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

Complexity in Prefix-Free Regular Languages

We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next provide the tight bounds on state complexity of symmetric difference, and deterministic and nondeterministic state complexity of difference and cyclic shift of prefix-free languages.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Quotient Complexity of Regular Languages

The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formal-language terms as the number of distinct quotients of the language, and to call it "quotient complexity". The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular languages and f is a regular operation, for example, union or concatenation. Since quotients can be represented by derivatives, one can find a formula for the typical quotient of f(K,L) in terms of the quotients of K and L. To obtain an upper bound on the number of quotients of f(K,L) all one has to do is count how many such quotients are possible, and this makes automaton constructions unnecessary. The advantages of this point of view are illustrated by many examples. Moreover, new general observations are presented to help in the estimation of the upper bounds on quotient complexity of regular operations

Most Complex Non-Returning Regular Languages

A regular language $L$ is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jir\'askov\'a derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each $n\ge 4$ there exists a ternary witness of state complexity $n$ that meets the bound for reversal and that at least three letters are needed to meet this bound. Moreover, the restrictions of this witness to binary alphabets meet the bounds for product, star, and boolean operations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has $(n-1)^n$ elements and requires at least $\binom{n}{2}$ generators. We find the maximal state complexities of atoms of non-returning languages. Finally, we show that there exists a most complex non-returning language that meets the bounds for all these complexity measures.Comment: 22 pages, 6 figure

Operations on Automata with All States Final

We study the complexity of basic regular operations on languages represented by incomplete deterministic or nondeterministic automata, in which all states are final. Such languages are known to be prefix-closed. We get tight bounds on both incomplete and nondeterministic state complexity of complement, intersection, union, concatenation, star, and reversal on prefix-closed languages.Comment: In Proceedings AFL 2014, arXiv:1405.527

Unrestricted State Complexity of Binary Operations on Regular and Ideal Languages

We study the state complexity of binary operations on regular languages over different alphabets. It is known that if $L'_m$ and $L_n$ are languages of state complexities $m$ and $n$, respectively, and restricted to the same alphabet, the state complexity of any binary boolean operation on $L'_m$ and $L_n$ is $mn$, and that of product (concatenation) is $m 2^n - 2^{n-1}$. In contrast to this, we show that if $L'_m$ and $L_n$ are over different alphabets, the state complexity of union and symmetric difference is $(m+1)(n+1)$, that of difference is $mn+m$, that of intersection is $mn$, and that of product is $m2^n+2^{n-1}$. We also study unrestricted complexity of binary operations in the classes of regular right, left, and two-sided ideals, and derive tight upper bounds. The bounds for product of the unrestricted cases (with the bounds for the restricted cases in parentheses) are as follows: right ideals $m+2^{n-2}+2^{n-1}$ ($m+2^{n-2}$); left ideals $mn+m+n$ ($m+n-1$); two-sided ideals $m+2n$ ($m+n-1$). The state complexities of boolean operations on all three types of ideals are the same as those of arbitrary regular languages, whereas that is not the case if the alphabets of the arguments are the same. Finally, we update the known results about most complex regular, right-ideal, left-ideal, and two-sided-ideal languages to include the unrestricted cases.Comment: 30 pages, 15 figures. This paper is a revised and expanded version of the DCFS 2016 conference paper, also posted previously as arXiv:1602.01387v3. The expanded version has appeared in J. Autom. Lang. Comb. 22 (1-3), 29-59, 2017, the issue of selected papers from DCFS 2016. This version corrects the proof of distinguishability of states in the difference operation on p. 12 in arXiv:1609.04439v