Analogy Theorem: A Formal Transfer Principle

• Analogy Theorem is a family of rigorous transfer principles that formalize the schema 'A is to B as C is to D' by defining transformation equivalences across domains.
• It encompasses universal algebra with rewrite rules, logic programming with modular forms, numerical analogies via power means, and computational classifiers preserving asymptotic complexity.
• The theorem’s variants provide actionable criteria for sound transfer, ensuring uniqueness, isomorphism preservation, and feasibility through structured model justification.

Searching arXiv for the cited Analogy-related papers to ground the article in current arXiv records. Search 1: universal algebra / analogical proportions. Search 2: logic-program analogy / proportions. Search 3: numerical analogy via generalized means. Search 4: computational analogy / transfer / classifiers. In recent arXiv literature, the expression Analogy Theorem does not denote a single classical theorem, but a family of formally specified results that make precise the schema “$A$ is to $B$ as $C$ is to $D$.” Across universal algebra, logic programming, numerical analysis, computation, learning theory, and program logic, these results replace informal similarity by explicit objects such as rewrite rules, forms, justifications, power transforms, preservation conditions, or metric-stability clauses. The common aim is to state when a transformation known in a source domain can be instantiated in a target domain, when the resulting target object is uniquely or characteristically justified, and when transfer is sound, feasible, or complexity-preserving (Antić, 2020, Antić, 2018, Lepage et al., 2024).

1. Universal-algebraic formulation

A general abstract basis is given in Antić’s framework of analogical proportions on algebras of a common language. For two $L$-algebras $\mathfrak A,\mathfrak B$, a rewrite rule is a pair of terms $s(\mathbf z)\to t(\mathbf z)$ in which every variable of $t$ occurs in $s$. For elements $a,b\in A$ and $B$0, the justification sets are defined by

$B$1

and similarly on $B$2. Arrow proportions, directed analogical proportions, and full analogical proportions are then built by intersecting justification sets and imposing maximality and symmetry conditions (Antić, 2020).

Within this setting, the central theorem is the Functional Proportion Theorem. If $B$3 is an $B$4-term, then for any $B$5 and $B$6,

$B$7

characteristically justified by $B$8. Under injectivity at the relevant points, this upgrades first to directed proportion and then to full analogical proportion:

$B$9

The same paper proves a First Isomorphism Theorem and a Second Isomorphism Theorem: analogical proportions are preserved by isomorphisms, but not in general by homomorphisms (Antić, 2020).

This formulation is significant because it makes analogy a local algebraic property controlled by common justifications rather than by surface resemblance. It also sharply limits the classical axiomatics. Symmetry, inner symmetry, reflexivity, inner reflexivity, and determinism hold in general, but central permutation, strong reflexivity, strong inner reflexivity, commutativity, transitivity, inner transitivity, central transitivity, and monotonicity fail in general. The framework therefore rejects the idea that analogical proportion is uniformly an equivalence-like relation; it is instead a structured but nontransitive relation driven by available rewrite justifications (Antić, 2020).

2. Logic-program proportions and the directed Analogy Theorem

In logic programming, the theorem is instantiated on Horn programs over an unranked first-order language. The key technical move is to represent programs modularly by forms, i.e. meta-terms generated from concrete programs, program variables, and operations such as union, composition $C$0, concatenation $C$1, substitution, $C$2, $C$3, $C$4, $C$5, and $C$6. A directed analogical proportion has the shape

$C$7

read “$C$8 transforms into $C$9 as $D$0 transforms into $D$1.” A justification is a pair of forms $D$2, with every variable in $D$3 occurring in $D$4, that simultaneously explains the source transformation and the target transformation (Antić, 2018).

The paper introduces two algebraic operations. Composition $D$5 is a sequential-resolution operation on programs, and is generally non-associative. Concatenation $D$6 is an argument-wise skeleton-preserving combination, associative on programs with matching skeletons. In both cases, least-model behavior is not modular in a straightforward way: neither $D$7 nor $D$8 factors simply through $D$9 and $L$0 (Antić, 2018).

The central result is again called the Functional Proportion Theorem, and in the logic-program setting it is the natural candidate for an Analogy Theorem:

$L$1

for any form $L$2 built over operations available in both domains, characteristically justified by the single justification $L$3. If $L$4, then $L$5 is a functional solution of $L$6. The Uniqueness Lemma adds a characteristic-justification criterion: if $L$7 has a unique witness in the target domain, then $L$8 uniquely pins down $L$9 (Antić, 2018).

