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Hendrik Jan Hoogeboom

Publications and source records attributed to Hendrik Jan Hoogeboom.

At least 19 recordsLinked to original sources
arXiv2022

Deterministic Non-cooperative Binding in Two-Dimensional Tile Assembly Systems Must Have Ultimately Periodic Paths

We consider non-cooperative binding, so-called 'temperature 1', in deterministic or directed (called here confluent) tile self-assembly systems in two dimensions and show a necessary and sufficient condition for such system to have an ultimately periodic assembly path. We prove that an infinite maximal assembly has an ultimately periodic assembly path if and only if it contains an infinite assembly path that does not intersect a periodic path in the Z2 grid. Moreover we show that every infinite assembly must satisfy this condition, and therefore, contains an ultimately periodic path. This result is obtained through a super-position and a combination of two paths that produce a new path with desired properties, a technique that we call co-grow of two paths. The paper is an updated and improved version of the first part of arXiv 1901.08575.

cs.CC↗
arXiv2019

Deterministic 2-Dimensional Temperature-1 Tile Assembly Systems Cannot Compute

We consider non cooperative binding in so called `temperature 1', in deterministic (here called {\it confluent}) tile self-assembly systems (1-TAS) and prove the standing conjecture that such systems do not have universal computational power. We call a TAS whose maximal assemblies contain at least one ultimately periodic assembly path {\it para-periodic}. We observe that a confluent 1-TAS has at most one maximal producible assembly, $α_{max}$, that can be considered a union of path assemblies, and we show that such a system is always para-periodic. This result is obtained through a superposition and a combination of two paths that produce a new path with desired properties, a technique that we call \emph{co-grow} of two paths. Moreover we provide a characterization of an $α_{max}$ of a confluent 1-TAS as one of two possible cases, so called, a grid or a disjoint union of combs. To a given $α_{max}$ we can associate a finite labeled graph, called \emph{quipu}, such that the union of all labels of paths in the quipu equals $α_{max}$, therefore giving a finite description for $α_{max}$. This finite description implies that $α_{max}$ is a union of semi-affine subsets of $\mathbb{Z}^2$ and since such a finite description can be algorithmicly generated from any 1-TAS, 1-TAS cannot have universal computational power.

cs.CC↗
arXiv2018

XML Navigation and Transformation by Tree-Walking Automata and Transducers with Visible and Invisible Pebbles

The pebble tree automaton and the pebble tree transducer are enhanced by additionally allowing an unbounded number of "invisible" pebbles (as opposed to the usual "visible" ones). The resulting pebble tree automata recognize the regular tree languages (i.e., can validate all generalized DTD's) and hence can find all matches of MSO definable patterns. Moreover, when viewed as a navigational device, they lead to an XPath-like formalism that has a path expression for every MSO definable binary pattern. The resulting pebble tree transducers can apply arbitrary MSO definable tests to (the observable part of) their configurations, they (still) have a decidable typechecking problem, and they can model the recursion mechanism of XSLT. The time complexity of the typechecking problem for conjunctive queries that use MSO definable patterns can often be reduced through the use of invisible pebbles.

cs.FL↗
arXiv2017

A Graph Isomorphism Condition and Equivalence of Reaction Systems

We consider global dynamics of reaction systems as introduced by Ehrenfeucht and Rozenberg. The dynamics is represented by a directed graph, the so-called transition graph, and two reaction systems are considered equivalent if their corresponding transition graphs are isomorphic. We introduce the notion of a skeleton (a one-out graph) that uniquely defines a directed graph. We provide the necessary and sufficient conditions for two skeletons to define isomorphic graphs. This provides a necessary and sufficient condition for two reactions systems to be equivalent, as well as a characterization of the directed graphs that correspond to the global dynamics of reaction systems.

