Aspects of shuffle and deletion on trajectories

Word and language operations on trajectories provide a general framework for the study of properties of sequential insertion and deletion operations. A trajectory gives a syntactical constraint on the scattered insertion (deletion) of a word into(from) another one, with an intuitive geometrical interpretation. Moreover, deletion on trajectories is an inverse of the shuffle on trajectories. These operations are a natural generalization of many binary word operations like catenation, quotient, insertion, deletion, shuffle, etc. Besides they were shown to be useful, e.g. in concurrent processes modelling and recently in biocomputing area. We begin with the study of algebraic properties of the deletion on trajectories. Then we focus on three standard decision problems concerning linear language equations with one variable, involving the above mentioned operations. We generalize previous results and obtain a sequence of new ones. Particularly, we characterize the class of binary word operations for which the validity of such a language equation is (un)decidable, for regular and context-free operands. © 2004 Elsevier B.V. All rights reserved.