Hierarchical Structure in Language and Action: A Formal Comparison

Since the cognitive revolution, language and action have been compared as cognitive systems, with cross-domain convergent views recently gaining renewed interest in biology, neuroscience, and cognitive science. Language and action are both combinatorial systems whose mode of combination has been argued to be hierarchical, combining elements into constituents of increasingly larger size. This structural similarity has led to the suggestion that they rely on shared cognitive and neural resources. In this article, we compare the conceptual and formal properties of hierarchy in language and action using set theory. We show that the strong compositionality of language requires a particular formalism, a magma, to describe the algebraic structure corresponding to the set of hierarchical structures underlying sentences. When this formalism is applied to actions, it appears to be both too strong and too weak. To overcome these limitations, which are related to the weak compositionality and sequential nature of action structures, we formalize the algebraic structure corresponding to the set of actions as a trace monoid. We aim to capture the different system properties of language and action in terms of the distinction between hierarchical sets and hierarchical sequences and discuss the implications for the way both systems could be represented in the brain.