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

Amorphous slicing of extended finite state machines

Slicing is useful for many Software Engineering applications and has been widely studied for three decades, but there has been comparatively little work on slicing Extended Finite State Machines (EFSMs). This paper introduces a set of dependency based EFSM slicing algorithms and an accompanying tool. We demonstrate that our algorithms are suitable for dependence based slicing. We use our tool to conduct experiments on ten EFSMs, including benchmarks and industrial EFSMs. Ours is the first empirical study of dependence based program slicing for EFSMs. Compared to the only previously published dependence based algorithm, our average slice is smaller 40% of the time and larger only 10% of the time, with an average slice size of 35% for termination insensitive slicing

NFA reduction via hypergraph vertex cover approximation

In this thesis, we study in minimum vertex cover problem on the class of k-partite k-uniform hypergraphs. This problem arises when reducing the size of nondeterministic finite automata (NFA) using preorders, as suggested by Champarnaud and Coulon. It has been shown that reducing NFAs using preorders is at least as hard as computing a minimal vertex cover on 3-partite 3-uniform hypergraphs, which is NP-hard. We present several classes of regular languages for which NFAs that recognize them can be optimally reduced via preorders. We introduce an algorithm for approximating vertex cover on k-partite k-uniform hypergraphs based on a theorem by Lovász and explore the use of fractional cover algorithms to improve the running time at the expense of a small increase in the approximation ratio

Syntactic Minimization Of Nondeterministic Finite Automata

Nondeterministic automata may be viewed as succinct programs implementing deterministic automata, i.e. complete specifications. Converting a given deterministic automaton into a small nondeterministic one is known to be computationally very hard; in fact, the ensuing decision problem is PSPACE-complete. This paper stands in stark contrast to the status quo. We restrict attention to subatomic nondeterministic automata, whose individual states accept unions of syntactic congruence classes. They are general enough to cover almost all structural results concerning nondeterministic state-minimality. We prove that converting a monoid recognizing a regular language into a small subatomic acceptor corresponds to an NP-complete problem. The NP certificates are solutions of simple equations involving relations over the syntactic monoid. We also consider the subclass of atomic nondeterministic automata introduced by Brzozowski and Tamm. Given a deterministic automaton and another one for the reversed language, computing small atomic acceptors is shown to be NP-complete with analogous certificates. Our complexity results emerge from an algebraic characterization of (sub)atomic acceptors in terms of deterministic automata with semilattice structure, combined with an equivalence of categories leading to succinct representations

On Minimality and Size Reduction of One-Tape and Multitape Finite Automata

In this thesis, we consider minimality and size reduction issues of one-tape and multitape automata. Although the topic of minimization of one-tape automata has been widely studied for many years, it seems that some issues have not gained attention. One of these issues concerns finding specific conditions on automata that imply their minimality in the class of nondeterministic finite automata (NFA) accepting the same language. Using the theory of NFA minimization developed by Kameda and Weiner in 1970, we show that any bideterministic automaton (that is, a deterministic automaton with its reversal also being deterministic) is a unique minimal automaton among all NFA accepting its language. In addition to the minimality in regard to the number of states, we also show its minimality in the number of transitions. Using the same theory of Kameda and Weiner, we also obtain a more general minimality result. We specify a set of sufficient conditions under which a minimal deterministic automaton (DFA

Combining Boolean Networks and Ordinary Differential Equations for Analysis and Comparison of Gene Regulatory Networks

