The Mathematical Colosseum
Ufinq discovers the source code of nature: small, readable programs that describe how data behaves. Built on Symbolic Regression, Ufinq treats the structure itself as the unknown, opening a vast, combinatorial search space of possible programs.
It is a mathematical colosseum. Generations of candidate programs enter the arena: some grow in complexity to capture more of the data; others stay small and elegant, trading detail for brevity. They compete on accuracy and simplicity. The fittest survive, recombine, and mutate; the weak fall away.
Ufinq scales the colosseum. Many evolutionary branches train in parallel, each a small population evolving in isolation until stagnation signals it is ready to enter the arena. There they meet equally mature opponents; their genetic material splits into an intensification path that exploits the strongest patterns and a diversification path that explores new ones. One arena, distributed across a cluster, where every node contributes to the same evolutionary contest.
What survives the colosseum is code: an explicit, line-by-line program written in the language of mathematics. Where deep learning compresses your data into millions of opaque weights, Ufinq returns a model deployable on any machine and readable by any human.
What survives the colosseum is code, not weights.
What makes Ufinq different
Algorithm
- AHISTER orchestration. Independent branches train in isolation, then meet in the arena; intensification and diversification happen simultaneously.
- DEBOHAC distributed coordination. The sister algorithm: one logical colosseum across a tree of cooperating instances, with capacity reported upward and budget granted downward.
- Two kinds of diversity. Structural and behavioral diversity are checked separately on every candidate, so a population cannot hide duplicates behind different spellings or different constants.
Universal applicability
- Every value type is first-class. Bool, integer, real, vector, matrix, tensor, text, set, tuple and record all take part in the symbolic search.
- Close to 200 elementary functions. Scalar arithmetic and analysis, comparison, boolean logic, conditionals, and vector, matrix and tensor operations, composable across value types.
- One search, several kinds of target. Real-valued regression, counts, binary and multiclass classification, and structured targets (vectors, tuples, records) are fitted by the same search, with the loss chosen from the type of the target.
Distributed evolutionary arena at scale
- One logical colosseum across many machines. Instances form a tree; work is delegated to wherever capacity is free, and the arena scales horizontally.
- Push-based cluster transport. Peers talk over TLS connections served by virtual threads; the messaging plane does not poll.
- Peer-local datastore. Datasets move between nodes on demand and are cached where they are used, without redundant transfers.
A refinement pipeline far beyond evolution alone
- Continuous simplification. Hundreds of declarative rules in a custom rule DSL, backed by an e-graph rewriter, simplify every candidate, every generation.
- Research-grade numeric optimization. Coordinate Descent, CMA-ES, L-BFGS, Nelder-Mead.
- Native symbolic differentiation. Exact gradients of evolved expressions, computed symbolically. Unusual for symbolic regression.
Output
- Code, not weights. Every model is an explicit, line-by-line program; readable by humans, executable by machines.
- One model, not a menu. The search keeps many candidates internally and returns exactly one: the model that best balances accuracy against size. The trade-off is the algorithm's job, not the reader's.
- Portable by construction. A model is a formula. It exports as readable mathematics, LaTeX, or source code in C, Java, JavaScript, Kotlin, Python, R and SQL, and runs wherever that code runs.
Underlying computation graphs
Solutions are symbolic expressions built from mathematical building blocks. Each solution consists of a computation graph containing one or more computation trees (abstract syntax trees), built up of function nodes, reference nodes, and constant nodes. The value spaces of all nodes, references, constants, and functions (domains and codomains) are precisely defined. A computation graph can process arbitrarily complex inputs, states, and targets, not restricted to real-valued vectors. It supports variables for intermediate computation results within a single computation and persistent states shared across multiple computations. Computation trees are evaluated sequentially and can build on previous results.
Value types
Universal Function Space
The algorithm searches for solutions in an approximated Universal Function Space, containing approximately all possible functions. Ufinq creates this effective function space from a fixed set of many predefined elementary functions and a comprehensive value system. Functions are built on value types and value spaces, with all characteristics (domain, codomain, …) defined for every elementary function. In computation graphs they are composed into more complex functions.
Elementary functions
Accuracy and size, on one scale
Models are found by searching the universal function space for functions that describe the relationship between inputs and targets. Accuracy alone is not the objective: a formula that is larger has to earn its size. Ufinq scores every candidate by description length, the number of bits needed to state the formula plus the bits needed to state what the formula gets wrong. A term stays only if it saves more bits in error than it costs to write down. There is no weight to tune between the two. The quality a reader is shown is on a familiar scale: R², discounted for the size of the formula, and converging to plain R² as the number of rows grows.
Sophisticated refinement
Each potential solution is verified and then refined before it joins the gene pool: simplified, linearly scaled, optimized, normalized and validated. Expressions are simplified by a declarative rule DSL, with an e-graph rewriter as a safety net. Linear scaling fits the output's slope and intercept directly, so the search does not spend generations finding them. Constants are then optimized with Coordinate Descent, CMA-ES, L-BFGS on exact symbolic gradients, and Nelder-Mead. Normalization rounds and names constants and orders operands. Validation rejects candidates that duplicate what the population already holds. It is an evolution of local optima: every individual is refined to its best form before it competes.
Evolution of local optima
The evolutionary process is built on many search strategies, genetic operators, and smart generations and modifications of computation graphs. Many specialized functions integrate techniques from across mathematics (symbolic differentiation, gradient descent, normalization, optimization, simplification, logic, algebra, analysis, statistics) to improve search efficiency. Many techniques limit the effective search space or guide the search process. Because the universal function space is infinite, smart and focused non-evolutionary methods are essential. Small surrogate models (a forest, a single-layer network, a kernel regression) are trained on the data and translated into symbolic starting points, so the search begins from structure the data already suggests. Hard targets are decomposed automatically: when one part of a formula is the bottleneck, the search opens a sub-search on that part alone and assembles the result.
Universal applicability
Ufinq is built for data of any structure. Targets can be real numbers, counts, classes, or structured values; inputs can be numeric, boolean, text or categorical. Rows can carry weights, and the confidence of every fitted constant is estimated from the single fit. Horizontal scaling across a cluster of instances lets the same search run on larger datasets and larger models.
Use Ufinq commercially
Ufinq is a closed technology distributed via Diafunc, the Symbolic AI platform. Diafunc bundles Ufinq with the computing resources, data infrastructure, and tooling you need to apply it directly in your projects.