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arxiv: 2604.10669 · v1 · submitted 2026-04-12 · 💻 cs.LO · math.LO

A Linear Temporal Logic of Frequencies on Series of Events

Melissa Antonelli , Leonardo Ceragioli , Alessandro Buda , Giuseppe Primiero This is my paper

Pith reviewed 2026-05-10 15:50 UTC · model grok-4.3

classification 💻 cs.LO math.LO
keywords linear temporal logicfrequency operatorsevent seriesmodal quantifiersKripke semanticsquantitative monitoringmachine learning classifiers
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The pith

LTLF adds modal quantifiers to linear temporal logic so frequencies of events in sequences can be expressed and compared to ideal distributions inside one formal system.

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

The paper introduces LTLF as a temporal logic equipped with new measure-sensitive operators. These operators work on top of ordinary Kripke semantics to let users write formulas that track how often events occur in an observed series and how those rates relate to a target distribution. A sympathetic reader would care because the same logical language can now be used both to describe what actually happened in data streams and to state requirements about what should happen, offering a uniform way to monitor and steer quantitative processes such as classifiers.

Core claim

LTLF is obtained by adjoining original modal quantifiers to linear temporal logic and equipping them with a standard Kripke-style semantics; the resulting structure lets a single formula simultaneously capture properties of an actual observed frequency and an ideal target distribution over the same series of events.

What carries the argument

The novel modal quantifiers that act on Kripke models to evaluate frequency measures of event occurrences.

If this is right

  • Formulas can now be written that assert both the actual rate at which an event has occurred and the rate at which it should occur in the future.
  • The same logic can be used to monitor quantitative systems by checking whether observed frequencies stay close to prescribed targets.
  • Prediction of future occurrences becomes expressible as a logical entailment from current frequency statements.
  • The relationship between empirical data and normative requirements can be investigated inside a single deductive system.

Where Pith is reading between the lines

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

  • The framework could be used for runtime verification of systems whose correctness depends on statistical properties rather than only on order of events.
  • Because the semantics is built on ordinary Kripke models, existing model-checking tools might be adapted with only local changes to handle frequency constraints.
  • The logic may serve as a specification language for data-driven controllers that must enforce both temporal order and quantitative bounds.

Load-bearing premise

The Kripke-style semantics together with the new modal quantifiers can be defined without internal contradictions while still distinguishing observed frequencies from ideal ones.

What would settle it

An example series of events together with an explicit frequency requirement for which the semantics assigns both true and false to the same formula, or for which the observed and ideal cases become indistinguishable.

Figures

Figures reproduced from arXiv: 2604.10669 by Alessandro Buda, Giuseppe Primiero, Leonardo Ceragioli, Melissa Antonelli.

Figure 1
Figure 1. Figure 1: A model representing four coin tosses resulting in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A model representing six coin tosses resulting in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A model representing four coin tosses resulting in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

This paper introduces LTLF, a temporal logic designed to express the frequency properties of event series in a natural but rigorous manner. By introducing novel, measure-sensitive operators, LTLF allows for the evaluation of frequencies and the prediction of future occurrences, thus providing a formal framework to monitor and control quantitative systems, such as machine learning classifiers. The core novelty lies in the introduction of original modal quantifiers associated with a standard Kripke-style semantics. These quantifiers enable the explicit formalization of event series properties and the investigation of the relationship between actual observed frequencies and ideal distributions within a single logical structure. This framework bridges the gap between formal logical reasoning and empirical observation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces LTLF, an extension of linear temporal logic that adds novel measure-sensitive modal quantifiers to a standard Kripke-style semantics. These operators are claimed to enable the formalization of frequency properties over series of events, allowing explicit distinction and reasoning between actual observed frequencies and ideal (normative or predicted) distributions within a single logical framework, with intended applications to monitoring and controlling quantitative systems such as machine learning classifiers.

