🤖 AI Summary
This study investigates the succinctness of chained co-Büchi automata (COCOA) for representing ω-regular languages and analyzes their state complexity under Boolean operations. By constructing specific families of languages, the work compares the expressive power of COCOA with deterministic parity automata and examines the state blowup induced by conjunction, disjunction, and complementation. It is shown for the first time that COCOA can be exponentially more succinct than deterministic parity automata—even when all component automata within the chain are deterministic—yet they inevitably suffer exponential state explosion under Boolean operations and complementation. These results highlight the inherent trade-off between the compact representational capacity of COCOA and their fundamental limitations in compositional system design.
📝 Abstract
Chains of co-Büchi automata (COCOA) have recently been introduced as a new canonical model for representing arbitrary omega-regular languages. They can be minimized in polynomial time and are hence an attractive language representation for applications in which normally, deterministic omega-automata are used. While it is known how to build COCOA from deterministic parity automata, little is currently known about their relationship to automaton models introduced earlier than COCOA.
In this paper, we analyze the conciseness of chains of co-Büchi automata. We provide three main results and give an overview of the implications of these results. First of all, we show that even in the case that all automata in the chain are deterministic, chains of co-Büchi automata can be exponentially more concise than deterministic parity automata. We then present two main results that together negatively answer the question if this conciseness is retained when performing Boolean operations (such as disjunction, conjunction, and complementation) over COCOA. For the binary operations, we show that there exist families of languages for which their application leads to an exponential growth of the sizes of the automata. The families have the property that when representing them using deterministic parity automata, taking the disjunction or conjunction of the family elements only requires a polynomial blow-up. We finally show that an exponential blow-up is also unavoidable when complementing a COCOA, as this operation can require redistributing with which colors words need to be recognized.