On the complexity of bidirected interleaved Dyck-reachability

Many program analyses need to reason about pairs of matching actions, such as call/return, lock/unlock, or set-field/get-field. The family Dyck languages { D k }, where has kinds parenthesis pairs, can be used model actions balanced parentheses. Consequently, many program-analysis problems formulated Dyck-reachability on edge-labeled digraphs. Interleaved (InterDyck-reachability), denoted by ⊙ -reachability, is a natural extension that allows one formulate involve multiple matching-action pairs. Unfortunately, the general InterDyck-reachability problem undecidable. In this paper, we study variants bidirected graphs , for each edge ⟨ p q ⟩ labeled an open ”( ”, there corresponding close ”) and vice versa . Language-reachability graph proven useful both (1) in its own right, way formalize problems, pointer analysis, (2) relaxation method uses fast algorithm over-approximate language-reachability directed graph. However, unlike counterpart, complexity still remains open. We establish first decidable variant (i.e., 1 -reachability) InterDyck-reachability. two restricted have only single kind pair. particular, show PTIME. also when extends language different parentheses -reachability with ≥ 2), NP-hard (and therefore much harder). implemented polynomial-time -reachability. provides new over-approximation sense relaxed then resulting solved precisely. compare -reachability-based approach against another known over-approximating algorithm. Surprisingly, found based computes more precise solutions, even though formalism inherently less expressive than formalism.