Range Avoidance Problem

• Range Avoidance Problem is the task of identifying an output that no system state can reach, with implications for both pseudorandom constructions and circuit lower bounds.
• In discrete complexity, efficient F-avoid algorithms for bounded circuits can yield explicit constructions of rigid matrices and hard Boolean functions, addressing key open questions.
• In control and robotics, range avoidance underpins safe navigation via barrier functions and decentralized optimization, ensuring collision-free operation in dynamic environments.

The range avoidance problem encompasses a diverse set of theoretical and algorithmic challenges across Boolean function theory, control theory, computational geometry, and safe autonomous systems. Fundamentally, it refers to the task of determining an output in some target space (typically, $\{0,1\}^m$ or a geometric configuration space) that is provably not reachable or attainable from a given system, circuit, controller, or dynamical process, over its entire allowed domain. This problem is instantiated both in discrete mathematics—especially in circuit complexity as the Range Avoidance, or $F$-avoid, problem for Boolean circuits—and in control/robotics as practical range or separation enforcement for collision avoidance. Solutions to the range avoidance problem have far-reaching consequences: in discrete mathematics via explicit constructions for pseudorandom objects and circuit lower bounds; in control, via provably safe controllers for agents with limited sensing, actuation, or communication.

1. Formal Definitions and Discrete Complexity of Range Avoidance

In discrete complexity, the canonical range avoidance problem is: given a circuit $C : \{0,1\}^n \to \{0,1\}^m$ from a class $F$ with $m > n$, find $y \in \{0,1\}^m$ such that $C(x) \ne y$ for all $x \in \{0,1\}^n$—denoted $F$-avoid. The guaranteed existence of such $y$ is a trivial consequence of the pigeonhole principle ($|\{0,1\}^n| < |\{0,1\}^m|$). However, the computational task of finding $y \notin \mathrm{Range}(C)$ is deeply linked to the explicit construction of pseudorandom objects: a deterministic polynomial-time algorithm for $F$-avoid is known to imply explicit hard Boolean functions, optimal linear codes, and pseudorandom generators (Kuntewar et al., 21 Mar 2025, Gajulapalli et al., 2023). For certain circuit classes—namely, $NC^0_k$, circuits of bounded depth with each output bit depending on at most $k$ input bits—the tractability of $F$-avoid shows threshold phenomena:

• $NC^0_2$-avoid is in P when $m > n$.
• $NC^0_3$-avoid is in FP$^{NP}$ for $m \ge n^2/\log n$.
• In contrast, if $NC^0_3$-avoid could be solved in polynomial time for $m = n+O(n^{2/3})$, it would yield explicit rigid matrices—implying superlinear lower bounds for linear circuits, a major open problem in complexity (Kuntewar et al., 21 Mar 2025, Gajulapalli et al., 2023).

Additionally, the existence of polynomial-time algorithms for $F$-avoid for circuits of constant degree corresponds to the existence of deterministic algorithms for constructing objects such as rigid matrices, hard functions, and PRGs, which are central open questions in explicit construction (Gajulapalli et al., 2023, Ren et al., 18 Nov 2025). Hardness results have been established under the assumption of "demi-bits generators"—cryptographic primitives hard for nondeterministic adversaries—showing that range avoidance remains hard even for constant-degree polynomial circuits over $\mathbb{F}_2$, and that it is not solvable by any nondeterministic polynomial-time (NP) algorithm under standard cryptographic conjectures (Ren et al., 18 Nov 2025).

2. Connections to Circuit Complexity, Explicit Constructions, and Proof Systems

The implications of tractable or intractable range avoidance reach into circuit theory and propositional proof complexity. Specifically, deterministic polynomial-time solutions for $F$-avoid in certain regimes would resolve entrenched open problems, such as:

• Explicit Construction of Rigid Matrices: Efficient algorithms for $NC^0_3$-avoid in the "small stretch" regime ($m = n+O(n^{2/3})$) would yield explicit constructions of rigid matrices, thereby proving the existence of Boolean functions requiring superlinear ($\Omega(n \log \log n)$) circuit size in $NC^1$ (Gajulapalli et al., 2023).
• Proof Complexity Generators and Dual Weak Pigeonhole Principle: The existence of demi-bits generators implies the unprovability (in Cook's $\mathsf{PV}_1$) of the dual weak pigeonhole principle (dwPHP) expressing totality of range avoidance. This result separates theories in bounded arithmetic, specifically $\mathsf{APC}_1 \supsetneq \mathsf{PV}_1$ (Ren et al., 18 Nov 2025).
• Hardness for Nondeterministic and Interactive Proofs: Range avoidance is provably hard even for nondeterministic algorithms and for students in $k$-round interactive (AM) protocols, given cryptographic hardness assumptions, using extractor-based reductions (Ren et al., 18 Nov 2025).

Conversely, intractability lends range avoidance its utility as a central primitive in establishing the fine-grained boundaries of total search, explicit construction, and proof complexity.

3. Hypergraph Methods and Algorithmic Regimes for Range Avoidance

A prominent methodological advance is the reduction of the range avoidance problem—especially for $NC^0_3$ circuits—to Turán-type questions in $k$-uniform hypergraphs, and the subsequent development of deterministic polynomial-time algorithms in specific parameter regimes (Kuntewar et al., 21 Mar 2025).

