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Disintegration Temporal Logic conditions probabilities on finite or infinite event sequences, making probabilistic hyperproperties like general non-interference expressible and, in two fragments, decidable.

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2026-08-01 23:48 UTC pith:Q6AVOCHH

load-bearing objection New disintegration-based probabilistic hyperproperty logic with a genuinely useful linear fragment, but the undecidability proof as printed fails and the qualitative fragment is oversold. the 3 major comments →

arxiv 2607.15223 v1 pith:Q6AVOCHH submitted 2026-07-16 cs.LO

Disintegration Temporal Logic for Probabilistic Hyperproperties

Mishel Carelli , Bernd Finkbeiner This is my paper
classification cs.LO MSC 03B7068Q6060A10
keywords Disintegration Temporal Logicprobabilistic hyperpropertiesprobabilistic non-interferenceperfect indistinguishabilitymeasure disintegrationMarkov chain model checkingundecidabilityautomata-theoretic verification
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper introduces Disintegration Temporal Logic (DTL), a probabilistic temporal logic that conditions probabilities on finite or infinite sequences of events through measure disintegration. Because disintegration handles probability-zero events, DTL can condition on individual infinite traces, which lets it express probabilistic hyperproperties that earlier logics could not capture, including general probabilistic non-interference and perfect indistinguishability. The paper shows that full DTL model checking is undecidable, but the linear fragment, which contains those security properties, is decidable in polynomial time. It also shows that the qualitative fragment, where inner probabilities compare only to 0 or 1, is decidable for precisely labeled Markov chains satisfying the cascade property, with complexity a tower of exponentials in alternation depth. The framework can distinguish systems that have the same overall failure probability but spread failures differently across environment executions.

Core claim

DTL evaluates formulas against a trace, a timestamp, a cut, and a probability measure. A cut divides each trace into a conditioned prefix and an unconstrained suffix, and disintegration yields the conditional distribution of the future given the prefix, even when the prefix has probability zero. The semantics are independent of the chosen disintegration kernel up to measure zero. The paper's main algorithmic contributions are two decidable fragments: the linear fragment, where probabilities are compared for equality or against 1, has a polynomial-time model-checking procedure via reduction to a quadratic form over paths of a multidynamic automaton; the qualitative fragment, where inner proba

What carries the argument

The central object is a cut—a function assigning each atomic proposition either a finite length or the symbol ω—splitting a trace into a conditioned prefix L(C) and an unconstrained suffix U(C). Disintegration kernels give conditional distributions of the suffix given the prefix even for probability-zero events, with Theorem 5 showing the kernel is a limit of finite-prefix conditionings. The cascade property, a conditional-independence condition, makes that limit stabilize after finitely many steps and drives the qualitative-fragment algorithm. The linear-fragment algorithm instead reduces conditional-probability equality to a quadratic form over a multidynamic automaton, using pointwise mod

Load-bearing premise

The qualitative-fragment decidability theorem requires the Markov chain's induced distribution to satisfy the cascade property—once a prefix is fixed, the future of the conditioned propositions must not depend on the rest of the prefix—and the chain must be precisely labeled; the correctness proof collapses if an arbitrary chain is fed to the algorithm.

