Flow Logic

Flow networks have attracted a lot of research in computer science. Indeed, many questions in numerous application areas can be reduced to questions about flow networks. Many of these applications would benefit from a framework in which one can formally reason about properties of flow networks that go beyond their maximal flow. We introduce Flow Logics: modal logics that treat flow functions as explicit first-order objects and enable the specification of rich properties of flow networks. The syntax of our logic BFL* (Branching Flow Logic) is similar to the syntax of the temporal logic CTL*, except that atomic assertions may be flow propositions, like $> \gamma$ or $\geq \gamma$, for $\gamma \in \mathbb{N}$, which refer to the value of the flow in a vertex, and that first-order quantification can be applied both to paths and to flow functions. We present an exhaustive study of the theoretical and practical aspects of BFL*, as well as extensions and fragments of it. Our extensions include flow quantifications that range over non-integral flow functions or over maximal flow functions, path quantification that ranges over paths along which non-zero flow travels, past operators, and first-order quantification of flow values. We focus on the model-checking problem and show that it is PSPACE-complete, as it is for CTL*. Handling of flow quantifiers, however, increases the complexity in terms of the network to ${\rm P}^{\rm NP}$, even for the LFL and BFL fragments, which are the flow-counterparts of LTL and CTL. We are still able to point to a useful fragment of BFL* for which the model-checking problem can be solved in polynomial time. Finally, we introduce and study the query-checking problem for BFL*, where under-specified BFL* formulas are used for network exploration

On the Implicit Bias in Deep-Learning Algorithms

Gradient-based deep-learning algorithms exhibit remarkable performance in practice, but it is not well-understood why they are able to generalize despite having more parameters than training examples. It is believed that implicit bias is a key factor in their ability to generalize, and hence it was widely studied in recent years. In this short survey, we explain the notion of implicit bias, review main results and discuss their implications.Comment: Some minor edit

Flow Games

In the traditional maximal-flow problem, the goal is to transfer maximum flow in a network by directing, in each vertex in the network, incoming flow into outgoing edges. While the problem has been extensively used in order to optimize the performance of networks in numerous application areas, it corresponds to a setting in which the authority has control on all vertices of the network. Today\u27s computing environment involves parties that should be considered adversarial. We introduce and study {em flow games}, which capture settings in which the authority can control only part of the vertices. In these games, the vertices are partitioned between two players: the authority and the environment. While the authority aims at maximizing the flow, the environment need not cooperate. We argue that flow games capture many modern settings, such as partially-controlled pipe or road systems or hybrid software-defined communication networks. We show that the problem of finding the maximal flow as well as an optimal strategy for the authority in an acyclic flow game is $Sigma_2^P$-complete, and is already $Sigma_2^P$-hard to approximate. We study variants of the game: a restriction to strategies that ensure no loss of flow, an extension to strategies that allow non-integral flows, which we prove to be stronger, and a dynamic setting in which a strategy for a vertex is chosen only once flow reaches the vertex. We discuss additional variants and their applications, and point to several interesting open problems

Eulerian Paths with Regular Constraints

Labeled graphs, in which edges are labeled by letters from some alphabet Sigma, are extensively used to model many types of relations associated with actions, costs, owners, or other properties. Each path in a labeled graph induces a word in Sigma^* -- the one obtained by concatenating the letters along the edges in the path. Classical graph-theory problems give rise to new problems that take these words into account. We introduce and study the constrained Eulerian path problem. The input to the problem is a Sigma-labeled graph G and a specification L subseteq Sigma^*. The goal is to find an Eulerian path in G that satisfies L. We consider several classes of the problem, defined by the classes of G and L. We focus on the case L is regular and show that while the problem is in general NP-complete, even for very simple graphs and specifications, there are classes that can be solved efficiently. Our results extend work on Eulerian paths with edge-order constraints. We also study the constrained Chinese postman problem, where edges have costs and the goal is to find a cheapest path that contains each edge at least once and satisfies the specification. Finally, we define and study the Eulerian language of a graph, namely the set of words along its Eulerian paths

The Unfortunate-Flow Problem

In the traditional maximum-flow problem, the goal is to transfer maximum flow in a network by directing, in each vertex in the network, incoming flow into outgoing edges. The problem is one of the most fundamental problems in TCS, with application in numerous domains. The fact a maximal-flow algorithm directs the flow in all the vertices of the network corresponds to a setting in which the authority has control in all vertices. Many applications in which the maximal-flow problem is applied involve an adversarial setting, where the authority does not have such a control. We introduce and study the unfortunate flow problem, which studies the flow that is guaranteed to reach the target when the edges that leave the source are saturated, yet the most unfortunate decisions are taken in the vertices. When the incoming flow to a vertex is greater than the outgoing capacity, flow is lost. The problem models evacuation scenarios where traffic is stuck due to jams in junctions and communication networks where packets are dropped in overloaded routers. We study the theoretical properties of unfortunate flows, show that the unfortunate-flow problem is co-NP-complete and point to polynomial fragments. We introduce and study interesting variants of the problem: integral unfortunate flow, where the flow along edges must be integral, controlled unfortunate flow, where the edges from the source need not be saturated and may be controlled, and no-loss controlled unfortunate flow, where the controlled flow must not be lost

Flow Logic

A flow network is a directed graph in which each edge has a capacity, bounding the amount of flow that can travel through it. Flow networks have attracted a lot of research in computer science. Indeed, many questions in numerous application areas can be reduced to questions about flow networks. This includes direct applications, namely a search for a maximal flow in networks, as well as less direct applications, like maximal matching or optimal scheduling. Many of these applications would benefit from a framework in which one can formally reason about properties of flow networks that go beyond their maximal flow. We introduce Flow Logics: modal logics that treat flow functions as explicit first-order objects and enable the specification of rich properties of flow networks. The syntax of our logic BFL* (Branching Flow Logic) is similar to the syntax of the temporal logic CTL*, except that atomic assertions may be flow propositions, like > gamma or geq gamma, for gamma in N, which refer to the value of the flow in a vertex, and that first-order quantification can be applied both to paths and to flow functions. For example, the BFL* formula Ef ((geq 100) wedge AG({it low} rightarrow (leq 20)) states that there is a legal flow function in which the flow is above 100 and in all paths, the amount of flow that travels through vertices with low security is at most 20. We present an exhaustive study of the theoretical and practical aspects of BFL*, as well as extensions and fragments of it. Our extensions include flow quantifications that range over non-integral flow functions or over maximal flow functions, path quantification that ranges over paths along which non-zero flow travels, past operators, and first-order quantification of flow values. We focus on the model-checking problem and show that it is PSPACE-complete, as it is for CTL*. Handling of flow quantifiers, however, increases the complexity in terms of the network to P^{NP}, even for the LFL and BFL fragments, which are the flow-counterparts of LTL and CTL. We are still able to point to a useful fragment of BFL* for which the model-checking problem can be solved in polynomial time