Formal Verification of Romanov's Triplet Logic: A Verified Filter for Sliding-window 3-CNF with Application to Structured Formulas

VeriTrans is a reliability-first ML system that compiles natural-language requirements into solver-ready logic with validator-gated reliability. The pipeline integrates an instruction-tuned NL$\!\to\!$PL translator, round-trip reconstruction (PL$\!\to\!$NL) used as a high-precision acceptance gate, and canonical PL$\!\to\!$CNF compilation, all executed via fixed API configuration (temperature$=0$; fine-tuning runs use seed$=42$) and per-item artifact logging (prompts, outputs, hashes) to support auditability and replay-driven debugging. On SatBench (2{,}100 specifications), VeriTrans achieves 94.46% SAT/UNSAT correctness and 87.73% median round-trip similarity. Compact fine-tuning on 100--150 curated examples improves fidelity by about 1--1.5,pp without increasing latency (mean 25.8,s/spec on our 201-spec runtime subset). A thresholded acceptance policy on the round-trip score exposes a reliability--coverage knob: at $τ{=}75$, roughly 68% of items are retained with $\sim$94% correctness on the accepted set. Validator overhead contributes $<15\%$ of end-to-end runtime, and all prompts/responses and timing metadata are logged to enable replay-driven debugging and regression testing. By separating learned translation from symbolic verification and enforcing deterministic, validator-gated acceptance, VeriTrans turns NL$\!\to\!$logic front-ends into auditable, reproducible components for reliability-critical workflows.

We introduce power term polynomial algebra, a representation language for Boolean formulae designed to bridge conjunctive normal form (CNF) and algebraic normal form (ANF). The language is motivated by the tiling mismatch between these representations: direct CNF<->ANF conversion may cause exponential blowup unless formulas are decomposed into smaller fragments, typically through auxiliary variables and side constraints. In contrast, our framework addresses this mismatch within the representation itself, compactly encoding structured families of monomials while representing CNF clauses directly, thereby avoiding auxiliary variables and constraints at the abstraction level. We formalize the language through power terms and power term polynomials, define their semantics, and show that they admit algebraic operations corresponding to Boolean polynomial addition and multiplication. We prove several key properties of the language: disjunctive clauses admit compact canonical representations; power terms support local shortening and expansion rewrite rules; and products of atomic terms can be systematically rewritten within the language. Together, these results yield a symbolic calculus that enables direct manipulation of formulas without expanding them into ordinary ANF. The resulting framework provides a new intermediate representation and rewriting calculus that bridges clause-based and algebraic reasoning and suggests new directions for structure-aware CNF<->ANF conversion and hybrid reasoning methods.

We introduce Sequent-Prover, a family of verifier-using language-model agents, and the post-training framework that produces them. Agents reason in an executable SMT-LIB scratchpad: they write complete SMT-LIB programs, query Z3 across turns, inspect solver feedback, repair formalizations, and terminate with an explicit answer. Our recipe, verifier-conditioned trajectory distillation, bootstraps from a noisy teacher proposal policy by sampling multi-turn trajectories, retaining the successful tail that is correct, non-trivial, grounded, and executable, then applying supervised fine-tuning and trajectory-level preference optimization. Because solver success alone does not guarantee semantic faithfulness, we evaluate task accuracy, k-sample reliability, and SMT program quality. Across 37 benchmarks and 7,254 problems per model family, post-training raises the 4B model's per-attempt accuracy from 85.6% to 97.0% and four-sample all-correct reliability (pass^4}) from 64.1% to 91.7%. It also reduces average SMT calls from 4.10 to 2.52 and tokens per attempt from 18.97k to 12.72k. Sequent-Prover agents give answers backed by executable formal artifacts.