What would settle it

Construct a small Markov chain where the future of a proposition b at step 2 depends on the full first-step prefix of another proposition a, not just on the b-projection, so the cascade property fails. Evaluate a qualitative formula such as P(φ | [{b}]) = 1, where φ asserts b at step 2, and compare the algorithm's output with the conditional probability computed as the limit of finite-prefix conditioning per Theorem 5; a mismatch would show the cascade assumption is load-bearing.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • General probabilistic non-interference and perfect indistinguishability are expressible as DTL formulas and model-checkable in polynomial time, something earlier probabilistic temporal logics could not do.
  • Any extension that retains full conditional-probability operators must respect the linear or qualitative fragment boundaries if it wants decidability.
  • The qualitative fragment provides a decidable soft alternative to existential scheduler quantification: instead of asking whether some scheduler achieves a goal, it asks whether a positive-measure set of schedulers does, turning undecidable probabilistic-automata problems into decidable ones.
  • DTL can separate systems with identical average failure probability according to how failures are distributed over environment executions, revealing concentrated failures that average-based reasoning misses.
  • Because the cascade property is expressible in the linear fragment, whether a chain qualifies for the qualitative algorithm can itself be checked in polynomial time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A practical implementation could first run the polynomial-time cascade-compatibility check and then apply the qualitative algorithm, rejecting inapplicable inputs before verification; this workflow is implied but not explicitly stated.
  • The linear-fragment quadratic-form reduction could plausibly be extended from exact equality to approximate or quantitative conditional-independence checking, yielding a broader set of tools for information-flow measurement.
  • The soft scheduler quantification suggests a more general pattern: probabilistic systems whose schedulers are drawn from a product measure may admit decidable hyperproperty verification in cases where strict existential quantification is undecidable, though that generalization remains unproven.
  • The framework's failure-concentration distinction could support a testable robustness metric: compute the maximum, over positive-measure environment traces, of the conditional probability of an undesirable behavior, which DTL can express but classical average-based logics cannot.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper introduces Disintegration Temporal Logic (DTL), a probabilistic temporal logic with a conditioning operator based on measure disintegration. DTL is designed to express probabilistic hyperproperties such as probabilistic non-interference, perfect indistinguishability, and properties of systems interacting with stochastic environments. The authors claim that model checking Markov chains against full DTL is undecidable (Theorem 14), that the linear fragment admits a polynomial-time model-checking algorithm (Theorem 18), and that the qualitative fragment is decidable for cascade-compatible, precisely labeled Markov chains with a non-elementary upper bound (Theorem 20). The technical development includes a measurability theorem for the semantics (Theorem 13), a constructive disintegration theorem (Theorem 5), and an automata-theoretic construction for the qualitative fragment.

Significance. If the technical results stand, DTL is a novel and well-motivated framework for conditional-independence hyperproperties, with a genuinely useful PTIME algorithm for the linear fragment and a nontrivial automata-theoretic decision procedure for a qualitative fragment. The measure-theoretic foundation, especially the measurability and a.e.-uniqueness theorem (Theorem 13), is careful and necessary. However, the undecidability proof is flawed as written, and the qualitative-fragment decidability claim is substantially narrower than the abstract suggests because it requires cascade-compatibility and precise labeling.

major comments (3)
  1. [§3.2.2, Theorem 14] The reduction is invalid as written. The formula ϕ := P( (P(af|[AP])>0.5) ) > 0 evaluates the inner probabilistic operator at timestamp n=0 under the semantics of §3.2.2; the outer P does not advance the timestamp, and C0(p)=0 makes the inner conditioning empty. Thus the inner value is the step-0 marginal of af, not the prefix-dependent probability. A one-state PFA with d=1, G={[1]}, π=[1], f=[0] is empty, yet the constructed Markov chain satisfies the formula because af holds with probability 1 at step 0. The correctness argument requires evaluating the inner operator at timestamp m (e.g., via F or an iterated next). This is load-bearing for the undecidability claim.
  2. [§5, linear fragment grammar] The production for θ is printed as 'θ ::= P(κ|[A1]) = P(κ|[A2]) | θ | θ | P(θ|[B]) = 1', with two bare θ alternatives; the connective symbols are missing, so the fragment is not well-defined. Moreover, the proof of Theorem 18 assumes θ is a chain of next operators and P(...)=1 operators around the atomic equality; it does not explain how Boolean combinations inside θ are treated. Please restore the full grammar and adjust the algorithm/proof accordingly, or state the intended fragment precisely.
  3. [Abstract / §6, Theorem 20] The abstract and introduction present the qualitative-fragment result without the hypotheses of Theorem 20: the Markov chain must be precisely labeled and the formula cascade-compatible. Corollary 23 and Lemma 25 rely on the cascade property for the finite-step stabilization, and on injective labeling to identify paths by their label. Without these assumptions the algorithm has no correctness guarantee. The scope should be disclosed in the abstract, or the theorem generalized if possible.
minor comments (6)
  1. [§2.1] The sentence 's_X denotes the projection of s on Y' should read 'on X'.
  2. [Theorem 14 proof] In the labeling definition, 'iff j = 1' should be 'if f_j = 1' (or the intended condition clarified).
  3. [Definition 21] The inequality 'C(a) ≤ N for all a ∈ AP′' should probably be 'a ∈ AP\AP′'.
  4. [§6.2] The encoding of finite timestamps and cuts via traces of the form 0^n ⊥ ... is under-specified; define the decoding from {0,⊥,⊤}-traces to N∪{ω} explicitly.
  5. [§5, Lemma 17] The complexity bound PTIME(|D|·n) is inconsistent with the vector-space dimension |Q|·n^2 used in the stabilization argument; the bound should be polynomial in |Q|·n^2 (still PTIME).
  6. [§6.3, Lemma 25] The proof uses 'µ-a.e. T' for individual n; the conclusion for all n relies on a countable union of null sets. This is acceptable but should be stated explicitly.

