arxiv: 2604.17967 · v1 · submitted 2026-04-20 · 💻 cs.AI · cs.LG

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A Sugeno Integral View of Binarized Neural Network Inference

Isma\"il Baaj , Henri Prade

Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:14 UTC · model grok-4.3

classification 💻 cs.AI cs.LG

keywords Binarized neural networksSugeno integralFuzzy measuresRule-based representationNeural network interpretabilityBinary inputsThreshold activationSet-function representation

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The pith

The activation threshold test in a binarized neural network neuron equals a Sugeno integral over its binary inputs.

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

The paper establishes that each hidden neuron's decision in a BNN at inference time is exactly equivalent to evaluating a Sugeno integral on binary inputs. This equivalence supplies an explicit set-function that encodes the importance of each input and their interactions, together with a direct translation into if-then rules. The same integral form is also given for the final-layer score. A sympathetic reader would care because the representation turns opaque threshold comparisons into inspectable aggregations that carry built-in rule semantics.

Core claim

We show that the activation threshold test of a hidden BNN neuron can be written as a Sugeno integral on binary inputs. This yields an explicit set-function representation of each neuron decision and an associated rule-based representation. We also provide a Sugeno-integral expression for the last-layer score.

What carries the argument

The Sugeno integral, which aggregates binary inputs according to a fuzzy measure that encodes importances and interactions, exactly reproducing the neuron's threshold test.

If this is right

Each neuron decision admits a set-function representation that makes input importances and pairwise interactions explicit.
The threshold test translates directly into a collection of if-then rules whose firing conditions are readable.
The final output score of the network receives the same integral representation, allowing uniform treatment of all layers.
The same construction can be adapted to richer input interactions or to non-binary inputs.

Where Pith is reading between the lines

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

The rule extraction might enable direct comparison of learned BNN decisions against domain knowledge expressed as fuzzy rules.
Interpretability tools built on the set-function view could be used to audit which input combinations drive a particular neuron.
Extending the same integral form to quantized or low-precision networks would test whether the equivalence survives modest relaxations of the binary assumption.

Load-bearing premise

The binarized neuron computation reduces precisely to a threshold test on binary inputs and the Sugeno integral definition matches this test without additional constraints or approximations.

What would settle it

Take any hidden BNN neuron with known binary weights and threshold, enumerate its output on all 2^n input vectors, and verify whether the Sugeno integral with the derived fuzzy measure produces identical outputs on every vector.

read the original abstract

In this article, we establish a precise connection between binarized neural networks (BNNs) and Sugeno integrals. The advantage of the Sugeno integral is that it provides a framework for representing the importance of inputs and their interactions, while being equivalent to a set of if-then rules. For a hidden BNN neuron at inference time, we show that the activation threshold test can be written as a Sugeno integral on binary inputs. This yields an explicit set-function representation of each neuron decision, and an associated rule-based representation. We also provide a Sugeno-integral expression for the last-layer score. Finally, we discuss how the same framework can be adapted to support richer input interactions and how it can be extended beyond the binary case induced by binarized neural networks.

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.

Desk Editor's Note private letter to a colleague

The paper equates BNN neuron thresholds to Sugeno integrals but the mapping only works for non-negative weights due to monotonicity.

