Attrunbounded

Attrunbounded is a theoretical property used in discussions of sequences and processes where the set of encountered attribute values is not bounded. It describes situations in which a process can accumulate attributes without an upper limit under a given attribute order. The term combines “attribute” with “unbounded,” emphasizing growth in tracked measures.

Formal definition often appears in a simple model: let A be a totally ordered set with order ≤. If a process generates a sequence f: N → A of attribute values, f is attrunbounded if for every a in A there exists n such that f(n) > a. Equivalently, the set {f(n) : n ∈ N} is unbounded in A. When A = the real numbers with the usual order, this means sup{f(n)} = +∞; for discrete orders, unboundedness is defined relative to that order.

Variants of attrunbounded consider different axes or constraints. Attrunbounded in time refers to attribute values that grow without bound as time progresses. Other variants examine unbounded growth with respect to space, resources, or other metrics, possibly under monotonicity requirements or bounded increments. The concept is typically contrasted with bounded or eventually bounded behavior.

Examples illustrate the idea. A process that outputs 0, 1, 2, 3, … is attrunbounded in the natural-number attribute. A system recording cumulative cost with nonnegative, unbounded increments is attrunbounded in the cost attribute.

Applications of the notion appear in theoretical computer science discussions of automata with unbounded attribute accumulation, in the study of attribute grammars, and in analyses of streaming or iterative systems where resources or signals may grow without bound. Related ideas include unbounded growth, divergence in dynamical systems, and unboundedness criteria in complexity theory.