Efficient Parallel Path Checking for Linear-Time Temporal Logic With Past and Bounds

Path checking, the special case of the model checking problem where the model under consideration is a single path, plays an important role in monitoring, testing, and verification. We prove that for linear-time temporal logic (LTL), path checking can be efficiently parallelized. In addition to the core logic, we consider the extensions of LTL with bounded-future (BLTL) and past-time (LTL+Past) operators. Even though both extensions improve the succinctness of the logic exponentially, path checking remains efficiently parallelizable: Our algorithm for LTL, LTL+Past, and BLTL+Past is in AC^1(logDCFL) \subseteq NC

On the Complexity of Temporal-Logic Path Checking

Given a formula in a temporal logic such as LTL or MTL, a fundamental problem is the complexity of evaluating the formula on a given finite word. For LTL, the complexity of this task was recently shown to be in NC. In this paper, we present an NC algorithm for MTL, a quantitative (or metric) extension of LTL, and give an NCC algorithm for UTL, the unary fragment of LTL. At the time of writing, MTL is the most expressive logic with an NC path-checking algorithm, and UTL is the most expressive fragment of LTL with a more efficient path-checking algorithm than for full LTL (subject to standard complexity-theoretic assumptions). We then establish a connection between LTL path checking and planar circuits, which we exploit to show that any further progress in determining the precise complexity of LTL path checking would immediately entail more efficient evaluation algorithms than are known for a certain class of planar circuits. The connection further implies that the complexity of LTL path checking depends on the Boolean connectives allowed: adding Boolean exclusive or yields a temporal logic with P-complete path-checking problem

Majority-Vote Cellular Automata, Ising Dynamics, and P-Completeness

We study cellular automata where the state at each site is decided by a majority vote of the sites in its neighborhood. These are equivalent, for a restricted set of initial conditions, to non-zero probability transitions in single spin-flip dynamics of the Ising model at zero temperature. We show that in three or more dimensions these systems can simulate Boolean circuits of AND and OR gates, and are therefore P-complete. That is, predicting their state t time-steps in the future is at least as hard as any other problem that takes polynomial time on a serial computer. Therefore, unless a widely believed conjecture in computer science is false, it is impossible even with parallel computation to predict majority-vote cellular automata, or zero-temperature single spin-flip Ising dynamics, qualitatively faster than by explicit simulation.Comment: 10 pages with figure

Parallel dynamics and computational complexity of the Bak-Sneppen model

The parallel computational complexity of the Bak-Sneppen evolution model is studied. It is shown that Bak-Sneppen histories can be generated by a massively parallel computer in a time that is polylogarithmic in the length of the history. In this parallel dynamics, histories are built up via a nested hierarchy of avalanches. Stated in another way, the main result is that the logical depth of producing a Bak-Sneppen history is exponentially less than the length of the history. This finding is surprising because the self-organized critical state of the Bak-Sneppen model has long range correlations in time and space that appear to imply that the dynamics is sequential and history dependent. The parallel dynamics for generating Bak-Sneppen histories is contrasted to standard Bak-Sneppen dynamics. Standard dynamics and an alternate method for generating histories, conditional dynamics, are both shown to be related to P-complete natural decision problems implying that they cannot be efficiently implemented in parallel.Comment: 37 pages, 12 figure

