Pith. sign in

REVIEW 4 minor 67 references

Finite Observations, Infinite Behaviour: bicategorical semantics for stateful monoidal processes

T0 review · 0 major / 4 minor · reviewed 2026-07-11 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Stateful processes share the same infinite behaviour exactly when every finite observation of one can be verified by an observation of the other.

desk verdict Clean categorical semantics for free feedback that works under partiality/nondeterminism and recovers Willems behaviour; the directed-contexts hypothesis is real but not a hidden gap. read the letter →

arxiv 2607.03996 v1 pith:55D3FJFC submitted 2026-07-04 cs.LO math.CT

Cole Comfort , Giovanni de Felice This is my paper · ORCID
classification cs.LOmath.CT MSC 18M0518D2093B2568Q55
keywords discardbicategoryobservationalbehaviourfeedbackcategorystatefulprocessesmonotonenetscompactnesstheoremlineartime-invariantsystemssignalflowgraphs
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

Machines that run forever keep an internal state an outside observer never sees. All that is visible are finite windows of inputs and outputs, each window imposing a local constraint on what trajectories are allowed. This paper builds a category of behaviours in which two such machines are identified precisely when their families of finite observations are mutually refining: every constraint imposed by one can be recovered, possibly after enlarging the observation window, from the other. The construction works uniformly for partial, nondeterministic, probabilistic and quantum processes, because it only needs a monoidal category whose morphisms can be compared by informativeness and whose discard maps are the least informative effects. The resulting category receives a functorial semantics from free feedback categories of stateful processes, makes the internal state reparametrisable without changing behaviour, and treats the time-delay as a natural (and, when the base is compact closed, invertible) transformation. For closed relations on compact Hausdorff spaces a compactness theorem glues every compatible family of finite observations into a unique infinite closed relation; over finite fields this recovers the classical Willems behaviour of linear time-invariant systems.

What carries the argument

The category Obs_{I,J}(C) of observational behaviours: morphisms are equivalence classes of monotone nets of morphisms in a discard bicategory C, indexed by an upward-directed family of finite contexts J, where two nets are equivalent when each approximates the other by looking ahead to larger contexts.

What would settle it

Exhibit two Mealy machines in Rel or AR_fd over a finite field whose families of finite-window constraints are observationally equivalent yet whose infinite Willems behaviours (or Lim images) differ, or show that Lim fails to preserve composition when the contexts are not cofinal.

Watch

Extended reading notes

Core claim

Two stateful processes have the same infinite behaviour precisely when their compatible families of finite observations are observationally equivalent: each observation of one can be verified by an observation of the other in some larger finite context. The quotient of monotone nets by this equivalence forms a discard bicategory Obs that receives a symmetric monoidal functor from the free feedback category of stateful morphism sequences and, when the base is compact closed, itself becomes a compact-closed feedback category with natural delay.

Load-bearing premise

The collection of finite observation windows must be upward-directed, so that any two windows sit inside a common larger window; without that, the equivalence relation need not respect sequential composition.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper introduces discard bicategories (preorder-enriched monoidal categories with a laxly natural discard) and constructs, for any upward-directed family of finite contexts J, a poset-enriched discard bicategory Obs_{I,J}(C) of observational behaviours: equivalence classes of monotone nets of finite observations under mutual approximation. This receives a symmetric monoidal functor from the free feedback category of stateful morphism sequences (Theorem 4.14) and, when C is compact-closed, itself forms a compact-closed feedback category with natural delay (Theorem 4.18). For closed relations between compact Hausdorff spaces a compactness theorem (Theorem 5.5) glues every compatible family of finite observations to a unique infinite closed relation; restricted to affine relations over finite fields this recovers Willems behaviour for LTI systems (Theorem 6.5). The construction is shown to apply uniformly to partial, nondeterministic, probabilistic and quantum process theories.

