Abstract
Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get even more compact representations of finite languages. DFCAs are extended by accepting either “do not care” symbols or considering multiple entry DFCAs. We study for each of the two models the existence of the minimization or simplification algorithms and their computational complexity, the state complexity of these representations compared with other representations of the same language, and the bounds for state complexity in case we perform a representation transformation. Minimization for both models proves to be NP-hard. A method is presented to transform minimization algorithms for deterministic automata into simplification algorithms applicable to these extended models. DFCAs with “do not care” symbols prove to have comparable state complexity as Nondeterministic Finite Cover Automata. Furthermore, for multiple entry DFCAs, we can have a tight estimate of the state complexity of the transformation into equivalent DFCA.
Highlights
Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages
We study for each of the two models the existence of the minimization or simplification algorithms and their computational complexity, the state complexity of these representations compared with other representations of the same language, and the bounds for state complexity in case we perform a representation transformation
The concept of Cover Automata was first presented at a conference paper of Câmpeanu et al at the Workshop on Implementations and Applications of Automata (WIAA) in Rouen
Summary
The number of elements of a set T is #T, an alphabet is usually denoted by Σ, and the set of words over Σ is Σ?. In the case of nondeterministic automata, an automaton is considered reduced if all its states are both reachable and useful. A cover automaton for a finite language L is a DFA recognizing a cover language L0 such that L = L0 ∩ Σ≤l , for l being the length of the longest word in L. In case of a finite language L, we can define the cover complexity variants: csc( L) = min{#Q | A = ( Q, Σ, δ, q0 , F ), deterministic, complete, and L = L( A) ∩ Σ≤l }. We have simplification algorithms that may reduce the number of states of the automaton used as input and produce an equivalent one with possibly fewer states. For these two types of automata, first, we give the new definitions, we analyze which results hold and which ones need to be adapted to the new concepts
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