arXiv subjects

Markus Lohrey

Publications and source records attributed to Markus Lohrey.

At least 55 records · Page 3Linked to original sources

arXiv2017

Knapsack Problems for Wreath Products

In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under wreath product. As a consequence, we obtain the decidability of knapsack for free solvable groups. Finally, it is shown that for every non-trivial abelian group $G$, knapsack (as well as the related subset sum problem) for the wreath product $G \wr \mathbb{Z}$ is NP-complete.

math.GR↗

arXiv2017

Path Checking for MTL and TPTL over Data Words

Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are quantitative extensions of linear temporal logic, which are prominent and widely used in the verification of real-timed systems. It was recently shown that the path checking problem for MTL, when evaluated over finite timed words, is in the parallel complexity class NC. In this paper, we derive precise complexity results for the path-checking problem for MTL and TPTL when evaluated over infinite data words over the non-negative integers. Such words may be seen as the behaviours of one-counter machines. For this setting, we give a complete analysis of the complexity of the path-checking problem depending on the number of register variables and the encoding of constraint numbers (unary or binary). As the two main results, we prove that the path-checking problem for MTL is P-complete, whereas the path-checking problem for TPTL is PSPACE-complete. The results yield the precise complexity of model checking deterministic one-counter machines against formulae of MTL and TPTL.

cs.LO↗

arXiv2017

Approximation ratio of RePair

In a seminal paper of Charikar et al.~on the smallest grammar problem, the authors derive upper and lower bounds on the approximation ratios for several grammar-based compressors. Here we improve the lower bound for the famous {\sf RePair} algorithm from $Ω(\sqrt{\log n})$ to $Ω(\log n/\log\log n)$. The family of words used in our proof is defined over a binary alphabet, while the lower bound from Charikar et al. needs an alphabet of logarithmic size in the length of the provided words.

cs.DS↗

arXiv2017

Universal Tree Source Coding Using Grammar-Based Compression

We apply so-called tree straight-line programs to the problem of lossless compression of binary trees. We derive upper bound on the maximal pointwise redundancy (or worst-case redundancy) that improve previous bounds obtained by Zhang, Yang, and Kieffer using directed acyclic graphs. Using this, we obtain universal codes for new classes of structered tree sources.

cs.IT↗

arXiv2016

The Complexity of Knapsack in Graph Groups

Myasnikov et al. have introduced the knapsack problem for arbitrary finitely generated groups. In previous work, the authors proved that for each graph group, the knapsack problem can be solved in $\mathsf{NP}$. Here, we determine the exact complexity of the problem for every graph group. While the problem is $\mathsf{TC}^0$-complete for complete graphs, it is $\mathsf{LogCFL}$-complete for each (non-complete) transitive forest. For every remaining graph, the problem is $\mathsf{NP}$-complete.

math.GR↗

arXiv2016

Circuit Evaluation for Finite Semirings

The computational complexity of the circuit evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or multiplicative identity. The following dichotomy is shown: If a finite semiring is such that (i) the multiplicative semigroup is solvable and (ii) it does not contain a subsemiring with an additive identity $0$ and a multiplicative identity $1 \neq 0$, then the circuit evaluation problem for the semiring is in $\mathsf{DET} \subseteq \mathsf{NC}^2$. In all other cases, the circuit evaluation problem is $\mathsf{P}$-complete.

cs.CC↗

arXiv2016

Efficient Quantile Computation in Markov Chains via Counting Problems for Parikh Images

A cost Markov chain is a Markov chain whose transitions are labelled with non-negative integer costs. A fundamental problem on this model, with applications in the verification of stochastic systems, is to compute information about the distribution of the total cost accumulated in a run. This includes the probability of large total costs, the median cost, and other quantiles. While expectations can be computed in polynomial time, previous work has demonstrated that the computation of cost quantiles is harder but can be done in PSPACE. In this paper we show that cost quantiles in cost Markov chains can be computed in the counting hierarchy, thus providing evidence that computing those quantiles is likely not PSPACE-hard. We obtain this result by exhibiting a tight link to a problem in formal language theory: counting the number of words that are both accepted by a given automaton and have a given Parikh image. Motivated by this link, we comprehensively investigate the complexity of the latter problem. Among other techniques, we rely on the so-called BEST theorem for efficiently computing the number of Eulerian circuits in a directed graph.

cs.FL↗

arXiv2015

Traversing Grammar-Compressed Trees with Constant Delay

A grammar-compressed ranked tree is represented with a linear space overhead so that a single traversal step, i.e., the move to the parent or the i-th child, can be carried out in constant time. Moreover, we extend our data structure such that equality of subtrees can be checked in constant time.

cs.DS↗

arXiv2015

Tree compression using string grammars

We study the compressed representation of a ranked tree by a (string) straight-line program (SLP) for its preorder traversal, and compare it with the well-studied representation by straight-line context free tree grammars (which are also known as tree straight-line programs or TSLPs). Although SLPs turn out to be exponentially more succinct than TSLPs, we show that many simple tree queries can still be performed efficiently on SLPs, such as computing the height and Horton-Strahler number of a tree, tree navigation, or evaluation of Boolean expressions. Other problems on tree traversals turn out to be intractable, e.g. pattern matching and evaluation of tree automata.