Model checking finite paths and trees

This thesis presents efficient parallel algorithms for checking temporal logic formulas over finite paths and trees. We show that LTL path checking is in AC1(logDCFL) and CTL tree checking is in AC2(logDCFL). For LTL with pastime and bounded modalities, which is an exponentially more succinct logic, we show that the path checking problem remains in AC1(logDCFL). Our results provide a foundation for efficient algorithms of various applications in monitoring, testing, and verification as well as for query processing for tree-datastructures, e.g. XML documents. The presented path and tree checking algorithms are based on efficient parallel evaluation strategies for monotone Boolean circuits. We reduce the evaluation of product circuits to the problem of evaluating one-input-face monotone planar Boolean circuits: for a monotone Boolean circuit that is a product of a tree and a path, we provide an AC1-reduction; for a monotone Boolean circuit that is a product of two trees, we provide an AC2-reduction. We develop a classification of Kripke structures with respect to the complexity of LTL model checking: Kripke structures for which the problem is PSPACE- complete, Kripke structures for which the problem is coNP-complete, and Kripke structures for which the problem is in NC.Wir präsentieren effiziente parallele Algorithmen zum Überprüfen der Erfülltheit von temporal logischen Formeln auf Pfaden und Bäumen. Wir zeigen, dass für die Logik LTL das Überprüfen von Ausführungspfaden in der Komplexitätsklasse AC1(logDCFL) liegt. Für die Logik CTL ist das Überprüfen von Bäumen in AC2(logDCFL). Für Erweiterungen von LTL mit Vergangenheit und beschränkten zeitlichen Modalitäten beweisen wir, dass Pfade ebenfalls in AC1(logDCFL) überprüft werden können, obwohl die Logik exponentiell kompakter ist als einfaches LTL. Unsere Resultate bielden eine Grundlage für effiziente Algorithmen für verschiedene Anwendungen in den Bereichen der Systemüberwachung, des Testens und der Verfikation sowie für die Anfragebearbeitung für Baumdatenstrukturen, wie zum Beispiel XML Dokumente. Die präsentierten Algorithmen zum Überprüfen von Pfaden und Bäumen basieren auf effizient parallelen Strategien zur Evaluierung von monotonen Boolschen Schaltkreisen. Wir reduzieren die Evaluierung von Produkt-Schaltkreisen auf das Problem der Evaluierung von monoton planaren Boolschen Schaltkreisen, bei denen sich alle Eingaben auf dem äußeren Rand befinden. Für monotone Boolsche Schaltkreise, die das Produkt von einem Baum und einem Pfad sind, geben wir eine AC1-Reduktion an. Für monotone Boolsche Schaltkreise, die das Produkt von zwei Bäumen sind, geben wir eine AC2-Reduktion an. Wir entwickeln eine Klassifizierung von Kripkestrukturen im Hinblick auf die Komplexität des Erfülltheitsproblems für LTL: Kripkestrukturen, für die das Problem PSPACE-vollständig ist, Kripkestrukturen, für die das Problem coNP- vollständig ist, und Kripkestrukturen, für die das Problem in NC liegt

Studies in Efficient Discrete Algorithms

This thesis consists of five papers within the design and analysis of efficient algorithms.In the first paper, we consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. We develop a combinatorial randomized algorithm that runs in subcubic time for a special class of graphs.In the second paper, we present a polynomial-time dynamic programming algorithm for optimal partitions of a complete edge-weighted graph, where the edges are weighted by the length of the unique shortest path connecting those vertices in the a priori given tree (shortest path metric induced by a tree). Our result resolves, in particular, the complexity status of the optimal partition problems in one-dimensional geometric (Euclidean) setting.In the third paper, we study the NP-hard problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles. We present an approximation algorithm with the approximation ratio 4 for the special case of the problem in which P is a so-called 3D histogram. We then apply it to compute the exact arithmetic matrix product of two matrices with non-negative integer entries. The computation is time-efficient if the 3D histograms induced by the input matrices can be partitioned into relatively few 3D rectangles.In the fourth paper, we present the first quasi-polynomial approximation schemes for the base of the number of triangulations of a planar point set and the base of the number of crossing-free spanning trees on a planar point set, respectively.In the fifth paper, we study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We present a fixed-parameter tractable algorithms for the detection of monomial having at least k distinct variables, parametrized with respect to k. Furthermore, we derive several hardness results on the detection of monomials with such properties within exact, parametrized and approximation complexity

Circuit Evaluation for Finite Semirings

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 R (i) has a solvable multiplicative semigroup and (ii) does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 != 0, then its circuit evaluation problem is in the complexity class DET (which is contained in NC^2). In all other cases, the circuit evaluation problem is P-complete