Modular Decomposition of Hierarchical Finite State Machines
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In this paper we develop an analogue of the graph-theoretic `modular decomposition' in automata theory. This decomposition allows us to identify hierarchical finite state machines (HFSMs) equivalent to a given finite state machine (FSM). We provide a definition of a module in an FSM, which is a collection of nodes which can be treated as a nested FSM. We identify a well-behaved subset of FSM modules called thin modules, and represent these using a linear-space directed graph we call a decomposition tree. We prove that every FSM has a unique decomposition tree which uniquely stores each thin module. We provide an O(n^2k) algorithm for finding the decomposition tree of an n-state k-alphabet FSM. The decomposition tree allows us to extend FSMs to equivalent HFSMs. For thin HFSMs, which are those where each nested FSM is a thin module, we can construct an equivalent maximally-hierarchical HFSM in polynomial time.
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