Wed 19 Jan 2022 15:30 - 15:55 at Salon I - Algorithmic Verification 1 Chair(s): Qirun Zhang

Dyck reachability is the standard formulation of a large domain of static analyses, as it achieves the sweet spot between precision and efficiency, and has thus been studied extensively. Interleaved Dyck reachability (denoted $D_k\odot D_k$) uses two Dyck languages for increased precision (e.g., context and field sensitivity) but is well-known to be undecidable. As many static analyses yield a certain type of bidirected graphs, they give rise to interleaved bidirected Dyck reachability problems. Although these problems have seen numerous applications, their decidability and complexity has largely remained open. In a recent work, Li \textit{et al}. made the first steps in this direction, showing that (i)~$D_1\odot D_1$ reachability (i.e., when both Dyck languages are over a single parenthesis and act as counters) is computable in $O(n^7)$ time, while (ii)~$D_k\odot D_k$ reachability is NP-hard. However, despite this recent progress, most natural questions about this intricate problem are open.

In this work we address the decidability and complexity of all variants of interleaved bidirected Dyck reachability. First, we show that $D_1\odot D_1$ reachability can be computed in $O(n^3\cdot \alpha(n))$ time, significantly improving over the existing $O(n^7)$ bound. Second, we show that $D_k\odot D_1$ reachability (i.e., when one language acts as a counter) is decidable, in contrast to the non-bidirected case where decidability is open. We further consider $D_k\odot D_1$ reachability where the counter remains linearly bounded. Our third result shows that this bounded variant can be solved in $O(n^2\cdot \alpha(n))$ time, while our fourth result shows that the problem has a (conditional) quadratic lower bound, and thus our upper bound is essentially optimal. Fifth, we show that full $D_k\odot D_k$ reachability is undecidable. This improves the recent NP-hardness lower-bound, and shows that the problem is equivalent to the non-bidirected case. Our experiments on standard benchmarks show that the new algorithms are very fast in practice, offering many orders-of-magnitude speedups over previous methods.

#### Wed 19 JanDisplayed time zone: Eastern Time (US & Canada) change

 15:05 - 16:20 Algorithmic Verification 1POPL at Salon I Chair(s): Qirun Zhang Georgia Institute of Technology 15:0525mResearch paper Efficient Algorithms for Dynamic Bidirected Dyck-ReachabilityRemotePOPLYuanbo Li Georgia Institute of Technology, Kris Satya Georgia Institute of Technology, Qirun Zhang Georgia Institute of Technology DOI Media Attached 15:3025mResearch paper The Decidability and Complexity of Interleaved Bidirected Dyck ReachabilityRemotePOPLAdam Husted Kjelstrøm Aarhus University, Andreas Pavlogiannis Aarhus University DOI Media Attached 15:5525mResearch paper Subcubic Certificates for CFL ReachabilityRemotePOPLDmitry Chistikov University of Warwick, Rupak Majumdar MPI-SWS, Philipp Schepper CISPA DOI Media Attached