Partiallength

Partiallength is a concept used in mathematics and computer science to denote a length measure associated with a partially constructed object. It captures how far an intended final object has progressed, rather than its full size. The notion is often applied in incremental processes, such as constructive algorithms, parsers, or data-structure construction, where a project evolves through discrete steps or partial views.

In a formal setting, consider a final object F built by a finite sequence of steps S = s1, s2, ..., sn. A partial object at stage i, where 0 ≤ i ≤ n, has a partiallength Li defined as the number of steps taken to reach stage i, or, equivalently, Li = i if each step contributes one unit of length. When a canonical final length L(F) is known, partiallength can be expressed as Li or as a normalized value Li/L(F). Some frameworks allow nonuniform step sizes, in which case Li is the sum of the lengths of the first i steps.

Example: building a list of n items; after k items, the partiallength is k. Example: depth-limited graph search; the partiallength at a node is its depth from the root. In text generation, partiallength can count the number of tokens produced so far.

Partiallength is typically monotone nondecreasing with extension of the partial object, and is bounded above by the final length. It is not unique across representations; different construction schemes can yield different Li for the same stage if step sizes vary. In some domains, a normalized partiallength between 0 and 1 is used to express progress.

See also: length, progress metric, incremental computation, partial order, complexity measure.