Syntactic Semigroup Research Articles

Most Complex Non-Returning Regular Languages

A regular language [Formula: see text] is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková 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 tight 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 [Formula: see text], there exists a ternary witness of state complexity [Formula: see text] that meets the bound for reversal, and restrictions of this witness to binary alphabets meet the bounds for star, product, and boolean operations. Hence all of these operations can be handled simultaneously with a single witness, using only three different transformations. 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 [Formula: see text] elements and requires at least [Formula: see text] generators. We find the maximal state complexities of atoms of non-returning languages. We show that there exists a most complex sequence of non-returning languages that meet the bounds for all of these complexity measures. Furthermore, we prove there is a most complex sequence that meets all the bounds using alphabets of minimal size.

Syntactic complexity of bifix-free regular languages

We study the properties of syntactic monoids of bifix-free regular languages. In particular, we solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a bifix-free language with state complexity n is at most (n−1)n−3+(n−2)n−3+(n−3)2n−3 for n⩾6. The main proof uses a large construction with the method of injective function. Since this bound is known to be reachable, and the values for n⩽5 are known, this completely settles the problem. We also prove that (n−2)n−3+(n−3)2n−3−1 is the minimal size of the alphabet required to meet the bound for n⩾6. Finally, we show that the largest transition semigroups of minimal DFAs which recognize bifix-free languages are unique up to renaming the states.

Complexity of proper prefix-convex regular languages

A language L over an alphabet Σ is prefix-convex if, for any words x,y,z∈Σ⁎, whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages, which were studied elsewhere. Here we concentrate on prefix-convex languages that do not belong to any one of these classes; we call such languages proper. We exhibit most complex proper prefix-convex languages, which meet the bounds for the size of the syntactic semigroup, reversal, complexity of atoms, star, product, and boolean operations.

Green’s Relations in Deterministic Finite Automata

Green’s relations are a fundamental tool in the structure theory of semigroups. They can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes of Green’s relations then correspond to the strongly connected components. We study the complexity of Green’s relations in semigroups generated by transformations on a finite set. We show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of strongly connected components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for a constant size alphabet is rather involved. We also investigate the special cases of unary and binary alphabets. All these results are extended to deterministic finite automata and their syntactic semigroups.

Syntactic complexity of suffix-free languages

We solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a suffix-free language with n left quotients (that is, with state complexity n) is at most (n−1)n−2+n−2 for n⩾6. Since this bound is known to be reachable, this settles the problem. We also reduce the alphabet of the witness languages reaching this bound to five letters instead of n+2, and show that it cannot be any smaller. Finally, we prove that the transition semigroup of a minimal deterministic automaton accepting a witness language is unique for each n.

Syntactic Complexity of Regular Ideals

The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn−1, nn−1 + n − 1, and nn−2 + (n − 2)2n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced.

Most Complex Regular Ideal Languages

A right ideal (left ideal, two-sided ideal) is a non-empty language $L$ over an alphabet $\Sigma$ such that $L=L\Sigma^*$ ($L=\Sigma^*L$, $L=\Sigma^*L\Sigma^*$). Let $k=3$ for right ideals, 4 for left ideals and 5 for two-sided ideals. We show that there exist sequences ($L_n \mid n \ge k $) of right, left, and two-sided regular ideals, where $L_n$ has quotient complexity (state complexity) $n$, such that $L_n$ is most complex in its class under the following measures of complexity: the size of the syntactic semigroup, the quotient complexities of the left quotients of $L_n$, the number of atoms (intersections of complemented and uncomplemented left quotients), the quotient complexities of the atoms, and the quotient complexities of reversal, star, product (concatenation), and all binary boolean operations. In that sense, these ideals are "most complex" languages in their classes, or "universal witnesses" to the complexity of the various operations.

Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups

The syntactic semigroup problem is to decide whether a given finite semigroup is syntactic or not. This work investigates the syntactic semigroup problem for both the semigroup reducts of [Formula: see text], the affine near-semiring over a Brandt semigroup [Formula: see text]. It is ascertained that both the semigroup reducts of [Formula: see text] are syntactic semigroups.

Large Aperiodic Semigroups

We search for the largest syntactic semigroups of star-free languages having n left quotients; equivalently, we look for the largest transition semigroups of aperiodic finite automata with n states. We first introduce unitary semigroups generated by transformations that change only one state. In particular, we study unitary-complete semigroups which have a special structure, and show that each maximal unitary semigroup is unitary-complete. For [Formula: see text] we exhibit a unitary-complete semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. We examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups.

