Vectoreach

Vectoreach is a theoretical framework for analyzing reachability in systems where states are represented by vectors rather than scalars. Each transition contributes a vector weight, and the goal is to determine whether a target state, typically a region in state space, can be reached from an initial state by following a sequence of transitions. The concept appears in robotics, control theory, and static software analysis, where vector-valued information such as resources, coordinates, or sensor readings must be tracked.

Formalism: A vectoreach system is modeled as a directed graph G=(V,E) with a labeling l:E -> R^d assigning a d-dimensional vector to each edge. A path p from s to t yields a path vector r(p) by combining the edge vectors along p, typically via addition. The question asks whether there exists a path whose r(p) lies in a target set T ⊆ R^d, encoding constraints or safety bounds.

Algorithms and complexity: Exact vectoreach is often intractable for large systems due to combinatorial path sums. Practical approaches use over-approximations, discretization, or convex geometry. Iterative methods compute reachable sets R_k from the initial state and expand by Minkowski sums with edge vectors, then test membership in T. Techniques include linear programming, polyhedral abstractions, and sampling.

Applications: In robotics and motion planning, vectoreach helps identify feasible trajectories under constraints on position, velocity, or energy. In software analysis, it supports vector-valued dataflow analyses that track resources or uncertainty across program points.

See also: reachability analysis, vector space, Minkowski sum, polyhedral computation, dataflow analysis.