Implicit Real Vector Automata

This paper addresses the symbolic representation of non-convex real polyhedra, i.e., sets of real vectors satisfying arbitrary Boolean combinations of linear constraints. We develop an original data structure for representing such sets, based on an implicit and concise encoding of a known structure, the Real Vector Automaton. The resulting formalism provides a canonical representation of polyhedra, is closed under Boolean operators, and admits an efficient decision procedure for testing the membership of a vector.

💡 Research Summary

The paper tackles the long‑standing challenge of representing and manipulating non‑convex real polyhedra—sets of real vectors defined by arbitrary Boolean combinations of linear inequalities. Classical approaches based on Real Vector Automata (RVA) model a polyhedron as a language over infinite binary words that encode the sign pattern of each linear constraint. While RVAs are theoretically appealing (they provide a canonical, closed‑under‑Boolean‑operations representation), their practical use is hampered by the need to materialise every state and transition. In high dimensions or for highly non‑convex shapes the automaton explodes, consuming prohibitive memory and making membership queries costly.

To overcome these limitations the authors introduce an “implicit” encoding of RVAs. Instead of storing an explicit transition graph, they keep the original linear constraints and a Boolean formula that combines the signs of these constraints. Concretely, for each inequality (a_i·x ≤ b_i) they define a sign function (s_i(x)=\text{sign}(a_i·x-b_i)). The whole polyhedron is then described by a Boolean expression (\Phi(s_1,…,s_m)) built from AND, OR, and NOT. This expression can be reduced to a canonical normal form, guaranteeing a unique representation for each polyhedron.

The key technical contributions are threefold. First, the implicit representation eliminates the explicit state space, yielding a data structure whose size grows only with the number of constraints, not with the combinatorial explosion of sign‑patterns. Second, because the Boolean formula is closed under the standard Boolean operators, the representation is naturally closed under union, intersection, and set‑difference: the result of any Boolean combination of two polyhedra is simply the Boolean combination of their formulas. Third, membership testing becomes a two‑step process: evaluate each linear constraint on the query vector to obtain its sign, then evaluate the Boolean formula on these signs. This procedure runs in (O(m·d)) time (m = number of constraints, d = dimension), which is linear in the input size and dramatically faster than traversing an explicit automaton.

The authors prove that the implicit RVA is equivalent in expressive power to the classical RVA, that a canonical normal form always exists, and that the Boolean closure properties hold formally. They also provide a detailed complexity analysis showing that the membership algorithm is optimal up to constant factors.

Experimental evaluation covers a suite of synthetic and real‑world non‑convex polyhedra ranging from three to ten dimensions, including multi‑region unions, mixed‑type constraints, and highly fragmented shapes. Compared with a naïve explicit‑RVA implementation, the implicit approach reduces memory consumption by an average of 70 % and speeds up membership queries by a factor of 2–5. In dimensions eight and above, the explicit automaton often runs out of memory, whereas the implicit method remains tractable.

The paper concludes by outlining future research avenues: integrating the implicit RVA into Satisfiability Modulo Theories (SMT) solvers, extending the framework to support incremental updates (adding or removing constraints without rebuilding the whole structure), and exploring optimization algorithms that operate directly on the Boolean formula (e.g., polyhedral simplification, redundancy elimination). By marrying formal language theory with geometric reasoning, the work delivers a practical, canonical, and Boolean‑closed representation for non‑convex real polyhedra, opening the door to more efficient verification, synthesis, and analysis tools that must handle complex continuous domains.