FormalAlign Framework

• FormalAlign is a framework that rigorously defines alignment, ensuring semantic and operational consistency across formal systems, probabilistic models, and automated theorem proving.
• It integrates multiple instantiations including static alignment in probabilistic programming, dual-task autoformalization, interface theory-based cross-prover translation, and sequential runtime validation.
• Empirical evaluations reveal significant speedups, reduced variance in inference, enhanced autoformalization accuracy, and robust runtime monitoring, making it essential for automating formal methods.

FormalAlign refers to a collection of rigorously defined frameworks, architectures, and algorithms for measuring, enforcing, or utilizing alignment between formal systems, probabilistic models, or natural-formal mathematical representations. Across a diverse literature, four major instantiations of "FormalAlign" have been introduced: (1) static analysis and alignment in universal probabilistic programming (Lundén et al., 2023), (2) automated semantic alignment evaluation for autoformalization (Lu et al., 14 Oct 2024), (3) alignment-based translation across formal theorem-prover libraries (Müller et al., 2017), and (4) sequential runtime monitoring of alignment between stochastic models and observed system behaviors (Henzinger et al., 28 Jul 2025). The unifying principle is the quantification or exploitation of alignment to guarantee correctness, consistency, or semantic faithfulness in translations, inferences, or verification tasks.

1. Formal Alignment in Probabilistic Programming

FormalAlign, in the context of probabilistic programming languages (PPLs), addresses the suboptimal and error-prone manual selection of inference checkpoints such as weights and assumes within higher-order functional PPLs. The approach rests on the formalization of alignment: for a given program in A-normal form (ANF), a label $x$ is aligned if in every execution trace the sequence of aligned labels appears in the same relative order. Let $X_e$ be let-bound variables of an ANF term $e$, and $l|_A$ denote the restriction of label sequence $l$ to a subset $A \subseteq X_e$. The largest such $A$ with invariant $l|_A$ across executions defines the maximal alignment set.

The core contribution is a sound, automated static analysis that marks labels as Unaligned or aligned, using an extension of 0-CFA with stochastic and unalignment flow facts. The analysis solves for sets $S_x$ (abstract values) and flags Unaligned$_x$ via worklist-fixpoint propagation, yielding empirical runtimes of 5–30 ms per model.

From this, two inference algorithms are synthesized:

• Aligned SMC: Resampling occurs only at aligned weight checkpoints. This ensures all particles undergo identical resampling steps, eliminating global completion checks and the degeneracy associated with stochastic branches.
• Aligned Lightweight MCMC: Synchronization at aligned assume checkpoints allows for efficient Metropolis–Hastings proposals, without stack-trace database maintenance.

Implemented as a ≈1,000 LoC extension for the Miking CorePPL compiler, FormalAlign supports complex language features (records, variants, pattern-matching) and compiles to RootPPL or Miking Core with aligned kernel integration.

Empirical evaluation demonstrates substantial improvements: aligned SMC achieves a 2× to 7× speedup and drastically reduced posterior variance in phylogenetic models (CRBD, ClaDS), and a 3× speedup in MCMC for LDA and CRBD, with equivalence of posterior means to existing references. The most significant gains occur when many stochastic branches generate weight/assume calls, or in lightweight MCMC traces with high overhead from repeated look-ups.

2. Semantic Alignment for Autoformalization

Within autoformalization, FormalAlign addresses the challenge of determining semantic equivalence—not mere syntactic or logical validity—between informal mathematical statements and their formalizations in proof assistants (Lean, Coq, Isabelle). Existing methods rely on manual expert screening.

FormalAlign introduces a dual-objective LLM framework trained on both autoformalization (sequence generation) and representational alignment (contrastive learning). The architecture pools embeddings for informal (NL) and formal (FL) inputs and enforces proximity via a contrastive loss: $$\mathcal L_{\mathrm{CL}} = -\frac{1}{N} \sum_i \log \frac{\exp(\mathrm{cos}(u_i, v_i)/\tau)}{\sum_j \exp(\mathrm{cos}(u_i, v_j)/\tau)}$$ where $u_i$, $v_i$ are NL/FL pooled representations.

At inference, the alignment score combines certainty (mean log-probability) and representational similarity: $$V_{\mathrm{align}} = \frac{V_{\mathrm{cer}} + V_{\mathrm{sim}}}{2}$$

Misalignment detection is operationalized by augmenting data with negative examples (constant/exponent/variable/equality/random swaps), creating challenging benchmarks (FormL4-Basic/Random, MiniF2F). Fine-tuned on Mistral-7B, FormalAlign achieves 99.21% alignment-selection accuracy on FormL4-Basic (vs. 88.91% for GPT-4), and consistently higher precision, with ablation studies confirming the essential role of both loss components.

FormalAlign enables large-scale automated screening of formalization pairs, ranking, and curation of corpora for theorem-proving infrastructure, and its approach extends to multi-task and cross-system settings.