Circularity Check

0 steps flagged

No significant circularity: the logic's semantics, decidability reductions, and fragment algorithms are built on external results, with the main caveat being a correctness gap in the printed undecidability reduction, not a self-referential derivation.

full rationale

I examined the claimed derivation chain: DTL is defined directly via a satisfaction relation and disintegration kernels; the expressiveness examples are semantic encodings rather than fitted predictions. The linear-fragment PTIME algorithm reduces model checking to a multidynamic-automaton quadratic form (Theorem 18, Lemma 17), using Courcoubetis–Yannakakis for LTL probabilities and Tzeng's equivalence algorithm as external tools. No parameter is fitted to data and then renamed as a prediction. The qualitative-fragment decidability result (Theorem 20) is explicitly conditional on cascade compatibility (Definition 21) and precise labeling; Corollary 23 is proved from the cascade property rather than assumed. The cascade property is itself checked by a reduction to the independently proven linear fragment, so this is an assumption stated in the theorem, not a hidden circular import. Self-citations to HyperLTL/HyperCTL* are background or extension points and are not used as an unverified uniqueness theorem. One non-circularity caveat: the printed Theorem 14 formula 'P( (P(af|[AP])>0.5)))>0' contains no explicit temporal operator to advance the timestamp, so as written the reduction to PFA strict cutpoint emptiness appears unsound; however, this is a correctness gap, not a circularity, because the proof does not use its own conclusion as an input.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 4 invented entities

The central claims rest on standard measure-theoretic and automata-theoretic tools, plus two domain assumptions for the qualitative fragment (cascade property and precise labeling). There are no fitted free parameters. The invented entities (cut, multidynamic automaton, cascade property) are new mathematical constructs introduced to define the logic and its algorithms, none with independent external evidence.

axioms (6)
  • standard math Disintegration theorem (Kallenberg, Theorem 1)
    Used to define conditional probabilities on null events (infinite traces), which underpins the DTL semantics.
  • standard math Conditioning Limits Theorem (Kallenberg, Theorem 9.24)
    Used in Theorem 5 to prove that finite-prefix conditioning converges to the disintegration measure.
  • standard math Limit Behaviour of Markov Chains (Baier-Katoen, Theorem 10.27)
    Used in Lemma 25 to argue that almost every path of a finite Markov chain visits every state of some BSCC infinitely often.
  • standard math Strict cutpoint emptiness for probabilistic finite automata is undecidable (Rabin)
    Basis of the undecidability reduction in Theorem 14.
  • domain assumption Cascade property (Definition 21)
    A conditional-independence requirement that the measure must satisfy for the qualitative-fragment algorithm to be correct. It is not true for all Markov chains and is not stated in the abstract.
  • domain assumption Precise labeling (l is injective)
    Assumed in Theorem 20 and used in Lemma 25 to identify finite paths by their label traces. Limits the applicability of the qualitative-fragment algorithm.
invented entities (4)
  • DTL logic no independent evidence
    purpose: The paper's central contribution: a logic for probabilistic hyperproperties with conditional-probability operators.
    The logic itself is the contribution; there is no external falsifiable handle beyond the definitions and theorems in the paper.
  • cut no independent evidence
    purpose: A formal device that separates the conditioned prefix from the unconditioned suffix of an infinite trace; underlies the semantics of P(φ|[A],B).
    A new mathematical construct defined in Section 2.4; no physical counterpart.
  • multidynamic automaton no independent evidence
    purpose: Auxiliary automaton used in the PTIME model-checking algorithm for the linear fragment (Section 5.1).
    A technical device introduced to encode the equality of two conditional probabilities; no external evidence.
  • cascade property no independent evidence
    purpose: A conditional-independence condition that makes the qualitative-fragment algorithm correct; holds for cascade products.
    Defined in Section 6.1; a property of measures, not an empirical entity.

