On the Balancedness of Tree-to-Word Transducers

A language over an alphabet [Formula: see text] of opening ([Formula: see text]) and closing ([Formula: see text]) brackets, is balanced if it is a subset of the Dyck language [Formula: see text] over [Formula: see text], and it is well-formed if all words are prefixes of words in [Formula: see text]. We show that well-formedness of a context-free language is decidable in polynomial time, and that the longest common reduced suffix can be computed in polynomial time. With this at a hand we decide for the class 2-TW of non-linear tree transducers with output alphabet [Formula: see text] whether or not the output language is balanced.

On the Commutative Equivalence of Algebraic Formal Series and Languages

The problem of the commutative equivalence of context-free and regular languages is studied. Conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated.

On restarting automata with auxiliary symbols and small window size

Here we show that for monotone RWW- (and RRWW-) automata, window size two is sufficient, both in the nondeterministic as well as in the deterministic case. For the former case, this is done by proving that each context-free language is already accepted by a monotone RWW-automaton of window size two. In the deterministic case, we first prove that each deterministic pushdown automaton can be simulated by a deterministic monotone RWW-automaton of window size three, and then we present a construction that transforms a deterministic monotone RWW-automaton of window size three into an equivalent automaton of the same type that has window size two. Furthermore, we study the expressive power of shrinking RWW- and RRWW-automata the window size of which is just one or two. We show that for shrinking RRWW-automata that are nondeterministic, window size one suffices, while for nondeterministic shrinking RWW-automata, we already need window size two to accept all growing context-sensitive languages. In the deterministic case, shrinking RWW- and RRWW-automata of window size one accept only regular languages, while those of window size two characterize the Church-Rosser languages.

COMPLEXITY OF THE INFINITARY LAMBEK CALCULUS WITH KLEENE STAR

We consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an $\omega $ -rule, and prove that the derivability problem in this calculus is $\Pi _1^0$ -hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007).

On the Prefix–Suffix Duplication Reduction

This work answers some questions proposed by Bottoni, Labella, and Mitrana (Theoretical Computer Science 682, 2017) regarding the prefix–suffix reduction on words. The operation is defined as a reduction by one half of every square that is present as either a prefix or a suffix of a word, leading thus to a finite set of words associated to the starting one. The iterated case considers consecutive applications of the operations, on all the resulting words. We show that the classes of linear and context-free language are closed under iterated bounded prefix–suffix square reduction, and that for a given word we can determine in [Formula: see text] time all of its primitive prefix–suffix square roots.

Groups whose word problems are not semilinear

Suppose that G is a finitely generated group and {\operatorname{WP}(G)} is the formal language of words defining the identity in G. We prove that if G is a virtually nilpotent group that is not virtually abelian, the fundamental group of a finite volume hyperbolic three-manifold, or a right-angled Artin group whose graph lies in a certain infinite class, then {\operatorname{WP}(G)} is not a multiple context-free language.

The word problem of ℤ n is a multiple context-free language

The word problem of a group {G=\langle\Sigma\rangle} can be defined as the set of formal words in {\Sigma^{*}} that represent the identity in G. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anīsīmov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Recently, Salvati showed that the word problem of {\mathbb{Z}^{2}} is a multiple context-free language, giving the first example of a natural word problem that is multiple context-free, but not context-free. We generalize Salvati’s result to show that the word problem of {\mathbb{Z}^{n}} is a multiple context-free language for any n.