arxiv: 2605.05840 · v2 · submitted 2026-05-07 · 💻 cs.LO

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Decidability Results for Fragments of First-Order Logic via a Symbolic Model Property

Neta Elad , Sharon Shoham

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Pith reviewed 2026-05-14 22:11 UTC · model grok-4.3

classification 💻 cs.LO

keywords first-order logicdecidabilitysymbolic modelsstratified formulasquantifier alternationself-looping functions

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The pith

Fragments of first-order logic extending stratified formulas with restricted self-looping functions on one sort are decidable.

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

The paper shows that several fragments of first-order logic, which extend stratified formulas by allowing self-looping functions on one sort under certain restrictions, satisfy the symbolic model property. This property guarantees that every satisfiable formula in the fragment has a finite symbolic model whose domain and interpretations are defined by terms and formulas over a base theory with a standard model. Since satisfiability checking for symbolic models is decidable, the fragments themselves have decidable satisfiability. The proof proceeds by a generic construction that turns an arbitrary model of a formula into a symbolic model while preserving satisfiability and correctness for the permitted restrictions.

Core claim

By generalizing symbolic structures to any base theory admitting a standard model and constructing a symbolic model directly from an arbitrary model, the targeted fragments extending stratified formulas with allowed self-loops on one sort satisfy the symbolic model property and are therefore decidable.

What carries the argument

The symbolic model property: every satisfiable formula in the fragment has a symbolic model defined using terms and formulas from a base theory.

If this is right

The satisfiability problem for these extended fragments is decidable.
Model checking reduces to a decidable procedure on the symbolic structures.
The decidability results cover relaxations of quantifier-alternation constraints beyond fully stratified formulas.
The same generic construction can be instantiated for other fragments obeying similar restrictions on self-looping functions.

Where Pith is reading between the lines

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

Changing the underlying base theory might yield analogous decidability results for structures defined over different domains such as reals or finite sets.
The fragments could serve as specification languages in automated verification when the system model naturally contains one sort with limited self-references.
Further relaxations allowing self-loops on multiple sorts might become decidable if additional syntactic conditions are identified.

Load-bearing premise

The generic construction of a symbolic model from an arbitrary model preserves satisfiability and is correct for formulas in the fragments that allow self-looping functions on one sort under the stated restrictions.

What would settle it

A satisfiable formula belonging to one of the targeted fragments for which the generic construction either produces no symbolic model or produces one that does not satisfy the formula.

read the original abstract

Recently, symbolic structures were proposed as finite representations of potentially infinite first-order structures, where Linear Integer Arithmetic terms and formulas define the domain and interpretations of a structure. We generalize symbolic structures to use any base theory that admits a standard model. Symbolic structures induce a symbolic model property, which holds for a fragment of first-order logic if every satisfiable formula in the fragment has a symbolic model. The symbolic model property implies decidability, since the model-checking problem for symbolic structures is decidable. We use the symbolic model property to prove decidability for several fragments that extend the fragment of stratified formulas, relaxing the quantifier-alternation constraints by allowing one sort to have self-looping functions, under certain restrictions. To establish the symbolic model property for these fragments we construct a symbolic model for a formula from an arbitrary model. The construction and its correctness are proved in a generic fashion, which may be instantiated to other similarly restricted fragments.

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

Generalizes symbolic structures to arbitrary base theories and gets decidability for FOL fragments allowing restricted self-loops on one sort.

read the letter

The main point is that this paper generalizes symbolic structures beyond linear integer arithmetic to any base theory with a standard model, then uses a generic construction to prove the symbolic model property for fragments extending stratified formulas. The extension relaxes quantifier alternation rules by permitting self-looping functions on exactly one sort under stated restrictions, which yields decidability via the fact that symbolic model checking is decidable. The construction turns an arbitrary model into a symbolic one built from base-theory terms and formulas, and the authors claim the correctness argument works generically so it can be reused for similar fragments. That generic step is the clearest advance over prior stratified-formula results. It avoids case-by-case proofs and keeps the focus on how the symbolic representation preserves satisfiability while respecting the syntactic limits. The approach looks formally grounded on the surface, with no obvious circularity or invented entities beyond the standard symbolic setup. The soft spots sit in the restrictions themselves and in the lack of visible detail on edge cases. The abstract only sketches how self-loops are encoded without introducing extra alternations or unsound interpretations, so it is hard to judge whether the restrictions are narrow enough to limit real use in verification tools or whether the model-checking procedure stays efficient after the generalization. If those points hold up in the full proofs, the central claim is fine; if not, the decidability results shrink. This is aimed at people already working on decidable fragments of first-order logic for infinite-state systems. Readers who know the stratified-formula literature will see the extension immediately and can judge whether the new fragments are useful. The thinking is clear and engaged with the existing results, so the paper deserves a serious referee to check the generic construction and the precise scope of the restrictions.

