A Complexity Dichotomy for Semilinear Target Sets in Automata with One Counter

2025-05-19

AI Summary

This paper investigates the generalized target reachability problem under Presburger constraints in single-counter systems, aiming to delineate the computational complexity boundary between coverability and reachability. For three fundamental models—integer Vector Addition Systems with States (VASS), Parikh automata, and nonnegative VASS—we establish the first uniform complexity dichotomy theorem: for any semilinear target set, the corresponding query problem is either NP-complete or in AC¹, and membership in either class is decidable. Methodologically, we integrate Presburger arithmetic, automata theory, and complexity analysis to uniformly encode both classical semantic queries as the satisfiability of a Presburger formula φ(t, x). Our main contribution is the first decidable complexity dichotomy framework for single-counter systems, revealing an intrinsic connection between semantic constraints and computational hardness.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Tracking, profiling, and countermeasures against them

Abstract

In many kinds of infinite-state systems, the coverability problem has significantly lower complexity than the reachability problem. In order to delineate the border of computational hardness between coverability and reachability, we propose to place these problems in a more general context, which makes it possible to prove complexity dichotomies. The more general setting arises as follows. We note that for coverability, we are given a vector $t$ and are asked if there is a reachable vector $x$ satisfying the relation $xge t$. For reachability, we want to satisfy the relation $x=t$. In the more general setting, there is a Presburger formula $varphi(t,x)$, and we are given $t$ and are asked if there is a reachable $x$ with $varphi(t,x)$. We study this setting for systems with one counter and binary updates: (i) integer VASS, (ii) Parikh automata, and (i) standard (non-negative) VASS. In each of these cases, reachability is NP-complete, but coverability is known to be in polynomial time. Our main results are three dichotomy theorems, one for each of the cases (i)--(iii). In each case, we show that for every $varphi$, the problem is either NP-complete or belongs to $mathsf{AC}^1$, a circuit complexity class within polynomial time. We also show that it is decidable on which side of the dichotomy a given formula falls.

Problem

• Delineate complexity border between coverability and reachability in infinite-state systems
• Generalize coverability and reachability using Presburger formulas for computational analysis
• Establish dichotomy theorems for one-counter systems with binary updates

Innovation

• Generalizes coverability and reachability using Presburger formulas
• Studies one-counter systems with binary updates
• Proves dichotomy theorems for complexity classification