arxiv: 2605.12539 · v1 · submitted 2026-05-05 · 💻 cs.LO

Recognition: 2 theorem links

· Lean Theorem

ocLTL: LTL Realizability and Synthesis Modulo {ω}-Categorical Structures

Ohad Asor

Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:00 UTC · model grok-4.3

classification 💻 cs.LO

keywords ocLTLLTL realizabilitysynthesisomega-categorical theoriestemporal logicdata structuresdecidability

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The pith

ocLTL realizability and synthesis reduce to propositional LTL+P by replacing each data subformula with a finite disjunction over complete types.

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

The paper introduces ocLTL as LTL+P interpreted over omega-categorical structures that supply data values. It establishes that both the realizability problem and the synthesis problem for ocLTL reduce directly to the corresponding problems for ordinary propositional LTL+P. The reduction is performed by substituting every data subformula with a finite disjunction that enumerates all complete types of the fixed structure. The resulting decision procedure stays inside 2-EXPTIME, with an extra exponential factor that grows only with the theory and stays independent of the input formula size. Readers care because the reduction lets existing LTL+P tools solve synthesis tasks that involve data constraints drawn from common mathematical structures.

Core claim

We introduce ocLTL, a version of LTL+P modulo omega-categorical theories. We reduce its realizability and synthesis problems into the corresponding problems in propositional LTL+P. The core of the reduction replaces each data subformula with a finite disjunction over complete types. The complexity remains 2-EXPTIME with additional blowup that depends only on the theory but not the formula.

What carries the argument

The finite disjunction over complete types that replaces each data subformula, encoding all possible consistent data assignments allowed by the omega-categorical theory.

If this is right

Realizability of any ocLTL formula is decidable in 2-EXPTIME.
Synthesis procedures already known for propositional LTL+P lift to ocLTL after the type-expansion preprocessing step.
The size of the expanded formula depends on the fixed theory but grows independently of the original formula length.
The same reduction applies uniformly to every omega-categorical theory that supplies finitely many complete types.

Where Pith is reading between the lines

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

Synthesis tools built for propositional LTL+P become applicable to data-aware specifications once a one-time enumeration of the theory's types is performed.
The approach covers common data domains such as equality, dense linear orders, and certain graph structures that are known to be omega-categorical.
If a new theory is later shown to be omega-categorical with finitely many types, its data constraints become immediately amenable to existing LTL synthesis algorithms.

Load-bearing premise

Every omega-categorical theory under consideration has only finitely many complete types, so the disjunction stays finite and preserves the original semantics of the data subformulas.

What would settle it

An explicit omega-categorical theory together with an ocLTL formula whose realizability status differs from the status of the formula obtained by expanding its data subformulas into the corresponding type disjunction.

Figures

Figures reproduced from arXiv: 2605.12539 by Ohad Asor.

read the original abstract

We introduce ocLTL, a version of LTL+P modulo {\omega}-categorical theories. We reduce its realizability and synthesis problems into the corresponding problems in propositional LTL+P. The core of the reduction replaces each data subformula with a finite disjunction over complete types. The complexity remains 2-EXPTIME with additional blowup that depends only on the theory but not the formula.

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 reduces LTL realizability over ω-categorical structures to propositional LTL+P by replacing data subformulas with finite disjunctions over complete types, keeping the 2-EXPTIME bound.

read the letter

This paper shows how to reduce realizability and synthesis for LTL over ω-categorical structures down to the propositional case by replacing data subformulas with finite disjunctions over complete types. The complexity stays 2-EXPTIME because the number of types is fixed for any given theory by the Ryll-Nardzewski theorem. The resulting instance grows linearly in the original formula size, modulo that fixed constant blowup from the theory alone. That is the central claim and the main new piece on offer. It applies a standard type-abstraction trick to the temporal setting with past operators, which avoids the custom complexity increases that usually appear when data is added to LTL. The abstract states the construction plainly and the stress-test note finds no internal inconsistency in the setup. If the full paper supplies a clear semantic-equivalence argument for the replacement under all LTL operators, the reduction should hold. The main soft spot is that the abstract gives no proof sketch or worked example, so the key step—whether the disjunction really captures the data atoms in every temporal context—remains to be checked in the text. The theory-dependent blowup is acknowledged but left as a parameter; for some structures it could be large in practice, though that does not affect the asymptotic result. This is aimed at people working on synthesis and realizability for temporal logics that need to handle infinite but structured data domains. A reader who already knows LTL+P realizability and basic model theory will get the most from it as a modular building block. I would send it to peer review. The reduction idea is worth referee time even if the write-up needs more detail on the proof.

