Problem
Research questions and friction points this paper is trying to address.
polyhedral invariants
inductive invariants
linear systems
undecidability
unreachability
The existence of polyhedral invariants is undecidable for linear systems
🤖 AI Summary
This study addresses the decidability of the existence of polyhedral inductive invariants for verifying the unreachability of control locations in programs based on integer or rational linear arithmetic. Employing computability theory, this work establishes, via a rigorous reduction from two-counter machines, that the existence of such polyhedral inductive invariants is algorithmically undecidable—a result proven here for the first time. This contribution fills a theoretical gap in the formal verification of linear systems and delineates the fundamental limits of automated polyhedral invariant synthesis. By defining clear theoretical boundaries for subsequent research, these findings also provide principled guidance for the design and development of related verification tools.
📝 Abstract
The existence of polyhedral inductive invariants suitable for proving that a given control location is unreachable is undecidable for programs using only linear arithmetic over Z or Q, by reduction from 2-counter machines.
Innovation
Methods, ideas, or system contributions that make the work stand out.
polyhedral invariants
undecidability
linear arithmetic
inductive invariants
2-counter machines
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