Invariant stream generators using automatic abstract transformers based on a decidable logic

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.

💡 Research Summary

The paper addresses two long‑standing obstacles in the practical deployment of abstract interpretation (AI) for static analysis: (i) the manual effort required to design abstract transfer functions for each program operation, which depends heavily on the chosen abstract domain, and (ii) the fact that precise but expensive domains typically yield useful invariants only after a post‑fix‑point is reached, a process that can be prohibitively slow for large programs or when widening is delayed for precision. To overcome these issues, the authors propose a fully automatic method for constructing abstract interpreters based on a decidable logical framework.

The core idea is to encode each program statement as a formula in a decidable fragment of first‑order logic (or an SMT‑supported theory). Because satisfiability in this fragment is decidable, a solver can be used to automatically synthesize the corresponding abstract transformer. In practice, the method proceeds as follows: (1) the source program is parsed into a control‑flow graph; (2) each edge (i.e., each primitive operation) is translated into a logical constraint that captures its concrete semantics; (3) the constraint is handed to a SAT/SMT engine, which either proves that the operation is compatible with the current abstract domain or produces a concrete model that can be abstracted. The result is an automatically generated abstract transfer function that requires no human‑written code.

A distinctive contribution of the work is the “on‑the‑fly” invariant generation capability. Traditional AI computes a sequence of abstract states, but only after convergence does it extract a final invariant. In the proposed framework, after each iteration the current abstract state is kept in logical form, and a lightweight query to the solver extracts a set of constraints that are guaranteed to hold in all reachable states up to that point. These intermediate constraints constitute sound invariants even though the global fix‑point has not yet been reached. Consequently, useful safety information becomes available early, which is especially valuable for large‑scale analyses, incremental verification, or runtime monitoring.

The authors evaluate their approach on a suite of benchmarks that include linear arithmetic loops, non‑linear updates, and mixed integer‑real programs. Two dimensions are measured: (a) precision, comparing the automatically generated transformers against hand‑crafted ones for the same abstract domain, and (b) performance, measuring the time to obtain a usable invariant. Results show that the automatically generated transformers achieve at least the same precision as manual implementations, and in many cases they are more precise because the logical encoding captures subtle relationships that hand‑written abstractions may overlook. Moreover, because useful invariants are emitted before convergence, the overall analysis time drops dramatically—up to 30 % faster on complex loops—while memory consumption remains comparable.

Beyond the experimental validation, the paper discusses several theoretical guarantees. Since the underlying logic is decidable, the synthesis of each transformer is guaranteed to terminate, and the soundness of the generated invariants follows directly from the soundness of the logical abstraction and the solver’s correctness. The method is also modular: extending the approach to new abstract domains simply requires providing a logical encoding of the domain’s operations, without rewriting the whole interpreter.

In conclusion, the work presents a novel, fully automated pipeline for building abstract interpreters that (1) eliminates the manual burden of writing abstract transfer functions, (2) leverages decidable logics to guarantee termination of the synthesis process, and (3) delivers sound invariants incrementally, before the fix‑point is reached. This combination of automation and early invariant generation broadens the applicability of abstract interpretation to large, safety‑critical software systems and opens avenues for integrating AI with other formal methods such as model checking and deductive verification. Future research directions include extending the logical encoding to richer theories (e.g., non‑linear arithmetic, arrays, heap structures), coupling the on‑the‑fly invariants with runtime assertion checking, and scaling the approach to distributed or embedded environments.