pith. machine review for the scientific record. sign in

arxiv: 2604.11245 · v1 · submitted 2026-04-13 · 💻 cs.LO

Recognition: unknown

Knowledge on a Budget

Ondrej Majer , Krishna Manoorkar , Wolfgang Poiger , Igor Sedl\'ar
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:47 UTC · model grok-4.3

classification 💻 cs.LO
keywords epistemic logictopological semanticsresource constraintssemiring annotationsaxiomatizationbisimulationevidence logic
0
0 comments X

The pith

Semiring-annotated topological spaces extend epistemic logic to track resource budgets required for evidence observation.

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

Standard epistemic logics treat evidence acquisition as free, but real systems face limits on time, memory or energy. The paper introduces seats as topological spaces where open sets representing evidence receive annotations from a semiring that encode the budgets sufficient to observe them. This yields a family of logics whose modalities are indexed by resource levels, so one can reason about what remains knowable once a concrete budget is imposed. Sound and strongly complete axiomatisations are given for these logics, together with bisimulations and disjoint-union constructions that fix their expressive power.

Core claim

Seats are topological spaces equipped with an annotation function that sends each open set of evidence to a semiring ideal; the ideal collects all resource values sufficient for observation of that evidence. Resource-indexed modalities then express statements such as 'under budget r one can know phi', and the resulting logics admit sound and strongly complete axiomatisations. Bisimulation and disjoint-union operations are defined so that the framework can compare models and combine them while respecting the resource annotations.

What carries the argument

Semiring-annotated topological spaces (seats) whose annotation function maps evidence open sets to semiring ideals that represent sufficient observation budgets.

If this is right

  • Reasoning can separate what is observable in principle from what is observable under a given resource bound.
  • Axiomatisations remain complete when modalities are indexed by different semiring elements.
  • Bisimilar models agree on all resource-indexed epistemic statements.
  • Disjoint unions of seat models combine independent systems while preserving the budget annotations.

Where Pith is reading between the lines

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

  • The same annotation technique could be applied to other topological or neighbourhood semantics to add cost tracking.
  • Implementations in sensor or robotic systems could test whether the resource-indexed modalities predict feasible knowledge acquisition under measured energy limits.
  • Extensions to dynamic or temporal versions would allow reasoning about how budgets change over sequences of observations.

Load-bearing premise

The chosen annotation function maps evidence to semiring ideals in a way that preserves the topological properties of evidence without creating inconsistencies under resource constraints.

What would settle it

A concrete model in which a formula valid under the intended resource semantics is not derivable from the axiomatisation, or an invalid formula is derivable.

read the original abstract

In various computational systems, accessing information incurs time, memory or energy costs. However, standard epistemic logics usually model the acquisition of evidence as a cost-free process, which restricts their applicability in environments with limited resources. In this paper, we bridge the gap between qualitative epistemic reasoning and quantitative resource constraints by introducing semiring-annotated topological spaces (seats). Building on Topological Evidence Logic (TEL), we extend the representation of evidence as open sets, adding an annotation function that maps evidence to semiring ideals, representing the resource budgets sufficient for observation. This framework allows us to reason not only about what is observable in principle, but also about what is affordable given a specific budget. We develop a family of seat-based epistemic logics with resource-indexed modalities and provide sound, strongly complete axiomatisations for these logics. Furthermore, we introduce suitable notions of bisimulation and disjoint union to delineate the expressive power of our framework.

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

0 major / 2 minor

Summary. The manuscript introduces semiring-annotated topological spaces (seats) extending Topological Evidence Logic (TEL), where an annotation function maps open sets (evidence) to semiring ideals representing resource budgets for observation. It develops a family of seat-based epistemic logics equipped with resource-indexed modalities, supplies sound and strongly complete axiomatizations, and defines bisimulations together with disjoint unions to analyze expressive power.