The worked examples show the constructive force of the theorem. A form extracted from unary naturals and the addition program yields an append program when instantiated with lists, giving the directed proportion “numbers are to addition what lists are to append.” A form

$\mathfrak A,\mathfrak B$0

transfers evenness from the unary-natural generator to a list-reversing program, producing a program that reverses lists of even length. A membership form built from lists, when instantiated with naturals, yields a program computing a less-than relation on numerals (Antić, 2018).

3. Quantitative and structural variants

A distinct numerical version appears in the theory of analogies on positive real numbers via generalized means. For $\mathfrak A,\mathfrak B$1, analogy in power $\mathfrak A,\mathfrak B$2 is defined by

$\mathfrak A,\mathfrak B$3

with $\mathfrak A,\mathfrak B$4 the two-variable power mean; for $\mathfrak A,\mathfrak B$5 this is equivalent to $\mathfrak A,\mathfrak B$6, while $\mathfrak A,\mathfrak B$7 gives $\mathfrak A,\mathfrak B$8. The theorem states that if $\mathfrak A,\mathfrak B$9 are positive, then there exists a unique $s(\mathbf z)\to t(\mathbf z)$0 such that

$s(\mathbf z)\to t(\mathbf z)$1

The same framework shows that every such analogy reduces to an arithmetic analogy through the transform $s(\mathbf z)\to t(\mathbf z)$2 for $s(\mathbf z)\to t(\mathbf z)$3 and $s(\mathbf z)\to t(\mathbf z)$4, so that

$s(\mathbf z)\to t(\mathbf z)$5

Analogical equations then admit explicit solutions:

$s(\mathbf z)\to t(\mathbf z)$6

and, for fixed real $s(\mathbf z)\to t(\mathbf z)$7, solvability extends to nonzero complex numbers (Lepage et al., 2024).

Another theorem explicitly framed as an analogy is the tensor version of Yuan’s theorem of the alternative. For even-order symmetric tensors $s(\mathbf z)\to t(\mathbf z)$8, assuming that a common linear transform makes them essentially nonpositive, exactly one of two alternatives holds: either there exists $s(\mathbf z)\to t(\mathbf z)$9 such that all homogeneous forms $t$0, or there is a convex combination $t$1 that lies in the SOS cone. This converts a quadratic alternative into a tensor/SOS certificate and leads to exact first-level SOS relaxation for polynomial optimization with essentially nonpositive coefficients (Hu et al., 2014).

These variants show that an Analogy Theorem need not be restricted to symbolic rewrite systems. In one case, analogy is parameterized by a unique power; in another, it is a structural transport from PSD-matrix certificates to SOS-tensor certificates. The unifying feature is that analogy is encoded by a formally checkable invariant rather than by informal similarity.

4. Computational analogy and analogy-preserving classifiers

In computability theory, Computational Analogy is a relation on computable functions $t$2 defined through Enumerating Turing Machines and approximations of E-Turing machines. Two functions are computationally analog, written $t$3, when E-Turing machines for one can serve as approximations for E-Turing machines of the other, with recovery overhead bounded by $t$4 or $t$5. The theorem-level consequences are strong: $t$6 is an equivalence relation; if $t$7, then

$t$8

and computational irreducibility is preserved in both the strong and ordinary senses. The set of computable functions is thereby partitioned into classes whose members share asymptotic complexity and irreducibility properties (Zwirn, 2013).

In analogical classification, the central result takes the form of a Galois connection. A classifier $t$9 is analogy-preserving relative to a pair $s$0 of 4-ary relations when componentwise $s$1-analogies in the feature space and solvability of the analogical equation in the label space force the label quadruple to satisfy $s$2. The key theorem is

$s$3

where $s$4 augments $s$5 by all unsolvable label quadruples. This turns analogical preservation into ordinary polymorphism preservation and permits a Galois theory of analogical classifiers. On the Boolean domain, the paper explicitly determines the closed classes for the five Antić relations $s$6: depending on the pair $s$7, the sound classifiers are exactly one of the affine clone $s$8, the projections-and-constants clone $s$9, the negations-and-constants clone $a,b\in A$0, or the constants-only clone $a,b\in A$1 (Couceiro et al., 2022).

Both theories replace vague claims that “computing one is like computing another” or that “a classifier respects analogy” by preservation theorems with explicit closure properties. In one case the preserved quantities are asymptotic computation time and irreducibility; in the other they are relational invariants under a Pol/Inv correspondence.

