The semantics of graphical languages

Visual notations are pervasive in circuit design, control systems, and increasingly in mainstream programming environments. Yet many of the foundational advances in programming language theory are taking place in the context of textual notations. In order to map such advances to the graphical world, and to take the concerns of the graphical world into account when working with textual formalisms, there is a need for rigorous connections between textual and graphical expressions of computation. To this end, this paper presents a graphical calculus called Uccello. Our key insight is that Ariola and Blom’s work on sharing in the cyclic lambda calculus provides an excellent foundation for formalizing the semantics of graphical languages. As an example of what can be done with this foundation, we use it to extend a graphical language with staging constructs. In doing so, we provide the first formal account of sharing in a multi-stage calculus. 1