Neural Algebraic Inequality Prover

• Neural Algebraic Inequality Prover is a system that combines deep learning with symbolic reasoning to automate the discovery and proof of complex algebraic inequalities.
• It leverages reinforcement learning and neuro-symbolic techniques, using LP and semidefinite programming to generate nonnegativity certificates for multivariate polynomials.
• Empirical results show state-of-the-art performance on both polynomial optimization benchmarks and Olympiad-grade problems, reducing search rounds significantly.

A Neural Algebraic Inequality Prover is a computational system that integrates deep neural networks, symbolic reasoning, and mathematical optimization to automate the discovery and proof of algebraic inequalities, particularly those involving multivariate polynomials or expressions with non-trivial structure. These provers leverage reinforcement learning, curriculum-guided neural heuristics, neuro-symbolic tactic synthesis, and fast algebraic operations to generate or verify inequalities, often at or beyond the level of mathematical olympiads or advanced combinatorial optimization. Recent developments have demonstrated that neural-guided approaches can achieve state-of-the-art performance across a range of polynomial and Olympiad-level algebraic inequality domains (Liu et al., 9 Mar 2025, Wei et al., 20 Jun 2024, Li et al., 19 Feb 2025, Fawzi et al., 2019).

1. Mathematical Formulation and Proof Systems

Neural algebraic inequality provers formalize the problem of certifying the nonnegativity of a target polynomial or more general algebraic expression over a semialgebraic domain (typically a box or simplex, e.g., $[0,1]^n$). The most common formalism relies on expressing a polynomial $f(x)$ as a nonnegative combination of basis polynomials, often using a Krivine or Handelman Positivstellensatz certificate: $$f(x) = \sum_{|\alpha|+|\beta|\leq D} \lambda_{\alpha,\beta} x^\alpha (1-x)^\beta, \quad \lambda_{\alpha,\beta} \geq 0$$ This reduces the inequality proving problem to a linear program (LP) over the coefficients $\lambda_{\alpha,\beta}$. Alternatively, semi-algebraic proof systems (Lovász–Schrijver, Sherali–Adams) define a set of elementary inference rules:

• Monotone multipliers: $g \geq 0 \Rightarrow x_i g \geq 0$ and $g \geq 0 \Rightarrow (1-x_i)g \geq 0$
• Nonnegative conic combinations: $$\{g_j \geq 0\} \Rightarrow \sum_j \lambda_j g_j \geq 0$$, $\lambda_j \geq 0$ Proof search then amounts to constructing a sequence of such inferences to establish $f(x) \geq 0$, with LP or semidefinite programming providing numeric certificates (Liu et al., 9 Mar 2025, Fawzi et al., 2019).

2. Neural and Neuro-Symbolic Architectures

Modern neural inequality provers combine neural policy/value networks with symbolic search:

• Reinforcement Learning (RL): The proof search process is modeled as a Markov Decision Process. At each proof round, the system considers the current set of basis polynomials or lemmas ($M_t$), and selects actions such as multiplying an existing basis by $x_i$ or $(1-x_i)$. The selection policy, typically realized via Deep Q-Learning (DQN), is optimized using reward signals defined by improvements in the LP objective (e.g., tighter lower bounds for nonnegativity certificates) (Liu et al., 9 Mar 2025, Fawzi et al., 2019).
• Value Networks and Curriculum Learning: Neural architectures (e.g., transformer encoders over LaTeX or structured representations) estimate the likelihood that a given proof state leads to a successful derivation. Value networks are pretrained on synthetic theorems using tree-depth or similar heuristics, then fine-tuned via reinforcement/curriculum learning on hard, machine-generated problems (Wei et al., 20 Jun 2024).
• Neuro-Symbolic Tactic Synthesis: Some systems divide inference responsibilities: LLMs or neural sequence models generate creative equivalence rewrites, while a symbolic engine applies domain-specific lemmas (AM-GM, Cauchy–Schwarz, Jensen, etc.), validates applicability (e.g., via SMT/CAD solvers), and prunes subgoals via heuristics and neural ranking (Li et al., 19 Feb 2025).