Significance. If the semantics can be rigorously defined without inconsistency and the operators shown to preserve useful LTL properties while adding frequency reasoning, the work could provide a valuable bridge between qualitative temporal logic and quantitative analysis. It offers a potential formal tool for specifying frequency-based requirements in event-driven systems, which is relevant for runtime verification and control in empirical settings. The absence of detailed derivations, consistency proofs, or concrete examples in the high-level presentation limits immediate assessment of its practical impact.

major comments (2)
  1. [§3] §3 (Semantics): The Kripke-style semantics with added modal quantifiers does not specify an explicit construction for assigning measures to paths or event sequences. Standard accessibility relations alone do not carry the required measure structure, leaving open whether formulas can consistently assert both observed frequencies (empirical) and ideal distributions (normative) without contradiction—for instance, when a path satisfies a high observed-frequency atom but violates an ideal-distribution quantifier under the same relation.
  2. [§4] §4 (Properties and examples): No derivations or proofs are provided to establish that the new operators preserve core LTL properties such as the semantics of the until operator or induction principles when mixed with frequency assertions. This undermines the claim that LTLF supports prediction of future occurrences in a rigorous manner.
minor comments (2)
  1. [Introduction] The abstract and introduction could benefit from a brief comparison table contrasting LTLF operators with standard LTL and existing quantitative extensions (e.g., those using probabilities or frequencies).
  2. [§2] Notation for the modal quantifiers should be introduced with an explicit example formula early in the paper to clarify their intended usage before the full semantics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight areas where the high-level presentation can be strengthened with explicit constructions and derivations. We address each major comment below and will incorporate the necessary clarifications and proofs in a revised version of the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Semantics): The Kripke-style semantics with added modal quantifiers does not specify an explicit construction for assigning measures to paths or event sequences. Standard accessibility relations alone do not carry the required measure structure, leaving open whether formulas can consistently assert both observed frequencies (empirical) and ideal distributions (normative) without contradiction—for instance, when a path satisfies a high observed-frequency atom but violates an ideal-distribution quantifier under the same relation.

    Authors: We agree that the current presentation does not provide a fully explicit measure construction. The semantics associates measure-sensitive quantifiers with paths in the Kripke frame, where empirical frequencies are derived from finite prefixes of event sequences and ideal distributions are given by a separate measure on the model (e.g., limiting frequencies or a normative probability measure). To eliminate any ambiguity regarding consistency, we will add an explicit definition of the measure space in the revised §3, including how the two types of measures are distinguished and combined without contradiction. This separation ensures that observed-frequency atoms evaluate path-specific empirical data while ideal-distribution quantifiers reference the model-level measure. revision: yes

  2. Referee: [§4] §4 (Properties and examples): No derivations or proofs are provided to establish that the new operators preserve core LTL properties such as the semantics of the until operator or induction principles when mixed with frequency assertions. This undermines the claim that LTLF supports prediction of future occurrences in a rigorous manner.

    Authors: The referee correctly notes the absence of explicit derivations in the current high-level exposition. While the manuscript claims that the frequency operators are conservative extensions that preserve LTL properties, we did not include the proofs. In the revision we will add a dedicated subsection in §4 containing (i) a proof that the until operator retains its standard inductive semantics when frequency quantifiers are present, and (ii) an extension of the standard LTL induction principle that accounts for frequency assertions. These derivations will directly support the rigorous prediction claims. revision: yes

Circularity Check

0 steps flagged

No circularity; new logic defined from first principles

full rationale

The paper introduces LTLF as a novel extension of LTL via new measure-sensitive modal quantifiers on Kripke frames. No derivation chain reduces a claimed result to its own inputs by construction, no fitted parameters are relabeled as predictions, and no load-bearing steps rely on self-citations or imported uniqueness theorems. The semantics and operators are presented as original definitions, with the relationship between observed frequencies and ideal distributions formalized inside the new structure rather than presupposed. This is a standard self-contained definitional contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on extending standard modal logic semantics with new frequency operators; no free parameters are mentioned, but domain assumptions about measures on event series are required.

