Minimaltransformation

Minimaltransformation is a term used in theoretical computer science and mathematics to refer to a least-cost sequence of operations that converts one object into another within a formal space. The exact meaning depends on the chosen domain, the set of allowed primitive transformations, and the associated cost function. In its common form, minimaltransformation denotes a sequence t1, t2, …, tk of elementary transformations from an initial object A to a target object B such that the sum of the costs c(ti) is minimal among all feasible sequences from A to B.

Formal framework: Let D be a domain of objects, P a finite or countable set of primitive transformations that map elements of D to other elements of D, and c: P → R≥0 a cost function. A transformation from A to B is a finite sequence p1, …, pn with applying consecutively to A yields B. A minimal transformation is a sequence with minimal total cost among all such sequences. If costs are uniform, a minimal transformation reduces to a shortest path in the state space; if costs vary, standard shortest-path algorithms apply.

Computational aspects and applications: Computing a minimal transformation is equivalent to a shortest-path problem and is tractable in some cases and intractable in others, depending on the structure of D and P. Applications appear in string editing (generalized edit distance), code refactoring and program synthesis, model-driven engineering, and automated planning where the objective is to minimize change cost.

Examples: The Levenshtein distance between strings is a classic minimal transformation under insert, delete, and substitute operations with unit costs. In graph transformations, transforming one graph into another with the fewest rewrites is another instance.

See also: edit distance, graph transformation, term rewriting, automated planning, model transformation.