Surface and Curve Languages

In the area of combinatorial studies of string languages under formal grammars, the concept of Generalized Parikh Vector (GPV) gives the positions of symbols in linear strings. It has been proved that GPVs of the strings of the same length lie on a hyper plane. The studies on GPV of strings over a binary alphabet gave rise to the concept of line languages. The concept has been extended to surface languages in this paper.

Approach for Flow Shop Scheduling Based on Petri Net

A flow shop scheduling problem based on the controlled Petri net and GASA is presented with multi-workstation operation. Firstly, the math model of this flow shop scheduling problem is constructed. Secondly, Petri net controller is designed based on Parikh vector, and the Petri net model is constructed. And then, GASA is applied based on the math model and the controlled Petri net model. Finally, one example is applied to test the effectiveness of the method.

Approach for Multi-Objective Flexible Job Shop Scheduling

A multi-objective scheduling method based on the controlled Petri net and GA is proposed to the flexible job shop scheduling problem (FJSP). Function objectives of the proposed method are to minimize the completion time and the total expense and workload of machines. Firstly, a Parikh vector based approach for Petri net controller is introduced, and based on this method, the Petri net model is constructed for FSP with machine breaking down. Then, the genetic algorithm (GA) is applied based on the controlled Petri net model and Pareto. Finally, simulation results based on an example show that the method is efficient.

ALGORITHMS FOR JUMBLED PATTERN MATCHING IN STRINGS

The Parikh vector p(s) of a string s over a finite ordered alphabet Σ = {a1, …, aσ} is defined as the vector of multiplicities of the characters, p(s) = (p1, …, pσ), where pi = |{j | sj = ai}|. Parikh vector q occurs in s if s has a substring t with p(t) = q. The problem of searching for a query q in a text s of length n can be solved simply and worst-case optimally with a sliding window approach in O(n) time. We present two novel algorithms for the case where the text is fixed and many queries arrive over time. The first algorithm only decides whether a given Parikh vector appears in a binary text. It uses a linear size data structure and decides each query in O(1) time. The preprocessing can be done trivially in Θ(n2) time. The second algorithm finds all occurrences of a given Parikh vector in a text over an arbitrary alphabet of size σ ≥ 2 and has sub-linear expected time complexity. More precisely, we present two variants of the algorithm, both using an O(n) size data structure, each of which can be constructed in O(n) time. The first solution is very simple and easy to implement and leads to an expected query time of [Formula: see text], where m = ∑i qi is the length of a string with Parikh vector q. The second uses wavelet trees and improves the expected runtime to [Formula: see text], i.e., by a factor of log m.

PRODUCT OF PARIKH MATRICES AND COMMUTATIVITY

The Parikh vector of a word enumerates the symbols of the alphabet that occur in the word. The Parikh matrix of a word which has been recently introduced, is an extension of the notion of Parikh vector and gives more numerical information about the word in terms of certain subwords. Intensive investigation on various theoretical properties of Parikh matrices has taken place. This paper deals with the problem of finding properties of words so that their Parikh matrices commute.

Controller Design for Petri Net with Uncontrollable and Unobservable Transitions

A controller design method for Petri net with uncontrollable and unobservable transitions that enforces the conjunction of a set of linear inequalities on the Parikh vector is proposed. The method is based on the theory that each place can be described with a Parikh vector inequality. Constraints are classified into admissible and inadmissible constraints. An inadmissible constraint cannot be directly enforced on a plant because of the uncontrollability or unobservability of certain transitions. Construct the controller though transforming the inadmissible constraint into admissible one. The method eases the design of controller, because it is based on part net design, and it only considers the direct or indirect transitions related to the constraints. So the computation required to find the Petri net controller is quite simple. Finally, the method is proved to be simple and efficient through one example.

ON PARIKH MATRICES

Mateescu et al (2000) introduced an interesting new tool, called Parikh matrix, to study in terms of subwords, the numerical properties of words over an alphabet. The Parikh matrix gives more information than the well-known Parikh vector of a word which counts only occurrences of symbols in a word. In this note a property of two words u, υ, called "ratio property", is introduced. This property is a sufficient condition for the words uυ and υu to have the same Parikh matrix. Thus the ratio property gives information on the M–ambiguity of certain words and certain sets of words. In fact certain regular, context-free and context-sensitive languages that have the same set of Parikh matrices are exhibited. In the study of fair words, Cerny (2006) introduced another kind of matrix, called the p–matrix of a word. Here a "weak-ratio property" of two words u, υ is introduced. This property is a sufficient condition for the words uυ and υu to have the same p–matrix. Also the words uυ and υu are fair whenever u, υ are fair and have the weak ratio property.

ON PARIKH MATRICES, AMBIGUITY, AND PRINTS

In the algebraic study of words and languages it is often convenient to analyze words as numerical quantities. One such now-famous example is the study of words by using their attached Parikh mapping (or vector). However, the Parikh vector abstracts away too much of the structure of the word. Parikh matrices were introduced as a means to obtain more than just the number of occurrences of single letters. When studying properties of words, an important property is that a word is uniquely determined by the number of occurrences of certain predetermined subwords. In the context of Parikh matrices, the problem is translated in finding which words are completely characterized by their associated Parikh matrix. This paper links this problem with that of the print of a word, i.e. the word obtained by considering consecutive occurrences of the same letter as only one letter. We obtain results regarding finiteness and context-freeness of such classes of words.