Decremental Single-Source Reachability in Planar Digraphs

In this paper we show a new algorithm for the decremental single-source reachability problem in directed planar graphs. It processes any sequence of edge deletions in $O(n\log^2{n}\log\log{n})$ total time and explicitly maintains the set of vertices reachable from a fixed source vertex. Hence, if all edges are eventually deleted, the amortized time of processing each edge deletion is only $O(\log^2 n \log \log n)$, which improves upon a previously known $O(\sqrt{n})$ solution. We also show an algorithm for decremental maintenance of strongly connected components in directed planar graphs with the same total update time. These results constitute the first almost optimal (up to polylogarithmic factors) algorithms for both problems. To the best of our knowledge, these are the first dynamic algorithms with polylogarithmic update times on general directed planar graphs for non-trivial reachability-type problems, for which only polynomial bounds are known in general graphs

Synthesising Strategy Improvement and Recursive Algorithms for Solving 2.5 Player Parity Games

2.5 player parity games combine the challenges posed by 2.5 player reachability games and the qualitative analysis of parity games. These two types of problems are best approached with different types of algorithms: strategy improvement algorithms for 2.5 player reachability games and recursive algorithms for the qualitative analysis of parity games. We present a method that - in contrast to existing techniques - tackles both aspects with the best suited approach and works exclusively on the 2.5 player game itself. The resulting technique is powerful enough to handle games with several million states

Model checking Branching-Time Properties of Multi-Pushdown Systems is Hard

We address the model checking problem for shared memory concurrent programs modeled as multi-pushdown systems. We consider here boolean programs with a finite number of threads and recursive procedures. It is well-known that the model checking problem is undecidable for this class of programs. In this paper, we investigate the decidability and the complexity of this problem under the assumption of bounded context-switching defined by Qadeer and Rehof, and of phase-boundedness proposed by La Torre et al. On the model checking of such systems against temporal logics and in particular branching time logics such as the modal $\mu$-calculus or CTL has received little attention. It is known that parity games, which are closely related to the modal $\mu$-calculus, are decidable for the class of bounded-phase systems (and hence for bounded-context switching as well), but with non-elementary complexity (Seth). A natural question is whether this high complexity is inevitable and what are the ways to get around it. This paper addresses these questions and unfortunately, and somewhat surprisingly, it shows that branching model checking for MPDSs is inherently an hard problem with no easy solution. We show that parity games on MPDS under phase-bounding restriction is non-elementary. Our main result shows that model checking a $k$ context bounded MPDS against a simple fragment of CTL, consisting of formulas that whose temporal operators come from the set {\EF, \EX}, has a non-elementary lower bound

Trains, Games, and Complexity: 0/1/2-Player Motion Planning through Input/Output Gadgets

We analyze the computational complexity of motion planning through local "input/output" gadgets with separate entrances and exits, and a subset of allowed traversals from entrances to exits, each of which changes the state of the gadget and thereby the allowed traversals. We study such gadgets in the 0-, 1-, and 2-player settings, in particular extending past motion-planning-through-gadgets work to 0-player games for the first time, by considering "branchless" connections between gadgets that route every gadget's exit to a unique gadget's entrance. Our complexity results include containment in L, NL, P, NP, and PSPACE; as well as hardness for NL, P, NP, and PSPACE. We apply these results to show PSPACE-completeness for certain mechanics in Factorio, [the Sequence], and a restricted version of Trainyard, improving prior results. This work strengthens prior results on switching graphs and reachability switching games.Comment: 37 pages, 36 figure

A semantic approach to reachability matrix computation

The Cyber Security is a crucial aspect of networks management. The Reachability Matrix computation is one of the main challenge in this field. This paper presents an intelligent solution in order to address the Reachability Matrix computational proble

Completeness Results for Parameterized Space Classes

The parameterized complexity of a problem is considered "settled" once it has been shown to lie in FPT or to be complete for a class in the W-hierarchy or a similar parameterized hierarchy. Several natural parameterized problems have, however, resisted such a classification. At least in some cases, the reason is that upper and lower bounds for their parameterized space complexity have recently been obtained that rule out completeness results for parameterized time classes. In this paper, we make progress in this direction by proving that the associative generability problem and the longest common subsequence problem are complete for parameterized space classes. These classes are defined in terms of different forms of bounded nondeterminism and in terms of simultaneous time--space bounds. As a technical tool we introduce a "union operation" that translates between problems complete for classical complexity classes and for W-classes.Comment: IPEC 201

Abstracting Asynchronous Multi-Valued Networks: An Initial Investigation

Multi-valued networks provide a simple yet expressive qualitative state based modelling approach for biological systems. In this paper we develop an abstraction theory for asynchronous multi-valued network models that allows the state space of a model to be reduced while preserving key properties of the model. The abstraction theory therefore provides a mechanism for coping with the state space explosion problem and supports the analysis and comparison of multi-valued networks. We take as our starting point the abstraction theory for synchronous multi-valued networks which is based on the finite set of traces that represent the behaviour of such a model. The problem with extending this approach to the asynchronous case is that we can now have an infinite set of traces associated with a model making a simple trace inclusion test infeasible. To address this we develop a decision procedure for checking asynchronous abstractions based on using the finite state graph of an asynchronous multi-valued network to reason about its trace semantics. We illustrate the abstraction techniques developed by considering a detailed case study based on a multi-valued network model of the regulation of tryptophan biosynthesis in Escherichia coli.Comment: Presented at MeCBIC 201

Weak Singular Hybrid Automata

The framework of Hybrid automata, introduced by Alur, Courcourbetis, Henzinger, and Ho, provides a formal modeling and analysis environment to analyze the interaction between the discrete and the continuous parts of cyber-physical systems. Hybrid automata can be considered as generalizations of finite state automata augmented with a finite set of real-valued variables whose dynamics in each state is governed by a system of ordinary differential equations. Moreover, the discrete transitions of hybrid automata are guarded by constraints over the values of these real-valued variables, and enable discontinuous jumps in the evolution of these variables. Singular hybrid automata are a subclass of hybrid automata where dynamics is specified by state-dependent constant vectors. Henzinger, Kopke, Puri, and Varaiya showed that for even very restricted subclasses of singular hybrid automata, the fundamental verification questions, like reachability and schedulability, are undecidable. In this paper we present \emph{weak singular hybrid automata} (WSHA), a previously unexplored subclass of singular hybrid automata, and show the decidability (and the exact complexity) of various verification questions for this class including reachability (NP-Complete) and LTL model-checking (PSPACE-Complete). We further show that extending WSHA with a single unrestricted clock or extending WSHA with unrestricted variable updates lead to undecidability of reachability problem