  Natural language syntax yields an unbounded array of hierarchically
structured expressions. We claim that these are used in the service of active
inference in accord with the free-energy principle (FEP). While conceptual
advances alongside modelling and simulation work have attempted to connect
speech segmentation and linguistic communication with the FEP, we extend this
program to the underlying computations responsible for generating syntactic
objects. We argue that recently proposed principles of economy in language
design - such as "minimal search" criteria from theoretical syntax - adhere to
the FEP. This affords a greater degree of explanatory power to the FEP - with
respect to higher language functions - and offers linguistics a grounding in
first principles with respect to computability. We show how both tree-geometric
depth and a Kolmogorov complexity estimate (recruiting a Lempel-Ziv compression
algorithm) can be used to accurately predict legal operations on syntactic
workspaces, directly in line with formulations of variational free energy
minimization. This is used to motivate a general principle of language design
that we term Turing-Chomsky Compression (TCC). We use TCC to align concerns of
linguists with the normative account of self-organization furnished by the FEP,
by marshalling evidence from theoretical linguistics and psycholinguistics to
ground core principles of efficient syntactic computation within active
inference.