math.CO↗
arXiv2016

Acyclic Constraint Logic and Games

Non-deterministic Constraint Logic is a family of graph games introduced by Demaine and Hearn that facilitates the construction of complexity proofs. It is convenient for the analysis of games, providing a uniform view. We focus on the acyclic version, apply this to Klondike, Mahjong Solitaire and Nonogram (that requires planarity), and discuss the more complicated game of Dou Shou Qi. While for the first three games we reobtain known characterizations in a simple and uniform manner, the result for Dou Shou Qi is new.

cs.CC↗
arXiv2014

Quaternary Bicycle Matroids and the Penrose Polynomial for Delta-Matroids

In contrast to matroids, vf-safe delta-matroids have three kinds of minors and are closed under the operations of twist and loop complementation. We show that the delta-matroids representable over GF(4) with respect to the nontrivial automorphism of GF(4) form a subclass of the vf-safe delta-matroids closed under twist and loop complementation. In particular, quaternary matroids are vf-safe. Using this result, we show that the matroid of a bicycle space of a quaternary matroid M is obtained from M by using loop complementation. As a consequence, the matroid of a bicycle space of a quaternary matroid M is independent of the chosen representation. This also leads to, e.g., an extension of a known parity-type characterization of the bicycle dimension, a generalization of the tripartition of Rosenstiehl and Read [Ann. Disc. Math. (1978)], and a suitable generalization of the dual notions of bipartite and Eulerian binary matroids to a vf-safe delta-matroids. Finally, we generalize a number of results concerning the Penrose polynomial from binary matroids to vf-safe delta-matroids. In this general setting the Penrose polynomial turns out to have a recursive relation much like the recursive relation of the Tutte polynomial.

math.CO↗
arXiv2014

Quaternary matroids are vf-safe

Binary delta-matroids are closed under vertex flips, which consist of the natural operations of twist and loop complementation. In this note we provide an extension of this result from GF(2) to GF(4). As a consequence, quaternary matroids are "safe" under vertex flips (vf-safe for short). As an application, we find that the matroid of a bicycle space of a quaternary matroid is independent of the chosen representation. This extends a result of Vertigan [J. Comb. Theory B (1998)] concerning the bicycle dimension of quaternary matroids.

math.CO↗
arXiv2014

Interlace Polynomials for Multimatroids and Delta-Matroids

We provide a unified framework in which the interlace polynomial and several related graph polynomials are defined more generally for multimatroids and delta-matroids. Using combinatorial properties of multimatroids rather than graph-theoretical arguments, we find that various known results about these polynomials, including their recursive relations, are both more efficiently and more generally obtained. In addition, we obtain several interrelationships and results for polynomials on multimatroids and delta-matroids that correspond to new interrelationships and results for the corresponding graphs polynomials. As a tool we prove the equivalence of tight 3-matroids and delta-matroids closed under the operations of twist and loop complementation, called vf-safe delta-matroids. This result is of independent interest and related to the equivalence between tight 2-matroids and even delta-matroids observed by Bouchet.

math.CO↗
arXiv2012

Nullity and Loop Complementation for Delta-Matroids

We show that the symmetric difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal "null space" while the third has a larger dimension.

math.CO↗
arXiv2012

The adjacency matroid of a graph

If $G$ is a looped graph, then its adjacency matrix represents a binary matroid $M_{A}(G)$ on $V(G)$. $M_{A}(G)$ may be obtained from the delta-matroid represented by the adjacency matrix of $G$, but $M_{A}(G)$ is less sensitive to the structure of $G$. Jaeger proved that every binary matroid is $M_{A}(G)$ for some $G$ [Ann. Discrete Math. 17 (1983), 371-376]. The relationship between the matroidal structure of $M_{A}(G)$ and the graphical structure of $G$ has many interesting features. For instance, the matroid minors $M_{A}(G)-v$ and $M_{A}(G)/v$ are both of the form $M_{A}(G^{\prime}-v)$ where $G^{\prime}$ may be obtained from $G$ using local complementation. In addition, matroidal considerations lead to a principal vertex tripartition, distinct from the principal edge tripartition of Rosenstiehl and Read [Ann. Discrete Math. 3 (1978), 195-226]. Several of these results are given two very different proofs, the first involving linear algebra and the second involving set systems or delta-matroids. Also, the Tutte polynomials of the adjacency matroids of $G$ and its full subgraphs are closely connected to the interlace polynomial of Arratia, Bollobás and Sorkin [Combinatorica 24 (2004), 567-584].

