[Paper Review] Linear-Time Verification of Data-Aware Dynamic Systems with Arithmetic

Paolo Felli, Marco Montali+1 more|arXiv (Cornell University)|Mar 15, 2022

Formal Methods in Verification4 citations

TL;DR

This paper introduces the finite summary property for data-aware dynamic systems with linear arithmetic, enabling decidable linear-time verification via finite-state abstraction. It proves that systems with finite summary allow efficient LTLf model checking using SMT solvers, generalizing prior decidability conditions and enabling modular verification of complex systems with arithmetic constraints.

ABSTRACT

Combined modeling and verification of dynamic systems and the data they operate on has gained momentum in AI and in several application domains. We investigate the expressive yet concise framework of data-aware dynamic systems (DDS), extending it with linear arithmetic, and provide the following contributions. First, we introduce a new, semantic property of "finite summary", which guarantees the existence of a faithful finite-state abstraction. We rely on this to show that checking whether a witness exists for a linear-time, finite-trace property is decidable for DDSs with finite summary. Second, we demonstrate that several decidability conditions studied in formal methods and database theory can be seen as concrete, checkable instances of this property. This also gives rise to new decidability results. Third, we show how the abstract, uniform property of finite summary leads to modularity results: a system enjoys finite summary if it can be partitioned appropriately into smaller systems that possess the property. Our results allow us to analyze systems that were out of reach in earlier approaches. Finally, we demonstrate the feasibility of our approach in a prototype implementation.

Motivation & Objective

• To address the challenge of verifying dynamic systems with arithmetic constraints, which are typically undecidable due to infinite state spaces.
• To develop a general, semantic property—finite summary—that guarantees the existence of a finite-state abstraction for such systems.
• To unify and generalize existing decidability conditions (e.g., monotonicity, gap-order, feedback freedom) under a single, checkable criterion.
• To enable modular verification of composite systems by showing that finite summary is preserved under sequential and variable-disjoint parallel composition.
• To demonstrate feasibility through a prototype tool, ada, that supports finite summary checking, abstraction, and LTLf model checking with SMT backends.

Proposed method

• Introduces the finite summary property as a semantic condition ensuring that the set of reachable states can be faithfully represented by a finite constraint-based abstraction.
• Uses logical constraint systems to represent state sets and constructs finite-state abstractions that preserve the truth of LTLf properties.
• Employs SMT solvers as a backend for model checking, enabling efficient witness search and satisfiability checking for LTLf formulas.
• Defines and checks four concrete instances of finite summary: monotonicity constraints, gap-order constraints, feedback freedom, and bounded lookback.
• Applies a modularity theorem showing that sequential or variable-disjoint parallel compositions of systems with finite summary also enjoy the property.
• Implements the approach in the tool ada, which tests for finite summary, computes abstractions, and performs LTLf verification via SMT solving.

Experimental results

Research questions

• RQ1 Can a general, checkable semantic condition be identified that ensures decidability of LTLf verification in data-aware dynamic systems with arithmetic?
• RQ2 How do existing decidability conditions—such as monotonicity constraints, gap-order constraints, and feedback freedom—relate to a unifying property?
• RQ3 Can the finite summary property be preserved under system composition, enabling modular verification of complex systems?
• RQ4 Is it possible to build an efficient, practical verification framework for systems with arithmetic using finite summary and SMT solving?
• RQ5 What is the empirical feasibility of the approach on real-world systems modeled as data-aware dynamic systems with arithmetic?

Key findings

• The finite summary property guarantees the existence of a faithful finite-state abstraction for DDSAs, enabling decidable LTLf model checking.
• Monotonicity constraints, gap-order constraints, feedback freedom, and bounded lookback are all instances of the finite summary property and can be checked algorithmically.
• Modular composition of DDSAs with finite summary—either sequentially or in parallel with disjoint variables—preserves the finite summary property.
• The prototype tool ada successfully verifies 43 properties across 14 real-world systems, including auction, road fine, and hospital billing processes, with analysis times ranging from 1 to 27 seconds.
• Unsatisfiability of negated LTLf formulas (e.g., ψ₁₅, ψ₂₆) confirms the absence of unwanted behaviors, while satisfiability of witness formulas (e.g., ψ₁₂, ψ₂₄) confirms the existence of desired behaviors.
• Performance varies with system complexity, with larger summary sets (e.g., in Bₕ) leading to longer analysis times, indicating room for optimization in abstraction construction.