Interpretive Equivalence in Formal Systems

• Interpretive equivalence is a concept that defines when two structures or theories share the same content through systematic translations.
• It encompasses frameworks like bi-interpretability, definitional, and trace equivalence, offering clear methodologies for comparing formal systems.
• Applications span model theory, philosophy of science, computability, and neural interpretability, providing actionable criteria across disciplines.

Interpretive equivalence is a multifaceted concept that formalizes when two structures, models, or theories are “the same” in content, meaning, or function, modulo a translation or mapping of interpretations. The typology, technical implementations, and philosophical significance of interpretive equivalence span model theory, categorical logic, the philosophy of science, theoretical computer science, and the empirical sciences. This article provides a structured survey of the principal technical frameworks and foundational results on interpretive equivalence, emphasizing definitions, interrelationships, illustrative examples, and applications across domains.

1. Fundamental Notions of Interpretive Equivalence

The notion of interpretive equivalence underpins comparisons of expressive or semantic content between formal systems. At its core, interpretive equivalence seeks to determine when two entities—be they first-order structures, formal theories, models in the physical sciences, or even neural networks—are “the same” after suitable translation of their terms, domains, and operations.

Interpretability Between Theories

Let $T_1$ and $T_2$ be first-order theories in respective languages $\mathcal{L}_1$ and $\mathcal{L}_2$. An interpretation of $T_2$ in $T_1$ consists of a definable domain and a translation of the symbols of $\mathcal{L}_2$ to formulae in $\mathcal{L}_1$ such that every axiom of $T_2$ is mapped to a theorem of $T_1$ under this translation. Interpretability is denoted $T_2$0 (Meadows, 3 Nov 2025).

Bi-interpretability and Definitional Equivalence

• Bi-interpretability requires interpretations both ways, along with definable isomorphisms between the compositions and the identities: $T_2$1 and $T_2$2 are bi-interpretable if there are interpretations $T_2$3 and $T_2$4, and the induced composites are definably isomorphic to the respective identities (Chen et al., 5 Aug 2025, Friedman et al., 1 Jun 2025, D'Arienzo et al., 2020).
• Definitional equivalence is a stricter notion: the reciprocal interpretations must be “direct,” i.e., one-dimensional, unrelativized, and identity-preserving, so that translation and back yields the original theory not only up to isomorphism but literally (Chen et al., 5 Aug 2025, Friedman et al., 1 Jun 2025).

Physical and Interpretational Equivalence

In philosophy of science and physics, interpretational equivalence is characterized by sameness of the ontological and dynamical content assigned to the world by two theories under their intended readings—not merely formal or empirical agreement (Weatherall, 2018, Haro, 2019).

2. Formal Frameworks and Model-Theoretic Realizations

Several frameworks for formalizing interpretive equivalence have been developed to suit specific domains.

a. Classical First-order and Categorical Settings

• Syntactic approach: Interpretations are given by definable domains, relations, equivalence classes, and translation formulas as explicit maps between first-order structures or theories (Rideau-Kikuchi et al., 2 May 2025).
• Categorical approach: Bi-interpretability corresponds to equivalence (or biequivalence) of syntactic categories. For coherent theories, $T_2$5 and $T_2$6 are bi-interpretable if and only if their syntactic categories' exact completions are equivalent as coherent categories (D'Arienzo et al., 2020).

b. Primitive Positive (pp) Bi-interpretability

In constraint satisfaction and infinite-domain CSP classification, pp-interpretations—interpretations using only existential conjunctions of atomic formulas—are essential. pp-bi-interpretability (mutual pp-interpretation with pp-definable round-trip maps) is equivalent to topological isomorphism of polymorphism clones and underpins algorithmic and complexity-theoretic equivalence (Feller et al., 2 Feb 2026).

c. Effective Interpretability in Computable Structure Theory

Effective interpretability strengthens classical interpretability to the computable setting. A structure $T_2$7 is effectively interpretable in $T_2$8 if the interpretation data (domain, relations, equivalence) are computably infinitary definable and uniform across all copies of $T_2$9 (Harrison-Trainor et al., 2015). The main theorem is the equivalence between effective interpretability and the existence of a computable functor between isomorphism categories of structures.