5. Feasibility, transfer gain, and verified transfer

A separate line of work studies when analogical transfer is feasible rather than merely definable. In a complexity-minimization framework based on model description length, a source model $a,b\in A$2 is weakly or strongly $a,b\in A$3-reusable for a target case depending on whether reusing $a,b\in A$4 reduces target description length by at least $a,b\in A$5. The associated Analogy Feasibility Theorem states that if a compatible source model is weakly $a,b\in A$6-reusable for some target output $a,b\in A$7, then the source compresses the target by at least $a,b\in A$8 bits relative to the best standalone target description; if the target MDL minimizer is unique and strong reusability holds, then the minimizing $a,b\in A$9 is a unique analogical solution. The same framework defines transferability coefficients $B$00 and $B$01 over compatible source models (Murena, 2022).

In evolutionary transfer optimization, analogical reasoning is decomposed into retrieval, mapping, and evaluation. The theoretical core consists of two performance-gain theorems. The Unconditionally Nonnegative Performance Gain Theorem states that the performance gain of an analogy-based knowledge transfer method is unconditionally nonnegative if and only if it includes the evaluation subprocess. The Conditionally Positive Performance Gain Theorem states that strictly positive gain can be guaranteed either by big-source retrieval, where useful source knowledge appears with probability tending to $B$02 as the source pool grows, or by appropriate mapping, where the usefulness improvement $B$03 is large enough to make the infimum gain positive. The framework relies on a monotonic relation between observable similarity $B$04 and usefulness $B$05 and gives a threshold

$B$06

for cancelling harmful transfer (Xue et al., 27 Mar 2025).

Program-logic work pushes the theorem into verified domain transfer. In a first-order and Hoare-logic formulation, one fixes predicates $B$07 and $B$08, a metric $B$09, a reference element $B$10, and thresholds $B$11. Under metric axioms, stability axioms, and a regularity condition, the FOL theorem proves

$B$12

together with the existence of a counterexample outside the $B$13-neighborhood. In Hoare logic, if a program $B$14 is $B$15-stable and preserves $B$16, then

$B$17

and a strengthened exterior precondition yields a violation postcondition. The practical condition is

$B$18

and the paper instantiates $B$19 with Wasserstein-type distances for MNIST$B$20USPS adaptation (Nikita, 4 Oct 2025).

6. Epistemic interpretation, limitations, and recurrent structure

Across these formulations, a recurring pattern is the passage from a source-side dependency to a target-side dependency by means of a shared formal device: a term $B$21, a form $B$22, a rewrite justification, a power transform $B$23, a translator between E-Turing machines, a polymorphism condition, a model-reuse code, or a stable program transformation. This suggests that the common content of an Analogy Theorem is not a particular formula but a proof principle: the same construction recipe is applied in both domains, and soundness depends on whether the recipe is characteristic, maximal, injective, or stability-preserving (Antić, 2020, Antić, 2018, Murena, 2022).

The limitations are equally systematic. In universal algebra, analogical proportion is local and generally nontransitive; many familiar axioms fail. In logic programming, composition is generally non-associative, least models do not distribute straightforwardly over composition or concatenation, trivial justifications may dominate, and computing generalization and justification sets is non-trivial. In complexity-based transfer, exact Kolmogorov complexity is unavailable and tractable practice depends on coding choices. In transfer optimization, the monotone similarity–usefulness relation, realizability of mappings, and IID source assumptions are substantive hypotheses. In Hoare-logic transfer, guarantees are local in the metric and narrow as $B$24 grows (Antić, 2020, Antić, 2018, Murena, 2022, Xue et al., 27 Mar 2025, Nikita, 4 Oct 2025).

A complementary philosophical account comes from the study of reasoning by analogy in mathematics. There, analogical support is said to be genuine only when three conditions are met: Materiality, Relevance in the form of a robust mathematical connection, and No-Essential-Difference. On this view, deep analogies function as “relay-results,” whereas superficial analogies are merely “hookings.” That account does not give a single symbolic theorem schema for all domains, but it does clarify why formal Analogy Theorems aim to exclude manufactured similarities and to preserve dependency structure rather than mere resemblance (Cangiotti et al., 2022).

Taken together, the literature treats an Analogy Theorem as a rigorous transfer principle. Its strongest forms assert existence, uniqueness, preservation, or nonnegative gain; its weakest forms provide only justified candidate solutions. The term therefore names a family of theorem schemas whose shared ambition is to make “transform in the same way” mathematically explicit.