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arxiv: 2506.13157 · v3 · submitted 2025-06-16 · 💻 cs.AI · cs.LG· cs.LO· cs.NE

Machine Learning as Iterated Belief Change a la Darwiche and Pearl

Theofanis Aravanis This is my paper

Pith reviewed 2026-05-19 09:58 UTC · model grok-4.3

classification 💻 cs.AI cs.LGcs.LOcs.NE
keywords binary neural networksbelief changeAGM frameworkDarwiche-Pearl frameworklexicographic revisionmoderate contractioniterated belief changemachine learning
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The pith

Training of binary neural networks corresponds to iterated belief change in the Darwiche-Pearl framework.

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

The paper extends prior work by showing that the training of binary artificial neural networks can be modeled as a sequence of belief set transitions using the Darwiche-Pearl framework for iterated belief change. Specifically, it replaces earlier full-meet AGM operations with lexicographic revision and moderate contraction to better capture the gradual changes in the network's input-output behavior. A sympathetic reader would care because this provides a symbolic, logical foundation for understanding how neural networks learn, potentially bridging machine learning with belief revision theory in AI.

Core claim

The training dynamics of binary ANNs can be more effectively modelled using robust AGM-style change operations -- namely, lexicographic revision and moderate contraction -- that align with the Darwiche-Pearl framework for iterated belief change.

What carries the argument

The Darwiche-Pearl framework for iterated belief change employing lexicographic revision and moderate contraction to represent the gradual evolution of belief sets that encode a binary ANN's input-output mappings.

If this is right

  • The input-output behavior of binary ANNs can be symbolically represented as propositional belief sets.
  • Training corresponds to sequences of belief set transitions using these operators.
  • This provides a more robust model than full-meet AGM belief change.
  • Gradual changes in network states align with the semantics of iterated revision and contraction.

Where Pith is reading between the lines

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

  • This framework might enable the development of belief-change-inspired algorithms for training neural networks.
  • It opens connections between connectionist learning and symbolic reasoning in AI.
  • Future work could test whether these operators predict specific training behaviors or convergence patterns.

Load-bearing premise

The input-output behavior of a binary ANN can be faithfully represented as a propositional belief set whose gradual changes during training correspond exactly to the semantics of lexicographic revision and moderate contraction.

What would settle it

A direct comparison between the sequence of input-output functions observed during actual binary ANN training and the belief sets generated by successive applications of lexicographic revision and moderate contraction; significant mismatch would disprove the modeling.

Figures

Figures reproduced from arXiv: 2506.13157 by Theofanis Aravanis.

Figure 1
Figure 1. Figure 1: A feed-forward ANN with a single hidden layer. cific AGM-style belief-change strategies: full-meet belief change [1] (Subsection 3.3) and Dalal’s approach [7] (Subsection 3.4). 3.1 Axiomatic Characterization Within the AGM framework [1], the process of belief revision is formalized through a revision func￾tion. A revision function ∗ is a binary function that takes as input a belief set K and a formula φ, a… view at source ↗
Figure 2
Figure 2. Figure 2: Graphical representation of lexicographic revision (a) and moderate contraction (b). A state of belief is modified (revised or contracted) by a sentence φ. φ-worlds are represented by green rectangles, while ¬φ-worlds are represented by red rectangles. The lower a rectangle is, the more plausible the represented worlds are relative to the state of belief — thus, the lowest rectangles represent the worlds o… view at source ↗
Figure 3
Figure 3. Figure 3: A binary ANN with a single hidden layer. The network maps binary inputs X1, . . . , Xn to binary outputs Y1, . . . , Ym. First, note that a binary ANN with a single binary output Y induces a Boolean function f, which we refer to as the Boolean function of Y . That is, Y = f [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representative MNIST images used in the experimental setup of Example 25, along with their corresponding (binary) labels. The architecture of the ANN is as follows: • Input Layer: 100 neurons, each corresponding to one binary pixel input, denoted 21 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sets of possible worlds illustrating whether [Ki ] ∩ [Ki+1] = ∅ or [Ki ] ∩ [Ki+1] ̸= ∅. Bold highlights indicate [φ i 1 ] in the former case, and [¬φ i 2 ] in the latter case, both playing a key role in the proof’s subsequent stages. Proof. Assume that [Ki ] ∩ [Ki+1] = ∅. By the specification of the sentence φ i 1 , we have that [φ i 1 ] = [Ki+1]. Let r ∈ [φ i 1 ]. Hence, r ∈ [Ki+1] and r /∈ [Ki ], meaning… view at source ↗
Figure 6
Figure 6. Figure 6: Successive belief-set transitions representing the evolving Boolean functions of Y during training. Solid arrows highlight changes in the satisfiability of worlds across adjacent belief sets. Thereafter, we examine separately the two transitions K1-to-K2 and K2-to-K3. • From K1 to K2 First, observe that [K1] ∩ [K2] = ∅. Let φ 2 1 , φ 2 2 be two sentences of L such that [φ 1 1 ] = [K2] = {r4, r5} and let [¬… view at source ↗
read the original abstract

