Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks

We are interested in fixed points in Boolean networks, {\em i.e.} functions $f$ from $\{0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the subcubes of $\{0,1\}^n$, and we characterizes a class $\mathcal{F}$ of Boolean networks satisfying the following property: Every subnetwork of $f$ has a unique fixed point if and only if $f$ has no subnetwork in $\mathcal{F}$. This characterization generalizes the fixed point theorem of Shih and Dong, which asserts that if for every $x$ in $\{0,1\}^n$ there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of $f$ evaluated at point $x$, then $f$ has a unique fixed point. Then, denoting by $\mathcal{C}^+$ (resp. $\mathcal{C}^-$) the networks whose the interaction graph is a positive (resp. negative) cycle, we show that the non-expansive networks of $\mathcal{F}$ are exactly the networks of $\mathcal{C}^+\cup \mathcal{C}^-$; and for the class of non-expansive networks we get a "dichotomization" of the previous forbidden subnetwork theorem: Every subnetwork of $f$ has at most (resp. at least) one fixed point if and only if $f$ has no subnetworks in $\mathcal{C}^+$ (resp. $\mathcal{C}^-$) subnetwork. Finally, we prove that if $f$ is a conjunctive network then every subnetwork of $f$ has at most one fixed point if and only if $f$ has no subnetwork in $\mathcal{C}^+$.Comment: 40 page

Computing Distances between Probabilistic Automata

We present relaxed notions of simulation and bisimulation on Probabilistic Automata (PA), that allow some error epsilon. When epsilon is zero we retrieve the usual notions of bisimulation and simulation on PAs. We give logical characterisations of these notions by choosing suitable logics which differ from the elementary ones, L with negation and L without negation, by the modal operator. Using flow networks, we show how to compute the relations in PTIME. This allows the definition of an efficiently computable non-discounted distance between the states of a PA. A natural modification of this distance is introduced, to obtain a discounted distance, which weakens the influence of long term transitions. We compare our notions of distance to others previously defined and illustrate our approach on various examples. We also show that our distance is not expansive with respect to process algebra operators. Although L without negation is a suitable logic to characterise epsilon-(bi)simulation on deterministic PAs, it is not for general PAs; interestingly, we prove that it does characterise weaker notions, called a priori epsilon-(bi)simulation, which we prove to be NP-difficult to decide.Comment: In Proceedings QAPL 2011, arXiv:1107.074

Computing Probabilistic Bisimilarity Distances for Probabilistic Automata

The probabilistic bisimilarity distance of Deng et al. has been proposed as a robust quantitative generalization of Segala and Lynch's probabilistic bisimilarity for probabilistic automata. In this paper, we present a characterization of the bisimilarity distance as the solution of a simple stochastic game. The characterization gives us an algorithm to compute the distances by applying Condon's simple policy iteration on these games. The correctness of Condon's approach, however, relies on the assumption that the games are stopping. Our games may be non-stopping in general, yet we are able to prove termination for this extended class of games. Already other algorithms have been proposed in the literature to compute these distances, with complexity in $\textbf{UP} \cap \textbf{coUP}$ and \textbf{PPAD}. Despite the theoretical relevance, these algorithms are inefficient in practice. To the best of our knowledge, our algorithm is the first practical solution. The characterization of the probabilistic bisimilarity distance mentioned above crucially uses a dual presentation of the Hausdorff distance due to M\'emoli. As an additional contribution, in this paper we show that M\'emoli's result can be used also to prove that the bisimilarity distance bounds the difference in the maximal (or minimal) probability of two states to satisfying arbitrary $\omega$-regular properties, expressed, eg., as LTL formulas