This thesis is concerned with different groups of qualitative models of gene regulatory networks. Four types of models will be considered: interaction graphs, Boolean networks, models based on differential equations and discrete abstractions of differential equations. We will investigate the relations between these modeling frameworks and how they can be used in the analysis of individual models. The focus lies on the mathematical analysis of these models. This thesis makes several contributions in relating these different modeling frameworks. The first approach concerns individual Boolean models and parametrized families of ordinary differential equations (ODEs). To construct ODE models systematically from Boolean models several automatic conversion algorithms have been proposed. In Chapter 2 several such closely related algorithms will be considered. It will be proven that certain invariant sets are preserved during the conversion from a Boolean network to a model based on ODEs. In the second approach the idea of abstracting the dynamics of individual models to relate structure and dynamics will be introduced. This approach will be applied to Boolean models and models based on differential equations. This allows to compare groups of models in these modeling frameworks which have the same structure. We demonstrate that this constitutes an approach to link the interaction graph to the dynamics of certain sets of Boolean networks and models based on differential equations. The abstracted dynamics – or more precisely the restrictions on the abstracted behavior – of such sets of Boolean networks or models based on differential equations will be represented as Boolean state transitions graphs themselves. We will show that these state transition graphs can be considered as asynchronous Boolean networks. Despite the rather theoretical question this thesis tries to answer there are many potential applications of the results. The results in Chapter 2 can be applied to network reduction of ODE models based on Hill kinetics. The results of the second approach in Chapter 4 can be applied to network inference and analysis of Boolean model sets. Furthermore, in the last chapter of this thesis several ideas for applications with respect to experiment design will be considered. This leads to the question how different asynchronous Boolean networks or different behaviours of a single asynchronous Boolean network can be distinguishedDiese Arbeit beschäftigt sich mit unterschiedlichen Typen von qualitativen Modellen genregulatorischer Netzwerke. Vier Typen von Modellen werden betrachtet: Interaktionsgraphen, Boolesche Netzwerke, Modelle, die auf Differentialgleichungen basieren und diskrete Abstraktionen von Differentialgleichungen. Wir werden mehr über die Beziehungen zwischen diesen Modellgruppen lernen und wie diese Beziehungen genutzt werden können, um einzelne Modelle zu analysieren. Der Schwerpunkt liegt hierbei auf der mathematischen Analyse dieser Modellgruppen. In dieser Hinsicht leistet diese Arbeit mehrere Beiträge. Zunächst betrachten wir Boolesche Netzwerke und parametrisierte Familien von gewöhnlichen Differentialgleichungen (ODEs). Um solche ODE-Modelle systematisch aus Booleschen Modellen abzuleiten, wurden in der Vergangenheit verschiedene automatische Konvertierungsalgorithmen vorgeschlagen. In Kapitel 2 werden einige dieser Algorithmen näher untersucht. Wir werden beweisen, dass bestimmte invariante Mengen bei der Konvertierung eines Booleschen Modells in ein ODE-Modell erhalten bleiben. Der zweite Ansatz, der in dieser Arbeit verfolgt wird, beschäftigt sich mit diskreten Abstraktionen der Dynamik von Modellen. Mit Hilfe dieser Abstraktionen ist es möglich, die Struktur – den Interaktionsgraphen – und die Dynamik der zugehörigen Modelle in Bezug zu setzen. Diese Methode wird sowohl auf Boolesche Modelle als auch auf ODE-Modelle angewandt. Gleichzeitig erlaubt dieser Ansatz Mengen von Modellen in unterschiedlichen Modellgruppen zu vergleichen, die dieselbe Struktur haben. Die abstrahierten Dynamiken (genauer die Einschränkungen der abstrahierten Dynamiken) der Booleschen Modellmengen oder ODE-Modellmengen können als Boolesche Zustandsübergangsgraphen repräsentiert werden. Wir werden zeigen, dass diese Zustandsübergangsgraphen wiederum selber als (asynchrone) Boolesche Netzwerke aufgefasst werden können. Trotz der theoretischen Ausgangsfrage werden in dieser Arbeit zahlreiche Anwendungen aufgezeigt. Die Ergebnisse aus Kapitel 2 können zur Modellreduktion benutzt werden, indem die Dynamik der ODE-Modelle auf den zu den Booleschen Netzwerken gehörigen “trap spaces” betrachtet wird. Die Resultate aus Kapitel 4 können zur Netzwerkinferenz oder zur Analyse von Modellmengen genutzt werden. Weiterhin werden im letzten Kapitel dieser Arbeit einige Anwendungsideen im Bezug auf Experimentdesign eingeführt. Dies führt zu der Fragestellung, wie verschiedene asynchrone Boolesche Netzwerke oder unterschiedliche Dynamiken, die mit einem einzelnen Modell vereinbar sind, unterschieden werden können