d. Trace Equivalence

Trace equivalence is a coarser equivalence: $\mathcal{L}_1$0 and $\mathcal{L}_1$1 are trace equivalent if each can “trace define” the other's definable sets via injective maps, without requiring a full interpretability scheme. This equivalence preserves a wide range of classification-theoretic invariants (e.g., NIP, stability, dp-rank) and is particularly relevant for the classification of tame theories (Walsberg, 2021).

e. Causal-Abstraction and Neural Interpretability

In machine learning, especially mechanistic interpretability, interpretive equivalence abstracts from explicit interpretation choice to the equivalence of all possible implementations: two (possibly unknown) interpretations are equivalent if the sets of their implementations (circuits) coincide up to isomorphism, measured via Hausdorff distance in the space of causal models (Sun et al., 31 Mar 2026).

3. Key Theorems and Structural Properties

a. Equivalence Hierarchies

The main relationships can be summarized as follows:

Level	Mutual/Invertible	Type of Translations	Structure Preserved
Definitional equivalence	Yes	Direct, 1-dim, identity-preserving	Literal identity after translation and back
Bi-interpretability	Yes	Arbitrary (possibly multi-dimensional)	Isomorphism after translation and back
Trace equivalence	Weak	Injection of definable relations	Coincidence of definable sets by tracing

Definitional equivalence $\mathcal{L}_1$2 bi-interpretability $\mathcal{L}_1$3 trace equivalence, but not conversely in general (Chen et al., 5 Aug 2025, Walsberg, 2021). Not all bi-interpretable theories are definitionally equivalent; second-order arithmetic and countable set theory provide a canonical counterexample (Chen et al., 5 Aug 2025).

b. Orey–Hájek Theorem (Bounded Arithmetic)

For sufficiently rich sequential theories $\mathcal{L}_1$4 and reflexive $\mathcal{L}_1$5, three key properties are inter-derivable:

• $\mathcal{L}_1$6 is interpretable in $\mathcal{L}_1$7.
• $\mathcal{L}_1$8 is $\mathcal{L}_1$9-conservative over $\mathcal{L}_2$0.
• $\mathcal{L}_2$1 proves the restricted consistency of $\mathcal{L}_2$2 for all finite fragments.

This equivalence, established under weak base metatheories, anchors interpretability as a fundamental measure of proof-theoretic strength (Joosten, 2016).

c. Effective Interpretability and Computable Functors

Effective interpretability and the existence of computable functors between isomorphism categories of (countable) structures are equivalent. The functorial direction guarantees computable uniformity in the realization of interpretations across copies; the converse reconstructs an effective interpretation from a computable functor, via definability of required domains and relations (Harrison-Trainor et al., 2015).

d. Decidability and Structure in Infinite-Domain CSPs

For $\mathcal{L}_2$3-categorical, transitive, no-algebraicity structures, pp-bi-interpretability is decidable and smooth as a Borel equivalence relation—meaning that equivalence classes can be classified up to real number equality (Feller et al., 2 Feb 2026).

e. Mechanistic/ML Equivalence

In model interpretability for neural networks, the main result is that representation similarity is a necessary and sufficient (up to technical slack) witness for interpretive equivalence of symbolic interpretations, laying the groundwork for automated equivalence checking (Sun et al., 31 Mar 2026).

4. Illustrative Examples and Applications

a. Physics: Duality, Gauge Theories, and Maxwell's Equations

Interpretive equivalence elucidates when distinct mathematical formulations—e.g., Faraday-tensor vs. gauge-potential vs. gauge-orbit models of electromagnetism—describe the same underlying physical content. The isomorphism of bare model roots combined with identity (or isomorphic) internal interpretations yields physical interpretive equivalence (Haro, 2019, Weatherall, 2018).

b. Logic: Boolean Algebras vs. Rings, Tilting Equivalence

Bi-interpretability is instantiated in the definable equivalence between Boolean algebras and Boolean rings, or between certain model-theoretic avatars of perfectoid rings and their tilts. In these cases, category-theoretic and syntactic formulations coincide (Rideau-Kikuchi et al., 2 May 2025, D'Arienzo et al., 2020).

