Partial Derivative Automaton for Regular Expressions with Shuffle

We generalize the partial derivative automaton to regular expressions with shuffle and study its size in the worst and in the average case. The number of states of the partial derivative automata is in the worst case at most 2^m, where m is the number of letters in the expression, while asymptotically and on average it is no more than (4/3)^m

Regular Combinators for String Transformations

We focus on (partial) functions that map input strings to a monoid such as the set of integers with addition and the set of output strings with concatenation. The notion of regularity for such functions has been defined using two-way finite-state transducers, (one-way) cost register automata, and MSO-definable graph transformations. In this paper, we give an algebraic and machine-independent characterization of this class analogous to the definition of regular languages by regular expressions. When the monoid is commutative, we prove that every regular function can be constructed from constant functions using the combinators of choice, split sum, and iterated sum, that are analogs of union, concatenation, and Kleene-*, respectively, but enforce unique (or unambiguous) parsing. Our main result is for the general case of non-commutative monoids, which is of particular interest for capturing regular string-to-string transformations for document processing. We prove that the following additional combinators suffice for constructing all regular functions: (1) the left-additive versions of split sum and iterated sum, which allow transformations such as string reversal; (2) sum of functions, which allows transformations such as copying of strings; and (3) function composition, or alternatively, a new concept of chained sum, which allows output values from adjacent blocks to mix.Comment: This is the full version, with omitted proofs and constructions, of the conference paper currently in submissio

Pairs of Languages Closed under Shuffle Projection

Shuffle projection is motivated by the verification of safety properties of special parameterized systems. Basic definitions and properties, especially related to alphabetic homomorphisms, are presented. The relation between iterated shuffle products and shuffle projections is shown. A special class of multi-counter automata is introduced, to formulate shuffle projection in terms of computations of these automata represented by transductions. This reformulation of shuffle projection leads to construction principles for pairs of languages closed under shuffle projection. Additionally, it is shown that under certain conditions these transductions are rational, which implies decidability of closure against shuffle projection. Decidability of these conditions is proven for regular languages. Finally, without additional conditions, decidability of the question, whether a pair of regular languages is closed under shuffle projection, is shown. In an appendix the relation between shuffle projection and the shuffle product of two languages is discussed. Additionally, a kind of shuffle product for computations in S-automata is defined

A regular viewpoint on processes and algebra

While different algebraic structures have been proposed for the treatment of concurrency, finding solutions for equations over these structures needs to be worked on further. This article is a survey of process algebra from a very narrow viewpoint, that of finite automata and regular languages. What have automata theorists learnt from process algebra about finite state concurrency? The title is stolen from [31]. There is a recent survey article [7] on finite state processes which deals extensively with rational expressions. The aim of the present article is different. How do standard notions such as Petri nets, Mazurkiewicz trace languages and Zielonka automata fare in the world of process algebra? This article has no original results, and the attempt is to raise questions rather than answer them

Decomposition and Descriptional Complexity of Shuffle on Words and Finite Languages

We investigate various questions related to the shuffle operation on words and finite languages. First we investigate a special variant of the shuffle decomposition problem for regular languages, namely, when the given regular language is the shuffle of finite languages. The shuffle decomposition into finite languages is, in general not unique. Thatis,therearelanguagesL^,L2,L3,L4withLiluL2= £3luT4but{L\,L2}^ {I/3, L4}. However, if all four languages are singletons (with at least two combined letters), it follows by a result of Berstel and Boasson [6], that the solution is unique; that is {L\,L2} = {L3,L4}. We extend this result to show that if L\ and L2 are arbitrary finite sets and Lz and Z-4 are singletons (with at least two letters in each), the solution is unique. This is as strong as it can be, since we provide examples showing that the solution can be non-unique already when (1) both L\ and L2 are singleton sets over different unary alphabets; or (2) L\ contains two words and L2 is singleton. We furthermore investigate the size of shuffle automata for words. It was shown by Campeanu, K. Salomaa and Yu in [11] that the minimal shuffle automaton of two regular languages requires 2mn states in the worst case (where the minimal automata of the two component languages had m and n states, respectively). It was also recently shown that there exist words u and v such that the minimal shuffle iii DFA for u and v requires an exponential number of states. We study the size of shuffle DFAs for restricted cases of words, namely when the words u and v are both periods of a common underlying word. We show that, when the underlying word obeys certain conditions, then the size of the minimal shuffle DFA for u and v is at most quadratic. Moreover we provide an efficient algorithm, which decides for a given DFA A and two words u and v, whether u lu u C L(A)