Computational adversarial risk analysis for general security games

David Rios Insua; Jose Manuel Camacho; Roi Naveiro

REVIEW 4 major objections 5 minor 42 references

Computational adversarial risk analysis for general security games

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that any proper bi-agent influence diagram security game, discrete or continuous, can be solved from the Defender's perspective by replacing every expected-utility maximization with a mode estimate of an augmented…

desk verdict Solid extension of APS to general security games with a useful case study, but the flagship reduction as written has a conditioning bug and the continuous-domain claims outrun the proofs. read the letter →

arxiv 2506.02603 v1 pith:MYLC24U6 submitted 2025-06-03 cs.GT

Jose Manuel Camacho , Roi Naveiro , David Rios Insua This is my paper

classification cs.GT

keywords adversarialriskanalysisaugmentedprobabilitysimulationbi-agentinfluencediagramssecuritygamessimultaneousdefend-attackdisinformationwarMarkovchainMonteCarlomodeestimation

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

This paper claims that a broad class of security games—any game that can be drawn as a proper bi-agent influence diagram with continuous or discrete decisions—can be solved from one decision-maker's point of view without assuming the opponents share her beliefs. The proposed algorithm works by replacing each expected-utility maximization step with a mode-finding step on an augmented probability distribution, alternately solving the Defender's decision nodes and simulating the Attacker's random optimal choices. If the claim holds, a defender facing an adversary whose utilities and probabilities are unknown can obtain an optimal policy and a probabilistic forecast of the adversary's actions in settings where previous methods required analytical reductions or discrete state spaces.

What carries the argument

The central mechanism is Augmented Probability Simulation (APS): instead of maximizing the Defender's expected utility directly, one samples from an augmented distribution proportional to the utility times the relevant probability factors and estimates the mode of the decision-variable marginal, which coincides with the optimal decision when the utility is positive. For the Attacker, the same construction applies to random utilities and probabilities, yielding a Random Augmented Distribution whose marginal mode produces a sample from the Attacker's random optimal action; repeating this supplies the Defender's probabilistic forecast of the attack. DAPS and AAPS reductions package these mode-finding steps so that Algorithm 2 works backward along the Defender Decision Path and Attacker Decision Path, the named reduction operations being the paper's main theoretical instruments.

What would settle it

Construct a small discrete proper BAID, compute the optimal Defender policy by exhaustive enumeration, then run Algorithm 2 with exact enumeration in place of sampling; if the output differs from the exhaustive optimum on any instance, Proposition 5 fails. For the computational pipeline, take a continuous instance where the true optimal second-stage decision has a known closed form and check whether the neural-network value-function metamodel has the same maximizer; a single case where the metamodel's mode differs from the true mode would falsify the approximation scheme.

Watch

Extended reading notes

Core claim

Proposition 5 states that in a proper BAID, Algorithm 2 provides the optimal policy for the Defender. The algorithm reads the game as two directed decision paths (the Defender's and the Attacker's), and at each backward step it either applies a DAPS reduction—defining an augmented distribution proportional to the Defender's utility times the relevant probability terms, so that the mode in the decision variable equals the optimal decision—or, when the Defender lacks the probabilities describing the Attacker's next move, an AAPS reduction that samples the Attacker's random optimal decision from his own augmented distribution. Because expected utilities are maximized, positive constants in Bayes' denominators can be ignored, so continuous and high-cardinality spaces are handled by sampling and mode estimation rather than arc inversions and analytic elimination. The paper also provides a full computational scheme, using neural-network metamodels trained on grids to carry approximate value functions and conditional attack densities backwards through the diagram, demonstrated on a disinformation-war case.

Load-bearing premise

Correctness depends on the informal availability check in Section 4.3: every Defender decision must be reachable either with all needed probabilities already available or after finitely many Attacker reductions, with no circular dependence on the very Defender decision being computed, and the metamodels used in continuous problems must preserve the modes they approximate.