Significance. If the claims hold, the work meaningfully connects qualitative epistemic logic with quantitative resource constraints, enabling reasoning about affordable evidence in limited-resource computational settings. The strong-completeness results and conservative extension of TEL are valuable technical contributions; the semiring annotation provides a flexible, parameter-light mechanism for budgets that could support applications in resource-bounded agents and verification.

minor comments (2)
  1. [Abstract / §1] The abstract and introduction refer to 'a family of seat-based epistemic logics' without immediately enumerating the members or the parameters that distinguish them; a short table or explicit list early in §2 would clarify the scope.
  2. [§3] Notation for the annotation function and the semiring ideals is introduced without a worked example using a concrete semiring (e.g., the min-plus semiring); adding one small example would aid readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the contributions of semiring-annotated topological spaces (seats), and recommendation for minor revision. The work extends Topological Evidence Logic with semiring annotations to model resource budgets for evidence observation, providing sound and strongly complete axiomatizations along with bisimulation and disjoint union results.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs a conservative extension of Topological Evidence Logic (TEL) by adding semiring annotations to topological spaces and resource-indexed modalities, then supplies independent axiomatizations claimed to be sound and strongly complete. No step reduces a derived result to a fitted parameter, self-definition, or load-bearing self-citation by construction; the completeness argument relies on standard canonical-model techniques applied to the new semantics rather than presupposing the target theorems. The framework introduces genuinely new structures (seats, annotation functions, budget-indexed operators) whose properties are not tautological with the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework relies on standard algebraic properties of semirings and topological spaces from prior work, while introducing new annotated structures for resources.

axioms (2)
  • standard math Semirings satisfy standard algebraic axioms for addition and multiplication with appropriate identities and absorption.
    Used to define ideals representing sufficient resource budgets.
  • domain assumption Evidence is represented as open sets in a topological space as in Topological Evidence Logic.
    Direct extension of TEL semantics.
invented entities (1)
  • semiring-annotated topological spaces (seats) no independent evidence
    purpose: To represent evidence together with resource budgets sufficient for its observation.
    New structure defined to bridge qualitative evidence and quantitative costs.

pith-pipeline@v0.9.0 · 5455 in / 1378 out tokens · 38890 ms · 2026-05-10T15:47:16.262513+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

47 extracted references · 31 canonical work pages

  1. [1]

    Abramsky

    S. Abramsky. Domain theory in logical form.Annals of Pure and Applied Logic, 51(1–2):1–77, Mar. 1991. ISSN 0168-0072. doi:10.1016/0168-0072(91)90065-t

    work page doi:10.1016/0168-0072(91)90065-t 1991

  2. [2]

    Aiello, J

    M. Aiello, J. van Benthem, and G. Bezhanishvili. Reasoning about space: The modal way.Journal of Logic and Computation, 13(6):889–920, Dec. 2003. ISSN 1465-363X. doi:10.1093/logcom/13.6.889

    work page doi:10.1093/logcom/13.6.889 2003

  3. [3]

    Balbiani, D

    P. Balbiani, D. Fernández-Duque, A. Herzig, and E. Lorini. Stratified evidence logics. InPro- ceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-2019, pages 1523–1529. International Joint Conferences on Artificial Intelligence Organization, Aug

    2019

  4. [4]

    doi:10.24963/ijcai.2019/211

    work page doi:10.24963/ijcai.2019/211 2019

  5. [5]

    Baltag, N

    A. Baltag, N. Bezhanishvili, A. Özgün, and S. Smets. Justified belief and the topology of evidence. InLogic, Language, Information, and Computation, pages 83–103. Springer Berlin Heidelberg,

  6. [6]

    doi:10.1007/978-3-662-52921-8_6

    ISBN 9783662529218. doi:10.1007/978-3-662-52921-8_6

    work page doi:10.1007/978-3-662-52921-8_6

  7. [7]

    Baltag, N

    A. Baltag, N. Gierasimczuk, and S. Smets. On the solvability of inductive problems: A study in epistemic topology. InTARK 2015, volume 215 ofElectronic Proceedings in Theoretical Computer Science, pages 81–98. Open Publishing Association, 2016. doi:10.4204/eptcs.215.7

    work page doi:10.4204/eptcs.215.7 2015

  8. [8]

    Baltag, N

    A. Baltag, N. Bezhanishvili, and S. Fernández González. The McKinsey-Tarski theorem for to- pological evidence logics. In R. Iemhoff, M. Moortgat, and R. de Queiroz, editors,Logic, Lan- guage, Information, and Computation, pages 177–194. Springer Berlin Heidelberg, 2019. ISBN 9783662595336. doi:10.1007/978-3-662-59533-6_11

    work page doi:10.1007/978-3-662-59533-6_11 2019

  9. [9]

