[Paper Review] On the Containment Problem for Unambiguous Single-Register Automata with Guessing

Antoine Mottet, Karin Quaas|arXiv (Cornell University)|May 28, 2019

semigroups and automata theory8 references4 citations

TL;DR

This paper establishes the decidability of the containment problem for unambiguous single-register register automata with guessing (GURA), proving that given a general GURA 𝒜 and an unambiguous single-register GURA 𝒷, the language containment L(𝒜) ⊆ L(𝒷) is decidable in EXPSPACE. The key insight is leveraging structural properties of unambiguous automata and a collapse mechanism on synchronized configurations to bound state space exploration.

ABSTRACT

Register automata extend classical finite automata with a finite set of registers that can store data from an infinite data domain for later equality comparisons with data from an input data word. While the registers in the original model of register automata, introduced in 1994 by Kaminski and Francez, can only store data occurring in the data word processed so far, we study here the more expressive class of register automata with guessing, where registers can nondeterministically take any value from the infinite data domain, even if this data does not occur in the input data word. It is well known that the containment problem, i.e., the problem of deciding for two given register automata with guessing A and B, whether the language L(A) accepted by A is contained in the language L(B) accepted by B, is undecidable, even if B only uses a single register. We prove that the problem is decidable if B is unambiguous and uses a single register.

Motivation & Objective

• To resolve the decidability of the containment problem L(𝒜) ⊆ L(𝒷) for register automata with guessing, where 𝒷 is unambiguous and uses a single register.
• To address a central open problem in formal verification of data languages, particularly concerning closure properties and containment in expressive automata models.
• To extend the frontier of decidable problems in register automata by identifying a non-trivial decidable fragment despite the general undecidability of containment in GURA.
• To provide a complexity upper bound (EXPSPACE) for the containment problem under unambiguity and single-register constraints.

Proposed method

• Introduces the concept of synchronized configurations to track joint behavior of two automata 𝒜 and 𝒷 during containment checking.
• Defines 'essentially coverable' configurations to capture reachable states that can be simulated by some data word, enabling bounded analysis.
• Applies a collapse mechanism to reduce configurations by merging data values under specific conditions, preserving reachability properties.
• Uses partial isomorphisms to simulate data value swaps, ensuring that coverability is preserved under structural transformations.
• Constructs a finite graph of maximally collapsed configurations, whose size is bounded by 2^{k log(k) 2^{|ℒ|}} for k registers and |ℒ| locations.
• Reduces the containment problem to a reachability problem in this finite graph, which is decidable in exponential space.

Experimental results

Research questions

• RQ1 Is the containment problem decidable for unambiguous single-register register automata with guessing, despite its undecidability in the general case?
• RQ2 Can the structural constraints of unambiguity and single-register usage enable a complexity-bound solution to the containment problem?
• RQ3 What is the precise complexity class of the containment problem under these restrictions?
• RQ4 Can a finite abstraction of configurations be constructed that preserves reachability and allows decidability?

Key findings

• The containment problem L(𝒜) ⊆ L(𝒷) is decidable when 𝒷 is unambiguous and uses a single register, even if 𝒜 is arbitrary.
• The problem is in EXPSPACE, establishing a tight complexity upper bound for this decidable fragment.
• The number of maximally collapsed configurations is asymptotically bounded by 2^{k log(k) 2^{|ℒ|}}, which is doubly exponential in the number of locations.
• The collapse mechanism ensures that reachability to a bad configuration is preserved through data value abstraction, enabling finite-state analysis.
• The proof relies on a novel characterization of coverability using partial isomorphisms and structural invariants on configurations.
• The result provides a positive answer to a long-standing open problem on the closure properties of unambiguous GURA, particularly regarding complementation.