Semantically-Preserving Transformations

• Semantically-preserving transformations are techniques that modify code, data, or models while maintaining their intrinsic meaning and observable behavior.
• They underpin compiler optimizations, formal verification, and automata theory through methods like rewriting systems, logical relations, and categorical approaches.
• Practical frameworks include machine-checked proofs and quantitative error analysis, ensuring robust and reliable transformations across diverse domains.

Semantically-preserving transformations are techniques and formal methods that alter the syntactic representation, structure, or presentation of code, data, or mathematical objects in a way that leaves their underlying semantics—meaning, observable behavior, or external effects—unchanged. Such transformations are foundational in compiler construction, program analysis, formal verification, optimization, and scientific computation, as well as in the paper of automata, physical processes, and machine learning. The precise formalization of what constitutes “semantic preservation” varies across domains but is always captured by a notion of behavioral equivalence or invariance under transformation.

1. Formal Definitions and Theoretical Foundations

In programming languages and formal systems, a transformation is called semantically-preserving if, for any original object (program, process, or model) $P$, the transformed object $T(P)$ is such that $P$ and $T(P)$ are indistinguishable under an appropriate equivalence relation. Key formalizations include:

• Contextual equivalence: Two programs are equivalent if their behavior is identical in all contexts; this underpins refactoring and compiler optimization correctness (Horpácsi et al., 2022).
• Logical relations and step-indexed logical relations: Relations parameterized by types (or step indexes) capture deeper semantic equivalence, especially for higher-order and recursively defined computations.
• Quantitative generalization: Approximate program transformations generalize semantic preservation by introducing error bounds, denoted $e \in \llbracket a \rrbracket_{q}$, indicating that $e$ and $a$ are within error $q$ of each other, where $q$ may itself be structured (e.g., function-valued) (Westbrook et al., 2013).

In data-centric and mathematical contexts, structure-preserving transformations map mathematical objects (e.g., automata, programs, graphs) to new representations without disturbing their accepted language, observational outcomes, or core algebraic structures (Casares et al., 2023, Gaeta et al., 2015).

2. Methodological Frameworks and Domain-Specific Realizations

The construction and verification of semantically-preserving transformations employ various frameworks tailored to the domain:

Domain	Semantic Equivalence Notion	Key Approaches
Programming Langs	Contextual/Logical Eq.	Frame stack semantics, logical relations
Compilers	Value & type preservation	Rewriting systems, normalization
Graph Rewriting	Functorial semantics	Double-pushout (DPO) approach
Automata Theory	Language preservation	State duplication, morphisms
Physics, Quantum	Enrichment preservation	Monoidal enrichment, Grothendieck constr.

Programming Languages: Rewriting systems in IR normalization (e.g., Diderot’s EIN) are proven type-preserving and value-preserving by inductive arguments on the rewrite rules and evaluation semantics (Chiw et al., 2017). Logical relations enforce that, for all contexts and substitutions, semantic outcomes are preserved under transformations such as function inlining, loop transformations, or beta-reduction (Horpácsi et al., 2022).

Term Graphs: In the double-pushout (DPO) approach, term graphs are rewritten via categorical pushout constructions. Semantic preservation is guaranteed by “context decomposition,” where the transformation replaces a subgraph $L$ with $R$ within larger structures, provided $\llbracket L \rrbracket = \llbracket R \rrbracket$, thus ensuring $\llbracket A \rrbracket = \llbracket B \rrbracket$ for the host graphs $A, B$ (Kahl et al., 2019).

Automata/Transition Systems: Transformations from Muller to parity or Rabin automata are defined using algorithms based on the Zielonka tree or the alternating cycle decomposition (ACD). Correctness is ensured by locally bijective morphisms (for deterministic parity) or history-deterministic mappings (for Rabin automata), which are structure-preserving and guarantee that the accepted language (or winning region) is unchanged (Casares et al., 2023).

Higher-Order Physics: Semantic structure in physical process transformations is formalized via enriched category theory, where structure-preserving functors between V-enriched monoidal categories, specified by the Grothendieck construction, guarantee that the operational semantics and composition laws are invariant (Wilson et al., 2022).

