interval arithmetic

Traditional refinement type systems are difficult to adopt in mainstream languages due to their heavy annotation overhead, particularly when handling common properties such as integer ranges, which often require extensive manual annotations. This work proposes Ranger, a bidirectional type system for integer range refinements that integrates type inference with lightweight, flow-sensitive static analysis. Ranger supports imperative constructs—including variables and loops—while substantially reducing the annotation burden on users. Experimental evaluation using the Licorne language demonstrates that Ranger can concisely verify properties beyond the reach of standard type systems, such as index safety, and achieves greater annotation succinctness compared to both the Java Checker Framework and Liquid Java.

To address rounding errors in floating-point arithmetic that compromise the correctness of MDP model checking, this paper presents the first end-to-end formal verification of interval iteration under IEEE 754 floating-point semantics. Leveraging Isabelle/HOL and the Refinement Framework, we extend the refinement calculus to support directed rounding modes, enabling rigorous correctness proofs spanning abstract semantics, algorithmic implementation, and LLVM-level code. Our approach bridges theoretical analysis with actual hardware floating-point behavior, ensuring mathematical precision guarantees while achieving competitive performance on standard MDP benchmarks—matching state-of-the-art tools without false positives or negatives induced by rounding errors. The core contributions are: (i) the first complete formal verification of interval iteration in a realistic floating-point setting; and (ii) a novel extension of the refinement framework to reason about directed rounding, thereby enabling sound and scalable quantitative verification of probabilistic systems under IEEE 754 constraints.

This work addresses the scalability bottleneck in symbolic model checking caused by state-space explosion by proposing a novel encoding method for Petri net markings based on vector intervals—distinct from conventional interval vectors—that supports global verification of CTL formulas. The approach innovatively introduces generalized vector intervals, defines their homomorphic operations and canonical forms, and thereby overcomes the expressiveness limitations of interval decision diagrams. By integrating saturation and clustering-based optimization techniques, the method achieves significant gains in verification efficiency. Empirical evaluation on the MCC 2022 benchmark suite demonstrates that the proposed technique substantially enhances both the scalability and performance of symbolic model checking.

This work addresses the lack of formal guarantees regarding semantic preservation during problem reformulation and solver correctness in constraint programming. It presents the first end-to-end verified framework implemented in the Lean theorem prover, enabling formal proofs of parameterized equivalence, equisatisfiability, and symmetry-breaking correctness for entire families of problems. The approach combines general, parameterized proofs with instance-level certificate checking, thereby eliminating the need to trust external solvers. Verified certificates are produced via backend transformations, and a single high-level proof suffices for arbitrarily large instances. This methodology achieves dramatic search-space reductions—up to a factor of twenty million—and enables full verification of the largest instances in just a few minutes.

While session types with arithmetic refinements enable precise specification of concurrent communication protocols, their heavy annotation burden has hindered practical adoption. This work proposes the first practical type inference algorithm for such refined session types, leveraging a semantic subtyping theory based on type simulation, combined with a constraint generation mechanism and the Z3 SMT solver to automatically infer both types and associated arithmetic constraints. We introduce three key optimizations that substantially improve inference efficiency. A fully implemented inference engine in the Rast language demonstrates the approach’s effectiveness across six complex benchmarks—including unary/binary natural numbers and the linear λ-calculus—significantly reducing solving times and establishing the practical feasibility of refined session type inference.

This work presents the first complete formal verification in Lean 4 of the informal Euclidean domain algorithms originally described in the 1986 ICON language. By separating concerns into mathematical definitions, computable implementations, and output formatting, the project constructs a computable mirror atop Mathlib’s `EuclideanDomain` hierarchy and integrates a regression testing infrastructure to reproduce the original outputs. All 14 algorithms are formally specified, with core procedures such as integer GCD and the extended Euclidean algorithm accompanied by machine-checked correctness proofs. The formalization precisely delineates the boundaries between computability and mathematical correctness while fully replicating the benchmark results reported in Ericson’s technical report.

Existing reaction system models lack a unified semantic foundation in handling multiplicity, resource management, concurrency, and state evolution. This work proposes a unified framework based on interval structures and interval transformation systems, decoupling operational semantics into four orthogonal components: resources, generation, update, and execution strategies, thereby enabling flexible instantiation of diverse reaction system variants. The approach is the first to uniformly characterize multiple classes of reaction systems using interval structures and naturally extends to computational models such as Petri nets. By incorporating a preprocessing mechanism, it further supports quantitative reaction systems. Experimental results reproduce classical models and their variants, demonstrating the framework’s superior expressiveness and generality.

This work addresses the challenge that unrestricted combinations of nested data types and arrays often give rise to non-standard models, thereby undermining the reliability of automated reasoning. To resolve this issue, the paper proposes a carefully restricted yet sufficiently expressive theory of nested data types, which systematically eliminates non-standard models by constraining how such types may interact with arrays. Building upon this theoretical foundation, the authors design and implement a decision procedure that integrates SMT-solving techniques. The effectiveness and practicality of the approach are demonstrated through evaluations on both real-world and synthetic benchmarks, establishing a robust basis for verifying programs involving complex nested structures.

This work addresses the limitations of existing SMT-based program verification tools, which suffer from insufficient expressiveness and low solver trustworthiness. To overcome these challenges, the authors propose FLEX—the first end-to-end foundational Constrained Horn Clause (CHC) solver implemented entirely within Lean. FLEX encodes CHCs as propositions verifiable by Lean’s trusted kernel and integrates metaprogramming tactics with Floyd-Hoare semantics to enable composable strategies for verification condition generation and solving. By leveraging Lean’s expressive logic, this approach transcends the representational constraints of SMT solvers and supports correctness proofs for arbitrary functions within Lean’s rich ecosystem. Evaluated on the FLUX benchmark, FLEX automatically solves 95.7% of CHC problems and successfully verifies multiple low-level Rust libraries.

This work addresses the challenge of synthesizing modular specifications for client programs that use complex libraries—specifications that must simultaneously satisfy formal verification requirements and be amenable to empirical testing. To this end, the paper introduces a novel form of specification called *contextual contracts* and develops a counterexample-guided learning framework that integrates constraint solving with black-box testing. The approach leverages an ICE-learning-based generalization engine within a CHC (Constrained Horn Clauses) solver to co-synthesize concise, verifiable, and testable contracts for library methods along with inductive invariants. Experimental evaluation demonstrates that the implemented tool, vmtlc, successfully generates effective contracts meeting both verification and testing criteria on real-world client benchmarks.