pith-pipeline@v1.3.0-alltime-deepseek · 26883 in / 18805 out tokens · 151134 ms · 2026-08-01T23:48:32.429055+00:00 · methodology

0 comments
read the original abstract

We introduce Disintegration Temporal Logic (DTL), a new probabilistic temporal logic that can express a wide range of probabilistic hyperproperties, including probabilistic non-interference and perfect indistinguishability. DTL is based on the notion of measure disintegration from probability theory, which allows for conditioning probabilities on a finite or infinite sequence of events occurring during a program execution. This naturally supports reasoning about interacting stochastic systems, where complete executions of one component induce conditional probability distributions over another. We illustrate applications of DTL to systems interacting with stochastic environments, distributional properties of Markov decision processes, and probabilistic automata on infinite words, and discuss its relationship to existing probabilistic logics. While model checking Markov chains against full DTL is undecidable, we identify two decidable fragments that capture many hyperproperties of interest. The linear fragment admits a polynomial-time model-checking procedure based on linear-algebraic techniques and captures probabilistic information-flow properties such as perfect indistinguishability and history-based probabilistic non-interference. The qualitative fragment admits an automata-theoretic model-checking procedure that extends the standard algorithm for $\mathit{HyperCTL}^*$ with reasoning about bottom strongly connected components.

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Reference graph

Works this paper leans on

24 extracted references · 1 canonical work pages

  1. [4]

    5 Miguel E

    URL:https://api.semanticscholar.org/CorpusID:13743458. 5 Miguel E. Andrés and Peter van Rossum. Conditional probabilities over probabilistic and nondeterministic systems. In C. R. Ramakrishnan and Jakob Rehof, editors,Tools and Algorithms for the Construction and Analysis of Systems, 14th International Conference, TACAS 2008, Held as Part of the Joint Eur...

    2008

  2. [8]

    Computingconditional probabilities in markovian models efficiently

    9 ChristelBaier, JoachimKlein, SaschaKlüppelholz, andSteffenMärcker. Computingconditional probabilities in markovian models efficiently. In Erika Ábrahám and Klaus Havelund, editors, Tools and Algorithms for the Construction and Analysis of Systems - 20th International Conference, TACAS 2014, Held as Part of the European Joint Conferences on Theory and Pr...

    2014

  3. [10]

    11 Gilles Barthe, Benjamin Grégoire, and Santiago Zanella Béguelin

    URL:http://dx.doi.org/10.1145/3704894, doi:10.1145/3704894. 11 Gilles Barthe, Benjamin Grégoire, and Santiago Zanella Béguelin. Formal certification of code-based cryptographic proofs.SIGPLAN Not., 44(1):90–101, January

    doi:10.1145/3704894

  4. [11]

    12 Gilles Barthe, Justin Hsu, and Kevin Liao

    doi: 10.1145/1594834.1480894. 12 Gilles Barthe, Justin Hsu, and Kevin Liao. A probabilistic separation logic.Proc. ACM Program. Lang., 4(POPL), December 2019.doi:10.1145/3371123. 13 R. Baumann. Radiation-induced soft errors in advanced semiconductor technologies.IEEE Transactions on Device and Materials Reliability, 5:305–316, 10 2005.doi:10.1109/TDMR. 20...