3. System Workflow and Algorithmic Structure

A typical workflow in a neural algebraic inequality prover consists of the following steps (detailed for the RL-based APPIRL system (Liu et al., 9 Mar 2025); the symbolic–neural "AIPS" and neuro-symbolic LLM approaches follow analogous routines):

1. Initialization:
   • Start with the target $f(x)$, domain $S$ (often $[0,1]^n$), and an initial basis $M_0$ (all $x^\alpha (1-x)^\beta$ up to $\deg f$).
2. State and Action Space Construction:
   • The state encodes current LP optimal bound and history (stagnation count, basis size).
   • Actions correspond to basis extension via multipliers ($x_i$, $1-x_i$) applied to current basis elements.
3. Stepwise Proof Search:
   • At each round, select an action using an $\epsilon$-greedy policy over the Q-network (states, actions).
   • Extend the basis, solve the updated LP for the best nonnegativity bound, compute the reward.
   • Update experience buffer for off-policy Q-learning.
4. Action Space Acceleration:
   • Basis updates and candidate generation are optimized using fast multivariate polynomial multiplication via FFT-based techniques, encoding monomials as univariate polynomials for convolution (Liu et al., 9 Mar 2025).
5. Proof Termination:
   • Search halts and outputs a Krivine-basis certificate if LP nonnegativity is achieved ($\gamma^* \geq 0$), or fails after a fixed number of steps.

Neuro-symbolic tactics systems follow an alternating pipeline between symbolic candidate enumeration/pruning and neural/LMM-guided transformation and subgoal ranking, leveraging formal proof assistants for tactic instantiation (Li et al., 19 Feb 2025).

4. Theorem Generation and Synthetic Datasets

Autonomous theorem generation is a signature capability of recent neural algebraic inequality provers:

• Synthetic Theorem Generation: Using a pool of cyclically symmetric premises, deduction engines iteratively apply transformation rules and algebraic lemmas, enforce equality condition checks, and filter by syntactic complexity and inference depth. This yields datasets of $10^5$–$10^6$ nontrivial, structurally diverse Olympiad-grade theorems within hours on modern compute (Wei et al., 20 Jun 2024).
• Dataset Construction: Algebraic expressions are indexed by tree-depth, variable arity, and syntactic length, supporting large-scale pretraining of value networks and enabling curriculum strategies for hard problem discovery and proof search.
• Sample Output: Theorems often exhibit high algebraic symmetry and are competitive with, or surpass, human benchmarks in complexity and novelty.

5. Empirical Performance and Benchmarks

Neural algebraic inequality provers have been evaluated on both polynomial optimization tasks and Olympiad-grade inequalities:

• Polynomial Benchmarks: RL-based provers (APPIRL) achieve $\sim 6\times$ fewer search rounds than random or uninformed search, match or exceed dynamic proof systems such as LDPP and S2V-DQN on classical combinatorial problems (e.g., maximum stable set in graphs) (Liu et al., 9 Mar 2025).
• Olympiad Inequality Benchmarks: Value curriculum–trained systems and neuro-symbolic LLM combiners (Lips) solve 50–80% of MO-INT-20 and ChenNEQ challenge suites, outperforming LLMs, SMT/CAD stacks, and earlier symbolic engines by large margins (Wei et al., 20 Jun 2024, Li et al., 19 Feb 2025). AIPS (curriculum) solves 10/20, while pure LLMs solve at most 1/20.
• Efficiency and Goal Pruning: Neuro-symbolic integration with LLM-driven subgoal ranking drastically reduces tree width and search iterations compared to either neural-only or symbolic-only baselines. Ablation studies confirm the necessity of both tactic types and neural ranking (Li et al., 19 Feb 2025).

System	MO-INT-20 (%)	Major Advantages
AIPS (curr.)	50	No human data, curriculum value net
Lips	80	Neuro-symbolic, LLM goal ranking
CAD/SMT stack	60	Symbolic only
LLMs (GPT-4)	5	Pure prompt reasoning

6. Limitations and Future Directions

Current neural algebraic inequality provers face structural and algorithmic constraints:

• Expressivity: State-of-the-art systems focus on cyclic ternary/quaternary or polynomial forms. Generalization to arbitrary $n$-variate non-symmetric, non-homogeneous, or piecewise-defined domains is not fully addressed (Wei et al., 20 Jun 2024).
• Manual Lemma Encoding: In symbolic engines, core algebraic theorems are manually coded and pattern matching remains expensive.
• Search Depth: Proofs longer than 5–7 steps or those involving deeper combinatorics are challenging for current curriculum-trained heuristics.
• Automated Lemma Discovery: While theorem enumeration is algorithmic, real-time discovery of useful intermediate lemmas during proof search is a target for future research.

Ongoing research aims to broaden variable arity, incorporate automatic lemma synthesis, hybridize with neural conjecture generation, and optimize symbolic pattern-matching using neural or structural embeddings. Application domains may extend to trigonometric, matrix-valued, or functional inequalities, and further integration with program synthesis is anticipated (Wei et al., 20 Jun 2024).