read the letter

The central new thing is the explicit rewriting of a binarized neuron's threshold test as a Sugeno integral over binary inputs, which immediately gives a set-function representation and a rule-based view of the decision. They also extend the same idea to the final-layer score. This is a clean theoretical link between efficient binary inference and the Sugeno framework, and it is not just a restatement of earlier fuzzy-neural work cited in the abstract. For readers who already work with rule extraction or fuzzy measures, the construction is straightforward and potentially useful for interpretability in the binary setting. The authors lay out the equivalence and the rule translation without unnecessary machinery, which keeps the note short and focused. The math appears to be a direct application of the Sugeno definition rather than a fitted or circular argument. No free parameters or invented entities show up in the summary. The main limitation is the monotonicity mismatch. Sugeno integrals are non-decreasing, but standard BNN neurons allow negative weights, so increasing an input can deactivate the neuron. The abstract gives no sign that the construction is restricted to positive weights or uses a non-monotone variant. If the full paper does not add that constraint or handle signed weights explicitly, the claimed equivalence holds only for a narrower class of networks than the title suggests. This is a moderate rather than fatal gap, but it needs to be stated clearly. The piece is aimed at people working at the intersection of fuzzy systems and binarized networks who want an alternative explanation tool. It is not broad enough or empirically tested enough to interest a general ML audience. I would send it to peer review so the authors can supply the full derivation, concrete examples, and a precise statement of the weight assumptions. That would let referees check whether the equivalence survives the signed-weight case or needs to be scoped down.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to establish a precise equivalence between binarized neural network (BNN) inference and Sugeno integrals. For a hidden-layer neuron, the standard threshold test on binary inputs is rewritten as a Sugeno integral with respect to a suitable fuzzy measure; this yields an explicit set-function representation of the neuron's decision together with an associated collection of if-then rules. A parallel Sugeno-integral expression is given for the final-layer score. The paper also sketches extensions to richer input interactions and to non-binary inputs.

Significance. If the claimed equivalence holds under the stated conditions, the work supplies a mathematically grounded, rule-extractable view of BNN decisions that links neural-network computation to fuzzy-measure theory. This could facilitate interpretability analyses and rule extraction without post-hoc approximation. The manuscript does not supply machine-checked proofs or reproducible code, but the direct rewriting of the threshold test constitutes a parameter-free derivation when the construction is valid.

major comments (2)

[Abstract / §3] Abstract and the central claim in §3: the asserted equivalence between the BNN activation test sign(∑ w_i x_i − t) (w_i ∈ {−1,+1}, x_i binary) and a Sugeno integral does not hold in general. Sugeno integrals are monotone non-decreasing by definition (they are taken with respect to a monotone fuzzy measure), yet a negative weight w_j = −1 renders the threshold function non-monotone: raising x_j from 0 to 1 can decrease the weighted sum and flip the neuron from active to inactive. The manuscript gives no indication that weights are restricted to be non-negative or that a non-monotone variant of the Sugeno integral is employed.
[§4] §4 (last-layer score): the Sugeno-integral expression for the output score inherits the same monotonicity requirement. If the preceding hidden-layer neurons already incorporate signed weights, the overall mapping from input vector to final score is not monotone, contradicting the properties required for the integral representation.

minor comments (2)

[§3] The definition of the fuzzy measure associated with each neuron (presumably in §3) should be stated explicitly as a function of the weight vector and threshold; the current presentation leaves the construction implicit.
Notation for the binary inputs and the threshold t is used without a dedicated table or running example; a small worked example with both positive and negative weights would clarify the scope of the claimed equivalence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The points raised regarding monotonicity are important for ensuring the correctness of the claimed equivalence. We respond to each major comment below and indicate the revisions that will be incorporated in the next version of the paper.

read point-by-point responses

Referee: [Abstract / §3] Abstract and the central claim in §3: the asserted equivalence between the BNN activation test sign(∑ w_i x_i − t) (w_i ∈ {−1,+1}, x_i binary) and a Sugeno integral does not hold in general. Sugeno integrals are monotone non-decreasing by definition (they are taken with respect to a monotone fuzzy measure), yet a negative weight w_j = −1 renders the threshold function non-monotone: raising x_j from 0 to 1 can decrease the weighted sum and flip the neuron from active to inactive. The manuscript gives no indication that weights are restricted to be non-negative or that a non-monotone variant of the Sugeno integral is employed.

Authors: We acknowledge that the referee is correct: the standard Sugeno integral is defined with respect to a monotone fuzzy measure and thus represents only monotone functions. The manuscript's central claim in §3 is presented for the general case including signed weights, which is not always monotone. To correct this, we will revise the abstract and §3 to specify that the equivalence holds for non-negative weights. We will also add an explanation that negative weights can be accommodated by complementing the input (x_j := 1 - x_j) and modifying the threshold, thereby restoring monotonicity in the new input variables while keeping the representation as a Sugeno integral. This approach is consistent with the binary nature of the inputs and will be detailed in the revised manuscript. revision: yes
Referee: [§4] §4 (last-layer score): the Sugeno-integral expression for the output score inherits the same monotonicity requirement. If the preceding hidden-layer neurons already incorporate signed weights, the overall mapping from input vector to final score is not monotone, contradicting the properties required for the integral representation.