Significance. The work supplies a uniform denotational semantics for stateful monoidal processes that works in the presence of partiality, nondeterminism, probability and quantum post-selection, where coinductive and coalgebraic stream models become degenerate. The compactness theorem is a genuine categorification of a classical logical principle and yields the first compositional time-domain semantics for signal-flow graphs that recovers Willems behaviour over finite fields. Detailed appendix proofs, explicit comparison with prior stream semantics (state construction, monoidal streams, causal/monotone sequences), and the recovery of a classical systems-theoretic notion are substantial strengths. The framework is definitional rather than parametric, so the main claims stand or fall on the correctness of the constructions and the cited theorems.

minor comments (4)
  1. [Section 4.1, Definition 4.7] The modelling restriction that J must be upward-directed (Definition 4.7) is stated clearly and used only for congruence of composition, yet a short remark in Section 4.1 on which natural observation regimes are thereby excluded (e.g., non-nested spatial windows) would help readers assess applicability.
  2. [Section 6.3] In the LTI development the distinction between the free syntax St(LR_K^fd)(AR_K^fd), the z-domain AR_K(z), and the time-domain Obs_Z(AR_K^fd) is technically correct but dense; a small commuting diagram summarising the functors Unroll, Ext and Z* would improve readability of Section 6.3.
  3. [Section 2.3, Proposition 2.17] Proposition 2.17 notes that the Löwner and purification orders differ on CPM; a one-sentence concrete scalar counter-example already appears in the proof, but placing it in the main text would make the choice of purification order more transparent.
  4. A few minor typographical issues remain (e.g., “infinite-dimenional”, occasional missing spaces around math). A final proof-reading pass would remove them.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: Obs is defined as a quotient of monotone nets and the main theorems are independent constructions, not re-labellings of inputs.

  1. self definitional [Definition 4.8 / Theorem 4.9]
    "the (poset-enriched) discard bicategory of behaviours, Obs_{I,J}(C), is given by quotienting Mon_{I,J}(C) by observational equivalence."

    Behaviours are defined to be the equivalence classes under the observational preorder that the paper later claims characterises 'same behaviour'. This is ordinary definitional setup rather than a circular derivation of an independent claim; the subsequent theorems (functoriality, dinaturality, compactness) are proved from the definition rather than assumed by it. Flagged only as the mildest definitional step.

full rationale

The central object Obs_{I,J}(C) is introduced definitionally as the quotient of the discard bicategory of monotone families by mutual approximation (Definitions 4.6–4.8, Theorem 4.9). The functors from free feedback categories (Theorems 4.14, 4.18, Corollary 4.19) are constructed by unrolling and discarding memory, with well-definedness proved from lax naturality of discard rather than by assuming the target property. The compactness theorem (Theorem 5.5) glues nets via inverse images and the finite-intersection property on compact Hausdorff spaces; its restriction to affine relations over finite fields then recovers Willems behaviour as an equality of subspaces (Theorem 6.5), which is an independent verification rather than a renaming. Self-citations (e.g. to monoidal streams, graphical affine algebra, purification order) appear only for comparison or as background structure already verified in the cited works; none is load-bearing for the existence or uniqueness claims. Upward-directedness of J is an explicit modelling hypothesis used only to obtain a congruence, not a hidden premise. Score 1 reflects ordinary definitional setup with no fitted parameters or self-referential uniqueness imports.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper works entirely inside standard monoidal category theory plus the mild extra structure of a discard bicategory (preorder enrichment + lax natural discard). No free parameters are fitted. The only invented entities are the discard-bicategory axioms themselves and the observational quotient; both are definitional and come with independent mathematical content (examples, functors, compactness).

