Kleene Algebra (KA) is a useful tool for proving that two programs are equivalent. Because KA's equational theory is decidable, it integrates well with interactive theorem provers. This raises the question: which equations can we (not) prove using the laws of KA? Moreover, which models of KA are complete, in the sense that they satisfy exactly the provable equations? Kozen (1994) answered these questions by characterizing KA in terms of its language model. Concretely, equivalences provable in KA are exactly those that hold for regular expressions. Pratt (1980) observed that KA is complete w.r.t. relational models, i.e., that its provable equations are those that hold for any relational interpretation. A less known result due to Palka (2005) says that finite models are complete for KA, i.e., that provable equivalences coincide with equations satisfied by all finite KAs. Phrased contrapositively, the latter is a finite model property (FMP): any unprovable equation is falsified by a finite KA. Both results can be argued using Kozen's theorem, but the implication is mutual: given that KA is complete w.r.t. finite (resp. relational) models, Palka's (resp. Pratt's) arguments show that it is complete w.r.t. the language model. We embark on a study of the different complete models of KA, and the connections between them. This yields a novel result subsuming those of Palka and Pratt, namely that KA is complete w.r.t. finite relational models. Next, we put an algebraic spin on Palka's techniques, which yield a new elementary proof of the finite model property, and by extension, of Kozen's and Pratt's theorems. In contrast with earlier approaches, this proof relies not on minimality or bisimilarity of automata, but rather on representing the regular expressions involved in terms of transformation automata.

We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) $\mathbf A = (A,\le, \cdot, \sim,-)$, and in particular every locally integral involutive semiring, decomposes in a unique way into a family $\{\mathbf A_p : p\in A^+\}$ of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms $Φ= \{φ_{pq}: \mathbf A_p\to \mathbf A_q : p\le q\}$, indexed over the positive cone $(A^+,\le)$, so that the structure of $\mathbf A$ can be recovered as a glueing $\int_Φ\mathbf A_p$ of its integral components along $Φ$. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family of integral ipo-monoids $\{\mathbf A_p : p\in D\}$, indexed over a join-semilattice $(D,\lor)$ along a family of monoid homomorphisms $Φ$ is an ipo-semigroup.

A set C of vertices in a graph G = (V, E) is an identifying code if it is dominating and any two vertices of V are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala’s contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes.

Logical modeling is a powerful tool in biology, offering a system-level understanding of the complex interactions that govern biological processes. A gap that hinders the scalability of logical models is the need to specify the update function of every vertex in the network depending on the status of its predecessors. To address this, we introduce in this paper the concept of strong regulation, where a vertex is only updated to active/inactive if all its predecessors agree in their influences; otherwise, it is set to ambiguous. We explore the interplay between active, inactive, and ambiguous influences in a network. We discuss the existence of phenotype attractors in such networks, where the status of some of the variables is fixed to active/inactive, while the others can have an arbitrary status, including ambiguous.

Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Given a graph G = ( V( G), E( G)), a set M ⊆ V( G) is a distance-edge-monitoring set if for every edge e ∈ E( G), there is a vertex x ∈ M and a vertex y ∈ V( G) such that the edge e belongs to all shortest paths between x and y. The smallest size of such a set in G is denoted by dem( G). Denoted by G – e (resp. G\ u) the subgraph of G obtained by removing the edge e from G (resp. a vertex u together with all its incident edges from G). In this paper, we first show that dem( G – e) – dem( G) ≤ 2 for any graph G and edge e ∈ E( G). Moreover, the bound is sharp. Next, we construct two graphs G and H to show that dem( G) – dem( G\ u) and dem( H \ v) – dem( H) can be arbitrarily large, where u ∈ V( G) and v ∈ V( H). We also study the relation between dem( H) and dem( G), where H is a subgraph of G. In the end, we give an algorithm to judge whether the distance-edge-monitoring set still remain in the resulting graph when any edge of a graph G is deleted.

