[Paper Review] Invariant stream generators using automatic abstract transformers based on a decidable logic

Pierre-Loïc Garoche, Temesghen Kahsai+1 more|arXiv (Cornell University)|May 16, 2012

Formal Methods in Verification16 references3 citations

TL;DR

This paper presents a fully automatic method to generate abstract interpreters using SMT solvers and decidable logic, enabling on-the-fly invariant discovery during fixpoint computation. By encoding transition systems in a decidable logic and leveraging abstract domain representations, the approach produces sound numerical invariants incrementally—before convergence—significantly improving efficiency in program analysis and model checking workflows.

ABSTRACT

The use of formal analysis tools on models or source code often requires the availability of auxiliary invariants about the studied system. Abstract interpretation is currently one of the best approaches to discover useful invariants, especially numerical ones. However, its application is limited by two orthogonal issues: (i) developing an abstract interpretation is often non-trivial; each transfer function of the system has to be represented at the abstract level, depending on the abstract domain used; (ii) with precise but costly abstract domains, the information computed by the abstract interpreter can be used only once a post fix point has been reached; something that may take a long time for very large system analysis or with delayed widening to improve precision. This paper proposes a new, completely automatic, method to build abstract interpreters. One of its nice features is that its produced interpreters can provide sound invariants of the analyzed system before reaching the end of the post fix point computation, and so act as on-the-fly invariant generators.

Motivation & Objective

• To address the challenge of manually constructing abstract interpreters, which is time-consuming and error-prone.
• To overcome the delay in obtaining useful invariants in abstract interpretation, which typically requires full fixpoint convergence.
• To enable incremental invariant generation during fixpoint computation, providing early, sound invariants for downstream analysis.
• To integrate abstract interpretation with logic-based model checking via SMT solvers, enhancing precision and automation.
• To support a wide range of programming languages and abstract domains without requiring language-specific transformer definitions.

Proposed method

• The method encodes the transition system of a program in a decidable logic, such as those used in SMT solvers.
• It uses SMT solvers to automatically compute abstract transformers for a chosen abstract domain, eliminating manual derivation.
• The abstract interpreter is constructed iteratively, with invariants extracted at each step of the fixpoint computation.
• The framework supports both forward and backward semantics, and can be applied to least or greatest fixpoint computations.
• Abstract domain elements are effectively representable in the logic, enabling sound and efficient abstraction.
• The approach supports big-step semantics on the logical side, preserving precision compared to small-step abstractions.

Experimental results

Research questions

• RQ1 Can abstract interpreters be generated automatically using SMT solvers and decidable logic?
• RQ2 Can useful invariants be extracted during fixpoint computation, before convergence is reached?
• RQ3 How does on-the-fly invariant generation improve performance in model checking workflows?
• RQ4 To what extent can this method be generalized across different programming languages and abstract domains?
• RQ5 Can this approach reduce the need for manual annotations in program analysis tools?

Key findings

• The method enables the generation of sound numerical invariants incrementally, even before the fixpoint is fully computed.
• In a case study with a 10,000-inductive property, the system produced a critical invariant (y < n² ⇒ x ≤ y) by the fourth iteration, enabling instant proof with k-induction.
• The prototype implementation successfully generated invariants on-the-fly, demonstrating feasibility and performance gains in analysis.
• The approach avoids precision loss associated with small-step semantics by supporting big-step logical encodings.
• The method is fully automatic and requires no manual derivation of abstract transformers for each language construct.
• The framework is general and can be applied to any system expressible in a decidable logic with a sound SMT solver.