    Baltag, N

    A. Baltag, N. Bezhanishvili, and S. Fernández González. Topological evidence logics: Multi- agent setting. InLanguage, Logic, and Computation (TbiLLC 2019), pages 237–257. Springer International Publishing, 2022. ISBN 9783030984793. doi:10.1007/978-3-030-98479-3_12

    work page doi:10.1007/978-3-030-98479-3_12 2019

  10. [10]

    Baltag, N

    A. Baltag, N. Bezhanishvili, A. Özgün, and S. Smets. Justified belief, knowledge, and the topology of evidence.Synthese, 200(6), Dec. 2022. ISSN 1573-0964. doi:10.1007/s11229-022-03967-6

    work page doi:10.1007/s11229-022-03967-6 2022

  11. [11]

    Baltag, N

    A. Baltag, N. Bezhanishvili, and D. Fernández-Duque. The topology of surprise.Artificial Intel- ligence, 349:104423, Dec. 2025. ISSN 0004-3702. doi:10.1016/j.artint.2025.104423

    work page doi:10.1016/j.artint.2025.104423 2025

  12. [12]

    Baltag, M

    A. Baltag, M. Gattinger, and D. Gomes. Virtual group knowledge and group belief in topological evidence models (extended version). Technical report, 2025

    2025

  13. [13]

    Baur and T

    M. Baur and T. Studer. Semirings of evidence.Journal of Logic and Computation, 31(8):2084–2106, Feb. 2021. ISSN 1465-363X. doi:10.1093/logcom/exab007

    work page doi:10.1093/logcom/exab007 2084

  14. [14]

    Belardinelli and R

    G. Belardinelli and R. K. Rendsvig. Epistemic planning with attention as a bounded resource. In Logic, Rationality, and Interaction (LORI 2021), pages 14–30. Springer International Publishing,

    2021

  15. [15]

    doi:10.1007/978-3-030-88708-7_2

    ISBN 9783030887087. doi:10.1007/978-3-030-88708-7_2

    work page doi:10.1007/978-3-030-88708-7_2

  16. [16]

    Bistarelli, U

    S. Bistarelli, U. Montanari, and F. Rossi. Semiring-based constraint satisfaction and optimization. Journal of the ACM, 44(2):201–236, Mar. 1997. ISSN 1557-735X. doi:10.1145/256303.256306

    work page doi:10.1145/256303.256306 1997

  17. [17]

    Bjorndahl and A

    A. Bjorndahl and A. Özgün. Logic and topology for knowledge, knowability, and belief.The Review of Symbolic Logic, 13(4):748–775, Oct. 2019. ISSN 1755-0211. doi:10.1017/s1755020319000509. 22

    work page doi:10.1017/s1755020319000509 2019

  18. [18]

    Blackburn, M

    P. Blackburn, M. de Rijke, and Y. Venema.Modal Logic. Cambridge University Press, Cambridge,

  19. [19]

    doi:10.1017/cbo9781107050884

    work page doi:10.1017/cbo9781107050884

  20. [20]

    Bonatti, S

    P. Bonatti, S. De Capitani di Vimercati, and P. Samarati. An algebra for composing access control policies.ACM Transactions on Information and System Security, 5(1):1–35, Feb. 2002. ISSN 1557-

    2002

  21. [21]

    doi:10.1145/504909.504910

    work page doi:10.1145/504909.504910

  22. [22]

    Costantini, A

    S. Costantini, A. Formisano, and V. Pitoni. An epistemic logic for multi-agent systems with budget and costs. InLogics in Artificial Intelligence (JELIA‘ 2021), pages 101–115. Springer Inter- national Publishing, 2021. ISBN 9783030757755. doi:10.1007/978-3-030-75775-5_8

    work page doi:10.1007/978-3-030-75775-5_8 2021

  23. [23]

    Dolgorukov, R

    V. Dolgorukov, R. Galimullin, and M. Gladyshev. Dynamic epistemic logic of resource bounded information mining agents. InProceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems, AAMAS ’24, page 481–489, Richland, SC, 2024. International Foundation for Autonomous Agents and Multiagent Systems. ISBN 9798400704864

    2024

  24. [24]

    Dudek, M

    G. Dudek, M. Jenkin, E. Milios, and D. Wilkes. Robotic exploration as graph construction.IEEE Transactions on Robotics and Automation, 7(6):859–865, 1991. doi:10.1109/70.105395

    work page doi:10.1109/70.105395 1991

  25. [25]

