftmost depicts mode collapse where [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Tempered distributions of pX: the first and third panels depict histograms of X, with 2- and 8-component mixtures respectively, and the next of each represents Qα . than dD(pS c t,ϵ , pG (t) ϵ (Z) ) particularly for the case without scaling data. The same arguments follow for dW1 as well. Simply adding noise may not effectively decrease the multimodality in view of the neural distance because it just increases the w… view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: Comparison of Lˆα b (D(t) , G(t) ) and Lˆ1 b (D(t) , G(t) ) on the toy example in [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: Variance reduction: the logarithm of Var[Lˆα b ] on ND over training iterations. Shaded areas indicate one standard deviation from the straight average lines. (a) 16 Real/Generated images of CIFAR10 (b) 16 Real/Generated images of BloodMnist (c) 8 Real/Generated images…

**Figure 6.** Figure 6: Real/Generated images are randomly picked from the original images and the [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]

**Figure 7.** Figure 7: The trade-off curves between AUC and SP for the downstream classifiers. [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

**Figure 8.** Figure 8: The left plot shows the logarithm of the 1-Wasserstein distance between [PITH_FULL_IMAGE:figures/full_fig_p038_8.png]

**Figure 9.** Figure 9: The left plot shows the logarithm of the 1-Wasserstein distance between the data [PITH_FULL_IMAGE:figures/full_fig_p043_9.png]

**Figure 10.** Figure 10: The unimodal components p1 and p2 are generated from normal distribution with µ1 = 3, µ2 = −3, and the variance σ 2 = 0.01. so |µ ∗ 1 − µ ∗ 2 | = 3|µ1 − µ2|/4 and the variance is (σ ∗ ) 2 = Var(αu + (1 − α)X2), = E[Var(αu + (1 − α)X2|α)] + Var(E[αu + (1 − α)X2|α]), = …

**Figure 11.** Figure 11: Plots illustrate the logarithm of the 1-Wasserstein distance between [PITH_FULL_IMAGE:figures/full_fig_p092_11.png]

**Figure 12.** Figure 12: Randomly selected real images of CIFAR10 [PITH_FULL_IMAGE:figures/full_fig_p098_12.png]

**Figure 13.** Figure 13: Randomly generated images for CIFAR10 63 [PITH_FULL_IMAGE:figures/full_fig_p099_13.png]

**Figure 14.** Figure 14: Randomly selected real images of BloodMnist [PITH_FULL_IMAGE:figures/full_fig_p100_14.png]

**Figure 15.** Figure 15: Randomly generated images for BloodMnist [PITH_FULL_IMAGE:figures/full_fig_p101_15.png]

**Figure 16.** Figure 16: Randomly selected real images of CelebA-HQ [PITH_FULL_IMAGE:figures/full_fig_p102_16.png]

**Figure 17.** Figure 17: Randomly generated images for CelebA-HQ 66 [PITH_FULL_IMAGE:figures/full_fig_p102_17.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

[3] [3]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

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, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

[5] [5]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

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