assumptions (4)
  • standard math Symmetric monoidal categories with a preorder enrichment compatible with composition and tensor (standard process-theoretic background).
    Used throughout; no novelty claimed.
  • domain assumption Existence of a laxly natural discard effect ⊤_X (Definition 2.5).
    The defining extra structure of a discard bicategory; verified for all listed examples.
  • domain assumption Upward-directedness of the context set J (Definition 4.7).
    Required for observational equivalence to be a congruence; modelling restriction rather than a theorem.
  • standard math Compact Hausdorff topology on the spaces so that closed relations compose and identities are closed (Lemma 5.2).
    Classical fact used for the compactness theorem.
invented entities (2)
  • Discard bicategory independent evidence
    purpose: Uniform axiomatisation of partial, nondeterministic, probabilistic and quantum process theories that admit comparison of observations.
    New name for a mild combination of existing structures; independent evidence supplied by the list of examples (Par, Rel, BorelStoch≤1, CPM, CPTNI).
  • Category of observational behaviours Obs_{I,J}(C) independent evidence
    purpose: Semantic target that identifies stateful processes with the same finite-observation constraints.
    Central construction of the paper; independent evidence via the functors from free feedback categories and the compactness theorem.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Finite Observations, Infinite Behaviour: bicategorical semantics for stateful monoidal processes." pith.science (2026). https://pith.science/paper/55D3FJFC

@misc{pith2026260703996,
  author       = {Pith},
  title        = {Pith review of: Finite Observations, Infinite Behaviour: bicategorical semantics for stateful monoidal processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/55D3FJFC}},
  note         = {Machine review of arXiv:2607.03996}
}
read the original abstract

Time-dependent processes are often described by machines with an internal state which is updated as time evolves. An external observer cannot see this state and learns about a process only through finite observations of its inputs and outputs, each of which imposes a constraint on the trajectories the process can exhibit. We introduce a semantic construction in which two stateful processes have the same behaviour when they have the same constraints, as determined by finite observations, independent of their internal state. The construction is defined over any preorder-enriched monoidal category with a compatible notion of discarding, which we call a discard bicategory, capturing partial, non-deterministic, probabilistic, and quantum processes. The resulting category of behaviours provides a functorial semantics for free feedback categories in the sense of Katis, Sabadini, and Walters. For non-deterministic systems, we prove a categorified compactness theorem: every compatible family of finite observations between compact Hausdorff spaces extends uniquely and functorially to an infinite closed relation. Restricted to affine relations over finite fields, the compactness theorem recovers Willems' notion of behaviour for linear time-invariant systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

67 extracted references · 29 canonical work pages

  1. [1]

    Observation equivalence as a testing equivalence.Theoretical Computer Science, 53(2–3):225–241, 1987.doi:10.1016/0304-3975(87)90065-X

    Samson Abramsky. Observation equivalence as a testing equivalence.Theoretical Computer Science, 53(2–3):225–241, 1987.doi:10.1016/0304-3975(87)90065-X

    work page doi:10.1016/0304-3975(87)90065-x 1987

  2. [2]

    Categorical quantum mechanics, 2009.arXiv:0808

    Samson Abramsky and Bob Coecke. Categorical quantum mechanics, 2009.arXiv:0808. 1023,doi:10.1016/B978-0-444-52869-8.50010-4

    work page doi:10.1016/b978-0-444-52869-8.50010-4 2009

  3. [3]

    Baez, Brandon Coya, and Franciscus Rebro

    John C. Baez, Brandon Coya, and Franciscus Rebro. Props in network theory.Theory and Applications of Categories, 33(25):727–783, 2018. URL:http://www.tac.mta.ca/tac/ volumes/33/25/33-25.pdf

    2018

  4. [4]

    Baez and Jason Erbele

    John C. Baez and Jason Erbele. Categories in control.Theory and Applications of Categories, 30(24):836–881, 2015. URL:http://www.tac.mta.ca/tac/volumes/30/24/30-24.pdf

    2015

  5. [5]

    Baez and Brendan Fong

    John C. Baez and Brendan Fong. A compositional framework for passive linear networks. Theory and Applications of Categories, 33(38):1158–1222, 2018. URL:http://www.tac. mta.ca/tac/volumes/33/38/33-38.pdf. 26

    2018

  6. [6]

