Function-Preserving Transformation

• Function-Preserving Transformation is a mathematical procedure that alters functions, programs, or algebraic structures while maintaining core properties such as output equivalence and structural integrity.
• These transformations are applied across domains—algebra, cryptography, signal processing, compilation, and quantum mechanics—to optimize computations and ensure invariant behavior under change.
• By preserving semantics and structural invariants, these methods enable safe transfers between representations and facilitate rigorous system analysis in both theory and practical applications.

A function-preserving transformation is any mathematically formal manipulation of functions, programs, or algebraic structures that maintains their key semantic or structural properties—such as output values, equivalence classes, autocorrelation spectra, or integrals—under transformation. These transformations play a pivotal role across algebra, programming language theory, cryptography, signal processing, and mathematical physics, serving for optimization, expansion, normalization, or transfer between domains without altering the essential functional mapping.

1. Algebraic Function Preservation: Congruence-Preserving Transformations

In universal algebra, a function $f:A^k\to A$ is said to be congruence-preserving (CP) if it maps tuples of elements that are equivalent with respect to any congruence $\sim$ to equivalent outputs, i.e., $x_i \sim y_i$ for $$i=1,\dots,k \implies f(x_1,\dots,x_k) \sim f(y_1,\dots,y_k)$$. This property is central for studying morphisms between algebraic structures such as free monoids $E^*$ generated by an alphabet $E$.

For $|E| \geq 3$, every CP function from $E^*$ to itself is characterized algebraically by composition with fixed words—specifically, any $f:E^*\to E^*$ can be uniquely decomposed as

$f(x) = w_0\,x\,w_1\,x\,\cdots\,x\,w_n,$

where $w_0, ..., w_n \in E^*$; for higher arity,

$$f(x_1, ..., x_k) = w_0\,x_{i(1)}\,w_1\,x_{i(2)}\,w_2\,\cdots\,x_{i(n)}\,w_n,$$

with $i(j)\in\{1,...,k\}$ (Cégielski et al., 2016).

This rigidity—where all congruence-preserving maps are term operations in the monoid signature—implies affine completeness ($\Pol(A) = \CP(A)$), with the free monoid on at least three generators serving as the first noncommutative affine-complete algebra.

2. Signal Processing: Odd-Periodic Autocorrelation-Preserving Operations

In binary sequence design, the preservation of special correlation metrics under transformation is vital. An operation is called OACF-preserving if it leaves invariant the multiset of odd-periodic autocorrelation function values (OACF),

$$\widehat R_s(τ) = \sum_{i=0}^{N-1} (-1)^{s(i) - s(i+\tau) + \lfloor\frac{i+\tau}{N}\rfloor},$$

for all shifts $\tau$.

Nega-decimation, introduced via Parker's transformation, performs a decimation followed by conditional negation on the doubled sequence $u = s \| (s\oplus1)$. Given $d$ coprime to $2N$,

$$[\widetilde D_d(s)](\tau) = s(d\tau) + \lfloor d\tau/N \rfloor \pmod{2},$$

preserves the multiset $\{\widehat R_s(\tau)\}$ for $\tau = 1,\ldots,N-1$ (Wang et al., 2020). This, together with nega-cyclic shifts and negation, classifies Parker's 16 case families into 8 OACF-equivalent classes, elucidating structure in binary sequence construction.

3. Compiler Theory: Program Function-Preserving Transformations

In programming languages and term rewriting systems, function-preserving transformations systematically encode imperative programs (with global variables and call stacks) into logically constrained TRSs (LCTRS) while maintaining semantics.

The transformation introduces explicit environment symbols $\Env$, packaging global variables and stack frames:

$\Env(g_1, ..., g_k, \mathit{stack}(f(a_1,...,a_m), \bot))$

and rewrite rules for function calls (push onto stack), returns (pop from stack), and global variable updates. Correctness—semantic equivalence between the original and transformed program—is established by showing that

$$\langle f_i(\vec a), \sigma_0, \sigma_1 \rangle \Downarrow_P \langle n, \sigma_0' \rangle$$

if and only if the corresponding $\Env$ term rewrites to

$\Env(\sigma_0'(x_1),..., \sigma_0'(x_k), \mathit{stack}(\mathit{return}(n), \bot))$

(Kanazawa et al., 2019), ensuring each function's behavior is preserved step-by-step via one-to-one correspondence with rewrite rules.

4. Stream Cipher Hardware: NLFSR Equivalence-Preserving Transformations

In hardware cryptography, Non-Linear Feedback Shift Registers (NLFSRs) can be reconfigured from a Fibonacci form (single feedback) to a Galois form (distributed feedback), maintaining sequence generation.

A function-preserving transformation algorithm redistributes the monomials in the original ANF feedback $f_{n-1}$ through a "shifting" procedure such that each new feedback $g_i$ satisfies the singularity and uniformity conditions. The transformed system's output sequence remains identical, proven via recurrence relation analysis and structural induction on the feedback graphs (0801.4079).

This enables stream-cipher designers to increase throughput and parallelism with no penalty in resource utilization, provided the conditions for uniformity and equivalence are satisfied.

5. Dynamical Systems: Form-Preserving Transformations in Quantum Mechanics

Form-preserving transformations transfer solutions of partial differential equations—such as Schrödinger's equation—to other potentials or coordinate frames without altering the analytic structure of the wavefunction or its associated phase-space distributions.

For the time-dependent Schrödinger equation, time-dependent coordinate and amplitude changes

$$X = \frac{x}{\gamma(t)} + \beta(t), \quad T = \int^t \frac{ds}{\gamma(s)^2},$$

with corresponding amplitude and phase factors,

$$\Psi(X,T) = e^{iS(x,t)/\hbar} \sqrt{\gamma(t)}\,\psi(x,t),$$

can be selected so that $\Psi$ solves a new equation with transformed potential $V'(X,T)$ satisfying a specific algebraic relation (see Eq. (5) in (Amin et al., 24 Oct 2025)). The Wigner function transforms as a true probability under this linear canonical map,

$W'(X,P;T) = W(x(X,P), p(X,P), t(T)),$

preserving phase-space structure through the evolution.

Classic examples include the mapping of Airy beams and coherent-excited states; these transformations reveal deep rigidity of quantum evolution under certain coordinate changes.

6. Compiler IR: Normalization as a Function-Preserving Transformation

Normalization transformations in compiler intermediate representations (IRs), such as the Diderot EIN IR for tensor calculus, apply systematic rewrite rules to expressions—pushing probes, distributing derivatives, flattening sums/fractions—while strictly preserving value (for tensor-valued expressions), types, and termination.

A well-founded size measure and rule-by-rule type and value inversion guarantee that every normalization step produces an expression equivalent in function to its predecessor. The rewrite system is total, deterministic, and semantics equivalent:

$$e \to_{norm} e' \implies \text{Value}(e) = \text{Value}(e')$$

(Chiw et al., 2017).

7. Overall Significance and Applications

Function-preserving and equivalence-preserving transformations underpin robust methodologies in mathematical and computational domains, from universal algebraic classification and stream cipher optimization to program compilation and quantum dynamics analysis. Their rigorous properties—semantic preservation, invertibility, and invariance transfer—enable safe migration between representations, efficient expansions or reductions, and theoretical insights into the structure and rigidity of functional mappings. Rigorous proofs and case analyses in the cited literature confirm their indispensability and wide applicability.