    Fagin, J

    R. Fagin, J. Y. Halpern, Y. Moses, and M. Y. Vardi.Reasoning About Knowledge. MIT Press, 1995. doi:10.7551/mitpress/5803.001.0001

    work page doi:10.7551/mitpress/5803.001.0001 1995

  26. [26]

    Fernández González

    S. Fernández González. Generic models for topological evidence logics. MSc. Thesis, ILLC, University of Amsterdam, Amsterdam, 2018

    2018

  27. [27]

    Galmiche, P

    D. Galmiche, P. Kimmel, and D. Pym. A substructural epistemic resource logic: theory and modelling applications.Journal of Logic and Computation, 29(8):1251–1287, Dec. 2019. ISSN 1465-363X. doi:10.1093/logcom/exz024

    work page doi:10.1093/logcom/exz024 2019

  28. [28]

    T. J. Green, G. Karvounarakis, and V. Tannen. Provenance semirings. InPODS 2007, pages 31–40. ACM, 2007. doi:10.1145/1265530.1265535

    work page doi:10.1145/1265530.1265535 2007

  29. [29]

    Heifetz and P

    A. Heifetz and P. Mongin. Probability logic for type spaces.Games and Economic Behavior, 35 (1–2):31–53, Apr. 2001. ISSN 0899-8256. doi:10.1006/game.1999.0788

    work page doi:10.1006/game.1999.0788 2001

  30. [30]

    Kozen, R

    D. Kozen, R. Mardare, and P. Panangaden. Strong completeness for markovian logics. In K. Chat- terjee and J. Sgall, editors,Mathematical Foundations of Computer Science 2013. MFCS 2013., number 8087 in Lecture Notes in Computer Science, pages 655–666, Berlin, Heidelberg, 2013. Springer. ISBN 9783642403132. doi:10.1007/978-3-642-40313-2_58

    work page doi:10.1007/978-3-642-40313-2_58 2013

  31. [31]

    Kremer and G

    P. Kremer and G. Mints. Dynamic topological logic.Annals of Pure and Applied Logic, 131(1–3): 133–158, Jan. 2005. ISSN 0168-0072. doi:10.1016/j.apal.2004.06.004

    work page doi:10.1016/j.apal.2004.06.004 2005

  32. [33]

    J. C. C. McKinsey and A. Tarski. The algebra of topology.The Annals of Mathematics, 45(1):141, Jan. 1944. ISSN 0003-486X. doi:10.2307/1969080

    work page doi:10.2307/1969080 1944

  33. [34]

    J. R. Munkres.Topology. Prentice Hall, 2nd edition edition, 2000

    2000

  34. [35]

    Naumov and J

    P. Naumov and J. Tao. Budget-constrained knowledge in multiagent systems. InProceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems, AAMAS ’15, page 219–226, Richland, SC, 2015. International Foundation for Autonomous Agents and Mul- tiagent Systems. ISBN 9781450334136

    2015

  35. [36]

    A. I. Ramírez Abarca. Topological models for group knowledge and belief. Msc thesis, ILLC, University of Amsterdam, 2015. URLhttps://eprints.illc.uva.nl/id/eprint/2250. 23

    2015

  36. [37]

    Sandhu, E

    R. Sandhu, E. Coyne, H. Feinstein, and C. Youman. Role-based access control models.Computer, 29(2):38–47, 1996. ISSN 0018-9162. doi:10.1109/2.485845

    work page doi:10.1109/2.485845 1996

  37. [38]

    M. B. Smyth. Topology. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors,Handbook of Logic in Computer Science, volume 1, pages 641–761. Oxford University Press, Oxford, 1992. ISBN 9781383026023. doi:10.1093/oso/9780198537359.003.0005

    work page doi:10.1093/oso/9780198537359.003.0005 1992

  38. [39]