cs.FL↗

arXiv2015

Constructing small tree grammars and small circuits for formulas

It is shown that every tree of size $n$ over a fixed set of $σ$ different ranked symbols can be decomposed (in linear time as well as in logspace) into $O\big(\frac{n}{\log_σn}\big) = O\big(\frac{n \log σ}{\log n}\big)$ many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size $O\big(\frac{n}{\log_σn}\big)$, which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case size of the computed grammar, except for the top dag of Bille et al., for which only the weaker upper bound of $O\big(\frac{n}{\log_σ^{0.19} n}\big)$ (which was very recently improved to $O\big(\frac{n \cdot \log \log_σn}{\log_σn}\big)$ by Hübschle-Schneider and Raman) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size $n$, in which only $m \leq n$ different variables occur, can be transformed (in linear time as well as in logspace) into an arithmetical circuit of size $O\big(\frac{n \cdot \log m}{\log n}\big)$ and depth $O(\log n)$. This refines a classical result of Brent from 1974, according to which an arithmetical formula of size $n$ can be transformed into a logarithmic depth circuit of size $O(n)$.

cs.DS↗

arXiv2015

Knapsack in graph groups, HNN-extensions and amalgamated products

It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for graph groups. This result even holds if the group elements are represented in a compressed form by SLPs, which generalizes the classical NP-completeness result of the integer knapsack problem. We also prove general transfer results: NP-membership of the knapsack problem is passed on to finite extensions, HNN-extensions over finite associated subgroups, and amalgamated products with finite identified subgroups.

math.GR↗

arXiv2015

Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups

It is shown that the knapsack problem (introduced by Myasnikov, Nikolaev, and Ushakov) is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2). Moreover, for the discrete Heisenberg group itself, the knapsack problem is decidable. Hence, decidability of the knapsack problem is not preserved under direct products. It is also shown that for every co-context-free group, the knapsack problem is decidable. For the subset sum problem (also introduced by Myasnikov, Nikolaev, and Ushakov) we show that it belongs to the class NL (nondeterministic logspace) for every finitely generated virtually nilpotent group and that there exists a polycyclic group with an NP-complete subset sum problem.

math.GR↗

arXiv2015

Satisfiability of ECTL* with tree constraints

Recently, we have shown that satisfiability for $\mathsf{ECTL}^*$ with constraints over $\mathbb{Z}$ is decidable using a new technique. This approach reduces the satisfiability problem of $\mathsf{ECTL}^*$ with constraints over some structure A (or class of structures) to the problem whether A has a certain model theoretic property that we called EHD (for "existence of homomorphisms is decidable"). Here we apply this approach to concrete domains that are tree-like and obtain several results. We show that satisfiability of $\mathsf{ECTL}^*$ with constraints is decidable over (i) semi-linear orders (i.e., tree-like structures where branches form arbitrary linear orders), (ii) ordinal trees (semi-linear orders where the branches form ordinals), and (iii) infinitely branching trees of height h for each fixed $h\in \mathbb{N}$. We prove that all these classes of structures have the property EHD. In contrast, we introduce Ehrenfeucht-Fraisse-games for $\mathsf{WMSO}+\mathsf{B}$ (weak $\mathsf{MSO}$ with the bounding quantifier) and use them to show that the infinite (order) tree does not have property EHD. As a consequence, a different approach has to be taken in order to settle the question whether satisfiability of $\mathsf{ECTL}^*$ (or even $\mathsf{LTL}$) with constraints over the infinite (order) tree is decidable.

cs.LO↗

arXiv2015

Parallel Identity Testing for Skew Circuits with Big Powers and Applications

Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labelled with powers $x^n$ for binary encoded numbers $n$. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to $\mathsf{coRNC}^2$, which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented: (i) Equivalence of higher-dimensional straight-line programs can be tested in $\mathsf{coRNC}^2$; this result is even new in the one-dimensional case, where the straight-line programs produce strings. (ii) The compressed word problem (or circuit evaluation problem) for certain wreath products of finitely generated abelian groups belongs to $\mathsf{coRNC}^2$.

cs.CC↗

arXiv2015

Compressed Tree Canonization

Straight-line (linear) context-free tree (SLT) grammars have been used to compactly represent ordered trees. It is well known that equivalence of SLT grammars is decidable in polynomial time. Here we extend this result and show that isomorphism of unordered trees given as SLT grammars is decidable in polynomial time. The proof constructs a compressed version of the canonical form of the tree represented by the input SLT grammar. The result is generalized to unrooted trees by "re-rooting" the compressed trees in polynomial time. We further show that bisimulation equivalence of unrooted unordered trees represented by SLT grammars is decidable in polynomial time. For non-linear SLT grammars which can have double-exponential compression ratios, we prove that unordered isomorphism is PSPACE-hard and in EXPTIME. The same complexity bounds are shown for bisimulation equivalence.

cs.DS↗

arXiv2015

Evaluating Matrix Circuits

The circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups is studied. The best upper bound for this problem is $\mathsf{coRP}$, which is shown by a reduction to polynomial identity testing. Conversely, the compressed word problem for the linear group $\mathsf{SL}_3(\mathbb{Z})$ is equivalent to polynomial identity testing. In the paper, it is shown that the compressed word problem for every finitely generated nilpotent group is in $\mathsf{DET} \subseteq \mathsf{NC}^2$. Within the larger class of polycyclic groups we find examples where the compressed word problem is at least as hard as polynomial identity testing for skew arithmetic circuits.

cs.CC↗

arXiv2014

Processing Succinct Matrices and Vectors

We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for MTDD_+-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is #P-complete.

cs.DS↗

arXiv2014

The First-Order Theory of Ground Tree Rewrite Graphs

We prove that the complexity of the uniform first-order theory of ground tree rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n)). Providing a matching lower bound, we show that there is some fixed ground tree rewrite graph whose first-order theory is hard for ATIME(2^{2^{poly(n)}},poly(n)) with respect to logspace reductions. Finally, we prove that there exists a fixed ground tree rewrite graph together with a single unary predicate in form of a regular tree language such that the resulting structure has a non-elementary first-order theory.

cs.LO↗