Authors: We agree that the last-layer score expression in §4 is subject to the same monotonicity constraint. We will revise §4 to incorporate the same clarification on weight signs and the input complementation technique for handling negative contributions from hidden-layer neurons. This will ensure that the overall mapping can be expressed using the Sugeno integral framework where applicable. We will also include a short discussion on the implications for the full network inference. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct definitional rewriting

full rationale

The paper's core derivation rewrites the BNN hidden-neuron threshold test (sign of weighted sum minus threshold on binary inputs) as a Sugeno integral with respect to a suitable set function. This is a direct equivalence obtained from the definitions of each, without fitted parameters renamed as predictions, self-citation load-bearing steps, uniqueness theorems imported from the authors' prior work, or smuggling of ansatzes. The last-layer score expression follows the same pattern. No load-bearing step reduces to its own input by construction; the result is an explicit set-function and rule-based view that is independent of the input representation once the equivalence is shown. The monotonicity concern raised externally is a question of correctness or scope (whether the construction applies to signed weights), not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of the Sugeno integral as an aggregation operator and the standard binarized neuron model; no free parameters or new entities are introduced.

axioms (2)

standard math Sugeno integral is defined via a fuzzy measure and satisfies the required monotonicity and boundary properties
Invoked to represent the threshold test as an integral expression
domain assumption BNN hidden neuron activation is exactly a threshold comparison on the dot product of binary inputs and binary weights
Standard inference-time model for binarized networks

pith-pipeline@v0.9.0 · 5424 in / 1277 out tokens · 46915 ms · 2026-05-10T05:14:39.930585+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 2 canonical work pages

[1]

Synergies between machine learning and reasoning - an introduction by the Kay R

Ismaïl Baaj, Zied Bouraoui, Antoine Cornuéjols, Thierry Denœux, Sébastien Destercke, Didier Dubois, Marie-Jeanne Lesot, João Marques-Silva, Jérôme Mengin, Henri Prade, Steven Schockaert, Mathieu Serrurier, Olivier Strauss, and Christel Vrain. Synergies between machine learning and reasoning - an introduction by the Kay R. Amel group.Int. J. of Approximate...

2024
[2]

d’Avila Garcez, Marco Gori, Luis Lamb, Luciano Serafini, Michael Spranger, and Son N

Artur S. d’Avila Garcez, Marco Gori, Luis Lamb, Luciano Serafini, Michael Spranger, and Son N. Tran. Neural-symbolic computing: An effective methodology for principled integration of machine learning and reasoning.Journal of Applied Logics - IfCoLog Journal of Logics and their Applications, 6(4):611–632, 2019

2019
[3]

Neuro-symbolic artificial intelligence: The state of the art

Pascal Hitzler and Md Kamruzzaman Sarker. Neuro-symbolic artificial intelligence: The state of the art. InNeuro-Symbolic Artificial Intelligence, 2021

2021
[4]

From statistical relational to neurosymbolic artificial intelligence: A survey.Artificial Intelligence, 328:104062, 2024

Giuseppe Marra, Sebastijan Dumančić, Robin Manhaeve, and Luc De Raedt. From statistical relational to neurosymbolic artificial intelligence: A survey.Artificial Intelligence, 328:104062, 2024

2024
[5]

Logistic regression, neural networks and Dempster–Shafer theory: A new perspective

Thierry Denœux. Logistic regression, neural networks and Dempster–Shafer theory: A new perspective. Knowledge-Based Systems, 176:54–67, 2019

2019
[6]

Princeton University Press, 1976

Glenn Shafer.A Mathematical Theory of Evidence. Princeton University Press, 1976

1976
[7]