math.CO↗
arXiv2011

The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems

We study the interplay between principal pivot transform (pivot) and loop complementation for graphs. This is done by generalizing loop complementation (in addition to pivot) to set systems. We show that the operations together, when restricted to single vertices, form the permutation group S_3. This leads, e.g., to a normal form for sequences of pivots and loop complementation on graphs. The results have consequences for the operations of local complementation and edge complementation on simple graphs: an alternative proof of a classic result involving local and edge complementation is obtained, and the effect of sequences of local complementations on simple graphs is characterized.

cs.DM↗
arXiv2010

Maximal Pivots on Graphs with an Application to Gene Assembly

We consider principal pivot transform (pivot) on graphs. We define a natural variant of this operation, called dual pivot, and show that both the kernel and the set of maximally applicable pivots of a graph are invariant under this operation. The result is motivated by and applicable to the theory of gene assembly in ciliates.

math.CO↗
arXiv2010

Nullity Invariance for Pivot and the Interlace Polynomial

We show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given (square) matrix is described by the symmetric difference operator (for sets). We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof.

cs.DM↗
arXiv2008

Pivots, Determinants, and Perfect Matchings of Graphs

We give a characterization of the effect of sequences of pivot operations on a graph by relating it to determinants of adjacency matrices. This allows us to deduce that two sequences of pivot operations are equivalent iff they contain the same set S of vertices (modulo two). Moreover, given a set of vertices S, we characterize whether or not such a sequence using precisely the vertices of S exists. We also relate pivots to perfect matchings to obtain a graph-theoretical characterization. Finally, we consider graphs with self-loops to carry over the results to sequences containing both pivots and local complementation operations.

math.CO↗
arXiv2007

Extending the Overlap Graph for Gene Assembly in Ciliates

Gene assembly is an intricate biological process that has been studied formally and modeled through string and graph rewriting systems. Recently, a restriction of the general (intramolecular) model, called simple gene assembly, has been introduced. This restriction has subsequently been defined as a string rewriting system. We show that by extending the notion of overlap graph it is possible to define a graph rewriting system for two of the three types of rules that make up simple gene assembly. It turns out that this graph rewriting system is less involved than its corresponding string rewriting system. Finally, we give characterizations of the `power' of both types of graph rewriting rules. Because of the equivalence of these string and graph rewriting systems, the given characterizations can be carried over to the string rewriting system.

cs.LO↗
arXiv2007

Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure

String languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first-order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the transitive closure. More precisely, first-order logic with k-ary deterministic transitive closure has the same power as deterministic automata walking on their input with k heads, additionally using a finite set of nested pebbles. This result is valid for strings, ordered trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toruses, and rectangular mazes. For nondeterministic automata, the logic is restricted to positive occurrences of transitive closure. The special case of k=1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary deterministic transitive closure. This refines our earlier result that placed these automata between first-order and monadic second-order logic on trees.

cs.LO↗
arXiv2007

How Overlap Determines the Macronuclear Genes in Ciliates

Formal models for gene assembly in ciliates have been developed, in particular the string pointer reduction system (SPRS) and the graph pointer reduction system (GPRS). The reduction graph is a valuable tool within the SPRS, revealing much information about how gene assembly is performed for a given gene. The GPRS is more abstract than the SPRS and not all information present in the SPRS is retained in the GPRS. As a consequence the reduction graph cannot be defined for the GPRS in general, but we show that it can be defined (in an equivalent manner as defined for the SPRS) if we restrict ourselves to so-called realistic overlap graphs. Fortunately, only these graphs correspond to genes occurring in nature. Defining the reduction graph within the GPRS allows one to carry over several results within the SPRS that rely on the reduction graph.

cs.LO↗