• A circuit $C:{0,1}^n\to{0,1}^m$ in $NC^0_3$ is associated to a 3-uniform linear hypergraph $H_C$ on the variable set $[n]$, with each edge corresponding to the support of an output function. The range of $C$ is linked to the set of feasible edge-colorings induced by input assignments.
• Quadratic Stretch Regime: For monotone $NC^0_3$ circuits with $m > c n^2$, $H_C$ must contain a "wicket" (a small fixed subgraph with a non-$\varphi$-colorable structure). This enables direct extraction of $y \notin \mathrm{Range}(C)$ in polynomial time (Kuntewar et al., 21 Mar 2025).
• Linear Stretch (Monotone Case): For $m > n$, any connected $H_C$ contains a "loose $\chi$-cycle," which again admits a non-$\varphi$-colorable edge-coloring, allowing for a polynomial-time construction of $y \notin \mathrm{Range}(C)$. This improves prior bounds and demonstrates the tractability in the monotone case only, not for general $NC^0_3$ circuits.

The connection to extremal combinatorics is formalized via Turán numbers, ex$_k(n,H')$, controlling the threshold number of outputs for guaranteed hardness/easiness transitions.

4. Continuous and Geometric Range Avoidance: Control, Robotics, and Safe Navigation

In control and robotics, the range avoidance problem translates to maintaining a safety margin (minimum separation) between agents or between an agent and obstacles, typically framed in terms of barrier functions, model predictive control, or reactive feedback laws.

• Control Barrier Function Framework: The system safety certificate is based on maintaining the minimum distance between state-dependent (possibly non-smooth) convex regions associated to robots and obstacles. Under strong convexity, the distance function is $C^1$ and can be enforced through a CBF constraint embedded in a real-time QP. Both primal and dual (KKT-multiplier) characterizations are employed, allowing for rapid online sensitivity updates and real-time safety guarantees for convex sets (polytopes or strictly convex bodies) (Thirugnanam et al., 2023, Thirugnanam et al., 2021).
• Range-Only Feedback Laws: For unicycle robots with only radial measurements to a target or obstacle, dynamic output feedback laws using joint quadratic and barrier Lyapunov functions (e.g., logarithmic-type barriers) guarantee both convergence to a desired radius and strict adherence to safety constraints, with all stability and boundedness properties established via Lyapunov theory (Singh et al., 14 Jan 2025, Bhati et al., 11 Jan 2025).
• Decentralized Safety under Limited Sensing: In multi-robot (e.g., quadrotor) teams, decentralized NMPC schemes embedding high-order (e.g., exponential) control barrier functions enable safe coordination with explicit, theoretically derived lower bounds on required sensing range. Conservative and practical bounds are derived in closed-form to ensure feasibility and forward invariance of the barrier constraints even under adversarial relative velocities and actuation limits (Goarin et al., 2024).

Geometric approaches also include HJ-based reachability and level-set PDEs for the offline/online computation of avoidance regions in vehicle systems, with efficient, high-dimensional numerical solvers validated on non-convex, time-dependent, and moving obstacle scenarios (Xausa et al., 2019).

5. Optimization, Learning-Based, and Communication-Driven Approaches

Beyond analytical controllers, range avoidance has been tackled using optimization, learning, and communication protocols:

• Convex Optimization Embeddings: Duality-based convex programming enables real-time, nonsmooth barrier enforcement between polytopes for complex geometries (e.g., L-shaped or "sofa" robots). All time-derivatives and barrier constraints admit convex QP formulations, robust to nonsmooth active set changes (Thirugnanam et al., 2021).
• Learning Detection-Range Frugal Policies: Neuroevolutionary algorithms (e.g., AGENT) learn NN-based policies for reciprocal collision avoidance between UAVs that minimize the maximum detection range required for strict safety across a wide distribution of encounter scenarios. The resulting controllers are compact, generalize well, and allow for efficient feasibility filtering (Behjat et al., 2019).
• Communication Protocols for Safety: In networked UAV systems, layered architectures with specified range and timing thresholds map to wireless communication technologies. High-rate SSID-beaconing enables sub-100ms update cycles for tactical deconfliction and last-resort avoidance, validated with field measurements to enforce well-clear and collision avoidance requirements in both urban and suburban contexts (Vinogradov et al., 2019).

6. Future Directions and Open Problems

Critical open directions include:

• Full Characterization of Tractability: For $NC^0_3$-avoid, whether deterministic polynomial-time algorithms exist for general (non-monotone) circuits, and sharp thresholds for $NC^0_k$-avoid with $k > 3$, remain unresolved (Kuntewar et al., 21 Mar 2025).
• Proof Complexity and Uniformity: The construction of uniform pseudo-surjective proof complexity generators and the possibility of basing hardness of range avoidance on more foundational cryptographic primitives (e.g., from one-way functions instead of demi-bits generators) are open (Ren et al., 18 Nov 2025).
• Integration of Hypergraph- and Control-Theoretic Methods: Adapting hypergraph coloring and Turán-type extremal arguments to hybrid or continuous state spaces, or integrating model-predictive and barrier-based controllers for high-dimensional, nonconvex environments with explicit performance and feasibility certificates, are ongoing challenges.
• Scalability and Real-Time Guarantees: Ensuring real-time solution of embedded optimization problems or learning-based policies under stringent actuation, sensing, and communication limits, especially in large teams or highly dynamic environments, is a persistent technical focus (Goarin et al., 2024, Thirugnanam et al., 2023, Behjat et al., 2019).

The range avoidance problem thus occupies a central and unifying role across computational complexity, coding theory, control and robotics, and distributed systems, driving both deep theoretical breakthroughs and practical safety-critical applications.