Nondeterministic Syntactic Complexity

We introduce a new measure on regular languages: their nondeterministic syntactic complexity. It is the least degree of any extension of the ‘canonical boolean representation’ of the syntactic monoid. Equivalently, it is the least number of states of any subatomic nondeterministic acceptor. It turns out that essentially all previous structural work on nondeterministic state-minimality computes this measure. Our approach rests on an algebraic interpretation of nondeterministic finite automata as deterministic finite automata endowed with semilattice structure. Crucially, the latter form a self-dual category.

Characterizing classes of regular languages using prefix codes of bounded synchronization delay

This paper is motivated by the work of Schützenberger on codes with bounded synchronization delay. He was interested in characterizing those regular languages where groups in the syntactic monoid belong to a variety [Formula: see text]. On the language side he allowed the operations union, intersection, concatenation and modified Kleene-star involving a mapping of a prefix code of bounded synchronization delay to a group [Formula: see text], but no complementation. In our notation, this leads to the language classes [Formula: see text]. Our main result shows that [Formula: see text] coincides with the class of languages having syntactic monoids where all subgroups are in [Formula: see text]. We show that this statement holds for all varieties [Formula: see text] of finite groups, whereas Schützenberger proved this result for varieties [Formula: see text] containing Abelian groups, only. Our method shows the result for all [Formula: see text] simultaneously on finite and infinite words. Furthermore, we introduce the notion of local Rees extension which refers to a restricted type of the classical Rees extension. We give a decomposition of a monoid in terms of its groups and local Rees extensions. This gives a somewhat similar, but simpler decomposition than in the synthesis theorem of Rhodes and Allen. Moreover, we need a singly exponential number of operations, only. Finally, our decomposition yields an answer to a question in a recent paper of Almeida and Klíma about varieties that are closed under Rees extensions.

On the Syntactic Monoids Associated with a Class of Synchronized Codes

A complete codeCover an alphabetAis calledsynchronizedif there existx,y∈C*such thatxA*∩A*y⊆C*. In this paper we describe the syntactic monoidSyn(C+)ofC+for a complete synchronized codeCoverAsuch thatC+, the semigroup generated byC, is a single class of its syntactic congruencePC+. In particular, we prove that, for such a codeC, eitherC=AorSyn(C+)is isomorphic to a special submonoid of𝒯l(I)×𝒯r(Λ), where𝒯l(I)and𝒯r(Λ)are the full transformation semigroups on the nonempty setsIandΛ, respectively.

GENERALIZED CONTEXTS AND n-ARY SYNTACTIC SEMIGROUPS OF TREE LANGUAGES

A new type of syntactic monoid and semigroup of tree languages is introduced. For each n ≥ 1, we define for any tree language T its n-ary syntactic monoid Mn(T) and its n-ary syntactic semigroup Sn(T) as quotients of the monoid or semigroup, respectively, formed by certain new generalized contexts. For n = 1 these contexts are just the ordinary contexts (or 'special trees') and M1(T) is the syntactic monoid introduced by W. Thomas (1982,1984). Several properties of these monoids and semigroups are proved. For example, it is shown that Mn(T) and Sn(T) are isomorphic to certain monoids and semigroups associated with the minimal tree recognizer of T. Using these syntactic monoids or semigroups, we can associate with any variety of finite monoids or semigroups, respectively, a variety of tree languages. Although there are varieties of tree languages that cannot be obtained this way, we prove that the definite tree languages can be characterized by the syntactic semigroups S2(T), which is not possible using the classical syntactic monoids or semigroups.

WATSON-CRICK BORDERED WORDS AND THEIR SYNTACTIC MONOID

DNA strands that, mathematically speaking, are finite strings over the alphabet {A, G, C, T} are used in DNA computing to encode information. Due to the fact that A is Watson-Crick complementary to T and G to C, DNA single strands that are Watson-Crick complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: If some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper studies a mathematical formalization of a particular case of hairpins, the Watson-Crick bordered words. A Watson-Crick bordered word is a word with the property that it has a prefix that is Watson-Crick complementary to its suffix. We namely study algebraic properties of Watson-Crick bordered and unbordered words. We also give a complete characterization of the syntactic monoid of the language consisting of all Watson-Crick bordered words over a given alphabet. Our results hold for the more general case where the Watson-Crick complement function is replaced by an arbitrary antimorphic involution.