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Stateful processes share the same infinite behaviour exactly when every finite observation of one can be verified by an observation of the other.

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2026-07-11 22:25 UTC pith:55D3FJFC

load-bearing objection 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.

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

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

Cole Comfort , Giovanni de Felice This is my paper
classification cs.LO math.CT MSC 18M0518D2093B2568Q55
keywords discard bicategoryobservational behaviourfeedback categorystateful processesmonotone netscompactness theoremlinear time-invariant systemssignal flow graphs
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

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.

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.

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.

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 this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, 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

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

specific steps
  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.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 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).

axioms (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.

pith-pipeline@v1.1.0-grok45 · 51787 in / 2414 out tokens · 19055 ms · 2026-07-11T22:25:28.089644+00:00 · methodology

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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.

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