ML-Driven Problem Rewrite Module

• Problem Rewrite Module is a reusable, ML-driven component that automates semantics-preserving transformations of structured problems via rule-based and neural methods.
• It leverages a graph encoder and sequence decoder to learn and apply explicit rewrite rules, ensuring fast, verifiable equivalence of dataflow graphs.
• The module integrates synthetic data generation, rigorous training, and real-time API interfaces, enabling scalable use in compiler optimization and program analysis.

A Problem Rewrite Module is a reusable, machine-learning-driven software component designed for automated, semantics‐preserving transformation of structured problems, particularly code or computation graphs, via rule‐based and neural methods. In modern computational research and systems—especially in program analysis, compiler optimization, and machine learning for code—the module provides a principled means to verify equivalence of programs or dataflow graphs by learning explicit sequences of rewrite rules, with correctness verifiable in negligible time. Its application is central to modern systems for algebraic simplification, program equivalence, and optimizing intermediate representations using large collections of rewrite rules.

1. Formal Definition of Dataflow Graph Equivalence

The foundational abstraction is the dataflow graph $G = (V, E, op, inPorts, outPorts)$, where:

• $V$ is a finite set of nodes,
• $E \subseteq V \times V$ indicates directed dataflow edges,
• $op(v)$ assigns an operator (e.g., MatMul, Add, Transpose) to each node,
• $inPorts, outPorts$ specify tensor shape indexing.

Formally, each graph $G$ denotes a function $\llbracket G \rrbracket$ mapping input tensors to output tensors under conventional dataflow semantics. Two graphs $G_1, G_2$ are semantically equivalent, denoted $G_1 \equiv G_2$, iff $$\llbracket G_1 \rrbracket(x) = \llbracket G_2 \rrbracket(x)$$ for all admissible $x$.

Because general semantic equivalence is undecidable, the Problem Rewrite Module operationalizes equivalence via a finite set $R$ of semantics-preserving rewrite rules. Rule application is $G \xrightarrow{r} G'$, where $r \in R$ is matched and applied to a subgraph. Rewrite-based equivalence holds if there exists a finite sequence $r_1, \ldots, r_k$ such that $$G_1 \xrightarrow{r_1} \ldots \xrightarrow{r_k} G_2'$$ and $G_2'$ is isomorphic to $G_2$ up to node renaming and port reordering.

2. Rewrite Rule Framework and Rule Application

The module uses a library of approximately 120 hand-coded axioms for linear algebra and dataflow transformations, each modeled as a pattern pair $(\mathrm{LHS}_r, \mathrm{RHS}_r)$ with optional variables over tensor shapes. Representative rules include distributivity, associativity, transposition, zero/identity elimination, and reshape fusion. Each rule is applied by:

• Pattern-matching $\mathrm{LHS}_r$ into host graph $G$ (via a VF2-style matcher adapted for acyclic graphs),
• Recording variable bindings $\sigma$,
• Deleting the matched subgraph,
• Instantiating and inserting $\mathrm{RHS}_r$ under $\sigma$,
• Reconnecting preserved edges.

For practical integration, the module exposes APIs for applying rules, verifying equivalence, and producing visual explanations of rewrites.

3. Automatic Training Data Generation

To train the rewrite sequence generator, the module uses a synthetic data generation process. The routine samples small random DAGs (10–30 nodes) constructed from the available operator vocabulary. For each graph, a random-length ($k$) sequence of rewrite rules is sampled and successively applied to create an output graph and corresponding rule trace. Only well-formed outputs are retained. This process yields a large corpus of aligned triples $(G, G', \text{seq})$ for supervised learning.

Training Set Generation Workflow

Step	Description	Output
Graph Sampling	Build random DAG (ops, shape-consistent)	$G$
Rule Sequencing	Randomly pick sequence of rewrite rules $r_1,..$	Transformation
Rule Application	Successively apply $(r,\; m, \;\sigma)$	$G'$ and $\text{seq}$
Filtering	Discard degenerate/ill-formed graphs	Final example set

This methodology generates datasets on the order of $N\approx 150,000$ pairs, with an 80/10/10 train/dev/test split.