From possibilistic rule-based systems to machine learning - A discussion paper

Didier Dubois and Henri Prade. From possibilistic rule-based systems to machine learning - A discussion paper. In J. Davis and K. Tabia, editors,Proc. 14th Int. Conf. on Scalable Uncertainty Management (SUM’20), Bozen-Bolzano, Sept. 23-25, volume 12322 ofLNCS, pages 35–51. Springer, 2020

2020
[8]

Min-max inference for possibilistic rule-based system

Ismaïl Baaj, Jean-Philippe Poli, Wassila Ouerdane, and Nicolas Maudet. Min-max inference for possibilistic rule-based system. InFUZZ-IEEE, pages 1–6. IEEE, 2021

2021
[9]

Reasoning and learning in the setting of possibility theory - overview and perspectives.Int

Didier Dubois and Henri Prade. Reasoning and learning in the setting of possibility theory - overview and perspectives.Int. J. of Approximate Reasoning, 171:109028, 2024

2024
[10]

Ismaïl Baaj and Pierre Marquis.Π-NeSy: A possibilistic neuro-symbolic approach.CoRR, abs/2504.07055, 2025

work page arXiv 2025
[11]

PhD thesis, Tokyo Institute of Technology, 1974

Michio Sugeno.Theory of Fuzzy Integrals and its Applications. PhD thesis, Tokyo Institute of Technology, 1974

1974
[12]

Binarized neural networks.Advances in Neural Information Processing Systems, 29, 2016

Itay Hubara, Matthieu Courbariaux, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Binarized neural networks.Advances in Neural Information Processing Systems, 29, 2016. 8 A Sugeno Integral View of Binarized Neural Network InferenceA Preprint

2016
[13]

On tractable representations of binary neural networks

Weijia Shi, Andy Shih, Adnan Darwiche, and Arthur Choi. On tractable representations of binary neural networks. InProc. of the Int. Conf. on Principles of Knowledge Representation and Reasoning, volume 17, pages 882–892, 2020

2020
[14]

Fraser, Giulio Gambardella, Michaela Blott, Philip Leong, Magnus Jahre, and Kees Vissers

Yaman Umuroglu, Nicholas J. Fraser, Giulio Gambardella, Michaela Blott, Philip Leong, Magnus Jahre, and Kees Vissers. Finn: A framework for fast, scalable binarized neural network inference. InProc. of the 2017 ACM/SIGDA Int. Symposium on Field-Programmable Gate Arrays, pages 65–74, 2017

2017
[15]

Binary neural networks: A survey.Pattern Recognition, 105:107281, 2020

Haotong Qin, Ruihao Gong, Xianglong Liu, Xiao Bai, Jingkuan Song, and Nicu Sebe. Binary neural networks: A survey.Pattern Recognition, 105:107281, 2020

2020
[16]

A comprehensive review of binary neural network.Artificial Intelligence Review, 56(11):12949–13013, 2023

Chunyu Yuan and Sos S Agaian. A comprehensive review of binary neural network.Artificial Intelligence Review, 56(11):12949–13013, 2023

2023
[17]

A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid.Annals of Operations Research, 175:247–286, 2010

Michel Grabisch and Christophe Labreuche. A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid.Annals of Operations Research, 175:247–286, 2010

2010
[18]

Springer, 2016

Michel Grabisch.Set Functions, Games and Capacities in Decision Making, volume 46. Springer, 2016

2016
[19]

Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules

Salvatore Greco, Benedetto Matarazzo, and Roman Slowinski. Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules. Europ. J. of Operational Research, 158:271–292, 2004

2004
[20]

Application of the Sugeno integral in fuzzy rule-based classification

Jonata Wieczynski, Giancarlo Lucca, Eduardo Borges, Asier Urio-Larrea, Carlos López Molina, Humberto Bustince, and Graçaliz Dimuro. Application of the Sugeno integral in fuzzy rule-based classification. Applied Soft Computing, 167:112265, 2024

2024
[21]