SYNTACTIC COMPLEXITY OF ℛ- AND 𝒥-TRIVIAL REGULAR LANGUAGES

The syntactic complexity of a subclass of the class of regular languages is the maximal cardinality of syntactic semigroups of languages in that class, taken as a function of the state complexity n of these languages. We prove that n! and ⌊e(n − 1)⌋. are tight upper bounds for the syntactic complexity of ℛ- and 𝒥-trivial regular languages, respectively.

A categorical invariant of flow equivalence of shifts

We prove that the Karoubi envelope of a shift—defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift—is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants—some new and some old—are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov–Dyck and Markov–Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property ($\mathscr{A}$) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property ($\mathscr{A}$) is decidable for sofic shifts.

Closures of regular languages for profinite topologies

The Pin-Reutenauer algorithm gives a method, that can be viewed as a descriptive procedure, to compute the closure in the free group of a regular language with respect to the Hall topology. A similar descriptive procedure is shown to hold for the pseudovariety $$\mathsf{A}$$ of aperiodic semigroups, where the closure is taken in the free aperiodic $$\omega $$ -semigroup. It is inherited by a subpseudovariety of a given pseudovariety if both of them enjoy the property of being full. The pseudovariety $$\mathsf{A}$$ , as well as some of its subpseudovarieties are shown to be full. The interest in such descriptions stems from the fact that, for each of the main pseudovarieties $$\mathsf{V}$$ in our examples, the closures of two regular languages are disjoint if and only if the languages can be separated by a language whose syntactic semigroup lies in $$\mathsf{V}$$ . In the cases of $$\mathsf{A}$$ and of the pseudovariety $$\mathsf{DA}$$ of semigroups in which all regular elements are idempotents, this is a new result.

Burrows–Wheeler transformations and de Bruijn words

We formulate and explain the extended Burrows–Wheeler transform of Mantaci et al. from the viewpoint of permutations on a chain taken as a union of partial order-preserving mappings. In so doing we establish a link with syntactic semigroups of languages that are themselves cyclic semigroups. We apply the extended transform with a view to generating de Bruijn words through inverting the transform. We also make use of de Bruijn words to facilitate a proof that the maximum number of distinct factors of a word of length n has the form 12n2−O(nlogn).

Syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that nn−2 is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be (n−1)n−2+(n−2), (n−1)n−3+(n−2)n−3+(n−3)2n−3, and (n−1)n−3+(n−3)2n−3+1, respectively, and exhibit languages with these syntactic complexities.

ON THE TRANSITION SEMIGROUPS OF CENTRALLY LABELED RAUZY GRAPHS

Rauzy graphs of subshifts are endowed with an automaton structure. For Sturmian subshifts, it is shown that its transition semigroup is the syntactic semigroup of the language recognized by the automaton. An inverse limit of the partial semigroups of nonzero regular elements of their transition semigroups is described. If the subshift is minimal, then this inverse limit is isomorphic, as a partial semigroup, to the [Formula: see text]-class associated to it in the free pro-aperiodic semigroup.

Semigroups with I-Quasi Length and Syntactic Semigroups

In this article, we first generalize the method of [13] and give some further characterizations of semigroups with [0-]quasi length. Then we give a very simple necessary and sufficient condition for a class of semigroups, including finitely generated semigroups with [0-]quasi length, finite π-groups, finite nil-extensions of finite inverse semigroups, to be syntactic. This work completely answers the following questions: “How to characterize the syntactic semigroup (monoid) of f-disjunctive languages?” or “What kind of f-disjunctive congruences is syntactic?” These questions are asked forward and studied in [6, 12, 13].

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.

A theorem on anti-ordered factor-semigroups

Let K be an anti-ideal of a semigroup (S,=, ?,?,?) with apartness. A construction of the anti-congruence Q(K) and the quasi-antiorder ?, generated by K, are presented. Besides, a construction of the anti-order relation ? on syntactic semigroup S/Q(K), generated by ?, is given in Bishop?s constructive mathematics.

Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts

For a pseudovariety V of ordered semigroups, let S ( V ) be the class of sofic subshifts whose syntactic semigroup lies in V . It is proved that if V contains Sl − then S ( V ∗ D ) is closed under taking shift equivalent subshifts, and conversely, if S ( V ) is closed under taking conjugate subshifts then V contains LSl − and S ( V ) = S ( V ∗ D ) . Almost finite type subshifts are characterized as the irreducible elements of S ( LInv ) , which gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.

THE SYNTACTIC GRAPH OF A SOFIC SHIFT IS INVARIANT UNDER SHIFT EQUIVALENCE

We define a new invariant for shift equivalence of sofic shifts. This invariant, which we call the syntactic graph of a sofic shift, is the directed acyclic graph of characteristic groups of the non-null regular [Formula: see text]-classes of the syntactic semigroup of the shift.