    arXiv 2019

  5. [14]

    org/abs/1401.4492,arXiv:1401.4492

    URL:https://arxiv. org/abs/1401.4492,arXiv:1401.4492. 18 Michael R. Clarkson and Fred B. Schneider. Hyperproperties. In2008 21st IEEE Computer Security Foundations Symposium, pages 51–65, 2008.doi:10.1109/CSF.2008.7. 19 Michael R. Clarkson and Fred B. Schneider. Hyperproperties.J. Comput. Secur., 18(6):1157– 1210,

    Pith/arXiv arXiv 2008

  6. [23]

    49 Yu Wang, Siddhartha Nalluri, Borzoo Bonakdarpour, and Miroslav Pajic

    arXiv:https://doi.org/10.1137/ 0221017,doi:10.1137/0221017. 49 Yu Wang, Siddhartha Nalluri, Borzoo Bonakdarpour, and Miroslav Pajic. Statistical model checking for hyperproperties,

    doi:10.1137/0221017

  7. [24]

    URL:https://arxiv.org/abs/1902.04111, arXiv:1902. 04111

    Pith/arXiv arXiv 1902

  8. [67]

    URL:http://mi.mathnet.ru/eng/sm5995. 45 C. E. Shannon. Communication theory of secrecy systems.The Bell System Technical Journal, 28(4):656–715, 1949.doi:10.1002/j.1538-7305.1949.tb00928.x. 46 G. Smith. Probabilistic noninterference through weak probabilistic bisimulation. In16th IEEE Computer Security Foundations Workshop,

    arXiv 1949

  9. [1963]

    URL: https://www.sciencedirect.com/science/article/pii/S0019995863902900, doi:10.1016/S0019-9958(63)90290-0. 44 V. A. Rokhlin. On the fundamental ideas of measure theory.Matematicheskii Sbornik, 25(1):107–150,

    doi:10.1016/s0019-9958(63)90290-0

  10. [1992]

    30 J.W. Gray. Probabilistic interference. InProceedings. 1990 IEEE Computer Society Symposium on Research in Security and Privacy, pages 170–179, 1990.doi:10.1109/RISP.1990.63848. 31 J.W. Gray. Toward a mathematical foundation for information flow security. InProceedings. 1991 IEEE Computer Society Symposium on Research in Security and Privacy, pages 21–3...

    arXiv 1990

  11. [1997]

    15 Krishnendu Chatterjee and Mathieu Tracol

    URL:https://api.semanticscholar.org/CorpusID:16749932. 15 Krishnendu Chatterjee and Mathieu Tracol. Decidable problems for probabilistic automata on infinite words.2012 27th Annual IEEE Symposium on Logic in Computer Science, pages 185–194,

    2012

  12. [2003]

    47 Terrence Tao

    Proceedings., pages 3–13, 2003.doi: 10.1109/CSFW.2003.1212701. 47 Terrence Tao. An introduction to measure theory

    Pith/arXiv arXiv 2003

  13. [2007]

    43 Michael O

    URL: http://dx.doi.org/10.2168/LMCS-3(3:5)2007,doi:10.2168/lmcs-3(3:5)2007. 43 Michael O. Rabin. Probabilistic automata.Information and Control, 6(3):230–245,

    doi:10.2168/lmcs-3(3:5)2007 2007

  14. [2008]

    Springer, 2008.doi:10.1007/ 978-3-540-78800-3\_12

    Proceedings, volume 4963 ofLecture Notes in Computer Science, pages 157–172. Springer, 2008.doi:10.1007/ 978-3-540-78800-3\_12. 6 Martin Avanzini, Gilles Barthe, Davide Davoli, and Benjamin Grégoire. A quantitative probabilistic relational hoare logic,

    2008

  15. [2009]

    8Christel Baier and Joost-Pieter Katoen.Principles of model checking

    doi: 10.4204/EPTCS.3.1. 8Christel Baier and Joost-Pieter Katoen.Principles of model checking. MIT Press,

    work page doi:10.4204/eptcs.3.1

  16. [2010]