Editorial extensions

If this is right

Security games with continuous action spaces and no common knowledge can be solved from the Defender's perspective, not just discrete or low-cardinality ones.
The Attacker's decisions come with probabilistic forecasts that feed naturally into the Defender's optimization and could also be used for attack awareness.
The approach provides a general resolution of the longstanding query about solving influence diagrams with multiple decision-makers, within the adversarial risk analysis solution concept.
Multi-stage games are handled by backward induction with metamodels, as illustrated by the disinformation-war study producing a first-stage defense allocation of 0.7 and a defensive response policy that deploys full reactive resources only when a high-intensity attack is expected.

Reading between the lines

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

The informal availability check in Section 4.3 is where the general claim is most exposed; a formal characterization of when repeated AAPS reductions terminate without circular dependence would turn the computational scheme into a fully stated theorem.
The same APS mode-finding construction could plausibly extend to games with more than two agents, since each additional strategic agent could be represented by another layer of random augmented distributions, though computational cost and confluence issues would likely grow.
The reliance on mode-consistent estimators suggests that multimodal augmented distributions could be a practical weak point; a testable extension is to compare the algorithm's output with exact enumeration on small multimodal instances.
Using adaptive experimental design instead of fixed grids, as the authors note, could reduce the cost of training the metamodels that approximate value functions and attack densities.

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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

Useful extension of Ekin et al.'s APS machinery to simultaneous and multi-stage security games; the disinformation case is a solid worked example. But the paper's flagship claim—first general solution—rests on a reduction that is mis-stated as written. Proposition 3's augmented distribution conditions on ant(Xc(D)) but drops utility antecessors that remain in the policy (e.g., d1), and the inherited-set formula adds the eliminated decision back when ant(Xc(D)) contains it. The stress-test counterexample is real: with uD = C-(d1-d2)^2, the d1-marginalized AD has mode d2=1/2, not pointwise d*2(d1)=d1. The implementation in Sec. 5.2.1 conditions each DAPS1 run on the grid point, which is the correct construction, so the method is salvageable; the formal statement and Prop 5's proof just don't match what the code does.

Other things worth saying. The extension is genuinely new: Ekin et al. treated two-stage sequential defend-attack; here they cover simultaneous single-stage, n-stage, and general proper BAIDs, with a sensible MLP/metamodel pipeline for continuous spaces. Props 3 and 4 are close to right and the mode-to-argmax equivalence is sound when all surviving policy arguments are conditioned on. The case study is detailed and the sensitivity analyses are reasonable. The paper also honestly concedes in Sec. 4 that Algorithm 2 is exact only for low-cardinality discrete problems.

Soft spots, in proportion. First, the formal gap above is load-bearing for Prop 5, though fixable. Second, the 'availability check' in Sec. 4.3 is informal; termination/confluence is asserted rather than proved. Third, the MLP approximations carry no mode-preservation or error-propagation guarantee, so 'continuous domains' is a heuristic claim, not a theorem. Fourth, d*1=0.7 in the case study equals the prior mean mu_d1 used in PA(D1), and sensitivity analysis does not vary mu_d1, so the headline number is partly inherited. Finally, 'first general solution to Shachter's query' overstates: Gonzalez-Ortega et al. (2019) solved discrete BAIDs under ARA, and Koller-Milch (2003) under common knowledge; the claim needs qualification. The code link as printed is also malformed.

Who this is for: people working in ARA, security games, or computational decision analysis. I'd send it to review—the core idea is promising and the worked example is useful—but require a major revision tightening Props 3-5 and re-calibrating the novelty claims. My own citation: probably not in the next 12 months until the reduction is corrected.

Referee Report

4 major / 5 minor

Summary. The paper proposes a computational scheme for solving general security games, modeled as bi-agent influence diagrams (BAIDs), from an adversarial risk analysis (ARA) perspective. The method replaces Shachter-style influence diagram reductions with augmented probability simulation (APS) reductions, so that defender decisions are obtained by finding modes of augmented distributions and attacker decisions are sampled from random augmented distributions. The authors claim this provides the first general solution to Shachter's query for possibly continuous BAIDs. The paper includes a formal algorithmic description (Algorithm 2, Proposition 5), a discussion of computational approximations using neural-network metamodels, and a disinformation-war case study with a reported optimal first-stage defense d*1 = 0.7.