Referee Report

2 major / 2 minor

Summary. The paper generalizes symbolic structures—finite representations of potentially infinite first-order structures defined via terms and formulas from a base theory—to arbitrary base theories that admit a standard model. It establishes that the resulting symbolic model property holds for several fragments extending the stratified formulas fragment, by relaxing quantifier-alternation constraints to allow self-looping functions on exactly one sort under stated restrictions. Decidability follows because model checking for symbolic structures is decidable. The key technical step is a generic construction that, given any model of a formula in these fragments, produces a symbolic model satisfying the same formula; the construction and its correctness are proved generically for instantiation to other similarly restricted fragments.

Significance. If the generic construction is correct, the results meaningfully enlarge the class of decidable fragments of first-order logic beyond stratified formulas, with potential applications in automated reasoning and verification. The generic proof strategy and use of arbitrary base theories are strengths that could support further extensions, provided the preservation of syntactic restrictions and satisfiability is fully verified.

major comments (2)

[Abstract and construction section] The load-bearing claim is that the generic construction from an arbitrary model produces a symbolic model that preserves both satisfiability and the syntactic restrictions (including exactly one sort with self-looping functions) without introducing spurious quantifier alternations. This must be shown explicitly for the relaxed fragments; the abstract asserts genericity but the correctness argument needs to address how base-theory terms encode self-loops without violating the fragment definition.
[Proof of symbolic model property] For the fragments with self-looping functions, the construction must ensure the resulting symbolic structure remains amenable to decidable model checking. Any encoding of loops via base-theory terms risks altering the quantifier structure or the single-sort restriction; the proof should contain a lemma verifying that the output satisfies the same syntactic constraints as the input formula.

minor comments (2)

[Introduction] Clarify the precise definition of 'self-looping functions' and the 'stated restrictions' in the opening paragraphs, including whether the base theory must be decidable or have additional properties beyond admitting a standard model.
[Main results] Provide at least one concrete instantiation of the generic construction for a specific fragment (e.g., the extension of stratified formulas) to illustrate how the symbolic model is built and verified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on the generic construction. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract and construction section] The load-bearing claim is that the generic construction from an arbitrary model produces a symbolic model that preserves both satisfiability and the syntactic restrictions (including exactly one sort with self-looping functions) without introducing spurious quantifier alternations. This must be shown explicitly for the relaxed fragments; the abstract asserts genericity but the correctness argument needs to address how base-theory terms encode self-loops without violating the fragment definition.

Authors: The generic construction encodes interpretations using base-theory terms that are chosen to respect the designated single sort for self-looping functions and the original quantifier structure of the formula. This ensures preservation by design, as the terms are drawn directly from the model without adding alternations. To address the request for explicit verification, we will add a dedicated lemma in the construction section of the revised manuscript that formally proves the output symbolic model satisfies the fragment's syntactic constraints, including the single-sort restriction and absence of spurious alternations. revision: yes
Referee: [Proof of symbolic model property] For the fragments with self-looping functions, the construction must ensure the resulting symbolic structure remains amenable to decidable model checking. Any encoding of loops via base-theory terms risks altering the quantifier structure or the single-sort restriction; the proof should contain a lemma verifying that the output satisfies the same syntactic constraints as the input formula.

Authors: The construction maintains amenability to decidable model checking precisely because it inherits the syntactic form (including the single-sort self-looping restriction) from the input, and model checking for symbolic structures depends only on this form plus decidability of the base theory. We agree a separate lemma would improve clarity and will insert one in the revised proof of the symbolic model property to explicitly verify that the output satisfies the same syntactic constraints as the input formula. revision: yes

Circularity Check

0 steps flagged

No significant circularity; generic construction from arbitrary models is independent

full rationale

The paper proves the symbolic model property for the extended fragments by exhibiting a generic construction that, given any model of a formula, produces a symbolic model using terms from the base theory while preserving satisfiability and the syntactic restrictions. This step is a standard model-theoretic argument with an explicit correctness proof, not a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equations or claims reduce the result to its inputs by construction, and the derivation remains self-contained against external model-checking decidability.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the existence of standard models for base theories and the ability to construct symbolic models that preserve satisfiability under the stated restrictions.

axioms (2)

domain assumption Base theory admits a standard model
Invoked to generalize symbolic structures beyond linear integer arithmetic.
standard math First-order logic semantics and model theory hold in the standard way
Background assumption for defining satisfiability and models.

invented entities (1)

Symbolic structure no independent evidence
purpose: Finite representation of potentially infinite first-order structures using base-theory terms and formulas
Core new representation enabling the symbolic model property and decidability.

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Works this paper leans on

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locations

A Proofs Proof of Theorem 4.4.Given a symbolic structureS and formulaφ, we deriveφS by the same transformation as in [15]. The theorem is proved by induction on the structure ofφ, following [15]. ◀ Proof of Theorem 6.2.LetF be a fragment that admits a symbolic model property. Thus, there exists a semi-decision procedure for satisfiability inF: enumerating...

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