4. Graph-to-Sequence Neural Model

The module implements a neural sequence generator with a graph encoder and sequence decoder:

• Encoder: Multi-layer Graph Convolutional Network (GCN). For each node $v$:

$$h_v^{(0)} = \text{EmbedOpType}(op(v)) \parallel \text{PosEmb}(v)$$

$$h_v^{(\ell+1)} = \operatorname{ReLU} \left( W_{self} h_v^{(\ell)} + \sum_{u \rightarrow v} W_{in} h_u^{(\ell)} + \sum_{v\rightarrow w} W_{out} h_w^{(\ell)} + b \right)$$

After $L$ layers, node embeddings are aggregated via mean-pooling.

• Decoder: LSTM emits sequence of rule tokens, including:
  • Rule identifier,
  • Match-location specifier (BFS-order index),
  • Special tokens \texttt{<END>}, \texttt{<PAD>}.

At each decoding step,

$$s_t = \mathrm{LSTM}(\mathrm{EmbedToken}(y_{t-1}), s_{t-1}, c_{t-1})$$

$$\alpha_{t,i} = \operatorname{Softmax}(h_i^T W_{att} s_t)$$

$c_t = \sum_i \alpha_{t,i} h_i$

$o_t = W_o [s_t \parallel c_t] + b_o$

$P(y_t | y_{<t}, G) = \operatorname{Softmax}(o_t)$

The network is trained to minimize cross-entropy over the ground-truth rule sequence.

5. Training Regimen, Inference, and Validation

The training regimen uses:

• Batch size: 32 graphs,
• Optimizer: Adam with $\beta_1=0.9$, $\beta_2=0.999$,
• Initial learning rate: $10^{-3}$, decayed by $0.9$ every 5,000 steps,
• 50 epochs with early stopping on dev set.

Metrics:

• Top-1 sequence accuracy,
• Graph equivalence accuracy after predicted rewrites ($G'_{pred} \equiv_{iso} G'_{true}$),
• Top-$k$ rule prediction accuracy.

Inference generates $B=5$ beam candidates, applies sequences to $G$ to produce $G_{pred}$, and verifies equivalence using canonical hashing—enabling validation in 10–50 ms per example. For input graphs with <$200$ nodes, rewrite and canonicalization times are $0.1$ ms and $0.5$ ms respectively.

6. Software Integration and Performance

The module is deployed as a reusable software artifact with both Python API and gRPC/REST interfaces. The core class:

class ProblemRewriteModule: def __init__(self, model_path: str, rule_set: RuleSet): … def rewrite(self, graph: DataflowGraph, target_graph: Optional[DataflowGraph] = None, max_steps: int = 20, beam_size: int = 5) -> RewriteResult: """Returns the rule sequence and transformed graph; verifies equivalence if target provided."""

RewriteResult provides:

• seq: list of applied RuleApplications,
• final_graph: resulting DataflowGraph,
• success: semantic equivalence verified,
• log_probs: per-step output probabilities.

Utility functions include:

• verify_equiv(G1, G2) → bool: graph equivalence checker,
• visualize_rewrite(G, seq): Graphviz diagram output.

Batch encoding (32 graphs) on commodity GPUs completes in 8 ms, while CPU-side beam-search decode and validation per graph require 30–50 ms. The module is thereby suitable for real-time deployment in compilers, optimizers, or interactive transformation systems.

7. Context and Impact

The Problem Rewrite Module, as instantiated in the context of "Equivalence of Dataflow Graphs via Rewrite Rules Using a Graph-to-Sequence Neural Model" (Kommrusch et al., 2020), demonstrates 96% correctness in producing rewrite sequences for 30-term programs on a test set of 10,000 graph pairs, with equivalence verifiable in negligible time. The framework enables principled integration of symbolic rewrite rules with neural learning for synthesis, explanation, and verification tasks. This establishes a robust approach for neural-augmented program analysis, offering scalable and certifiable equivalence checking for complex expression languages and custom DSLs. It enables deployment in compilers, differentiable programming systems, and algebraic transformation environments.