Significance. If the theoretical core were fully correct, the paper would be a valuable contribution: it extends prior discrete ARA solutions (González-Ortega et al., 2019) and sequential APS methods (Ekin et al., 2023) to general, possibly continuous BAIDs, and it provides an implemented, reproducible case study with publicly available code. The mode-to-argmax idea for APS is attractive and, for unconditional defender decisions, the paper correctly proves the required equivalence under positivity and continuity assumptions. However, the formal statements of the conditional-policy reductions contain genuine defects, the termination/confluence argument for Algorithm 2 is incomplete, and the approximation error introduced by the MLP metamodels is not accounted for. These issues are load-bearing for the central claim, so the paper requires major revision before the contribution can be accepted as stated.

major comments (4)

[§3.1 (DAPS1), §3.2 (Prop. 3), §5.2.1] The mode-to-argmax equivalence for conditional policies is misstated. In Section 3.1, DAPS1 defines π(d1,d2,θ2|a2,θ1) ∝ uD(d1,d2,θ2)pD(θ2|d2,a2,θ1) and then says that estimating mode(π(d1|a2,θ1)) approximates d*2(d1,a2,θ1). The joint distribution contains d1, but the conditioning set does not, so the marginal over d2 averages over d1 without a specified weighting measure; its mode is not the pointwise argmax d*2(d1,a2,θ1). A simple counterexample is uD = C − (d1−d2)^2, for which the true conditional argmax is d2 = d1 but the unconditional marginal mode is an averaged value. The correct construction, which Section 5.2.1 actually uses, conditions on d1 as well: π(d2,θ2|d1,a2,θ1) ∝ uD(d1,d2,θ2)pD(θ2|d2,a2,θ1). Proposition 3 has the same defect: conditioning on ant(Xc(D)) can include the very decision being maximized (e.g., D2 ∈ ant(Θ2) in the case study), and it omits utility antecessors such as d1 that survive in the policy argument y′. As written, Proposition 5's proof, which cites Propositions 3 and 4, does not establish the conditional-policy claims. This is a correctable but central error in the formal development.
[§3.2 (Prop. 3 proof) and Section 3.2 discussion] The proof of Proposition 3, and the surrounding discussion, assert that denominators arising from Bayes' rule are 'positive constants' and can therefore be ignored. Positivity alone is insufficient: a positive factor that depends on the decision being maximized changes the argmax. In Example 2 the denominator pD(x2) is independent of d, so the argument works, but the general proposition does not state or prove that the ignored marginals are constant with respect to the decision variable. Arc inversions in a general proper BAID can produce marginals that depend on remaining decision variables. The paper should either prove that the Shachter elimination ordering in a proper BAID always makes such marginals independent of the optimized decision, or add the required condition to Propositions 3 and 4.
[§3.2 (Prop. 5 proof) and §4.3] Proposition 5's proof concludes termination from the finiteness of the DDP and ADP, but correctness and termination of Algorithm 2 also depend on the informal availability check in Section 4.3. The proof does not show that, whenever a Defender decision Di is reached, the required attacker distributions are actually available after finitely many AAPS reductions, nor that these reductions do not depend circularly on the same Di. The statement 'reduce the required Aj nodes in the ADP inverse sequence order until deducing the corresponding attack distributions' describes a heuristic rather than a proven property of proper BAIDs. Finiteness of the two decision paths alone does not rule out the possibility that the else branch reduces a wrong attacker node and the loop fails to make progress toward the needed distribution. A rigorous proof of the availability/confluence property is needed for Proposition 5.
[§4.1, §4.2, §5.2] The implemented algorithm replaces the exact ingredients of Algorithm 2 with MLP metamodels: ψD(d1,a2,θ1) in Section 4.1, pD(a2|d1,a1) and ΨA(d1,a1) in Section 4.2, and subsequently pD(a1) in Section 5.2.3. Proposition 5, however, states optimality for Algorithm 2 under exact distributions and utilities. The paper does not provide any error propagation, consistency, or approximation bound for the MLP-based scheme, nor does it state conditions under which the approximate d*1 approaches the exact ARA solution. This gap is load-bearing because the case study's final recommendation d*1 = 0.7 is computed entirely through the approximate pipeline. The authors should either prove a convergence or error bound under explicit metamodel accuracy assumptions, or clearly restrict the theoretical optimality claim to the exact version and present the MLP scheme as a heuristic with empirical validation.