Artificial Neural Networks (ANNs) are powerful machine-learning models capable of capturing intricate non-linear relationships. They are widely used nowadays across numerous scientific and engineering domains, driving advancements in both research and real-world applications. In our recent work, we focused on the statics and dynamics of a particular subclass of ANNs, which we refer to as binary ANNs. A binary ANN is a feed-forward network in which both inputs and outputs are restricted to binary values, making it particularly suitable for a variety of practical use cases. Our previous study approached binary ANNs through the lens of belief-change theory, specifically the Alchourron, Gardenfors and Makinson (AGM) framework, yielding several key insights. Most notably, we demonstrated that the knowledge embodied in a binary ANN (expressed through its input-output behaviour) can be symbolically represented using a propositional logic language. Moreover, the process of modifying a belief set (through revision or contraction) was mapped onto a gradual transition through a series of intermediate belief sets. Analogously, the training of binary ANNs was conceptualized as a sequence of such belief-set transitions, which we showed can be formalized using full-meet AGM-style belief change. In the present article, we extend this line of investigation by addressing some critical limitations of our previous study. Specifically, we show that Dalal's method for belief change provides a natural basis for a structured, gradual evolution of states of belief. More importantly, given the known shortcomings of full-meet belief change, we demonstrate that the training dynamics of binary ANNs can be more effectively modelled using robust AGM-style change operations -- namely, lexicographic revision and moderate contraction -- that align with the Darwiche-Pearl framework for iterated belief change.

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 extends the authors' prior AGM-based analysis of binary ANNs by arguing that their training dynamics are more effectively captured as iterated belief change operations—specifically lexicographic revision and moderate contraction—within the Darwiche-Pearl framework, rather than full-meet AGM operators.

Significance. If the semantic correspondence between discrete weight updates and the chosen belief-change operators is rigorously established, the work could supply a symbolic, postulate-driven account of gradual learning in binary networks, potentially aiding interpretability and connecting neural training to iterated revision theory.

major comments (2)
  1. [Abstract and modeling sections] The central claim (abstract) that lexicographic revision and moderate contraction 'more effectively model' training requires an explicit construction showing that the total preorder induced by a binary ANN's input-output table yields exactly the same transitions as the DP operators under weight updates; without this, the alignment remains an existence claim rather than a faithful simulation.
  2. [Sections defining the belief-set representation] The weakest assumption—that the I/O behavior of a binary ANN can be faithfully encoded as a propositional epistemic state whose changes match the semantics of moderate contraction and lexicographic revision—needs a concrete verification that the encoding is unique up to the observed training trajectory and not chosen post-hoc.
minor comments (2)
  1. [Introduction] Clarify the precise Darwiche-Pearl postulates invoked for each operator and how they improve upon the full-meet case used in prior work.
  2. [Throughout] Ensure all notation for epistemic states and preorders is introduced before use and is consistent with standard DP literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We agree that strengthening the explicit correspondence between the ANN weight updates and the Darwiche-Pearl operators will improve the manuscript. We address each major comment below and will incorporate revisions as indicated.

read point-by-point responses

  1. Referee: [Abstract and modeling sections] The central claim (abstract) that lexicographic revision and moderate contraction 'more effectively model' training requires an explicit construction showing that the total preorder induced by a binary ANN's input-output table yields exactly the same transitions as the DP operators under weight updates; without this, the alignment remains an existence claim rather than a faithful simulation.

