On the Optimality of Kinship Naming: an Information-theoretic Approach

Human languages expand vocabularies by combining existing morphemes rather than inventing arbitrary forms. Communicative efficiency shapes lexical systems at multiple levels (Gibson et al., 2019), yet morphological composition -- combining morphemes through compounding or affixation -- has rarely been modeled as a historically situated speaker choice among competing morpheme sequences, leaving unanswered why a language settles on one morpheme combination over other plausible alternatives. We ask whether a trade-off between listener recoverability and speaker production cost can predict attested compositions over contemporaneously available alternatives. Here we show, within the Rational Speech Act (RSA) framework (Frank & Goodman, 2012; Goodman & Frank, 2016) using a time-indexed lexicon constructed from Corpus of Historical American English (COHA) and Corpus of Contemporary American English (COCA), that across 4323 naturally occurring English compounds and derivations spanning 1820--2019, attested compositions are systematically ranked above unattested alternatives generated from contemporaneously available morphemes. Models integrating semantic informativeness with production cost outperform semantic-only and cost-only baselines on Mean Reciprocal Rank (MRR) and top-k accuracy (Acc@k), with the advantage of the Pragmatic Speaker model ($S_1$) over the semantic-only baseline growing as the candidate set expands, where meaning alone leaves morphological choice underdetermined. These findings suggest that lexicalization reflects a communicative trade-off between expressiveness and efficiency, extending rational accounts of communication from utterance-level choice to the internal structure of words.

We propose a mathematical theory of pragmatic information that connects communication, control, and decision-making. Its central notion is the isoteleia mapping, which formalizes equifinality: distinct semantic paths that lead to the same optimal action are treated as pragmatically equivalent. This mapping yields a three-tier hierarchy of syntactic, semantic, and pragmatic information, in which each successive abstraction removes distinctions that are irrelevant to the task. We then define pragmatic entropy, up/down pragmatic mutual information, channel capacity, and rate-distortion, and prove lossless source coding, channel coding, and rate-distortion theorems that extend Shannon's results. These measures quantify decision uncertainty, reliable transmission, and task-oriented compression at the level of terminal actions. We further introduce pragmatic value of information (VoI) and pragmatic cost of information (CoI) as decision-theoretic duals to rate-distortion and capacity, and develop a Lagrangian dual framework for cross-layer optimization. The resulting pragmatic efficiency bound $\mathcal{E}_p(λ)=\sup_R[Φ_p(R)-λ\mathrm{CoI}_p(R)]$ characterizes the maximum net utility attainable by a resource-constrained intelligent system under a given resource price, yielding a behavioral capacity that extends Shannon's symbol-level capacity to goal-directed action. Extensions to continuous messages provide closed-form expressions for Gaussian channels and sources, while dynamic settings are addressed through a Bellman equation for sequential decision-making. The framework supports task-oriented communication, networked control, autonomous systems, and embodied AI by shifting emphasis from symbol fidelity to the effectiveness of information in guiding actions. In this way, it offers a common language for systems that extract value from information under resource constraints.

We introduce the ladderpath index as a measure of language complexity grounded in algorithmic information theory. It counts the minimum steps needed to reconstruct a sequence through hierarchical reuse of repeated substructures, capturing an exactly computable but constrained form of algorithmic compressibility related to, but distinct from, Kolmogorov complexity. We apply the ladderpath approach to 21 parallel corpora from the Parallel Universal Dependencies dataset. The ladderpath index is approximately invariant across the languages, and varies much less than the corpus length. This is more pronounced when all corpora are mapped to a unified binary representation, providing evidence for the equi-complexity hypothesis from a representation-independent perspective. We also observe trade-offs between character inventory size and corpus length, and between vocabulary-level and corpus-level reconstruction complexity, supporting the trade-off hypothesis that total complexity is conserved and redistributed across linguistic levels. The reusable substructures identified by the ladderpath approach, without any linguistic input, overlap with words and morphological components attested in the natural vocabulary. The hierarchical reuse captured by the ladderpath approach parallels the chunking mechanisms proposed in cognitive science, where the human cognitive system compresses linguistic input into nested, reusable units under shared memory and processing constraints. This connection between cognitive chunking and the ladderpath approach provides a new interpretation for the equi-complexity and trade-off hypotheses, grounding both in the shared cognitive architecture that underlies language processing across human languages.