minor comments (5)

[§3.1 (DAPS1)] The notation 'mode(π(d1|a2,θ1))' appears to be a typo; the intended quantity is the mode of π(d2|d1,a2,θ1). This typo is not merely cosmetic because it obscures the conditioning set required for the conditional argmax.
[Algorithm 1c] In the AAPS loop, the MH call is written as MH(gD, uD, ...) but should be MH(gA, uA, ...), since the proposal and utility are the attacker's.
[§5.2.1] The text says 'J = 9621 = 213' values for the grid (d1,a2,θ1). With splits of 0.05 on [0,1], the correct count is 21^3 = 9261. The number 9621 appears to be a typo.
[§5.2.4 and SM2.2] The reported optimal first-stage defense d*1 = 0.7 coincides exactly with the mean µd1 = 0.7 used to center the attacker's assessed distribution PA(D1) in the supplementary materials. Since the output is partly inherited from this input prior, the case study would be more convincing if the sensitivity of d*1 to µd1 were reported.
[§4.3] The term 'DP P' in the sentence 'the last decision node Di in DP P' is a typo for 'DDP'.

Circularity Check

3 steps flagged · score 6.0 of 10

The single-stage APS mode identities are not circular, but the general BAID reduction as written conditions on the wrong information set, so the claimed conditional policy is, by construction, a d1-averaged decision; Proposition 5 inherits this gap.

self definitional [Section 3.1, DAPS1]
"DAPS1. Aggregate steps D1-D2, corresponding to d∗2(d1,a2,θ1)=arg maxd2 ∫ uD(d1,d2,θ2)pD(θ2|d2,a2,θ1) dθ2. For this, define the AD π(d1,d2,θ2|a2,θ1) ∝ uD(d1,d2,θ2)pD(θ2|d2,a2,θ1). By sampling from this distribution and consistently estimating mode(π(d1|a2,θ1)), we approximate d∗2(d1,a2,θ1)."

The augmented distribution is conditioned on (a2,θ1) only, not on d1. Its marginal in d2 integrates uD·pD over θ2 and d1, so its mode is argmax_{d2} ∫ ψD(d1,d2,a2,θ1) dd1, the d1-averaged second-stage optimum, not the pointwise policy d*2(d1,a2,θ1). If the text instead means the mode over d1, that maximizes a d2-averaged function of d1 and is not a conditional policy at all. Either way, the written AD removes the conditioning on d1 that would make its mode equal the claimed policy; Section 5.2.1 silently restores this conditioning by invoking DAPS1 on a grid of (d1,a2,θ1) values.
self definitional [Appendix A, proof of Proposition 3; SM1, proof of Proposition 4]
"Consider the AD π(d, x1, . . . , xnD |ant(Xc(D))) ∝ ψD(y) ∏_{Xi∈Xc(D)} pD(xi|ant(Xi)). If all nodes participating in arc inversions are eliminated, then d∗(y′) = mode(π(d|ant(Xc(D))))"

The general reduction conditions the AD on ant(Xc(D)) and asserts the mode equals d*(y′), where y′ = (y \ (Xc(D) ∪ D)) ∪ ant(Xc(D)). But y′ can contain utility antecessors not in ant(Xc(D)): in DAPS1, y={d1,d2,a2,θ1}, Xc(D)={Θ2}, ant(Θ2)={d2,a2,θ1}, so y′ contains d1 while the AD conditions only on {a2,θ1}. The marginal mode over d then integrates over d1, yielding an averaged optimum rather than the pointwise policy. The proof's assertion that the conditional marginal is proportional to the expected utility holds only after conditioning on all of y′; the written construction defines a different functional. Proposition 4 and its SM proof repeat the same conditioning error for the attacker.

1 more flagged steps

other [Appendix A, proof of Proposition 5]
"Then, reduce the corresponding Di from the DDP, with the same optimality preserving features. As both the DDP and the ADP are finite, the algorithm necessarily terminates providing the required output as deduced from Propositions 3 and 4."