Using neural networks to determine sugeno measures by statistics

Wang Jia and Wang Zhenyuan. Using neural networks to determine sugeno measures by statistics. Neural Networks, 10(1):183–195, 1997

1997
[22]

Verifying properties of binarized deep neural networks

Nina Narodytska, Shiva Kasiviswanathan, Leonid Ryzhyk, Mooly Sagiv, and Toby Walsh. Verifying properties of binarized deep neural networks. InProc. of the AAAI Conf. on Artificial Intelligence, volume 32, 2018

2018
[23]

The Möbius transform on symmetric ordered structures and its application to capacities on finite sets.Discrete Mathematics, 287(1-3):17–34, 2004

Michel Grabisch. The Möbius transform on symmetric ordered structures and its application to capacities on finite sets.Discrete Mathematics, 287(1-3):17–34, 2004

2004
[24]

Possibilistic logic - an overview

Didier Dubois and Henri Prade. Possibilistic logic - an overview. InComputational Logic, volume 9 of Handbook of the History of Logic, pages 283–342. Elsevier, 2014

2014
[25]

The logical encoding of Sugeno integrals.Fuzzy Sets and Systems, 241:61–75, 2014

Didier Dubois, Henri Prade, and Agnès Rico. The logical encoding of Sugeno integrals.Fuzzy Sets and Systems, 241:61–75, 2014

2014
[26]

The use of the discrete Sugeno integral in decision-making: A survey.Int

Didier Dubois, Jean-Luc Marichal, Henri Prade, Marc Roubens, and Régis Sabbadin. The use of the discrete Sugeno integral in decision-making: A survey.Int. J. of Uncertainty, Fuzziness and Knowledge-based Systems, 9(5):539–561, 2001

2001
[27]

Robust fitting for the Sugeno integral with respect to general fuzzy measures.Information Sciences, 514:449–461, 2020

Gleb Beliakov, Marek Gagolewski, and Simon James. Robust fitting for the Sugeno integral with respect to general fuzzy measures.Information Sciences, 514:449–461, 2020

2020
[28]

Machine learning with the Sugeno integral: The case of binary classification.IEEE Trans

Sadegh Abbaszadeh and Eyke Hüllermeier. Machine learning with the Sugeno integral: The case of binary classification.IEEE Trans. on Fuzzy Systems, 29(12):3723–3733, 2020

2020
[29]

On learning capacities of Sugeno integrals with systems of fuzzy relational equations

Ismaïl Baaj. On learning capacities of Sugeno integrals with systems of fuzzy relational equations. In 2025 IEEE Int. Conf. on Fuzzy Systems (FUZZ), pages 1–6. IEEE, 2025

2025
[30]

Quantization and training of neural networks for efficient integer- arithmetic-only inference

Benoit Jacob, Skirmantas Kligys, Bo Chen, Menglong Zhu, Matthew Tang, Andrew Howard, Hartwig Adam, and Dmitry Kalenichenko. Quantization and training of neural networks for efficient integer- arithmetic-only inference. InProc. of the IEEE Conf. on computer vision and pattern recognition, pages 2704–2713, 2018

2018
[31]

Towards a reconciliation between reasoning and learning-a position paper

Didier Dubois and Henri Prade. Towards a reconciliation between reasoning and learning-a position paper. InInt. Conf. on Scalable Uncertainty Management, pages 153–168. Springer, 2019

2019
[32]

Réseau de neurones et logique: un cadre qualitatif

Ismaïl Baaj, Didier Dubois, Francis Faux, Henri Prade, Agnès Rico, and Olivier Strauss. Réseau de neurones et logique: un cadre qualitatif. InLFA 2022-31es Rencontres francophones sur la Logique Floue et ses Applications, pages 127–134. Cépaduès, Toulouse, 2022

2022
[33]

arXiv preprint arXiv:2507.11127 , year=

Lennert De Smet and Luc De Raedt. Defining neurosymbolic AI.arXiv e-prints, art. arXiv:2507.11127, July 2025. doi: 10.48550/arXiv.2507.11127. 9

work page doi:10.48550/arxiv.2507.11127 2025