Self-Evolutionary Math Reasoning

Updated 8 August 2025

The paper presents a self-evolutionary framework that iteratively refines mathematical reasoning through cognitive primitives, algebraic abstraction, and automated tactic induction.
It employs a blend of procedural, algebraic, and probabilistic techniques to autonomously enhance problem-solving and evolve reasoning strategies over time.
The approach co-evolves data and model complexity via reinforcement learning and explicit knowledge graphs, leading to robust, adaptive, and verifiable reasoning processes.

A self-evolutionary framework for mathematical reasoning refers to algorithmic and cognitive architectures that can autonomously refine, expand, and adapt their methods of mathematical reasoning over time, through iterative self-improvement processes rather than solely relying on static external supervision or human intervention. This paradigm has increasingly shaped both the formal modeling of mathematical cognition and the computational design of autonomous reasoning agents.

1. Foundational Cognitive Mechanisms

Several self-evolutionary frameworks ground mathematical reasoning in models inspired by cognition and the structure of logic. Early work (Tuncer, 2011) established that both qualitative comparisons (detecting similarity—equivalence relations) and quantitative comparisons (detecting order—order relations) serve as universal primitives:

Equivalence Relations (~): The grouping of objects or instances via shared characteristics (e.g., $x_i \sim x_j$ if they share all features) yields equivalence classes, fundamental for classification and naive logic.
Order Relations (<, ≤): Comparisons on magnitude or sequence (e.g., $x_i < x_j$ ) impose structure necessary for the development of arithmetic and induction.
Association as Inference ( $\Rightarrow$ ): Linking objects or classes to predicates or properties provides the basis for inference and learning; association rules such as $x_i \in [X] \Rightarrow P(x_i)$ underpin naive logic.

This model demonstrates that natural Peano arithmetic emerges from the successor structure implied by total orderings, connecting cognitive primitives to the axiomatic foundations of numbers.

Additionally, the association process is treated as a Markov chain over predicates, yielding a stationary distribution that encodes long-term cognitive activation patterns and equivalence classes among predicates—a formalization of the "world view" that supports iterative refinement and inference (Tuncer, 2011).

2. Algebraic and Structural Abstraction Mechanisms

Autonomous problem solvers benefit from algebraic and graph-based abstraction mechanisms to realize self-evolution (Tirri, 2013):

Knowledge as Nets: States and knowledge components are represented as nets (graphs with nodes, edges, and arity), supporting rich syntactic and semantic manipulation.
Net Block Homomorphism (NBH): These are algebraic transformations that abstract away internal structure while preserving essential linkage information, crucial for universal abstraction. NBH enables grouping nets into equivalence classes (abstract "sisters") facilitating higher-order reasoning.
Renetting Systems: Advanced rewriting systems that handle position, environment, and context, enabling the transformation and normalization of nets into operationally powerful forms.
Saturation via Equivalence Relations: By iteratively closing the solution set under groups of equivalence relations, the framework ensures broad coverage and the evolution of solution classes.

The framework introduces iterative closures and quotient transducer algebras, providing the mathematical infrastructure for transferring solutions across equivalence classes. This enables the autonomous development of higher-order abstractions, concept generalization, and problem-solving efficiency beyond single-instance adaptation.

3. Procedural Abstraction and Inductive Tactic Discovery

The development and reuse of procedural abstractions ("tactics") underpins rapid self-improvement in both human and artificial mathematical reasoners (Poesia et al., 2022):

Finite-Action Theorem Environments: Languages such as Peano are constructed so that at every proof step, the set of valid actions is finite, supporting tractable search and abstraction mining.
Abstraction Induction: Sub-sequences in successful solution traces are generalized into tactics via anti-unification, quantifying their compression utility as

$u(t) = \frac{m(t,\mathcal{S})( |t| - 1 ) }{p(t)}$

where $m(t,\mathcal{S})$ is the coverage, $|t|$ is sequence length, and $p(t)$ is parameter count.

Self-Ordering Curriculum: The dependency graph of induced tactics defines a partial order on problems, enabling the formation of curricula which significantly improve learning efficiency among successive agent generations.

The iterative process—starting from axiomatization, tactic discovery from solutions, to curriculum induction—realizes a pipeline where each cycle enhances both the repertoire and sequencing of mathematical knowledge.

