State complexity of union and intersection of star on k regular languages

AbstractIn this paper, we continue our study on state complexity of combined operations. We study the state complexities of L1∗∪L2∗, ⋃i=1kLi∗, L1∗∩L2∗, and ⋂i=1kLi∗ for regular languages Li, 1≤i≤k. We obtain the exact bounds for these combined operations and show that the bounds are different from the mathematical compositions of the state complexities of their component individual operations

Some Single and Combined Operations on Formal Languages: Algebraic Properties and Complexity

In this thesis, we consider several research questions related to language operations in the following areas of automata and formal language theory: reversibility of operations, generalizations of (comma-free) codes, generalizations of basic operations, language equations, and state complexity. Motivated by cryptography applications, we investigate several reversibility questions with respect to the parallel insertion and deletion operations. Among the results we obtained, the following result is of particular interest. For languages L1, L2 ⊆ Σ∗, if L2 satisfies the condition L2ΣL2 ∩ Σ+L2Σ+ = ∅, then any language L1 can be recovered after first parallel-inserting L2 into L1 and then parallel-deleting L2 from the result. This property reminds us of the definition of comma-free codes. Following this observation, we define the notions of comma codes and k-comma codes, and then generalize them to comma intercodes and k-comma intercodes, respectively. Besides proving all these new codes are indeed codes, we obtain some interesting properties, as well as several hierarchical results among the families of the new codes and some existing codes such as comma-free codes, infix codes, and bifix codes. Another topic considered in this thesis are some natural generalizations of basic language operations. We introduce block insertion on trajectories and block deletion on trajectories, which properly generalize several sequential as well as parallel binary language operations such as catenation, sequential insertion, k-insertion, parallel insertion, quotient, sequential deletion, k-deletion, etc. We obtain several closure properties of the families of regular and context-free languages under the new operations by using some relationships between these new operations and shuffle and deletion on trajectories. Also, we obtain several decidability results of language equation problems with respect to the new operations. Lastly, we study the state complexity of the following combined operations: L1L2∗, L1L2R, L1(L2 ∩ L3), L1(L2 ∪ L3), (L1L2)R, L1∗L2, L1RL2, (L1 ∩ L2)L3, (L1 ∪ L2)L3, L1L2 ∩ L3, and L1L2 ∪ L3 for regular languages L1, L2, and L3. These are all the combinations of two basic operations whose state complexities have not been studied in the literature

Derivative Based Extended Regular Expression Matching Supporting Intersection, Complement and Lookarounds

Regular expressions are widely used in software. Various regular expression engines support different combinations of extensions to classical regular constructs such as Kleene star, concatenation, nondeterministic choice (union in terms of match semantics). The extensions include e.g. anchors, lookarounds, counters, backreferences. The properties of combinations of such extensions have been subject of active recent research. In the current paper we present a symbolic derivatives based approach to finding matches to regular expressions that, in addition to the classical regular constructs, also support complement, intersection and lookarounds (both negative and positive lookaheads and lookbacks). The theory of computing symbolic derivatives and determining nullability given an input string is presented that shows that such a combination of extensions yields a match semantics that corresponds to an effective Boolean algebra, which in turn opens up possibilities of applying various Boolean logic rewrite rules to optimize the search for matches. In addition to the theoretical framework we present an implementation of the combination of extensions to demonstrate the efficacy of the approach accompanied with practical examples

Phase Retrieval for Sparse Signals: Uniqueness Conditions

In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: "Under which conditions can we uniquely recover the signal of interest from its measured magnitudes?" In this paper, we assume the measured signal to be sparse. This is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. In this work, we derive a sufficient condition for the uniqueness of the solution of the phase retrieval (PR) problem for both discrete and continuous domains, and for one and multi-dimensional domains. More precisely, we show that there is a strong connection between PR and the turnpike problem, a classic combinatorial problem. We also prove that the existence of collisions in the autocorrelation function of the signal may preclude the uniqueness of the solution of PR. Then, assuming the absence of collisions, we prove that the solution is almost surely unique on 1-dimensional domains. Finally, we extend this result to multi-dimensional signals by solving a set of 1-dimensional problems. We show that the solution of the multi-dimensional problem is unique when the autocorrelation function has no collisions, significantly improving upon a previously known result.Comment: submitted to IEEE TI