Efficient Minimization of DFAs with Partial Transition Functions

Let PT-DFA mean a deterministic finite automaton whose transition relation is a partial function. We present an algorithm for minimizing a PT-DFA in $O(m \lg n)$ time and $O(m+n+\alpha)$ memory, where $n$ is the number of states, $m$ is the number of defined transitions, and $\alpha$ is the size of the alphabet. Time consumption does not depend on $\alpha$, because the $\alpha$ term arises from an array that is accessed at random and never initialized. It is not needed, if transitions are in a suitable order in the input. The algorithm uses two instances of an array-based data structure for maintaining a refinable partition. Its operations are all amortized constant time. One instance represents the classical blocks and the other a partition of transitions. Our measurements demonstrate the speed advantage of our algorithm on PT-DFAs over an $O(\alpha n \lg n)$ time, $O(\alpha n)$ memory algorithm

On one-way cellular automata with a fixed number of cells

We investigate a restricted one-way cellular automaton (OCA) model where the number of cells is bounded by a constant number k, so-called kC-OCAs. In contrast to the general model, the generative capacity of the restricted model is reduced to the set of regular languages. A kC-OCA can be algorithmically converted to a deterministic finite automaton (DFA). The blow-up in the number of states is bounded by a polynomial of degree k. We can exhibit a family of unary languages which shows that this upper bound is tight in order of magnitude. We then study upper and lower bounds for the trade-off when converting DFAs to kC-OCAs. We show that there are regular languages where the use of kC-OCAs cannot reduce the number of states when compared to DFAs. We then investigate trade-offs between kC-OCAs with different numbers of cells and finally treat the problem of minimizing a given kC-OCA

Small NFAs from Regular Expressions: Some Experimental Results

Regular expressions (res), because of their succinctness and clear syntax, are the common choice to represent regular languages. However, efficient pattern matching or word recognition depend on the size of the equivalent nondeterministic finite automata (NFA). We present the implementation of several algorithms for constructing small epsilon-free NFAss from res within the FAdo system, and a comparison of regular expression measures and NFA sizes based on experimental results obtained from uniform random generated res. For this analysis, nonredundant res and reduced res in star normal form were considered.Comment: Proceedings of 6th Conference on Computability in Europe (CIE 2010), pages 194-203, Ponta Delgada, Azores, Portugal, June/July 201

On minimizing deterministic tree automata

We present two algorithms for minimizing deterministic frontier-to-root tree automata (dfrtas) and compare them with their string counterparts. The presentation is incremental, starting out from definitions of minimality of automata and state equivalence, in the style of earlier algorithm taxonomies by the authors. The first algorithm is the classical one, initially presented by Brainerd in the 1960s and presented (sometimes imprecisely) in standard texts on tree language theory ever since. The second algorithm is completely new. This algorithm, essentially representing the generalization to ranked trees of the string algorithm presented by Watson and Daciuk, incrementally minimizes a dfrta. As a result, intermediate results of the algorithm can be used to reduce the initial automaton’s size. This makes the algorithm useful in situations where running time is restricted (for example, in real-time applications). We also briefly sketch how a concurrent specification of the algorithm in CSP can be obtained from an existing specification for the dfa case

On minimizing deterministic tree automata

We present two algorithms for minimizing deterministic frontier-to-root tree automata (dfrtas) and compare them with their string counterparts. The presentation is incremental, starting out from definitions of minimality of automata and state equivalence, in the style of earlier algorithm taxonomies by the authors. The first algorithm is the classical one, initially presented by Brainerd in the 1960s and presented (sometimes imprecisely) in standard texts on tree language theory ever since. The second algorithm is completely new. This algorithm, essentially representing the generalization to ranked trees of the string algorithm presented by Watson and Daciuk, incrementally minimizes a dfrta. As a result, intermediate results of the algorithm can be used to reduce the initial automaton’s size. This makes the algorithm useful in situations where running time is restricted (for example, in real-time applications). We also briefly sketch how a concurrent specification of the algorithm in CSP can be obtained from an existing specification for the dfa case