Complexity of Prioritized Default Logics

In default reasoning, usually not all possible ways of resolving conflicts between default rules are acceptable. Criteria expressing acceptable ways of resolving the conflicts may be hardwired in the inference mechanism, for example specificity in inheritance reasoning can be handled this way, or they may be given abstractly as an ordering on the default rules. In this article we investigate formalizations of the latter approach in Reiter's default logic. Our goal is to analyze and compare the computational properties of three such formalizations in terms of their computational complexity: the prioritized default logics of Baader and Hollunder, and Brewka, and a prioritized default logic that is based on lexicographic comparison. The analysis locates the propositional variants of these logics on the second and third levels of the polynomial hierarchy, and identifies the boundary between tractable and intractable inference for restricted classes of prioritized default theories

Answer Set Planning Under Action Costs

Recently, planning based on answer set programming has been proposed as an approach towards realizing declarative planning systems. In this paper, we present the language Kc, which extends the declarative planning language K by action costs. Kc provides the notion of admissible and optimal plans, which are plans whose overall action costs are within a given limit resp. minimum over all plans (i.e., cheapest plans). As we demonstrate, this novel language allows for expressing some nontrivial planning tasks in a declarative way. Furthermore, it can be utilized for representing planning problems under other optimality criteria, such as computing ``shortest'' plans (with the least number of steps), and refinement combinations of cheapest and fastest plans. We study complexity aspects of the language Kc and provide a transformation to logic programs, such that planning problems are solved via answer set programming. Furthermore, we report experimental results on selected problems. Our experience is encouraging that answer set planning may be a valuable approach to expressive planning systems in which intricate planning problems can be naturally specified and solved

On Polynomially Many Queries to NP or QMA Oracles

We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as $P^{NP}$ and $P^{QMA}$, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes $P^{NP[\log]}$ and $P^{QMA[\log]}$, defined identically to $P^{NP}$ and $P^{QMA}$, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a $P^{NP}$ machine have a "query graph" which is a tree, then this computation can be simulated in $P^{NP[\log]}$. In this work, we first show that for any verification class $C\in\{NP,MA,QCMA,QMA,QMA(2),NEXP,QMA_{\exp}\}$, any $P^C$ machine with a query graph of "separator number" $s$ can be simulated using deterministic time $\exp(s\log n)$ and $s\log n$ queries to a $C$-oracle. When $s\in O(1)$ (which includes the case of $O(1)$-treewidth, and thus also of trees), this gives an upper bound of $P^{C[\log]}$, and when $s\in O(\log^k(n))$, this yields bound $QP^{C[\log^{k+1}]}$ (QP meaning quasi-polynomial time). We next show how to combine Gottlob's "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding $P^C$ computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial $p$ specified via an arithmetic circuit, if one can "weakly compress" $p$ so that its optimal value requires $m$ bits to represent, then $P^{NP}$ can be decided with only $m$ queries to an NP-oracle.Comment: 46 pages pages, 5 figures, to appear in ITCS 202

Efficient traffic trajectory error detection

Our recent survey on publicly reported router bugs shows that many router bugs, once triggered, can cause various traffic trajectory errors including traffic deviating from its intended forwarding paths, traffic being mistakenly dropped and unauthorized traffic bypassing packet filters. These traffic trajectory errors are serious problems because they may cause network applications to fail and create security loopholes for network intruders to exploit. Therefore, traffic trajectory errors must be quickly and efficiently detected so that the corrective action can be performed in a timely fashion. Detecting traffic trajectory errors requires the real-time tracking of the control states (e.g., forwarding tables, packet filters) of routers and the scalable monitoring of the actual traffic trajectories in the network. Traffic trajectory errors can then be detected by efficiently comparing the observed traffic trajectories against the intended control states. Making such trajectory error detection efficient and practical for large-scale high speed networks requires us to address many challenges. First, existing traffic trajectory monitoring algorithms require the simultaneously monitoring of all network interfaces in a network for the packets of interest, which will cause a daunting monitoring overhead. To improve the efficiency of traffic trajectory monitoring, we propose the router group monitoring technique that only monitors the periphery interfaces of a set of selected router groups. We analyze a large number of real network topologies and show that effective router groups with high trajectory error detection rates exist in all cases. We then develop an analytical model for quickly and accurately estimating the detection rates of different router groups. Based on this model, we propose an algorithm to select a set of router groups that can achieve complete error detection and low monitoring overhead. Second, maintaining the control states of all the routers in the network requires a significant amount of memory. However, there exist no studies on how to efficiently store multiple complex packet filters. We propose to store multiple packet filters using a shared Hyper- Cuts decision tree. To help decide which subset of packet filters should share a HyperCuts decision tree, we first identify a number of important factors that collectively impact the efficiency of the resulting shared HyperCuts decision tree. Based on the identified factors, we then propose to use machine learning techniques to predict whether any pair of packet filters should share a tree. Given the pair-wise prediction matrix, a greedy heuristic algorithm is used to classify packet filters into a number of shared HyperCuts decision trees. Our experiments using both real packet filters and synthetic packet filters show that our shared HyperCuts decision trees require considerably less memory while having the same or a slightly higher average height than separate trees. In addition, the shared HyperCuts decision trees enable concurrent lookup of multiple packet filters sharing the same tree. Finally, based on the two proposed techniques, we have implemented a complete prototype system that is compatible with Juniper's JUNOS. We have shown in the thesis that, to detect traffic trajectory errors, it is sufficient to only selectively implement a small set of key functions of a full-fletched router on our prototype, which makes our prototype simpler and less error prone. We conduct both Emulab experiments and micro-benchmark experiments to show that the system can efficiently track router control states, monitor traffic trajectories and detect traffic trajectory errors