    20 Arthur Correnson and Bernd Finkbeiner

    URL:https://doi.org/10.3233/JCS-2009-0393. 20 Arthur Correnson and Bernd Finkbeiner. Coinductive proofs for temporal hyperliveness.Proc. ACM Program. Lang., 9(POPL), January 2025.doi:10.1145/3704889. 21 Oswaldo Costa, Marcelo Fragoso, and Ricardo Marques. Discrete-time markov jump linear systems. 01 2005.doi:10.1007/b138575. 22 Costas Courcoubetis and Mih...

    arXiv 2009

  17. [2011]

    30 Disintegration Temporal Logic for Probabilistic Hyperproperties 16Michael R

    URL:https://api.semanticscholar.org/CorpusID:14146404. 30 Disintegration Temporal Logic for Probabilistic Hyperproperties 16Michael R. Clarkson, Bernd Finkbeiner, Masoud Koleini, Kristopher K. Micinski, Markus N. Rabe, and César Sánchez. Temporal logics for hyperproperties. In Martín Abadi and Steve Kremer, editors,Principles of Security and Trust - Third...

    2014

  18. [2013]

    35 Vijay Anand Korthikanti, Mahesh Viswanathan, Gul Agha, and YoungMin Kwon

    URL:https://books.google.de/books?id=KeObBgAAQBAJ. 35 Vijay Anand Korthikanti, Mahesh Viswanathan, Gul Agha, and YoungMin Kwon. Reasoning about mdps as transformers of probability distributions. In2010 Seventh International Conference on the Quantitative Evaluation of Systems, pages 199–208, 2010.doi:10.1109/ QEST.2010.35. 36 Orna Kupferman and Moshe Y. V...

    arXiv 2010

  19. [2014]

    Springer, 2014.doi:10.1007/ 978-3-642-54862-8\_43

    Proceedings, volume 8413 ofLecture Notes in Computer Science, pages 515–530. Springer, 2014.doi:10.1007/ 978-3-642-54862-8\_43. 10 Jialu Bao, Emanuele D’Osualdo, and Azadeh Farzan. Bluebell: An alliance of relational lifting and independence for probabilistic reasoning.Proceedings of the ACM on Programming Languages, 9(POPL):1719–1749, January

    2014

  20. [2018]

    URL:https://arxiv.org/abs/1804.01853,arXiv:1804.01853. 2S. Akshay, Krishnendu Chatterjee, Tobias Meggendorfer, and Ðorđe Žikelić. Certified policy verification and synthesis for mdps under distributional reach-avoidance properties,

    Pith/arXiv arXiv

  21. [2020]

    URL: https://arxiv.org/abs/2005.03362, arXiv:2005. 03362. 24 E. O. Elliott. Estimates of error rates for codes on burst-noise channels.The Bell System Technical Journal, 42(5):1977–1997, 1963.doi:10.1002/j.1538-7305.1963.tb00955.x. 25 Bernd Finkbeiner. Logics and algorithms for hyperproperties.ACM SIGLOG News, 10(2):4–23, 2023.doi:10.1145/3610392.3610394....

    Pith/arXiv arXiv 2005

  22. [2023]

    Springer-Verlag.doi: 10.1007/978-3-031-37709-9_5. 4 S. Akshay, Blaise Genest, and Nikhil Vyas. Distribution-based objectives for markov decision processes.Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science,

    doi:10.1007/978-3-031-37709-9_5

  23. [2024]

    URL: https://arxiv.org/abs/2405.04015,arXiv:2405.04015. 3 S. Akshay, Krishnendu Chatterjee, Tobias Meggendorfer, and Ðorđe Žikelić. MDPs as distribution transformers: Affine invariant synthesis for safety objectives. InComputer Aided Verification: 35th International Conference, CAV 2023, Paris, France, July 17–22, 2023, Proceedings, Part III, page 86–112,...

    Pith/arXiv arXiv 2023

  24. [2025]

    7 Christel Baier, Nathalie Bertrand, and Marcus Größer

    URL:https://arxiv.org/abs/2407.17127, arXiv: 2407.17127. 7 Christel Baier, Nathalie Bertrand, and Marcus Größer. Probabilistic automata over infinite words: Expressiveness, efficiency, and decidability. volume 3, pages 3–16, 07

    Pith/arXiv arXiv