Higher-Order Causal Stream Functions in Sig from First Principles

The Sig programming language is a total functional, clocked synchronous data-flow language. Its core has been designed to admit concise coalgebraic semantics. Universal coalgebra is an expressive theoretical framework for behavioral semantics, but traditionally phrased in abstract categorical language, and generally considered inaccessible. In the present paper, we rephrase the coalgebraic concepts relevant for the Sig language semantics in basic mathematical notation. We demonstrate how the language features characteristic of its paradigms, namely sequential and parallel composition for applicative style, delay for data flow, and apply for higher-order functional programming, are shaped naturally by the semantic structure. Thus the present paper serves two purposes, as a gentle, self-contained and applied introduction to coalgebraic semantics, and as an explication of the Sig core language denotational and operational design.