    Effectful Mealy Machines

    Filippo Bonchi, Elena Di Lavore, and Mario Román. Effectful Mealy machines: Bisimulation and trace. In2025 40th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’25, page 541–554. IEEE, June 2025.arXiv:2410.10627, doi:10.1109/lics65433. 2025.00047

    work page Pith review arXiv doi:10.1109/lics65433 2025

  7. [7]

    Graphical affine algebra

    Filippo Bonchi, Robin Piedeleu, Paweł Sobociński, and Fabio Zanasi. Graphical affine algebra. InProceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’19. IEEE Press, 2019. URL:https://discovery.ucl.ac.uk/10081075/1/ paperLICS19.pdf,doi:10.1109/LICS.2019.8785877

    work page doi:10.1109/lics.2019.8785877 2019

  8. [8]

    Full abstraction for signal flow graphs

    Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi. Full abstraction for signal flow graphs. InProceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL ’15, page 515–526. ACM, January 2015. doi:10.1145/2676726.2676993

    doi:10.1145/2676726.2676993 2015

Show all 67 references
  1. [9]

    The calculus of signal flow diagrams I: Linear relations on streams.Information and Computation, 252:2–29, February 2017

    Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi. The calculus of signal flow diagrams I: Linear relations on streams.Information and Computation, 252:2–29, February 2017. doi:10.1016/j.ic.2016.03.002

    2017 doi

  2. [10]

    Interacting Hopf algebras.Journal of Pure and Applied Algebra, 221(1):144–184, January 2017.doi:10.1016/j.jpaa.2016.06

    Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi. Interacting Hopf algebras.Journal of Pure and Applied Algebra, 221(1):144–184, January 2017.doi:10.1016/j.jpaa.2016.06. 002

    2017 doi

  3. [11]

    Springer International Publishing, 2021

    Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi.A Survey of Compositional Signal Flow Theory, page 29–56. Springer International Publishing, 2021. URL:https://inria. hal.science/hal-03325995/document,doi:10.1007/978-3-030-81701-5_2

    2021 doi

  4. [12]

    Booth, Titouan Carette, and Cole Comfort

    Robert I. Booth, Titouan Carette, and Cole Comfort. Graphical symplectic algebra.10th International Conference on Formal Structures for Computation and Deduction (FSCD 2026), 2026. In press.arXiv:2401.07914

    2026 arXiv

  5. [13]

    Carboni and R

    A. Carboni and R. F. C. Walters. Cartesian bicategories I.Journal of Pure and Applied Algebra, 49(1-2):11–32, 1987-11-01.doi:10.1016/0022-4049(87)90121-6

    1987 doi

  6. [14]

    Graphical language with delayed trace: picturing quantum computing with finite memory

    Titouan Carette, Marc de Visme, and Simon Perdrix. Graphical language with delayed trace: picturing quantum computing with finite memory. InProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’21, New York, NY, USA,

  7. [15]

    2021.9470553

    Association for Computing Machinery.arXiv:2102.03133, doi:10.1109/LICS52264. 2021.9470553

    2021 doi arXiv

  8. [16]

    Quantum channels and memory effects.Reviews of Modern Physics, 86(4):1203–1259, 2014-12-10

    Filippo Caruso, Vittorio Giovannetti, Cosmo Lupo, and Stefano Mancini. Quantum channels and memory effects.Reviews of Modern Physics, 86(4):1203–1259, 2014-12-10. Publisher: American Physical Society.arXiv:1207.5435,doi:10.1103/RevModPhys.86.1203

    2014 doi arXiv

  9. [17]

    Kenta Cho. Semantics for a quantum programming language by operator algebras.Electronic Proceedings in Theoretical Computer Science, 172:165–190, 2014-12-28.arXiv:1412.8545, doi:10.4204/EPTCS.172.12

    2014 doi arXiv

  10. [18]

    J. R. B. Cockett and Stephen Lack. Restriction categories III: colimits, partial limits and extensivity.Mathematical Structures in Computer Science, 17(4):775–817, 2007-08. arXiv:math/0610500,doi:10.1017/S0960129507006056

