Tough-constructions (e.g., 'This problem is tough to solve') involve apparent extraction of an underlying object to the subject position of an evaluative adjective. Analysis remains debated: raising, movement, or control. The construction is restricted to English and similar languages and interacts with complex predicate structure.
You have already studied raising (where a noun phrase moves from an embedded clause to the matrix subject position without acquiring a thematic role there) and control (where the matrix subject provides the interpretation of a silent PRO subject in the embedded clause). Tough-movement sits at the intersection of these, and what makes it so theoretically interesting is that neither analysis is fully satisfactory.
Start with the surface form: "This problem is tough to solve." The noun phrase *this problem* is the matrix subject of *is tough*. But semantically, *this problem* is not the one experiencing toughness — it is the entity being solved. The toughness belongs to the *solving*. Contrast this with "It is tough to solve this problem," where *this problem* sits in its thematically natural position as the object of *solve*. Tough-movement takes that object and fronts it to the matrix subject position, leaving a gap inside the infinitival clause: "This problem is tough [to solve \_\_\_]."
The movement analysis treats this exactly as it looks: *this problem* has been moved (extracted) from inside the embedded infinitival and left-dislocated to the matrix subject position, leaving a trace or gap behind. Evidence for movement includes the fact that the object position is empty — you cannot say "This problem is tough to solve it" (the pronoun and the fronted NP compete for the same thematic role). Movement analyses handle the gap naturally.
The control analysis argues instead that the matrix subject is a controller, and the embedded subject is PRO (silent, controlled by the matrix NP). But this runs into trouble: *tough*-adjectives are not normal control predicates. In standard object control ("I persuaded John [PRO to leave]"), the controller bears a thematic role in the matrix clause. In "*This problem* is tough," *this problem* has no matrix thematic role — it is not experiencing toughness, being evaluated, or doing anything in the matrix. This looks more like raising, where the subject is semantically inert in the matrix and derives its interpretation from the embedded clause.
The raising analysis faces its own challenge: raising predicates (like *seem*) allow expletive subjects ("It seems that she left" → "She seems to have left"). But tough-adjectives resist expletives: "*It is tough to solve" is grammatical only as an impersonal, not as the source of a raising derivation where *it* is semantically the object of *solve*. The construction appears to require a specific, referential NP in subject position — which is more characteristic of control.
This is why the debate persists: tough-movement looks like movement in its gap-leaving behavior, like raising in its lack of a matrix thematic role, and like control in its requirement for a referential subject. Current minimalist analyses generally treat it as a form of A-movement with special properties, but the construction remains a productive test case for the boundary between raising and control — two categories you can now see have a fuzzy empirical frontier.