Computing the rotation distance between two binary trees with n internal nodes efficiently (in poly( n) time) is a long standing open question in the study of height balancing in tree data structures. In this paper, we initiate the study of this problem bounding the rank of the trees given at the input (defined in [1] in the context of decision trees). We define the rank-bounded rotation distance between two given full binary trees T1 and T2 (with n internal nodes) of rank at most r = max{rank( T1), rank( T2)}, denoted by dR( T1, T2), as the length of the shortest sequence of rotations that transforms T1 to T2 with the restriction that the intermediate trees must be of rank at most r. We show that the rotation distance problem reduces in polynomial time to the rank bounded rotation distance problem. This motivates the study of the problem in the combinatorial and algorithmic frontiers. Observing that trees with rank 1 coincide exactly with skew trees (full binary trees where every internal node has at least one leaf as a child), we show the following results in this frontier: We present an O( n2) time algorithm for computing dR( T1, T2). That is, when the given full binary trees are skew trees (we call this variant the skew rotation distance problem) - where the intermediate trees are restricted to be skew as well. In particular, our techniques imply that for any two skew trees dR( T1, T2) ≤ n2. We show the following upper bound: for any two full binary trees T1 and T2 of rank r1 and r2 respectively, we have that: dR( T1, T2) ≤ n2(1 + (2 n + 1)( r1 + r2 – 2)) where r = max{ r1, r2}. This bound is asymptotically tight for r = 1. En route to our proof of the above theorems, we associate full binary trees to permutations and relate the rotation operation on trees to transpositions in the corresponding permutations. We give exact combinatorial characterizations of permutations that correspond to full binary trees and full skew binary trees under this association. We also precisely characterize the transpositions that correspond to the set of rotations in full binary trees. We also study bi-variate polynomials associated with binary trees (introduced by [2]), and show characterizations and algorithms for computing rotation distances for the case of full skew trees using them.

The study of strings is an important combinatorial field that precedes the digital computer. Strings can be very long, trillions of letters, so it is important to find compact representations. Here we first survey various forms of one potential compaction methodology, the cover of a given string x, initially proposed in a simple form in 1990, but increasingly of interest as more sophisticated variants have been discovered. We then consider covering by a seed; that is, a cover of a superstring of x. We conclude with many proposals for research directions that could make significant contributions to string processing in future.

In a former paper [1] the concept of Bipartite PageRank was introduced and a theorem on the limit of authority flowing between nodes for personalized PageRank has been generalized. In this paper we want to extend those results to multimodal networks. In particular we deal with a hypergraph type that may be used for describing multimodal network where a hyperlink connects nodes from each of the modalities. We introduce a generalisation of PageRank for such graphs and define the respective random walk model that can be used for computations. We state and prove theorems on the limit of outflow of authority for cases where individual modalities have identical and distinct damping factors.

We study minimization problems for deterministic ω-automata in the presence of don’t care words. We prove that the number of priorities in deterministic parity automata can be efficiently minimized under an arbitrary set of don’t care words. We derive that from a more general result from which one also obtains an efficient minimization algorithm for deterministic parity automata with informative right-congruence (without don’t care words). We then analyze languages of don’t care words with a trivial right-congruence. For such sets of don’t care words it is known that weak deterministic Büchi automata (WDBA) have a unique minimal automaton that can be efficiently computed from a given WDBA (Eisinger, Klaedtke 2006). We give a congruence-based characterization of the corresponding minimal WDBA, and show that the don’t care minimization results for WDBA do not extend to deterministic ω-automata with informative right-congruence: for this class there is no unique minimal automaton for a given don’t care set with trivial right congruence, and the minimization problem is NP-hard. Finally, we extend an active learning algorithm for WDBA (Maler, Pnueli 1995) to the setting with an additional set of don’t care words with trivial right-congruence.

Graph embeddings play a significant role in the design and analysis of parallel algorithms. It is a mapping of the topological structure of a guest graph G into a host graph H, which is represented as a one-to-one mapping from the vertex set of the guest graph to the vertex set of the host graph. In multiprocessing systems, the interconnection networks enhance the efficient communication between the components in the system. Obtaining minimum wirelength in embedding problems is significant in the designing of networks and simulating one architecture by another. In this paper, we determine the wirelength of embedding 3-ary n-cubes into cylinders and certain trees.