Merge on workspaces as Hopf algebra Markov chain

We study the dynamical properties of a Hopf algebra Markov chain with state space the binary rooted forests with labelled leaves. This Markovian dynamical system describes the core computational process of structure formation and transformation in syntax via the Merge operation, according to Chomsky's Minimalism model of generative linguistics. The dynamics decomposes into an ergodic dynamical system with uniform stationary distribution, given by the action of Internal Merge, while the contributions of External Merge and (a minimal form of) Sideward Merge reduce to a simpler Markov chain with state space the set of partitions and with combinatorial weights. The Sideward Merge part of the dynamics prevents convergence to fully formed connected structures (trees), unless the different forms of Merge are weighted by a cost function, as predicted by linguistic theory. Results on the asymptotic behavior of the Perron-Frobenius eigenvalue and eigenvector in this weighted case, obtained in terms of an associated Perron-Frobenius problem in the tropical semiring, show that the usual cost functions (Minimal Search and Resource Restrictions) proposed in the linguistic literature do not suffice to obtain convergence to the tree structures, while an additional optimization property based on the Shannon entropy achieves the expected result for the dynamics. We also comment on the introduction of continuous parameters related to semantic embedding and other computational models, and also on some filtering of the dynamics by coloring rules that model the linguistic filtering by theta roles and phase structure, and on parametric variation and the process of parameter setting in Externalization.