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Register a new userTransition Complexity of Incomplete DFAs
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EPTCS
Publication date
2010
fields
Informatics Engineering
and research's language is
English
Authors
Yuan Gao
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In this paper, we consider the transition complexity of regular languages based on the incomplete deterministic finite automata. A number of results on Boolean operations have been obtained. It is shown that the transition complexity results for union and complementation are very different from the state complexity results for the same operations. However, for intersection, the transition complexity result is similar to that of state complexity.
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We compute all synchronizing DFAs with 7 states and synchronization length >= 29. Furthermore, we compute alphabet size ranges for maximal, minimal and semi-minimal synchronizing DFAs with up to 7 states.
We prove the following theorem. Suppose that $M$ is a trim DFA on the Boolean alphabet $0,1$. The language $L(M)$ is well-ordered by the lexicographic order $slex$ iff whenever the non sink states $q,q.0$ are in the same strong component, then $q.1$ is a sink. It is easy to see that this property is sufficient. In order to show the necessity, we analyze the behavior of a $slex$-descending sequence of words. This property is used to obtain a polynomial time algorithm to determine, given a DFA $M$, whether $L(M)$ is well-ordered by the lexicographic order. Last, we apply an argument in cite{BE,BEa} to give a proof that the least nonregular ordinal is $omega^omega $.
This paper contains results which arose from the research which led to arXiv:1801.10436, but which did not fit in arXiv:1801.10436. So arXiv:1801.10436 contains the highlight results, but there are more results which are interesting enough to be shared.
We consider Markov decision processes (MDP) as generators of sequences of probability distributions over states. A probability distribution is p-synchronizing if the probability mass is at least p in a single state, or in a given set of states. We consider four temporal synchronizing modes: a sequence of probability distributions is always p-synchronizing, eventually p-synchronizing, weakly p-synchronizing, or strongly p-synchronizing if, respectively, all, some, infinitely many, or all but finitely many distributions in the sequence are p-synchronizing. For each synchronizing mode, an MDP can be (i) sure winning if there is a strategy that produces a 1-synchronizing sequence; (ii) almost-sure winning if there is a strategy that produces a sequence that is, for all epsilon > 0, a (1-epsilon)-synchronizing sequence; (iii) limit-sure winning if for all epsilon > 0, there is a strategy that produces a (1-epsilon)-synchronizing sequence. We provide fundamental results on the expressiveness, decidability, and complexity of synchronizing properties for MDPs. For each synchronizing mode, we consider the problem of deciding whether an MDP is sure, almost-sure, or limit-sure winning, and we establish matching upper and lower complexity bounds of the problems: for all winning modes, we show that the problems are PSPACE-complete for eventually and weakly synchronizing, and PTIME-complete for always and strongly synchronizing. We establish the memory requirement for winning strategies, and we show that all winning modes coincide for always synchronizing, and that the almost-sure and limit-sure winning modes coincide for weakly and strongly synchronizing.
We revisit the complexity of procedures on SFAs (such as intersection, emptiness, etc.) and analyze them according to the measures we find suitable for symbolic automata: the number of states, the maximal number of transitions exiting a state, and the size of the most complex transition predicate. We pay attention to the special forms of SFAs: {normalized SFAs} and {neat SFAs}, as well as to SFAs over a {monotonic} effective Boolean algebra.
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