axioms (2)
  • standard math Kripke-style semantics for the base temporal logic
    Invoked to define the modal structure for event series.
  • domain assumption Existence of suitable measures on paths for defining frequencies
    Needed to interpret the measure-sensitive operators.
invented entities (1)
  • Measure-sensitive modal quantifiers no independent evidence
    purpose: To express frequencies of events and relate observed to ideal distributions
    New operators introduced as the core novelty of LTLF.

pith-pipeline@v0.9.0 · 5411 in / 1235 out tokens · 40696 ms · 2026-05-10T15:50:06.604723+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Antonelli, U

    M. Antonelli, U. Dal Lago, and P. Pistone. On counting propositional logic and Wagner’s hierarchy.Theoretical Computer Science, 966-967, 2023

    work page 2023

  2. [2]

    Antonelli, U

    M. Antonelli, U. Dal Lago, and P. Pistone. Towards logical foundations for probabilistic computation.Annals of Pure and Applied Logic, 175(9), 2024

    work page 2024

  3. [3]

    A. Aziz, K. Sanwal, V. Singhal, and R. Brayton. Verifying continuous time Markov chains. In R. Alur and T. A. Henzinger, editors,Computer Aided Verification, pages 269–276, Berlin, Heidelberg, 1996. Springer Berlin Heidelberg

    work page 1996

  4. [4]

    A. Aziz, V. Singhal, F. Balarin, R. K. Brayton, and A. L. Sangiovanni-Vincentelli. It usually works: The temporal logic of stochastic systems. In P. Wolper, editor,Computer Aided Verification, pages 155–165, Berlin, Heidelberg, 1995. Springer Berlin Heidelberg

    work page 1995

  5. [5]

    Billingsley.Probability and Measure

    P. Billingsley.Probability and Measure. John Wiley and sons, 1995

    work page 1995

  6. [6]

    Bollig, N

    B. Bollig, N. Decker, and M. Leucker. Frequency linear-time temporal logic. In2012 Sixth International Symposium on Theoretical Aspects of Software Engineering, pages 85–92, 2012

    work page 2012

  7. [7]

    A. Buda, G. Primiero, L. Ceragioli, and M. Antonelli. Counting worlds branching time semantics for post-hoc bias mitigation in generative ai. submitted (2026)

    work page 2026

  8. [8]

    Ceragioli and G

    L. Ceragioli and G. Primiero. Trustworthiness preservation by copies of machine learning systems.International Journal of Approximate Reasoning, 192:109638, 2026

    work page 2026

  9. [9]

    T. Chen, G. Primiero, F. Raimondi, and N. Rungta. A computationally grounded, weighted doxastic logic.Stud Logica, 104(4):679–703, 2016

    work page 2016

  10. [10]

    F. A. D’Asaro, F. A. Genco, and G. Primiero. Checking trustworthiness of probabilistic computations in a typed natural deduction system.Journal of Logic and Computation, 35(6), 2025

    work page 2025

  11. [11]

    Fagin, J

    R. Fagin, J. Halpern, and N. Megiddo. A logic for reasoning about probabilities.Information and Computation, 87(1-2):78–128, 1990. 34

    work page 1990

  12. [12]

    Finkbeiner, F

    B. Finkbeiner, F. Klein, and K. H. Rath. Counting models of linear-time temporal logic. In Language and Automata Theory and Applications - 8th International Conference (LATA ’14), pages 304–315, 2014

    work page 2014

  13. [13]

    Hansson and B

    H. Hansson and B. Jonsson. A logic for reasoning about time and reliability.Formal Aspects of Computing, 6(5):512–535, 1994

    work page 1994

  14. [14]