    Authors: We accept this observation. The current manuscript builds on the semantic mapping from our prior AGM work but does not supply a fully explicit, step-by-step construction equating the preorder induced by the I/O table with the transitions produced by lexicographic revision and moderate contraction. In the revised version we will add a dedicated subsection that (i) formally defines the total preorder ≼_w from the current binary ANN weights, (ii) states the precise conditions under which a weight update corresponds to a single application of the chosen DP operator, and (iii) proves that the resulting belief states coincide. A small illustrative network will be included to exhibit the exact transition matching. revision: yes

  2. Referee: [Sections defining the belief-set representation] The weakest assumption—that the I/O behavior of a binary ANN can be faithfully encoded as a propositional epistemic state whose changes match the semantics of moderate contraction and lexicographic revision—needs a concrete verification that the encoding is unique up to the observed training trajectory and not chosen post-hoc.

    Authors: We agree that an explicit uniqueness argument is required. The encoding used in the paper is the direct propositional representation of the ANN’s input-output table introduced in our earlier AGM study; each possible world corresponds to a complete input-output assignment consistent with the current weights. To remove any appearance of post-hoc selection, the revision will contain a short lemma showing that, for any finite training trajectory, the epistemic state at each step is uniquely recoverable from the observed I/O behavior and that the application of moderate contraction or lexicographic revision is determined solely by the semantics of the Darwiche-Pearl framework. This will be placed immediately after the definition of the belief-set representation. revision: yes

Circularity Check

1 steps flagged

ANN training dynamics as Darwiche-Pearl operators inherits encoding and transitions from self-cited prior AGM work

specific steps
  1. self citation load bearing [Abstract]
    "In our recent work, we focused on the statics and dynamics of a particular subclass of ANNs, which we refer to as binary ANNs. ... Our previous study approached binary ANNs through the lens of belief-change theory, specifically the Alchourron, Gardenfors and Makinson (AGM) framework, yielding several key insights. Most notably, we demonstrated that the knowledge embodied in a binary ANN (expressed through its input-output behaviour) can be symbolically represented using a propositional logic language. ... the training of binary ANNs was conceptualized as a sequence of such belief-set transiti"

    The claim that lexicographic revision and moderate contraction 'more effectively model' training dynamics and 'align with the Darwiche-Pearl framework' is presented as an extension that addresses limitations of the prior full-meet formalization. Because the underlying propositional encoding of ANN behavior and the mapping of training steps to belief-set transitions are taken directly from the self-cited previous study, the new operators are applied inside an already self-defined framework without a separate construction showing that their semantics coincide with actual discrete weight-update changes.

full rationale

The paper's central extension asserts that lexicographic revision and moderate contraction align with and more effectively model binary ANN training than the full-meet AGM operators from prior work. This rests on the self-cited representation of network I/O behavior as propositional belief sets whose training-induced changes are AGM-style transitions. The alignment claim therefore reduces to re-describing the same self-established encoding using different operators from the Darwiche-Pearl framework rather than an independent first-principles derivation or external verification of semantic coincidence with weight updates. The derivation chain is not fully self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on established belief-change theory rather than new free parameters or invented entities. Specific axioms are inherited from AGM and Darwiche-Pearl frameworks; details of any additional assumptions cannot be checked from the abstract alone.

axioms (2)
  • standard math AGM postulates for belief revision and contraction
    Invoked as the baseline framework being extended.
  • domain assumption Darwiche-Pearl postulates for iterated belief change
    Used to define the target modeling of training dynamics.

pith-pipeline@v0.9.0 · 5854 in / 1360 out tokens · 37129 ms · 2026-05-19T09:58:14.458671+00:00 · methodology

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Reference graph

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