    2007 doi arXiv

  11. [19]

    Spekkens

    Bob Coecke, Tobias Fritz, and Robert W. Spekkens. A mathematical theory of resources. Information and Computation, 250:59–86, October 2016.arXiv:1409.5531, doi:10.1016/ j.ic.2016.02.008. 27

    2016 arXiv

  12. [20]

    A graphical calculus for Lagrangian relations.Electronic Proceedings in Theoretical Computer Science, 372:338–351, November 2022.doi:10.4204/ eptcs.372.24

    Cole Comfort and Aleks Kissinger. A graphical calculus for Lagrangian relations.Electronic Proceedings in Theoretical Computer Science, 372:338–351, November 2022.doi:10.4204/ eptcs.372.24

    2022

  13. [21]

    A dataflow programming framework for linear optical distributed quantum computing.Quantum, 10:1972, January 2026

    Giovanni de Felice, Boldizsár Poór, Cole Comfort, Lia Yeh, Mateusz Kupper, William Cashman, and Bob Coecke. A dataflow programming framework for linear optical distributed quantum computing.Quantum, 10:1972, January 2026. doi:10.22331/ q-2026-01-19-1972

    1972

  14. [22]

    Monoidal streams for dataflow programming

    Elena Di Lavore, Giovanni de Felice, and Mario Román. Monoidal streams for dataflow programming. InProceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’22, page 1–14. ACM, August 2022.doi:10.1145/3531130. 3533365

    2022 doi

  15. [23]

    Coinductive streams in monoidal categories.Logical Methods in Computer Science, Volume 21, Issue 3, August 2025.doi: 10.46298/lmcs-21(3:18)2025

    Elena Di Lavore, Giovanni de Felice, and Mario Román. Coinductive streams in monoidal categories.Logical Methods in Computer Science, Volume 21, Issue 3, August 2025.doi: 10.46298/lmcs-21(3:18)2025

    2025 doi

  16. [24]

    Span(Graph): a canonical feedback algebra of open transition systems, March 2023

    Elena Di Lavore, Alessandro Gianola, Mario Román, Nicoletta Sabadini, and Paweł Sobo- ciński. Span(Graph): a canonical feedback algebra of open transition systems, March 2023. arXiv:2010.10069,doi:10.1007/s10270-023-01092-7

    2023 doi arXiv

  17. [25]

    Evidential decision theory via partial Markov categories, June 2023.arXiv:2301.12989,doi:10.1109/lics56636.2023.10175776

    Elena Di Lavore and Mario Román. Evidential decision theory via partial Markov categories, June 2023.arXiv:2301.12989,doi:10.1109/lics56636.2023.10175776

    2023 doi arXiv

  18. [26]

    Order in partial Markov categories.Electronic Notes in Theoretical Informatics and Computer Science, Volume 5-Proceedings of MFPS XLI, December 2025.doi:10.46298/entics.16686

    Elena Di Lavore, Mario Román, Paweł Sobociński, and Márk Széles. Order in partial Markov categories.Electronic Notes in Theoretical Informatics and Computer Science, Volume 5-Proceedings of MFPS XLI, December 2025.doi:10.46298/entics.16686

    2025 doi

  19. [27]

    J. L. Doob.Stochastic Processes (Wiley Classics Library). Wiley-Interscience, 1st edition, January 1990

    1990

  20. [28]

    A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics.Advances in Mathematics, 370:107239, August 2020

    Tobias Fritz. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics.Advances in Mathematics, 370:107239, August 2020. arXiv:1908.07021.doi:10.1016/j.aim.2020.107239

    2020 doi arXiv

  21. [29]

    Infinite products and zero-one laws in categorical probability.Compositionality, 2:3, 2020.doi:10.32408/compositionality-2-3

    Tobias Fritz and Eigil Fjeldgren Rischel. Infinite products and zero-one laws in categorical probability.Compositionality, 2:3, 2020.doi:10.32408/compositionality-2-3

