  In this paper we present a formal computational framework for modeling
manipulation actions. The introduced formalism leads to semantics of
manipulation action and has applications to both observing and understanding
human manipulation actions as well as executing them with a robotic mechanism
(e.g. a humanoid robot). It is based on a Combinatory Categorial Grammar. The
goal of the introduced framework is to: (1) represent manipulation actions with
both syntax and semantic parts, where the semantic part employs
$\lambda$-calculus; (2) enable a probabilistic semantic parsing schema to learn
the $\lambda$-calculus representation of manipulation action from an annotated
action corpus of videos; (3) use (1) and (2) to develop a system that visually
observes manipulation actions and understands their meaning while it can reason
beyond observations using propositional logic and axiom schemata. The
experiments conducted on a public available large manipulation action dataset
validate the theoretical framework and our implementation.
