At the core of intelligence is the capacity to solve new problems. To do so, the brain must generate novel thoughts and behaviors in a goal-directed manner. This creative capacity has been hypothesized to depend on cognitive operations resembling symbolic grammars, generative models with discrete atomic objects (i.e., symbols) that use rule-based operations to generate newly composed representations. Whether and how symbolic grammars are implemented in neural substrates remains unknown. Here we develop an interdisciplinary approach to study the neural implementation of a grammar for action, centered on a novel drawing task. Macaque monkeys first learn action grammars for drawing, and then compositionally generalize these grammars to draw new images. Behavioral analyses indicate that these learned grammars consist of symbolic action primitives (e.g., strokes such as circles or lines) and syntactic rules (e.g., “repeat three times”). To investigate neural substrates of these action grammars, we recorded unit activity across multiple areas of frontal cortex (sixteen 32-electrode arrays), spanning motor, premotor, prefrontal, and frontal polar cortices. We have found evidence of functional localization of action grammar components to specific areas of the frontal cortex, including motor primitives, action symbols, and syntactic rules. Here, we highlight the finding that action symbol representations localize to ventral premotor cortex (PMv), based on evidence that PMv activity encodes action primitives in a manner that is abstract and categorical, and is combined when producing novel compositional drawings. This finding may unify prior observations of abstract motor properties in PMv under the general principle that PMv encodes semantically meaningful action categories. More broadly, by combining analyses of neural activity with interrogation of task-optimized neural network models, we are pursuing a mechanistic understanding of how neural dynamics, in PMv and interconnected areas, implement cognitive operations consistent with symbol manipulation in an action grammar.