    Kontinen

    J. Kontinen. A logical characterization of the counting hierarchy.ACM Trans. Comput. Log., 10(1):7:1–7:21, 2009

    work page 2009

  15. [15]

    Kubyshkina and G

    E. Kubyshkina and G. Primiero. A possible worlds semantics for trustworthy non- deterministic computations.International Journal of Approximate Reasoning, 172:109212, 2024

    work page 2024

  16. [16]

    Kwiatkowska, G

    M. Kwiatkowska, G. Norman, and J. Sproston. Symbolic computation of maximal proba- bilisti reachability. In K. G. Larsen and M. Nielsen, editors,CONCUR 2001 — Concurrency Theory, pages 169–183, Berlin, Heidelberg, 2001. Springer Berlin Heidelberg

    work page 2001

  17. [17]

    Laroussinie, A

    F. Laroussinie, A. Meyer, and E. Petonnet. Counting CTL. In L. Ong, editor,Foundations of Software Science and Computational Structures, pages 206–220, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg

    work page 2010

  18. [18]

    Laroussinie, A

    F. Laroussinie, A. Meyer, and E. Petonnet. Counting LTL. In17th International Symposium on Temporal Representation and Reasoning, pages pages 51–58, Paris, France, Sept. 2010. IEEE Computer Society. The original publication is available at ieeexplore.ieee.org

    work page 2010

  19. [19]

    Legastelois, M.-J

    B. Legastelois, M.-J. Lesot, and A. R. d’Allonnes. Typology of axioms for a weighted modal logic.International Journal of Approximate Reasoning, 90:341–358, 2017

    work page 2017

  20. [20]

    Mostowski

    A. Mostowski. On a generalization of quantifiers. In A. Mostowski, editor,Foundational Studies, volume 93 ofStudies in Logic and the Foundations of Mathematics, pages 311–335. Elsevier, 1979

    work page 1979

  21. [21]

    Tomita, S

    T. Tomita, S. Hagihara, and N. Yonezaki. A probabilistic temporal logic with frequency operators and its model checking.Electronic Proceedings in Theoretical Computer Science, 73:79–93, Nov. 2011

    work page 2011

  22. [22]

    K. Wagner. The complexity of combinatorial problems with succinct input representation. Acta Informatica, 23(3):325–356, 1986

    work page 1986

  23. [23]

    W. Wan, J. Bentahar, and A. Ben Hamza. Model checking epistemic–probabilistic logic using probabilistic interpreted systems.Knowledge-Based Systems, 50:279–295, 2013. 35 Appendix A. The Interdefinability of Operators Appendix A.1. Definability of⊕ <q,⊕ =q, and⊕ q Notation A.1.We generalize Notation 4.4 to introduce⊞ ≤qϕto be defined as predictable, for in...

    work page 2013

  24. [24]

    Hence, since■ 1pHead, also■ <1pTail is satisfied atw 1

    Regardless ofa, both■ 1pHead and■ 1pTail are satisfied atw 1. Hence, since■ 1pHead, also■ <1pTail is satisfied atw 1. Moreover, since■ 1pTail, also■ <1pHead is satisfied atw 1. Hence,■ 1ϕand■ <1ϕcan both be satisfied at the same world, while■ 1ϕ and¬■ 1ϕcannot. 36 On the contrary, a Def.of■ =qϕ,■ <qϕand■ >qϕcan be given along the lines of Th. 5.2. Theorem...

    work page

  25. [25]

    current world

    Then, for everyMconstructed usingWandF Ω, and for everya∈F Ω ∪ {a}, it holds thatM, w 1 ⊨a ■1pHead, but clearly M̸⊨⋆ 1pHead. Notice also that Th.B.4 cannot be generalized to non-atomic formulas, as shown by the following example. Example B.3.Consider a model like the one in Ex. 4.9. We have seen that for every assignment a∈F Ω ∪ {a}, it holdsM, w 3 ⊨a ⋆1■...

    work page