What is cyclic optimization? What distinguishes it empirically from theories employing other mechanisms? How could the human sentence processing mechanism work, if based on a cyclically optimizing grammar?

Optimization provides a unifying perspective over many domains. From the principle of least action in physics, to the free energy principle in cognitive science, nature appears thrifty in all its actions. The grammars of languages define paths for the derivation of linguistic expressions, but these paths consist of multiple self-similar subpaths — language is recursive. The principle of cyclic optimization conjectures that language optimizes paths not globally, but rather locally: an optimal path is one whose sub-paths are minimal.

The aim of this project is to understand the mathematical and computational properties of formalisms incorporating cyclic optimization, with the underlying question being "what would language be like if it were truly cyclically optimizing?" This question encompasses both typological aspects of language ("what kinds of constructions should (not) exist") as well as more behavioural ones ("what should be the time course of linguistic processing in these constructions").

Although optimization is fairly well understood from a formal perspective, few formal treatments of cyclicity are available, and thus, it is not at all clear just what it is. Cyclicity is often described dynamically, in terms of interleaving the building and interpretation of structure. This description reveals cyclicity (at least this conception thereof) to be a form of compositional interpretation. In contrast to more familiar cases of compositional interpretation, cyclic interpretation involves (cyclically) reapplying operations to the output structure being incrementaly assembled. This general description is the basis of universal methods of computation, from the Turing machine to the Lambda calculus, and thus needs to be supplanted with formal restrictions in order to make empirically desirable linguistic predictions. Two such which have been explored in the mathematical linguistic literature are a no-self-reapplication restriction (which prohibits a rule from reapplying to its own output), and a length-preserving restriction (which requires input and output segments to be in a bijective relationship). Neither of these seems to be a good approximation to actual linguistic practice, and so a narrow goal of this project is to identify more linguistically adequate restrictions.

Another strategy for understanding cyclic optimization is to start, not from the most general formulation, and try to identify restrictions so as to arrive at the desired concept, but rather to start with specific formulations, and to try to work one's way up. Two specific cyclic approach (in syntax) include Chomsky's feature inheritance mechanism, and Heck and Müller's local optimization mechanism. Chomsy's feature inheritance proposal is a modern reimplementation of the old-school transformational assumption that S was a cyclic node; that the transformational rules re-applied at each subsequent S node. While the concrete proposals about which nodes are cyclic have changed, the fundamental idea is a counter-cyclic one: 'counter-cyclic' operations are permitted within the window between cyclic nodes. According to Heck and Müller, the choice of next derivational step is itself optimized over; whether the operation of Agree, or internal or external Merge, applies, is determined by which option immediately produces the most optimal outcome. Another narrow goal of this project is to formalize and implement these two specific formulations of (optimizing) cyclicity in syntax.