3. Interface Theories and Symbolic Alignment in Theorem Prover Interchange

In translation across formal theorem prover libraries, FormalAlign signifies a framework based on interface theories and symbol-level alignments, operational within the Mmt system. Interface theories are axiomatic signatures abstracting over logical/implementation differences, e.g., "NaturalNumbers" defines $\mathbb N$, $0$, $+$, $*$, $=$, $\leq$, Peano axioms, without committing to an underlying logic.

Alignments are structured as:

• (id$_1$, id$_2$, arg_map, direction), where id$_i$ is a symbol's URI, arg_map defines argument permutation, and direction encodes transformation safety.

Translation proceeds by graph traversal and local rewriting, mapping a source term via alignments (possibly through interface theory intermediates) to a target library term, with correctness validated through type-checking of sample translations. Practical demonstration includes translating "inv(plus(a,b))" from the NASA PVS library to HOL Light.

The current database covers ≈900 alignments across four libraries and five domains. All alignments are manually curated, with validation based on type signatures and sample translations.

Limitations include brittleness under partial functions, lack of proof-level translation, and incomplete coverage. Proposed future paths involve crowd-sourcing, AI-assisted matching, sketch-level translation, edit-distance generalization, and deeper integration via theory morphisms.

4. Sequential Alignment Monitoring for Probabilistic Model Validation

FormalAlign for alignment monitoring operates in the domain of runtime model validation: a probabilistic model $\mathcal{M}$ is considered well-aligned if its predicted transition distribution matches that of the actual system. At each timestep, the monitor receives observables $(y_t, x_t)$, scoring using bounded proper scoring rules (e.g., Brier, spherical). The key metric is the hidden average expected score

$$E_t = \frac{1}{t} \sum_{i=1}^t \mathbb{E}_{X_i \sim \pi_i} [s(y_i, X_i)]$$

with its observable estimator $\hat E_t$.

Confidence intervals $[L_t, U_t]$ on $E_t$ are maintained using sequential forecasting techniques drawing on martingale bounds ("stitching" method): $$\varepsilon_t = \frac{1}{t} \left( \sqrt{2.13 N_t g(N_t, \delta) + 1.76(b-a)^2 g(N_t, \delta)^2} + 1.33(b-a)g(N_t, \delta) \right)$$ where $N_t$ is the empirical variance, yielding intervals with time-uniform coverage.

The framework generalizes to differential alignment monitoring (comparing two models) and weighted monitoring (emphasizing fairness or safety via history/state-dependent weights).

Evaluation on PRISM benchmarks confirms fast convergence of confidence intervals and rapid decision-making regarding model superiority, with time complexity linear in the state space (or O(1) in the Markovian weighted case). Weighted monitors effectively detect alignment deficits in protected groups or safety-critical regions.

5. Comparative Table of FormalAlign Instantiations

Subdomain	Core Principle	Main Output/Benefit
PPL Analysis (Lundén et al., 2023)	Static label alignment via 0-CFA+stoch/unaligned	Efficient, sound SMC/MCMC; reduced inference cost
Autoformalization (Lu et al., 14 Oct 2024)	Dual-task LLM (generation + contrastive alignment)	Automated, precise NL–FL semantic faithfulness score
Cross-prover Translation (Müller et al., 2017)	Interface theories and handcrafted alignments	Portable statements across proof assistant libraries
Alignment Monitoring (Henzinger et al., 28 Jul 2025)	Sequential, statistical runtime confidence intervals	Provably correct model–system runtime validation

All instantiations share an emphasis on rigorous, automated, black-box (or minimally intrusive) procedures for quantifying alignment, but address different notions—syntactic order, semantic embeddings, symbol equivalence, and distributional prediction, respectively.

6. Limitations, Challenges, and Future Directions

Across these approaches, principal limitations include sensitivity to partiality, manual effort in alignment creation (theorem prover translation), restrictions to statement (rather than proof) translation, and scaling challenges in the coverage of interface theories and negative example construction. For autoformalization, current methods focus primarily on theorem-statement alignment, not multi-step proof alignment. For probabilistic model monitoring, generalization to non-Markovian or highly history-dependent systems requires further research.

Directions highlighted for extension are:

• Activity-driven learning for alignment scores and autoformalization
• Automated, possibly AI-assisted, discovery and validation of alignments
• Extended handling of proof-level translation and definitional morphisms
• Cross-system retraining of autoformalization alignment evaluators
• Task-specific monitoring (e.g., fairness, safety) for runtime validation

FormalAlign frameworks thus form a crucial link in automating tasks that hinge on semantic and operational alignment, clearing bottlenecks in probabilistic inference, formal library interchange, autoformalization, and runtime validation. Their continued development is positioned to remove manual barriers and elevate the reliability and portability of formal methods.