Single-Source-Single-Target Interleaved-Dyck Reachability with ℤ-VASS Reachability
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Di Leva, Ignacio B.
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Abstract
Interleaved-Dyck (InterDyck) languages consist of multiple interleaved Dyck languages, where each Dyck language represents a set of strings of balanced parentheses. InterDyck reachability is a specific instance of an L-transitive closure problem, or the L-reachability problem, which involves approximating the undecidable problem of determining if there is a path between a pair of nodes in a graph that represents an InterDyck language. The reachability problem allows static program analysis based on the tracking of pairs of actions in software, such as the call/return of a method.
The work of Li et al. presented the use of path expressions and integer linear programming (ILP) to approximate the single-source-single-target variant of the InterDyck reachability problem. They showed that parentheses counting can be more precise than alternatives based on the tabulation method, achieving better precision with an approximation of the counting problem. This work improves the precision of the path expression methodology, solving the parentheses-counting constraint exactly by modeling it as a ℤ-VASS reachability problem. The proposed approach solves ℤ-VASS reachability instances by computing the path expression and producing a combination of unconditional and conditional linear constraints based on it.
This contribution has the key benefit of not being prone to multi-path imprecision by determining if there is an exact path that meets one of the properties of InterDyck languages and ignoring other properties. This improvement results in an algorithm that is strictly more precise than the one presented by Li et al. This work also eliminates some requirements in the form that path expressions take, such that they are always polynomial-sized, significantly reducing the time and memory required to compute a path expression between two nodes before constraint generation. The benefits of these two improvements are noticeable in an empirical evaluation that compares this proposed solution to the previous work.
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Undergraduate Research Option Thesis