Automatic Equivalence Structures of Polynomial Growth

In this paper we study the class EqP of automatic equivalence structures of the form ?=(D, E) where the domain D is a regular language of polynomial growth and E is an equivalence relation on D. Our goal is to investigate the following two foundational problems (in the theory of automatic structures) aimed for the class EqP. The first is to find algebraic characterizations of structures from EqP, and the second is to investigate the isomorphism problem for the class EqP. We provide full solutions to these two problems. First, we produce a characterization of structures from EqP through multivariate polynomials. Second, we present two contrasting results. On the one hand, we prove that the isomorphism problem for structures from the class EqP is undecidable. On the other hand, we prove that the isomorphism problem is decidable for structures from EqP with domains of quadratic growth

Visibly Pushdown Languages over Sliding Windows

We investigate the class of visibly pushdown languages in the sliding window model. A sliding window algorithm for a language L receives a stream of symbols and has to decide at each time step whether the suffix of length n belongs to L or not. The window size n is either a fixed number (in the fixed-size model) or can be controlled by an adversary in a limited way (in the variable-size model). The main result of this paper states that for every visibly pushdown language the space complexity in the variable-size sliding window model is either constant, logarithmic or linear in the window size. This extends previous results for regular languages

The Complexity of Bisimulation and Simulation on Finite Systems

In this paper the computational complexity of the (bi)simulation problem over restricted graph classes is studied. For trees given as pointer structures or terms the (bi)simulation problem is complete for logarithmic space or NC$^1$, respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and S\'antha. Furthermore, if only one of the input graphs is required to be a tree, the bisimulation (simulation) problem is contained in AC$^1$ (LogCFL). In contrast, it is also shown that the simulation problem is P-complete already for graphs of bounded path-width

Compression by Contracting Straight-Line Programs

In grammar-based compression a string is represented by a context-free grammar, also called a straight-line program (SLP), that generates only that string. We refine a recent balancing result stating that one can transform an SLP of size $g$ in linear time into an equivalent SLP of size $O(g)$ so that the height of the unique derivation tree is $O(\log N)$ where $N$ is the length of the represented string (FOCS 2019). We introduce a new class of balanced SLPs, called contracting SLPs, where for every rule $A \to \beta_1 \dots \beta_k$ the string length of every variable $\beta_i$ on the right-hand side is smaller by a constant factor than the string length of $A$. In particular, the derivation tree of a contracting SLP has the property that every subtree has logarithmic height in its leaf size. We show that a given SLP of size $g$ can be transformed in linear time into an equivalent contracting SLP of size $O(g)$ with rules of constant length. We present an application to the navigation problem in compressed unranked trees, represented by forest straight-line programs (FSLPs). We extend a linear space data structure by Reh and Sieber (2020) by the operation of moving to the $i$-th child in time $O(\log d)$ where $d$ is the degree of the current node. Contracting SLPs are also applied to the finger search problem over SLP-compressed strings where one wants to access positions near to a pre-specified finger position, ideally in $O(\log d)$ time where $d$ is the distance between the accessed position and the finger. We give a linear space solution where one can access symbols or move the finger in time $O(\log d + \log^{(t)} N)$ for any constant $t$ where $\log^{(t)} N$ is the $t$-fold logarithm of $N$. This improves a previous solution by Bille, Christiansen, Cording, and G{\o}rtz (2018) with access/move time $O(\log d + \log \log N)$

Revisiting Membership Problems in Subclasses of Rational Relations

We revisit the membership problem for subclasses of rational relations over finite and infinite words: Given a relation R in a class C_2, does R belong to a smaller class C_1? The subclasses of rational relations that we consider are formed by the deterministic rational relations, synchronous (also called automatic or regular) relations, and recognizable relations. For almost all versions of the membership problem, determining the precise complexity or even decidability has remained an open problem for almost two decades. In this paper, we provide improved complexity and new decidability results. (i) Testing whether a synchronous relation over infinite words is recognizable is NL-complete (PSPACE-complete) if the relation is given by a deterministic (nondeterministic) omega-automaton. This fully settles the complexity of this recognizability problem, matching the complexity of the same problem over finite words. (ii) Testing whether a deterministic rational binary relation is recognizable is decidable in polynomial time, which improves a previously known double exponential time upper bound. For relations of higher arity, we present a randomized exponential time algorithm. (iii) We provide the first algorithm to decide whether a deterministic rational relation is synchronous. For binary relations the algorithm even runs in polynomial time

A Characterization of Wreath Products Where Knapsack Is Decidable

The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group $G$ and takes as input group elements $g_1,\ldots,g_n,g\in G$ and asks whether there are $x_1,\ldots,x_n\ge 0$ with $g_1^{x_1}\cdots g_n^{x_n}=g$. We study the knapsack problem for wreath products $G\wr H$ of groups $G$ and $H$. Our main result is a characterization of those wreath products $G\wr H$ for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors $G$ and $H$. To this end, we introduce two decision problems, the intersection knapsack problem and its restriction, the positive intersection knapsack problem. Moreover, we apply our main result to $H_3(\mathbb{Z})$, the discrete Heisenberg group, and to Baumslag-Solitar groups $\mathsf{BS}(1,q)$ for $q\ge 1$. First, we show that the knapsack problem is undecidable for $G\wr H_3(\mathbb{Z})$ for any $G\ne 1$. This implies that for $G\ne 1$ and for infinite and virtually nilpotent groups $H$, the knapsack problem for $G\wr H$ is decidable if and only if $H$ is virtually abelian and solvability of systems of exponent equations is decidable for $G$. Second, we show that the knapsack problem is decidable for $G\wr\mathsf{BS}(1,q)$ if and only if solvability of systems of exponent equations is decidable for $G$