Michael Domaratzki

Deletion along trajectories

We describe a new way to model deletion operations on formal languages, called deletion along trajectories. We examine its closure properties, which differ from those of shuffle on trajectories, previously introduced by Mateescu et al. In particular, we define classes of nonregular sets of trajectories such that the associated deletion operation preserves regularity. Our results give uniform proofs of closure properties of the regular languages for several deletion operations.We also show that deletion along trajectories serves as an inverse to shuffle on trajectories. This leads to results on the decidability of certain language equations, including those of the form LTX = R, where L,R are regular languages and X is unknown.

State complexity of power

The number of states in a deterministic finite automaton (DFA) recognizing the language L^k, where L is regular language recognized by an n-state DFA, and k>=2 is a constant, is shown to be at most n2^(^k^-^1^)^n and at least (n-k)2^(^k^-^1^)^(^n^-^k^) in the worst case, for every n>k and for every alphabet of at least six letters. Thus, the state complexity of L^k is @Q(n2^(^k^-^1^)^n). In the case k=3 the corresponding state complexity function for L^3 is determined as 6n-384^n-(n-1)2^n-n with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of L^k is demonstrated to be nk. This bound is shown to be tight over a two-letter alphabet.

Trajectory-based codes

Abstract.The notion of shuffle on trajectories is a natural generalization of many word operations considered in the literature. For a set of trajectories T, we define the notion of a T-code and examine its properties. Particular instances of T-codes are prefix-, suffix-, infix-, outfix- and hyper-codes, as well as other classes studied in the literature.

Transition complexity of language operations

The number of transitions required by a nondeterministic finite automaton (NFA) to accept a regular language is a natural measure of the size of that language. There has been a significant amount of work related to the trade-off between the number of transitions and other descriptional complexity measures for regular languages. In this paper, we consider the effect of language operations on the number of transitions required to accept a regular language. This work extends previous work on descriptional complexity of regular language operations, in particular, under the measures of deterministic state complexity, nondeterministic state complexity and regular expression size.

Semantic shuffle on and deletion along trajectories

We introduce semantic shuffle on trajectories (SST) and semantic deletion along trajectories (SDT). These operations generalize the notion of shuffle on trajectories, but add sufficient power to encompass various formal language operations used in applied areas. However, the added power given to SST and SDT does not destroy many desirable properties of shuffle on trajectories, especially with respect to solving language equations involving SST. We also investigate closure properties and decidability questions related to SST and SDT.

Simulating finite automata with context-free grammars

We consider simulating finite automata (both deterministic and nondeterministic) with context-free grammars in Chomsky normal form (CNF). We show that any unary DFA with n states can be simulated by a CNF grammar with O(n1/3) variables, and this bound is tight. We show that any unary NFA with n states can be simulated by a CNF grammar with O(n2/3) variables. Finally, for larger alphabets we show that there exist languages which can be accepted by an n-state DFA, but which require Ω(n/log n) variables in any equivalent CNF grammar.

Codes defined by multiple sets of trajectories

We investigate the use of shuffle on trajectories to model certain classes of languages arising in the theory of codes. In particular, for each finite set of sets of trajectories, which we call a hyperset of trajectories, we define a class of languages induced by that hyperset of trajectories. We investigate the properties of hypersets of trajectories and the associated classes of languages, including the problem of decidability of membership and the problem of equivalence of hypersets of trajectories.

Abelian primitive words

We investigate Abelian primitive words, which are words that are not Abelian powers. We show the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive in linear time. Also different from classical primitive words, we find that a word may have more than one Abelian root. We also consider enumeration of Abelian primitive words.

Hairpin Structures Defined by DNA Trajectories

Abstract We examine scattered hairpins, which are structures formed when a single strand of nucleotides folds into a partially hybridized stem and a loop. To specify different classes of hairpins, we use the concept of DNA trajectories, which allows precise descriptions of valid bonding patterns on the stem of the hairpin. DNA trajectories have previously been used to describe bonding between separate strands. We are interested in the mathematical properties of scattered hairpins described by DNA trajectories. We examine the complexity of the set of hairpin-free words described by a set of DNA trajectories. In particular, we consider the closure properties of language classes under sets of DNA trajectories of differing complexity. We address decidability of recognition problems for hairpin structures.

Minimality in template-guided recombination

Ciliates are unicellular organisms, some of which perform complicated rearrangements of their DNA. Template-guided recombination (TGR) is a formal model for the DNA recombination which occurs in ciliates. TGR has been the subject of much research in formal language theory, as it can be viewed as an operation on formal languages. In TGR, a set of templates serves as a parameter to a language operation which controls which rearrangements can take place; thus, a set of templates is itself a language. Recently, the concept of equivalence in TGR has been considered: given two sets of templates, do they define the same language operation? This paper considers the related question of minimality: given a set of templates T, what is the smallest set of templates (with respect to inclusion) equivalent to T? We show that the minimal set of templates is unique, and consider closure properties and decidability questions related to minimality. We define an operational characterization for equivalence which is useful for results on minimality.