Graphs Encoded by Regular Expressions

In the conversion of finite automata to regular expressions, an exponential blowup in size can generally not be avoided. This is due to graph-structural properties of automata which cannot be directly encoded by regular expressions and cause the blowup combinatorially. In order to identify these structures, we generalize the class of arc-series-parallel digraphs to the acyclic case. The resulting digraphs are shown to be reversibly encoded by linear-sized regular expressions. We further derive a characterization of our new class by a finite set of forbidden minors and argue that these minors constitute the primitives causing the blowup in the conversion from automata to expressions

Simplifying Regular Expressions A Quantitative Perspective

Abstract. We consider the efficient simplification of regular expressions and suggest a quantitative comparison of heuristics for simplifying regular expressions. To this end, we propose a new normal form for regular expressions, which outperforms previous heuristics while still being computable in linear time. This allows us to determine an exact bound for the relation between the two prevalent measures for regular expression- size: alphabetic width and reverse polish notation length. In addition, we show that every regular expression of alphabetic width n can be converted into a nondeterministic finite automaton with ε-transitions of size at most 4 2 n+1, and prove this bound to be optimal. This an-5 swers a question posed by Ilie and Yu, who had obtained lower and upper bounds of 4n − 1 and 9n − 1, respectively [15]. For reverse polish notation length a

Local Elimination-Strategies in Automata for Shorter Regular Expressions

Abstract. We propose a construction of regular expressions from particularly restricted NFA via extended automata. It proceeds in two main steps, elimination of cycles in the state graph followed by a recursive construction of the final regular expression. Inbetween these eliminations, series-parallel substructures are reduced to single transitions. The process gives rise to compact regular expressions by avoiding redundancies in the intermediate extended automata. Altough derived from state-elimination-techniques, the constructions are rather ’transitionoriented’.

An Optimal Constructionof Finite Automatafrom Regular Expressions

ABSTRACT. We consider the construction of finite automata from their corresponding regular expressionsbyaseriesofdigraph-transformationsalongtheexpression’sstructure. Eachintermediate graph represents an extended finite automaton accepting the same language. The character of our construction allows a fine-grained analysis of the emerging automaton’s size, eventually leading to an optimality result.

Applicability of Formal Methods for Safety-Critical Systems in the Context of ISO 26262

Formal methods are a means for verification and validation with the main advantage that a system property can be verified for the overall system (including all possible system states). The drawbacks of formal methods are the additional effort for the formalisation of the requirements and for building a model of the system, and, the limitations due to computational restrictions (handling the state-space explosion). ISO 26262 “Road Vehicles - Functional Safety” is a standard for the assessment of the development process for safety-relevant components in the automotive domain. The standard addresses formal methods for the specification of safety requirements and for the product development at software level. Formal methods for the hardware development or at system level are (by now) not explicitly foreseen by the standard. In this work we will give an overview on the basic principles and the state-of-the-art of formal methods (in detail, model checking). Then we will present different approaches for the application of formal methods at system level including some preliminary evaluation results for an industrial use case. Based on these experiences we will discuss the applicabi lity of formal methods in the context of ISO 26262 (i.e., for automotive components) in view of the limitations of formal techniques for applications inthe automotive domain