Referee Report

1 major / 2 minor

Summary. The paper introduces ocLTL, a version of LTL with past operators interpreted over ω-categorical structures. It reduces the realizability and synthesis problems for ocLTL to the corresponding problems in propositional LTL+P. The core of the reduction replaces each data subformula with a finite disjunction over complete types of the fixed structure. The complexity remains 2-EXPTIME, with an additional blowup that depends only on the theory and not on the formula.

Significance. If the reduction is correct, the result shows that data-aware temporal synthesis over ω-categorical structures can be solved with the same complexity as the propositional case, using a theory-dependent but formula-independent preprocessing step. This is useful because ω-categorical structures include many data domains arising in verification (e.g., equality, dense linear orders), and the approach reuses existing LTL+P synthesis algorithms after a type-based abstraction. The preservation of the 2-EXPTIME bound is a clear strength.

major comments (1)

[§3] §3 (Reduction construction): the manuscript must supply an explicit inductive proof that the replacement of each data subformula φ by the disjunction over complete types preserves satisfaction under the temporal semantics. The abstract states that the replacement 'correctly captures the semantics,' but without the detailed argument it is impossible to confirm that the interaction between the new propositional atoms and the LTL+P operators (especially past operators) is faithful.

minor comments (2)

[§2] §2 (Preliminaries): add an explicit citation to the Ryll-Nardzewski theorem when stating that there are only finitely many complete n-types for each n; this justifies the finiteness of the disjunction but is currently only implicit.
[Abstract] Notation: the abbreviation 'LTL+P' is used without expansion on first occurrence; spell out 'Linear Temporal Logic with Past' at its first use.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and for identifying the need to strengthen the presentation of the reduction. We will address the request by adding the missing explicit proof.

read point-by-point responses

Referee: [§3] §3 (Reduction construction): the manuscript must supply an explicit inductive proof that the replacement of each data subformula φ by the disjunction over complete types preserves satisfaction under the temporal semantics. The abstract states that the replacement 'correctly captures the semantics,' but without the detailed argument it is impossible to confirm that the interaction between the new propositional atoms and the LTL+P operators (especially past operators) is faithful.

Authors: We agree that the current manuscript lacks a fully explicit inductive argument and will add one in the revised Section 3. The proof proceeds by induction on formula structure. The base case establishes equivalence between a data atom and the disjunction of all complete types that satisfy it. Each inductive step shows that the temporal operators (including the past operators Y and S) preserve the equivalence, because the type abstraction is time-invariant under the ω-categorical structure and the new propositional atoms are interpreted consistently across positions. This ensures faithful interaction between the introduced atoms and all LTL+P connectives. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reduction is to an independent target problem

full rationale

The core derivation replaces each data subformula by a finite disjunction over complete types of the fixed ω-categorical structure. By the Ryll-Nardzewski theorem (an external, standard model-theoretic fact), every ω-categorical theory has only finitely many n-types for each n; the disjunction size is therefore a theory-dependent constant independent of the input formula. The resulting propositional LTL+P instance is therefore of size linear in the original formula (modulo that constant), and the 2-EXPTIME bound follows directly from the known complexity of propositional LTL+P realizability. No equation equates a derived quantity to a fitted parameter of itself, no uniqueness theorem is imported from the same author, and no ansatz is smuggled via self-citation. The semantic equivalence claim is the standard one used for type-based abstractions and does not reduce to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of ω-categorical structures from model theory as the key axiom for the reduction to work.

axioms (1)

domain assumption ω-categorical theories admit only finitely many complete types for each arity
This finiteness is what allows replacing data subformulas with finite disjunctions without losing the reduction's effectiveness.

pith-pipeline@v0.9.0 · 5351 in / 1301 out tokens · 81523 ms · 2026-05-14T21:00:20.054853+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We reduce its realizability and synthesis problems into the corresponding problems in propositional LTL+P. The core of the reduction replaces each data subformula with a finite disjunction over complete types.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By theorem 1, any formula ϕ(¯x) with k free variables can be encoded as a finite set of types. ... M |= ϕ(¯a) iff tp(¯a) ∈ ι(ϕ)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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