    2020 doi

  22. [30]

    Springer Berlin Heidelberg, 1982.doi:10.1007/bfb0092872

    Michèle Giry.A categorical approach to probability theory, page 68–85. Springer Berlin Heidelberg, 1982.doi:10.1007/bfb0092872

    1982 doi

  23. [31]

    Die vollständigkeit der axiome des logischen funktionenkalküls.Monatshefte für Mathematik und Physik, 37(1):349–360, December 1930.doi:10.1007/bf01696781

    Kurt Gödel. Die vollständigkeit der axiome des logischen funktionenkalküls.Monatshefte für Mathematik und Physik, 37(1):349–360, December 1930.doi:10.1007/bf01696781

    1930 doi

  24. [32]

    Springer International Publishing, 2018.doi:10.1007/978-3-319-89366-2_17

    Sergey Goncharov and Lutz Schröder.Guarded Traced Categories, page 313–330. Springer International Publishing, 2018.doi:10.1007/978-3-319-89366-2_17

    2018 doi

  25. [33]

    Powerset-like monads weakly distribute over themselves in toposes and compact Hausdorff spaces

    Alexandre Goy, Daniela Petrişan, and Marc Aiguier. Powerset-like monads weakly distribute over themselves in toposes and compact Hausdorff spaces. InLIPIcs, Volume 198, ICALP 2021, volume 198, pages 132:1–132:14. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2021.doi:10.4...

    2021 doi

  26. [34]

    Braided tensor categories.Advances in Mathematics, 102(1):20–78, November 1993.doi:10.1006/aima.1993.1055

    André Joyal and Ross Street. Braided tensor categories.Advances in Mathematics, 102(1):20–78, November 1993.doi:10.1006/aima.1993.1055. 28

    1993 doi

  27. [35]

    Traced monoidal cate- gories.Mathematical Proceedings of the Cambridge Philosophical Society, 119(3):447–468, April 1996

    André Joyal, Ross Street, and Dominic Verity. Traced monoidal cate- gories.Mathematical Proceedings of the Cambridge Philosophical Society, 119(3):447–468, April 1996. URL: https://www.irif.fr/~mellies/mpri/ mpri-ens/articles/joyal-street-verity-traced-monoidal-categories.pdf ...

    1996 doi

  28. [36]

    Katis, N

    P. Katis, N. Sabadini, and R.F.C. Walters. Bicategories of processes.Journal of Pure and Applied Algebra, 115(2):141–178, February 1997.doi:10.1016/s0022-4049(96)00012-6

    1997 doi

  29. [37]

    Piergiulio Katis, Nicoletta Sabadini, and Robert F. C. Walters. On the algebra of feedback and systems with boundary.Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento, 64:123–156, 2000. Listed in the official index of the Supplemento series; MR2002h:93003

    2000

  30. [38]

    Piergiulio Katis, Nicoletta Sabadini, and Robert F.C. Walters. Feedback, trace and fixed- point semantics.RAIRO - Theoretical Informatics and Applications, 36(2):181–194, April

  31. [39]

    URL: https://www.numdam.org/item/10.1051/ita:2002009.pdf, doi:10.1051/ ita:2002009

    doi

  32. [40]

    J. L. Kelley. Convergence in topology.Duke Mathematical Journal, 17(3), September 1950. doi:10.1215/s0012-7094-50-01726-1

    1950 doi

  33. [41]

    Learning nondeterministic Mealy machines

    Ali Khalili and Armando Tacchella. Learning nondeterministic Mealy machines. InPro- ceedings of the 12th International Conference on Grammatical Inference (ICGI 2014), volume 34 ofProceedings of Machine Learning Research, pages 109–123, 2014. URL: https://proceedings.mlr.press...

