A Coalgebraic View of ε-Transitions
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In automata theory, a machine transitions from one state to the next when it reads an input symbol. It is common to also allow an automaton to transition without input, via an ε-transition. These ε-transitions are convenient, e.g., when one defines the composition of automata. However, they are not necessary, and can be eliminated. Such ε-elimination procedures have been studied separately for different types of automata, including non-deterministic and weighted automata.
It has been noted by Hasuo that it is possible to give a coalgebraic account of ε-elimination for some automata using trace semantics (as defined by Hasuo, Jacobs and Sokolova).
In this paper, we give a detailed description of the ε-elimination procedure via trace semantics (missing in the literature). We apply this framework to several types of automata, and explore its boundary.
In particular, we show that is possible (by careful choice of a monad) to define an ε-removal procedure for all weighted automata over the positive reals (and certain other semirings). Our definition extends the recent proposals by Sakarovitch and Lombardy for these semirings.
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