formal language engineers

Abstract: In 2022 Dassow and Jecker started investigations on the operational complexity of the pumping constant w.r.t.~two pumping lemmata. Since then, the research on pumping lemmata emerged to several variants for regular as well as context-free languages. Within this work, we further investigate the \emph{witnesses} of regular languages (firstly introduced by Rauch) w.r.t.~different pumping lemmata: For a language $L$ and a minimal pumping constant $K$, a \emph{witness} of a language is a word that cannot be \emph{pumped} by $K$ w.r.t.~the given pumping lemma. Here, the pumping constant is minimal if for all smaller numbers, the pumping lemma condition is not satisfied. Alongside combinatorial investigations on (unary) witness languages, we introduce and study the measure of scattered factor distance, which is related to minimal pumping constants. We present bounds on minimal pumping constants by the scattered factor distance and show that deciding, given two regular languages, whether one of them is equal to the witness language of the second is \coNP-complete.

Abstract: Twisted equations in free groups are equations in which a finite group of symmetries acts on the free group by automorphisms and anti-automorphisms, such as inversion and reversal; moreover, the variables may be subject to rational constraints. Algorithms from the theory of equations in virtually free groups solve such equations in PSPACE, but they are intricate. Our main result is an elementary algorithm for quadratic twisted equations, which extends the algorithm of Comerford and Edmunds (1981) for quadratic equations in free groups to the twisted setting and to rational constraints. Besides deciding solvability in PSPACE and yielding EDT0L descriptions of solution sets, our algorithm computes, for every quadratic expression, finitely many parametric forms describing its values as freely reduced words. As an application, we study the palindrome length of elements of free groups, that is, the least number of palindromes whose product they are. Guided by the parametric forms that our algorithm computes for products of palindromes, we found, and then proved directly, a characterization of the freely reduced words of palindrome length at most $n$ in terms of patterns describing how palindromes are nested by twisted conjugations, thereby confirming a Catalan structure conjectured by the second author. Consequently, these words form a context-free language, and palindrome length is computable in cubic time as well as in LOGCFL, answering a question of Bardakov, Shpilrain, and Tolstykh (2005).

Abstract: We investigate the computational complexity of analyzing the structural and behavioral properties of deterministic k-pebble automata, which represent a natural framework for studying minimal programmable machines. First, we provide an explicit construction of a three-pebble automaton U capable of simulating any deterministic finite automaton (DFA) on a given input word, establishing a link between pebble automata capabilities, Kolmogorov complexity, and automatic complexity. Next, we restrict our architectural framework to two-pebble programmable machines U(2,N) simulating DFAs with at most N states. We analyze the decision problem ANAL(U(2,5)), which asks whether a five-state program can separate two distinct finite words. By establishing a polynomial-time reduction from the quasi-identity checking problem for finite semigroups, we prove that ANAL(U(2,5)) is NP-complete. Finally, we explore the complexity of separating binary strings under the Exponential Time Hypothesis (ETH), showing how computational hardness implies polynomial lower bounds for the size of the smallest DFAs separating binary strings.

Abstract: Ambiguity measures the number of accepting runs in nondeterministic finite automata (NFA). We consider finitely ambiguous NFA, where there exists a constant $N$ such that over every word $w$ there are at most $N$ accepting runs. In such a case we also say that the NFA is $N$-ambiguous. Importantly $N$ depends only on the NFA, it does not depend on the length of the word. Weber and Seidl showed that every NFA is $N$-ambiguous for $N = 2^{O(n \log n)}$, where $n$ is the number of states. We improve this to $N = 2^{O(n)}$, which is asymptotically tight.

Abstract: The language $\mathrm{MIX}$ consists of all words over a three-letter alphabet that have an equal number of occurrences of each letter. It is also the word problem of $\mathbb{Z}^2$ with respect to a suitable choice of generators. The Kanazawa--Salvati conjecture states that $\mathrm{MIX}$ is not a well-nested multiple context-free language. Every well-nested multiple context-free language is an indexed language. We reduce the conjecture to an explicit combinatorial problem about tuples of words, which is easier to state than the original formulation in terms of arbitrary well-nested multiple context-free grammars. More generally, for every surjective monoid homomorphism $ψ\colon Σ^* \to \mathbb{Z}^d$, we define a family of well-nested multiple context-free grammars $G_ψ[r]$ for $r \geq 1$, each of which generates a sublanguage of $ψ^{-1}(\mathbf{0})$. We prove that every well-nested multiple context-free sublanguage of $ψ^{-1}(\mathbf{0})$ is contained in $L(G_ψ[r])$ for some $r \geq 1$. Using this family, we prove that the four-letter analogue $\mathrm{MIX}_4$, which is a word problem of $\mathbb{Z}^3$, is not a well-nested multiple context-free language. The proof reduces this claim to a result of Bishop--Elder--Evetts--Gallot--Levine stating that the two-letter analogue $\mathrm{MIX}_2$ is not generated by any non-branching multiple context-free grammar. The Kanazawa--Salvati conjecture itself remains open.

Abstract: In 1980 Bucher asked whether, given context-free languages $L\subseteq U$ with $U\setminus L$ infinite, there must be a context-free language $K$ between them for which both $K\setminus L$ and $U\setminus K$ are infinite. We give a negative answer. We first construct an infinite language $D$ with context-free complement such that, for every regular language $R$, either $D\cap R$ or $D\setminus R$ is finite. The words of $D$ encode computations of factorials; repetition of letters ensures that each finite automaton either accepts all but finitely many words of $D$ or rejects all but finitely many words of $D$, while a one-counter automaton recognizes errors in the encodings. We then construct $L$ and $U$ from the complement of $D$. A grammar argument shows that any context-free intermediate language $K$ would divide $D$ in the same way as some regular language. This proves the required impossibility. Both $L$ and $U$ can be taken over a binary alphabet.