Proposition 5's optimality conclusion is deduced entirely from Propositions 3 and 4, and those propositions' mode identities fail for conditional policies whenever the policy's information set includes variables outside ant(Xc(·)). The central claim that Algorithm 2 provides the Defender's optimal policy therefore rests on identifying the policy with a mode that the written augmented distributions compute only after averaging out the policy's conditioning variables. This is a construction-level mismatch in the derivation chain, not an external benchmark or independent verification.

full rationale

The core single-stage machinery is not circular: πD(d,a,θ) ∝ uD(d,θ)pD(θ|d,a)pD(a) has marginal proportional to ψD(d), and the random-attacker analogue has marginal mode equal to the random optimal attack; these are mathematical identities that assume no target result. The case-study agreement d*1=0.7 with the prior mean µd1=0.7 is suggestive but no equation forces that equality, so I do not score it as formal circularity. The circularity lies in the general BAID extension: DAPS1 and Propositions 3-4 define augmented distributions conditioned on ant(Xc(·)) rather than on the full information set y′, so the mode they define is, by construction, an averaged decision rather than the claimed conditional policy d*(y′). Proposition 5 inherits this gap, making the paper's central 'first general solution' claim rest on a self-definitional identification. Since the single-stage identities are sound and the case-study implementation de facto conditions on the grid points, thereby correcting the written construction, the paper is partially circular rather than wholly forced; score 6.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The theoretical framework requires standard regularity (positive utilities, compact sets, continuous densities, full-support proposals), the proper-BAID graphical condition, and a level-2 bounded-rationality solution concept. The case study introduces many hand-set parameters; none of them are fitted to external data, but the illustrative optimum is not independent of them, notably the prior mean mu_d1=0.7.

free parameters (5)

mu_d1 (prior mean for PA(D1)) = 0.7
Hand-chosen expert input in SM2.2; the final d*1 equals this value, so the illustrative optimum is not independent of it.
omega_d2 = 0.9
Effectiveness parameter in the case study; varied in sensitivity analysis, showing the output is sensitive to it.
td, ta = td=1, ta=1.2
Relative effectiveness of defender and attacker investments in the case study.
gamma_D, gamma_A = 2616, 1280
Additive constants to ensure positive utilities; do not change the argmax, so they are harmless free choices.
augmentation parameter h per stage = 40, 80, 120, 20
Chosen heuristically in SM3 to achieve convergence; affects MCMC efficiency but not the theoretical mode.

assumptions (6)

domain assumption Defender and attacker utilities are positive (a.s. for random utilities).
Required for augmented distributions to be well-defined as probability densities; paper adds constants gamma_D, gamma_A to enforce it. Section 2.1 and Appendix A.
domain assumption Decision sets are compact and distributions continuous and positive; proposal distributions have full support.
Conditions of Proposition 1 for MH convergence and mode consistency.
domain assumption The game is a proper BAID with acyclic directed graph and directed decision paths (DDP, ADP).
From Banks et al. 2015; assumed throughout the general scheme, Section 3.2.
domain assumption Level-2 thinking: attacker maximizes expected utility while modeling the defender as non-strategic.
Restricts the ARA solution concept to a specific bounded-rationality hierarchy; Section 1.
ad hoc to paper Metamodel approximations (MLPs) are sufficiently accurate to preserve the mode of the augmented distribution.
Unquantified in the paper; the case study uses holdout errors but does not propagate them to the final decision. Sections 4.1-4.2, 5.2.
standard math Consistent mode estimators and MH convergence (Smith & Roberts 1993; Chacon 2020).
Used in proofs of Propositions 1, 3, 4.

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Pith. "Pith review of Computational adversarial risk analysis for general security games." pith.science (2026). https://pith.science/paper/MYLC24U6

@misc{pith2026250602603,
  author       = {Pith},
  title        = {Pith review of: Computational adversarial risk analysis for general security games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MYLC24U6}},
  note         = {Machine review of arXiv:2506.02603}
}

read the original abstract

This paper provides an efficient computational scheme to handle general security games from an adversarial risk analysis perspective. Two cases in relation to single-stage and multi-stage simultaneous defend-attack games motivate our approach to general setups which uses bi-agent influence diagrams as underlying problem structure and augmented probability simulation as core computational methodology. Theoretical convergence and numerical, modeling, and implementation issues are thoroughly discussed. A disinformation war case study illustrates the relevance of the proposed approach.

