Futuresets

Futuresets are a mathematical construct used to describe the collection of states that a system could occupy at a specified future time, given current information, a model of dynamics, and uncertainties. In formal terms, let S be the state space, s0 the current state, and M a model that defines a transition map T. For a horizon t, the futureset F_t(s0) is the set of all states s in S that can be reached after t transitions from s0 under some admissible sequence of disturbances. When disturbances are uncertain, F_t is usually treated as an over-approximation, capturing all states that could occur under any allowed evolution.

Futuresets are constructed through forward reachability analysis, set-valued maps, interval or polyhedral bounds, and stochastic sampling methods such as Monte Carlo simulations. For finite state spaces, they can be computed exactly; for continuous or high-dimensional spaces, practical representations use over- or under-approximations such as convex hulls, zonotopes, or ellipsoids.

They may be monotone with time in systems that allow persistence, and their geometry reflects system constraints and uncertainties. They are distinct from probability forecasts: futuresets describe what states are possible, not the likelihood of each state.

Applications span planning under uncertainty in robotics and autonomous systems, control engineering, risk management, and scenario planning in business and finance. In robotics, the set of positions reachable within a time horizon informs collision avoidance and path planning. For a simple grid world, starting at a cell, the futureset after three steps comprises all cells reachable within three moves, ignoring dynamic obstacles.

See also: reachable set; set-valued forecasting; scenario planning.