3. Quantitative and Approximate Notions

In approximate program transformation, the binary notion of semantic equivalence is relaxed. Correctness becomes quantitative: one demonstrates that an error measure induced by the transformation remains below a user-specified threshold. The key innovation is the introduction of structured “approximation types,” where error values can be real numbers for numerical data, functions for higher-order data, or more complex structures for polymorphic or parameterized types (Westbrook et al., 2013).

The semantic relation $e \in \llbracket a \rrbracket_{q}$ expresses that $a$ approximates $e$ within $q$. The framework supports modular reasoning: error bounds can be computed and combined using monoid structures $(Q, \leq, +, 0)$, allowing for compositional transformation correctness proofs. This approach handles both data-level (e.g., floating-point) approximations and control-level (e.g., loop perforation) transformations with explicit, structured error analysis.

4. Applications: Compiler Construction, Program Analysis, and Beyond

Compiler Optimizations: Trust in compiler passes stems from semantic preservation guarantees at each intermediate stage. In Diderot, normalization—as a rewriting system—proves type and value preservation and termination, ensuring correctness through compilation pipelines (Chiw et al., 2017).

Program Analysis and Neural Models: Semantically-preserving transformations expose the robustness or brittleness of neural program analyzers. Evaluations show that even innocuous transformations (e.g., variable renaming, statement permutation) can induce significant prediction changes, indicating overreliance on surface features rather than semantics (Rabin et al., 2020, Hort et al., 30 Mar 2025). Such transformations are analogues of “mutation operators” in metamorphic testing.

Automata Theory: Transformations between automata types (e.g., Muller to parity or Rabin) are foundational to model checking, synthesis, and verification, where the semantic preservation is essential for maintaining language acceptance properties (Casares et al., 2023).

Physical and Quantum Theories: In modeling transformations of processes or channels (including higher-order phenomena), semantic preservation ensures that operations, composition, and causal structures maintain their intended interpretive content—crucial in quantum information and categorical models of physics (Wilson et al., 2022).

5. Verification, Challenges, and Limitations

The verification of semantically-preserving transformations is conducted through a variety of approaches:

• Machine-checked proofs: In Core Erlang and similar languages, machine-checked Coq proofs establish that transformations preserve contextual equivalence, step-indexed logical relations, and termination behavior (Horpácsi et al., 2022).
• Compositionality: Approximate transformations leverage modularity so that correctness proofs for individual transformations can be reused and composed (Westbrook et al., 2013).
• Empirical validation: In neural program analysis and defect detection, empirical evidence shows that finding or creating correct semantic-preserving transformations is challenging, especially when reusing artifacts developed for other purposes. Manual validation is often required to weed out transformations that inadvertently alter semantics (Hort et al., 30 Mar 2025).

A persistent challenge is the guarantee of semantic preservation across language features, data types, and domains with complex operational semantics (e.g., concurrency, exceptions, or quantum noncommutativity). Even published transformations may fail to preserve semantics due to subtle scope, control-flow, or data-flow interactions.

6. Impact, Extensions, and Future Directions

Semantically-preserving transformations underpin correctness, reliability, and trust in a wide range of computational and mathematical systems. Their rigorous formalization and modular verification enable safe compiler optimization, robust program analysis tools, and trustworthy model transformation in automata and physical theories.

Possible future directions identified in recent research include:

• Developing unified frameworks and toolkits for semantic-preserving transformations that span multiple languages and domains, incorporating machine-checked proof support and automated validation (Hort et al., 30 Mar 2025).
• Extending compositional error analysis for quantitative and approximate transformations to settings with probabilistic, concurrent, or quantum features (Westbrook et al., 2013, Wilson et al., 2022).
• Systematically evaluating robustness both in learned models (adversarial/mutation testing using semantically neutral changes) and in classical codebases, with the aim of identifying fragility and guiding the design of invariance-aware representations (Rabin et al., 2020).
• Deepening categorical and algebraic approaches to capture increasingly rich process transformations in mathematical physics, quantum theory, and data science (Wilson et al., 2022, Gaeta et al., 2015).

By establishing the boundaries and mechanisms by which meaning is maintained under transformation, this field continues to expand the theoretical and practical frontiers of correctness, optimization, and adaptation in computational systems.