c. Trace Equivalence in NIP/Stable Theory

Trace equivalence captures broad preservation of “tameness” invariants and is the appropriate level of equivalence for certain classification dichotomies and combinatorial properties in model theory—for example, distinguishing between field-like, group-like, and vector-space-like behaviors via trichotomies (Walsberg, 2021).

d. Computable Reductions and Universality

Any countable structure is effectively bi-interpretable in a computable graph, underpinning universality results in computable structure theory. This aligns with the classification of bi-interpretability classes via computable functors (Harrison-Trainor et al., 2015).

e. LLMs and Interpretive Annotation

In evaluation of LLMs versus human experts, the degree of interpretive equivalence (in annotation tasks) is recoverable by calibration against human judgment in a triplet (odd-one-out) protocol, with LLM-facilitated metrics closing much of the gap to expert-defined equivalence (Nam et al., 2 Sep 2025).

5. Open Problems, Limitations, and Future Directions

• Limits of Categorical Generalizations: Generous generalizations (e.g., using all definable class equivalences in ZFCU) trivialize the bi-interpretability classification for categorical rigid theories unless one restricts definability resources (e.g., to computable or constructible definitions) (Meadows, 3 Nov 2025).
• Necessity of Identity Preservation: The strictness of definitional equivalence over bi-interpretability for sequential theories is characterized in the need for both interpretations to be one-dimensional and identity-preserving (Friedman et al., 1 Jun 2025).
• Trace Equivalence Classification: Full classification of structure classes (e.g., homogeneous structures) up to trace equivalence remains open, with conjectured manageability due to the coarseness of the equivalence (Walsberg, 2021).
• Algorithmic Discovery: For neural interpretability and beyond, automated search and validation of candidate interpretations up to equivalence pose challenging algorithmic and sample complexity constraints (Sun et al., 31 Mar 2026).
• Descriptive Set-Theoretic Structure: The smoothness/Borel complexity of various equivalence relations remains a central theme, with implications for effective classification and equivalence checking (Feller et al., 2 Feb 2026).

6. Comparative Table: Principal Notions of Interpretive Equivalence

Notion	Structural Requirement	Extent of Equivalence	Domain/Paradigm	Key References
Definitional equivalence	Direct (1-dim, unrelativized, preserves =)	Strict (literal identity)	Model theory, classical logic	(Chen et al., 5 Aug 2025, Friedman et al., 1 Jun 2025)
Bi-interpretability	Interp. both ways with definable isos	Moderate (up to iso)	Logic, categorical frameworks	(D'Arienzo et al., 2020, Friedman et al., 1 Jun 2025)
pp-bi-interpretability	Primitive positive interpretations	CSP complexity	CSP, infinite-domain logic	(Feller et al., 2 Feb 2026)
Trace equivalence	Tracing definable sets by injections	Coarse	Model theory (NIP, stability)	(Walsberg, 2021)
Effective bi-interpretability	Uniform computability/functors	Computability	Computable structure theory	(Harrison-Trainor et al., 2015)
Interpretational equivalence	Same interpretation/ontology	Semantic/metaphysical	Philosophy of science, physics	(Haro, 2019, Weatherall, 2018)
Mechanistic (NN)	Coincidence of all interpretations	Algorithmic	Neural interpretability, ML	(Sun et al., 31 Mar 2026)

7. Synthesis and Significance

Interpretive equivalence provides the mathematical infrastructure for understanding when two systems present the same underlying content, across logic, mathematics, and natural science. The fine gradations—from strict definitional equivalence through bi-interpretability and trace equivalence to semantic and mechanistic versions—enable precise calibrations of “sameness” suited to contextual constraints: descriptive rigidity in logic, operational preservation in computation, ontological unity in science, or behavioral indistinguishability in machine learning.

Ongoing developments engage the limits and breadth of these notions, the algorithmic tractability of deciding equivalence, and their implications for both theory classification and real-world model evaluation. As interpretive equivalence continues to be refined and applied, it anchors foundational discourse at the intersection of formal systems, computation, semantics, and scientific modeling.