DFA Minimization Research Articles

An Evaluation of Massively Parallel Algorithms for DFA Minimization

An Evaluation of Massively Parallel Algorithms for DFA Minimization

DFA minimization: Double reversal versus split minimization algorithms

In this paper, we show the relationship between the two most widely used approaches for the minimization of deterministic finite automata: minimization by split of partitions and minimization by double reversal. Even though the double reversal approach has usually been considered to be unconventional with respect to the more common split approach, we show that any double reversal minimization algorithm can be related to a split minimization algorithm and vice versa.

EFFICIENT DETERMINISTIC FINITE AUTOMATA SPLIT-MINIMIZATION DERIVED FROM BRZOZOWSKI'S ALGORITHM

Minimization of deterministic finite automata is a classic problem in Computer Science which is still studied nowadays. In this paper, we relate the different split-minimization methods proposed to date, or to be proposed, and the algorithm due to Brzozowski which has been usually set aside in any classification of DFA minimization algorithms. In our work, we first propose a polynomial minimization method derived from a paper by Champarnaud et al. We also show how the consideration of some efficiency improvements on this algorithm lead to obtain an algorithm similar to Hopcroft's classic algorithm. The results obtained lead us to propose a characterization of the set of possible splitters.

Minimization of symbolic automata

Symbolic Automata extend classical automata by using symbolic alphabets instead of finite ones. Most of the classical automata algorithms rely on the alphabet being finite, and generalizing them to the symbolic setting is not a trivial task. In this paper we study the problem of minimizing symbolic automata. We formally define and prove the basic properties of minimality in the symbolic setting, and lift classical minimization algorithms (Huffman-Moore's and Hopcroft's algorithms) to symbolic automata. While Hopcroft's algorithm is the fastest known algorithm for DFA minimization, we show how, in the presence of symbolic alphabets, it can incur an exponential blowup. To address this issue, we introduce a new algorithm that fully benefits from the symbolic representation of the alphabet and does not suffer from the exponential blowup. We provide comprehensive performance evaluation of all the algorithms over large benchmarks and against existing state-of-the-art implementations. The experiments show how the new symbolic algorithm is faster than previous implementations.

Compositional Analysis of Dynamic Bayesian Networks and Applications to Complex Dynamic System Decomposition

Dynamic Bayesian networks (DBNs) can be effectively used to model various problems in complex dynamic systems. We perform an empirical investigation on compositional analysis of DBNs using abstraction. In static systems and hidden Markov models, computation of a metric called treewidth induces a tree decomposition that can be used to perform logical or probabilistic inference and max+optimizations in time exponential in treewidth and linear in overall system size. Intuitively, the linear scaling means that very large systems can be analyzed as long as they are sufficiently sparse and well structured. In these simple cases, summary propagation, which uses two operations, summation (projection) and product (composition), suffices to perform the inference or optimization. Here, we begin an extension of this to structured networks of communicating dynamic systems.We define generalizations of projection and composition operators that treat labeled Markov chains as primitive objects. The projection operation, corresponding to summation, is implemented as label deletion followed by exact state reduction for Markov chains, similar to Hopcroft's DFA minimization algorithm, with O(nlogm) complexity. The composition operation is the product of state machines. We use canonical MDDs, similar to BDDs, to capture logical dependencies symbolically. Combining symbolic representations with Markov chain lumping algorithms is a novel contribution. Using this approach, we have created a tool lever- aging model based systems engineering technologies. The communicating Markov chains are specified using UML Statecharts via Papyrus extended using an ANTLR parsed domain specific language (DSL).The tool reduces the number of states in networks of Markov chains by several orders of magnitude. In one example, a network having a product state space of more than 600 million states is reduced to about 500 states. A feature of this technique is that the state space is examined incrementally, meaning that the full state space is never explicitly represented, even as an input to the reduction algorithm. The primary reduction appears to come from symmetry which is surprising because the technique includes no explicit symmetry handling. We note that composition is efficient at least for systems with high symmetry. We describe applications to a hospital intensive care unit (ICU) from a systems engineering perspective.

On size reduction techniques for multitape automata

We present a method for size reduction of two-way multitape automata. Our algorithm applies local transformations that change the order in which transitions concerning different tapes occur in the automaton graph, and merge suitable states into a single state. Our work is motivated by implementation of a language for string manipulation in database systems where string predicates are compiled into two-way multitape automata. Additionally, we present a (one-tape) NFA reduction algorithm that is based on a method proposed for DFA minimization by Kameda and Weiner, and apply this algorithm, combined with the multitape automata reduction algorithm, on our multitape automata. Empirical results on the performance of our method when applied on some multitape automata originating from string predicates are reported.

Weak minimization of DFA—an algorithm and applications

DFA minimization is an important problem in algorithm design and is based on the notion of DFA equivalence: Two DFA's are equivalent if and only if they accept the same set of strings. In this paper, we propose a new notion of DFA equivalence (that we call weak-equivalence): We say that two DFA's are weakly equivalent if they both accept the same number of strings of length k for every k. The motivation for this problem is as follows. A large number of counting problems can be solved by encoding the combinatorial objects we want to count as strings over a finite alphabet. If the collection of encoded strings is accepted by a DFA, then standard algorithms from computational linear algebra can be used to solve the counting problem efficiently. When applying this approach to large-scale applications, the bottleneck is the space complexity since the computation involves a matrix of order k × k if k is the size of the underlying DFA M. This leads to the natural question: Is there a smaller DFA that is weakly equivalent to M? We present an algorithm of time complexity O ( k 3 ) to find a compact DFA weakly equivalent to a given DFA. We illustrate, in the case of a tiling problem, that our algorithm reduces a (strongly minimal) DFA by a factor close to 1/2.

How to squeeze a lexicon

AbstractMinimal acyclic deterministic finite automata (ADFAs) can be used as a compact representation of finite string sets with fast access time. Creating them with traditional algorithms of DFA minimization is resource greedy when a large collection of strings is involved. This paper aims to popularize an efficient but little‐known algorithm for creating minimal ADFAs recognizing a finite language, invented independently by several authors. The algorithm is presented for three variants of ADFAs, its minor improvements are discussed, and minimal ADFAs are compared to competitive data structures. Copyright © 2001 John Wiley & Sons, Ltd.

Program derivation with verified transformations — a case study

AbstractA program development methodology based on verified program transformations is described and illustrated through derivations of a high level bisimulation algorithm and an improved minimum‐state DFA algorithm. Certain doubts that were raised about the correctness of an initial paper‐and‐pencil derivation of the DFA minimization algorithm were laid to rest by machine‐checked formal proofs of the most difficult derivational steps. Although the protracted labor involved in designing and checking these proofs was almost overwhelming, the expense was somewhat offset by a successful reuse of major portions of these proofs. In particular, the DFA minimization algorithm is obtained by specializing and then extending the last step in the derivation of the high level bisimulation algorithm.Our experience suggests that a major focus of future research should be aimed towards improving the technology of machine checkable proofs — their construction, presentation, and reuse. This paper demonstrates the importance of such a technology to the verification of programs and program transformations. We believe that the utility of transformational systems to program development will ultimately rest on a practical program correctness technology. © 1996 John Wiley & Sons, Inc.