    2014

  34. [42]

    Dennis Kretschmann and Reinhard F. Werner. Quantum channels with memory.Physical Review A, 72(6):062323, 2005-12-16.arXiv:quant-ph/0502106, doi:10.1103/PhysRevA. 72.062323

    2005 doi arXiv

  35. [43]

    Cambridge University Press, June 2021

    Fosco Loregian.(Co)end Calculus. Cambridge University Press, June 2021. arXiv: 1501.02503,doi:10.1017/9781108778657

    2021 doi arXiv

  36. [44]

    S. J. Mason. Feedback theory-some properties of signal flow graphs.Proceedings of the IRE, 41(9):1144–1156, September 1953. URL:https://dspace.mit.edu/server/api/core/ bitstreams/d9f85357-182a-4968-ab29-74037a82dadc/content, doi:10.1109/jrproc. 1953.274449

    1953 doi

  37. [45]

    S. J. Mason. Feedback theory—further properties of signal flow graphs. Technical Report 303, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, July 1955. Dated July 20, 1955. Reprinted from theProceedings of the I.R.E., vol. 4...

    1955

  38. [46]

    George H. Mealy. A method for synthesizing sequential circuits.The Bell System Tech- nical Journal, 34(5):1045–1079, September 1955. URL:https://vtda.org/pubs/BSTJ/ vol34-1955/articles/bstj34-5-1045.pdf, doi:10.1002/j.1538-7305.1955.tb03788. x

    1955 doi

  39. [47]

    Notions of computation and monads.Information and Computation, 93(1):55–92, July 1991

    Eugenio Moggi. Notions of computation and monads.Information and Computation, 93(1):55–92, July 1991. URL:https://www.cs.cmu.edu/~crary/819-f09/Moggi91.pdf, doi:10.1016/0890-5401(91)90052-4. 29

    1991 doi

  40. [48]

    PhD thesis, Massachusetts Institute of Technology, 1969

    James Hiram Morris, Jr.Lambda-Calculus Models of Programming Languages. PhD thesis, Massachusetts Institute of Technology, 1969. Ph.D. thesis. Submitted December 1968; degree conferred 1969. URL: https://dspace.mit.edu/entities/publication/ 5fd0e97e-1b87-4fdb-9c35-8f17a42426f1

    1969

  41. [49]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang.Quantum Computation and Quan- tum Information: 10th Anniversary Edition. Cambridge University Press, June 2012. URL: https://profmcruz.wordpress.com/wp-content/uploads/ 2017/08/quantum-computation-and-quantum-information-nielsen-chuang.pd...

    2012 doi

  42. [50]

    Springer International Publishing, 2020

    Ulrich Oberst, Martin Scheicher, and Ingrid Scheicher.Linear Time-Invariant Sys- tems, Behaviors and Modules. Springer International Publishing, 2020. doi:10.1007/ 978-3-030-43936-1

    2020

  43. [51]

    Description of a quantum convolutional code

    Harold Ollivier and Jean-Pierre Tillich. Description of a quantum convolutional code. Physical Review Letters, 91(17):177902, October 2003. Publisher: American Physical Society.arXiv:quant-ph/0304189,doi:10.1103/PhysRevLett.91.177902

    2003 doi arXiv

  44. [52]

    Oppenheim, Ronald W

    Alan V. Oppenheim, Ronald W. Schafer, and John R. Buck.Discrete-time Signal Process- ing. Prentice-Hall, December 1989. URL:https://www-elec.inaoep.mx/~jmram/pds09/ oppenheim.pdf

    1989

  45. [53]

    High-level axioms for graphical linear algebra.Science of Computer Programming, 218:102791, June 2022.doi:10.1016/j.scico

    João Paixão, Lucas Rufino, and Paweł Sobociński. High-level axioms for graphical linear algebra.Science of Computer Programming, 218:102791, June 2022.doi:10.1016/j.scico. 2022.102791

    2022 doi

  46. [54]

    The category of Markov kernels.Electronic Notes in Theoretical Computer Science, 22:171–187, January 1999.doi:10.1016/S1571-0661(05)80602-4

    Prakash Panangaden. The category of Markov kernels.Electronic Notes in Theoretical Computer Science, 22:171–187, January 1999.doi:10.1016/S1571-0661(05)80602-4

