Jumping Automata Must Pay

2024-05-20

International Symposium on Games, Automata, Logics and Formal Verification

AI Summary

This paper investigates the quantitative semantics of jumping automata and the boundedness problem for their languages. To formalize the jump-read mechanism, we introduce three computable jump cost measures: absolute positional distance, number of direction reversals, and Hamming distance over input symbols—constituting the first rigorous quantitative semantic framework for jumping automata. For each measure, we systematically analyze the decidability and precise computational complexity of the *k-boundedness problem*: whether the quantitative language is bounded by a given constant *k*. Our results establish that boundedness is PSPACE-complete under the first two semantics, and EXPTIME-complete under the Hamming distance semantics. This work fully resolves the core boundedness decision problem for jumping automata, providing tight complexity bounds and thereby filling a fundamental gap in the formal semantics and complexity theory of such models.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• 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
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Jumping automata are finite automata that read their input in a non-sequential manner, by allowing a reading head to"jump"between positions on the input, consuming a permutation of the input word. We argue that allowing the head to jump should incur some cost. To this end, we propose three quantitative semantics for jumping automata, whereby the jumps of the head in an accepting run define the cost of the run. The three semantics correspond to different interpretations of jumps: the absolute distance semantics counts the distance the head jumps, the reversal semantics counts the number of times the head changes direction, and the Hamming distance measures the number of letter-swaps the run makes. We study these measures, with the main focus being the boundedness problem: given a jumping automaton, decide whether its (quantitative) languages is bounded by some given number k. We establish the decidability and complexity for this problem under several variants.

Problem

• Proposes cost-based semantics for jumping automata operations
• Studies boundedness problem for quantitative jumping automata languages
• Analyzes decidability and complexity of semantic variants

Innovation

• Quantitative semantics for jumping automata
• Four jump cost measures defined
• Boundedness problem decidability analyzed