Modern self-evolutionary mathematical frameworks employ iterative self-critique and multi-path search mechanisms for robust reasoning:

Monte Carlo Tree Search (MCTS) with Self-Refine: At each node, candidate reasoning paths are repeatedly critiqued and rewritten, with paths selected using a combination of reward models and search heuristics (e.g., UCT, Nash Equilibrium policies) (Rabby et al., 23 Nov 2024, Zhang et al., 3 Oct 2024, Guan et al., 8 Jan 2025).
Preference and Reward Modeling: Pairwise preference models or process reward models, trained via RLHF or pairwise ranking loss, evaluate the local and global quality of reasoning steps or whole chains without fragile scalar rewards (Zhang et al., 3 Oct 2024, Guan et al., 8 Jan 2025).
Verification via Executable Code: Code-augmented chain-of-thought ensures that only solution paths with verifiable intermediate steps are propagated during search (Guan et al., 8 Jan 2025).

This blend of deep search, explicit self-correction, and RL-driven path selection allows even small models to exhibit "deep thinking" and state-of-the-art math reasoning, as shown in rStar-Math and LLaMA-Berry.

5. Autonomous and Explicit Knowledge Evolution

Recent frameworks emphasize explicit, interpretable knowledge evolution and diversification:

Explicit Knowledge Graph Evolution: Frameworks such as LeAp use a variational autoencoder to learn explicit, interpretable word-word and word-operator relationships in the form of knowledge graphs, bridging knowledge acquisition and application (Liu et al., 2023).
Self-Correction and Diversity Induction: Pipelines like SPHERE generate, critique, and diversify solution chains, using on-policy reward models to prefer promising steps and diversity induction (with auxiliary weaker models) to cover a spectrum of reasoning modes (Singh et al., 4 Mar 2025).
Fill-in-the-Middle Expansion: MathFimer expands reasoning chains by training models to fill in missing intermediate steps—recurrently increasing the depth and completeness of explanations, thereby enhancing self-supervised learning and iteratively refining data quality (Yan et al., 17 Feb 2025).

Such explicit graph-based or reward-driven approaches stabilize and sharpen the underlying knowledge representations throughout autonomous evolution cycles.

6. Data and Model Co-evolution Strategies

The evolution of model capability is paralleled by the co-evolution of data complexity:

Synthetic Problem Generation with RL: MathSmith generates entirely new, high-difficulty mathematical problems from random concept–explanation pairs sampled from PlanetMath. The RL objective jointly optimizes problem structure, reasoning complexity, and answer consistency, with cognitive complexity measured by the token length of model-generated solving traces (Zhan et al., 7 Aug 2025).
Multimodal and Cross-Modal Co-evolution: C²-Evo jointly evolves both textual and visual mathematical data in geometry by expanding diagrams and subproblem structure as model capabilities increase, employing closed-loop selection based on difficulty and supervised plus RL-based model training (Chen et al., 22 Jul 2025).

This co-evolutionary approach ensures that the reasoning agent is continually challenged with tasks commensurate to its capability, eliminating the stagnation caused by static datasets.

7. Normative and Probabilistic Models of Inquiry

Self-evolutionary frameworks can also take formally probabilistic perspectives:

Stochastic Mathematical Systems (SMS): SMS treat the process of mathematical inquiry as a stochastic generation of question–answer pairs, with calibration relations imposing normative conditions between the agent (reasoner) and an oracle (idealized mathematical community). This supports rational updates of belief and abductive inference under uncertainty (Wolpert et al., 2022).
Markovian Association: Modeling the long-term structure of associative reasoning as a stationary distribution of a Markov process clarifies how cognitive units stabilize on particular inference patterns (Tuncer, 2011).

These perspectives supplement deterministic, rule-based frameworks by capturing the epistemic and adaptive nature of inquiry.

In conclusion, self-evolutionary frameworks for mathematical reasoning integrate progressive abstraction, explicit knowledge evolution, iterative self-correction, preference-driven optimization, and adaptive data–model co-evolution. They formalize how mathematical reasoning can autonomously develop, reflecting foundational cognitive principles, deep abstraction, advanced search, and ensemble learning over solution traces. This multi-level, iterative design enables both algorithmic agents and cognitive systems to reliably extend their own reasoning repertoire and to generalize effectively, offering both theoretical insight and practical advances in automated mathematics and artificial intelligence.