State Complexity of Combined Operations on Finite Languages

State complexity is a descriptive complexity measure for regular languages. It is a fundamental topic in automata and formal language theory. The state complexity of a regular language is the number of states in the minimal complete deterministic finite automaton accepting the language. During the last few decades, many publications have focused and studied the state complexity of many individual as well as combined operations on regular languages. Also, the state complexity of some basic operations on finite languages has been studied. But until now there has been no study on the state complexity of combined operations on finite languages. In this thesis, we will first study the state complexity of the combined operation, star of union, on finite languages and give an exact bound. Then we will investigate the state complexity of star of catenation and show its approximation with a good ratio bound and finally, we will prove an upper bound for star of intersection

On Analytic Nonlinear Input-output Systems: Expanded Global Convergence and System Interconnections

Functional series representations of nonlinear systems first appeared in engineering in the early 1950’s. One common representation of a nonlinear input-output system are Chen-Fliess series or Fliess operators. Such operators are described by functional series indexed by words over a noncommutative alphabet. They can be viewed as a noncommutative generalization of a Taylor series. A Fliess operator is said to be globally convergent when its radius of convergence is infinite, in other words, when there is no a priori upper bound on both the L1-norm of an admissible input and the length of time over which the corresponding output is well defined. If such bounds are required to ensure convergence, then the Fliess operator is said to be locally convergent with a finite radius of convergence. However, in the literature, a Fliess operator is classified as locally convergent or globally convergent based solely on the growth rate of the coefficients in its generating series. The existing growth rate bounds provide sufficient conditions for global convergence which are very conservative. Therefore, the first main goal of this dissertation is to develop a more exact relationship between the coefficient growth rate and the nature of convergence of the corresponding Fliess operator. This first goal is accomplished by introducing a new topological space of formal power series which renders a Fréchet space instead of the more commonly used ultrametric space. Then, a direct relationship is developed between the nature of convergence of a Fliess operator and its generating series. The second main goal of this dissertation is to show that the global convergence of Fliess operators is preserved under the nonrecursive interconnections, namely the parallel sum and product connections and the cascade connection. This fact had only been understood previously in a narrow sense based on the more conservative tests for global convergence

A Formal Power Series Approach to Multiplicative Dynamic Feedback Interconnection

The goal of the paper is multi-fold. First, an explicit formula is derived to compute the non-commutative generating series of a closed-loop system when a (multi-input, multi-output) plant, given in Chen--Fliess series description is in multiplicative output feedback interconnection with another system, also given as Chen--Fliess series. Furthermore, it is shown that the multiplicative dynamic output feedback connection has a natural interpretation as a transformation group acting on the plant. A computational framework for computing the generating series for multiplicative dynamic output feedback is devised utilizing the Hopf algebras of the coordinate functions corresponding to the shuffle group and the multiplicative feedback group. The pre--Lie algebra in multiplicative feedback is shown to be an example of Foissy's com-pre-Lie algebras indexed by matrices with certain structure

Interconnections of Nonlinear Systems Driven by L₂-ITÔ Stochastic Processes

Fliess operators have been an object of study in connection with nonlinear systems acting on deterministic inputs since the early 1970\u27s. They describe a broad class of nonlinear input-output maps using a type of functional series expansion, but in most applications, a system\u27s inputs have noise components. In such circumstances, new mathematical machinery is needed to properly describe the input-output map via the Chen-Fliess algebraic formalism. In this dissertation, a class of L2-Itô stochastic processes is introduced specifically for this purpose. Then, an extension of the Fliess operator theory is presented and sufficient conditions are given under which these operators are convergent in the mean-square sense. Next, three types of system interconnections are considered in this context: the parallel, product and cascade connections. This is done by first introducing the notion of a formal Fliess operator driven by a formal stochastic process. Then the generating series induced by each interconnection is derived. Finally, sufficient conditions are given under which the generating series of each composite system is convergent. This allows one to determine when an interconnection of Fliess operators driven by a class of L2-Itô stochastic processes is well-defined

Commutative Languages and their Composition by Consensual Methods

Commutative languages with the semilinear property (SLIP) can be naturally recognized by real-time NLOG-SPACE multi-counter machines. We show that unions and concatenations of such languages can be similarly recognized, relying on -- and further developing, our recent results on the family of consensually regular (CREG) languages. A CREG language is defined by a regular language on the alphabet that includes the terminal alphabet and its marked copy. New conditions, for ensuring that the union or concatenation of CREG languages is closed, are presented and applied to the commutative SLIP languages. The paper contributes to the knowledge of the CREG family, and introduces novel techniques for language composition, based on arithmetic congruences that act as language signatures. Open problems are listed.Comment: In Proceedings AFL 2014, arXiv:1405.527