Figures

Figures reproduced from arXiv: 2506.02603 by the authors.

**Figure 2.** Template for basic n-stage simultaneous defend-attack game BAID. Let us briefly discuss how the single-stage approach in Section 2.1 extends to a 2-stage case [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** presents the (a) defender’s and (b) attacker’s problems in this case. Assuming that the utility function uD(d1, d2, θ1, θ2), and the distributions pD(θ2|d2, θ1, a2), pD(a2|a1), pD(a1), and pD(θ1|d1, a1) are available, we proceed as follows, using Shachter (1986)’s algorithm to solve the Defender’s problem. 1. Remove chance node θ2, computing ψD(d1, d2, θ1, a2)=R uD(d1, d2, θ1, θ2)pD(θ2|d2, θ1, a2)dθ2. 2. Remove chan… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: (a) BAIDS for DC case study, (b) D’s influence diagram, (c) A’s influence diagram. Figures 4b and 4c respectively reflect IDs for the Defender and Attacker problems. 3.1. Methodology motivation Let us first solve D’s problem conceptually. Assume for now that we have av…

**Figure 5.** Figure 5: ID for problem where all nodes affected by arc inversion are reduced before node D. Invert arc X2−X3, with updated probabilities pD(x3|x2) = pD(x2|x3)p(x3)/pD(x2) and pD(x2) = R pD(x2|x3)pD(x3) dx3. Eliminate X3, obtaining the expected utility ψD(x2, d) = R uD(x3, d)pD…

**Figure 6.** Figure 6: ID for problem where not all nodes affected by arc inversion are reduced before node D. Invert arc X1 → X2, computing pD(x2) = R pD(x2|x1)pD(x1) dx1 and pD(x1|x2) = pD(x2|x1) pD(x1)/pD(x2). Eliminate X1, computing ψD(x2, d) = R uD(d, x1)pD(x1|x2) dx1. Eliminate D, comp…

**Figure 7.** Figure 7: (Left) Mean ⟨d ∗ 2 (d1, a2, θ1)⟩d1 of d ∗ 2 (d1, a2, θ1) over d1. (Right) Expected number E[θ2 | d ∗ 2 (d1, a2, θ1), a2, θ1] of individuals to be infected. Best viewed in color [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: Expected random optimal attack A∗ 2 (d1, a1) in the second stage. Best viewed in color [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: Density of empirical distribution A∗ 1 . 5.2.4. DAPS2. Defender’s first stage decision With the approximations ψbD(d1, a2, θ1), pbD(a2|d1, a1), and pbD(a1), finding the Defender’s first stage decision involves applying APS by invoking DAPS2. The optimal decision is d ∗…

**Figure 10.** Figure 10: Expected degree of recognition E[θ1 | d ∗ 1 , a1, a2] for d ∗ 1 across different values of (a1, a2). Best viewed in color. 5.2.5. Summary of Defender’s optimal policy and Attacker’s random optimal policy. Defender’s optimal policy. In the first stage, the Defender inv…

**Figure 11.** Figure 11: (Top row) Mean ⟨d ∗ 2 (d1, a2, θ1)⟩d1 of d ∗ 2 (d1, a2, θ1) over d1 for different ωd2 . (Bottom row) Expected number E[θ2 | d ∗ 2 (d1, a2, θ1), a2, θ1] of individuals to be infected for different ωd2 . Best viewed in color. Observe the differences across different val…

**Figure 12.** Figure 12: A ∗ 2 (d1, a1) for several (td, ta). Best viewed in color. Although the three plots display similar patterns, we observe some differences between them. First, when ta/td decreases, the region A ∗ 2 (d1, a1) ≈ 0 (white) expands, suggesting that A perceives a lower prob…

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write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence '...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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