Obtaining combinatorial structures associated with low-dimensional Leibniz algebras

In this paper, we analyze the relation between Leibniz algebras and combinatorial structures. More concretely, we study the properties to be satisfied by (pseudo)digraphs so that they are associated with low-dimensional Leibniz algebras. We present some results related to this association and show an algorithmic method to obtain them, which has been implemented with Maple

Modular Decomposition of Hierarchical Finite State Machines

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.Comment: 38 pages, 11 figures. Submitted to Theoretical Computer Scienc