Classifications of Recognizable Infinitary Trace Languages and the Distributed Synthesis Problem

The classification of recognizable ω-word languages into Borel levels is the basis of many specialized solutions in the fields of formal verification and algorithmic controller synthesis. Each of these levels is characterized by a class of deterministic ω-automata, namely deterministic weak (reachability and safety), deterministic Büchi, and deterministic Muller automata. This thesis analyses the more general setting of infinitary Mazurkiewicz traces, which model nonterminating, concurrent computation of distributed systems. The study of finitary trace-languages generalizes that of word-languages, and numerous results have been extended to infinitary trace-languages (due to Gastin, Petit, Diekert, Muscholl, and others). However, the current definitions of asynchronous ω-automata fail to yield a classification of ω-trace languages that is compatible with the Borel hierarchy. We close this gap, which had been open since the 1990’s, by introducing the family of “synchronization aware” automata. We also demonstrate the semi-decidability of the problem to determine whether a given recognizable ω-trace language is also recognized by a deterministic synchronization aware Büchi automaton. Although asynchronous automata are useful in implementing distributed monitors and distributed controllers, their constructions are prohibitively expensive even by automata-theoretic standards. On the other hand, “linearizations” of infinitary trace languages, which invoke the framework of “trace-closed” ω-word languages, can find immediate applications in model checking and formal verification of distributed systems. This is because word automata recognizing trace-closed languages support more efficient analyses of most of the interesting properties pertaining to distributed computations. In this setting, we present another classification of ω-trace languages in terms of a Borel-like hierarchy of trace-closed ω-word languages. Finally, we introduce a distributed version of Church’s synthesis problem and compare it with two well known variants of distributed controller synthesis, viz. action-based control (due to Gastin, Lerman, and Zeitoun) and process-based control (due to Madhusudan, Thiagarajan, and Yang). While the algorithmic solution of these problems remains an open area of investigation, we build upon the work of Muscholl, Walukiewicz, and Zeitoun, and compare their problem classes with variants of distributed Church’s synthesis problem by demonstrating suitable reductions in the sense that a solution of any one problem can be converted into a solution of the other.