    A. Solaki. Actualizing distributed knowledge in bounded groups.Journal of Logic and Compu- tation, 33(6):1497–1525, Mar. 2022. ISSN 1465-363X. doi:10.1093/logcom/exac007

    work page doi:10.1093/logcom/exac007 2022

  39. [40]

    ten Cate, D

    B. ten Cate, D. Gabelaia, and D. Sustretov. Modal languages for topology: Expressivity and definability.Annals of Pure and Applied Logic, 159(1–2):146–170, May 2009. ISSN 0168-0072. doi:10.1016/j.apal.2008.11.001

    work page doi:10.1016/j.apal.2008.11.001 2009

  40. [41]

    van Benthem, D

    J. van Benthem, D. Fernández-Duque, and E. Pacuit. Evidence and plausibility in neighbor- hood structures.Annals of Pure and Applied Logic, 165(1):106–133, Jan. 2014. ISSN 0168-0072. doi:10.1016/j.apal.2013.07.007

    work page doi:10.1016/j.apal.2013.07.007 2014

  41. [42]

    Vickers.Topology via Logic

    S. Vickers.Topology via Logic. Cambridge University Press, 1989

    1989

  42. [43]

    Zhou.Complete Deductive Systems for Probability Logic with Application to Harsanyi Type Spaces

    C. Zhou.Complete Deductive Systems for Probability Logic with Application to Harsanyi Type Spaces. Phd thesis, Indiana University, 2007

    2007

  43. [44]

    C. Zhou. Probability logic for harsanyi type spaces.Logical Methods in Computer Science, 10(2),

  44. [45]

    doi:10.2168/lmcs-10(2:13)2014

    ISSN 1860-5974. doi:10.2168/lmcs-10(2:13)2014

    work page doi:10.2168/lmcs-10(2:13)2014 2014

  45. [46]

    Gilbarg and N

    A. Özgün, S. Smets, and T.-S. Zotescu. Evidence diffusion in social networks: a topological perspective. In V. Goranko, C. Shi, and W. Wang, editors,Logic, Rationality, and Interaction, pages 110–123. Springer Nature Singapore, Oct. 2025. ISBN 9789819524815. doi:10.1007/978- 981-95-2481-5_8. A Appendix: Full proofs and details This appendix contains the f...

    work page doi:10.1007/978- 2025

  46. [47]

    Let𝑋 1 ={𝑥 1},T 1 ={∅, 𝑋 1},A 1(∅, 𝑥1)=A 1(𝑋1, 𝑥1)=Q ∞ ≥0, and for all𝑝∈Prop,V 1(𝑝)=𝑋 1

    Let𝑴 1 =⟨𝑋 1,T 1,A 1,V 1⟩and𝑴 2 =⟨𝑋 2,T 2,A 2,V 2⟩be strong uniform bounded𝐾- models defined as follows. Let𝑋 1 ={𝑥 1},T 1 ={∅, 𝑋 1},A 1(∅, 𝑥1)=A 1(𝑋1, 𝑥1)=Q ∞ ≥0, and for all𝑝∈Prop,V 1(𝑝)=𝑋 1 . Let𝑋 2 ={𝑥 2, 𝑦2},T 2 ={∅,{𝑥 2}, 𝑋2},A 2(∅, 𝑥)=A 2𝐾 ({𝑥 2}, 𝑥)= A2(𝑋2, 𝑥)=Q ∞ ≥0 for all𝑥∈𝑋 2 andV 2(𝑝)={𝑥 2}for all𝑝∈Prop. Define𝑍={(𝑥 1, 𝑥2)}. It is straightfor...

  47. [48]

    Let𝑋 1 ={𝑥 1, 𝑦1, 𝑧1},T 1 ={∅,{𝑥 1, 𝑦1}, 𝑋1}, and for all𝑝∈Prop, V1(𝑝)=𝑋 1

    Let𝑴 1 =⟨𝑋 1,T 1,A 1,V 1⟩and𝑴 2 =⟨𝑋 2,T 2,A 2,V 2⟩be strong uniform bounded𝐾- models defined as follows. Let𝑋 1 ={𝑥 1, 𝑦1, 𝑧1},T 1 ={∅,{𝑥 1, 𝑦1}, 𝑋1}, and for all𝑝∈Prop, V1(𝑝)=𝑋 1 . For all𝑥∈𝑋 1,A 1(𝑋1, 𝑥)=[0,∞],A 1({𝑥 1, 𝑦1}, 𝑥)=(0,∞], andA 1(∅, 𝑥)={1}. Let𝑋 2 ={𝑥 2, 𝑦2},T 2 ={∅, 𝑋 2}, and for all𝑝∈Prop,V 2(𝑝)=𝑋 2. For all𝑥∈𝑋 2,A 2(𝑋2, 𝑥)= [0,∞], andA 2(...