Some Theoretical Results on Parallel Automata, Conflict, Complexity

The purpose of this work is to classify parallel automata conflicts, to develop techniques for handling conflicts a-priori and a-posteriori, and to analyze the complexity of conflicts and other problems concerning parallel automata. Synchronous parallel automata having inputs and outputs, and which are useful for designing real time systems will be discussed. A-priori conflict detection for parallel automata is shown to be P-space complete. In view of this, an approach based on potential conflicts, is developed. The complexity of detecting potential conflicts is shown to be NP complete, but if automata conditions are in disjunctive normal form, potential conflict detection is possible in polynomial time. We show that any parallel automaton can be converted in polynomial time into a nearly equivalent automaton with no potential conflicts, with size proportional to the size of the original automaton. The new automaton is equivalent to the original automaton, when the original automaton is free of conflicts. The complexity of liveness, looping and deadlock in parallel automata is discussed. Apart from parallel execution, rules are given ensuring well-defined execution in any sequential order. Some methods of handling conflicts a-posteriori at run time are discussed. Prioritized execution is discussed and concerning prioritized execution, conversion of a parallel automaton into a form where all conditions are conjunctions of primitive conditions is possible in polynomial time. The representation of parallel automata and conflict in the predicate calculus is discussed. Certain extensions to abstract parallel automata are also briefly discussed.