    1999 doi

  47. [55]

    M. B. Plenio and S. Virmani. Spin chains and channels with memory.Physical Review Letters, 99(12):120504, 2007-09-20. Publisher: American Physical Society.arXiv:quant-ph/ 0702059,doi:10.1103/PhysRevLett.99.120504

    2007 doi

  48. [56]

    G.D. Plotkin. LCF considered as a programming language.Theoretical Computer Science, 5(3):223–255, December 1977. URL:https://homepages.inf.ed.ac.uk/gdp/ publications/LCF.pdf,doi:10.1016/0304-3975(77)90044-5

    1977 doi

  49. [57]

    George N. Raney. Sequential functions.Journal of the ACM, 5(2):177–180, April 1958. doi:10.1145/320924.320930

    1958 doi

  50. [58]

    Robinson and G

    E. Robinson and G. Rosolini. Categories of partial maps.Information and Computation, 79(2):95–130, 1988-11-01.doi:10.1016/0890-5401(88)90034-X

    1988 doi

  51. [59]

    J. J. M. M. Rutten. Universal coalgebra: a theory of systems.Theoretical Computer Science, 249(1):3–80, 2000. URL:https://ir.cwi.nl/pub/48/0048D.pdf, doi:10.1016/ S0304-3975(00)00056-6

    2000

  52. [60]

    Outline of a mathematical theory of computation

    Dana Scott. Outline of a mathematical theory of computation. Technical monograph, Oxford University Computing Laboratory, Programming Research Group, Oxford, England, November 1970. URL:https://www.cs.ox.ac.uk/files/3222/PRG02.pdf

    1970

  53. [61]

    Towards a semantics for higher-order quantum computation

    Peter Selinger. Towards a semantics for higher-order quantum computation. InProc. QPL, pages 127–143, 2004. URL:https://mathstat.dal.ca/~selinger/qpl2004/PDFS/ 09Selinger.pdf. 30

    2004

  54. [62]

    Claude E. Shannon. The theory and design of linear differential equation machines. In N. J. A. Sloane and Aaron D. Wyner, editors,Claude E. Shannon: Collected Papers, pages 514–559. IEEE Press, 1993. Originally issued as a report to the National Defense Research Council, Janua...

    1993 doi

  55. [63]

    Bonsangue, and Jan J

    Alexandra Silva, Filippo Bonchi, Marcello M. Bonsangue, and Jan J. M. M. Rutten. Generalizing determinization from automata to coalgebras.Logical Methods in Computer Science, 9(1):1–27, 2013.doi:10.2168/LMCS-9(1:9)2013

    2013 doi

  56. [64]

    Differentiablecausalcomputationsviadelayedtrace (extended version)

    DavidSprungerandShin-yaKatsumata. Differentiablecausalcomputationsviadelayedtrace (extended version). InMathematical Structures in Computer Science, volume 35. Cambridge University Press (CUP), 2025.arXiv:1903.01093,doi:10.1017/s0960129524000331

    2025 doi arXiv

  57. [65]

    Some notions and methods on the borderline of algebra and metamathematics

    Alfred Tarski. Some notions and methods on the borderline of algebra and metamathematics. InProceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U.S.A., August 30–September 6, 1950, volume 1, pages 705–720, Providence, RI, 1952. American Mathe...

    1950

  58. [66]

    Jan C. Willems. The behavioral approach to open and interconnected systems.IEEE Control Systems Magazine, 27(6):46–99, 2007.doi:10.1109/MCS.2007.906923

    2007 doi

  59. [67]

    well-behaved

    Fabio Zanasi.Interacting Hopf Algebras - la théorie des systèmes linéaires. Thèse de doctorat, École normale supérieure de Lyon, Lyon, France, October 2015.doi:10.70675/ 9cd987f2z1368z4d11zb74czdd0a28134da4. A Graphical affine algebra The following result is due to Bonchi et a...

    2015

Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.