From d3e2d63a8525915e29034bf381495d616d3f8d85 Mon Sep 17 00:00:00 2001 From: Yann Orlarey Date: Fri, 18 Sep 2020 15:50:04 +0200 Subject: [PATCH] Minor edits --- README.md | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/README.md b/README.md index 42cd1ad..2f090f4 100644 --- a/README.md +++ b/README.md @@ -1,4 +1,4 @@ -# _On-demand_ computations in Faust +# _On-demand_ Computations in Faust **[YO, Preliminary draft v4]** @@ -10,7 +10,7 @@ The challenge is to introduce _on-demand_ computations while keeping the simple ## Semantics of Faust Expressions -Before describing the particular semantics of $\mathtt{ondemand}()$, let's take a look at the semantics of Faust in general. Faust semantics is based on _signals_ and _signal processors_. A _signal_ is a function of time and a _signal processor_ is a function of signals. A Faust program describes a _signal processor and programming in Faust is essentially combining _signal processors_ together, for example using composition operators like $\mathtt{:}$ or $\mathtt{\sim}$. +Before describing the particular semantics of $\mathtt{ondemand}()$, let's take a look at the semantics of Faust in general. The main notions are _signals_ and _signal processors_. A _signal_ is a function of time and a _signal processor_ is a function of signals. A Faust program describes a _signal processor_. Programming in Faust is essentially _combining signal processors_ together, for example using composition operators like $\mathtt{:}$ or $\mathtt{\sim}$. ### Time @@ -32,11 +32,11 @@ A *signal processors* $P:\mathbb{S}^{n}\rightarrow\mathbb{S}^{m}$, is a function ### Composition operators -The five composition operators of Faust ($<: \ :\ :>\ ,\ \sim$) are all binary operations on signal processors: $\mathbb{P}\times\mathbb{P}\rightarrow\mathbb{P}$. +The five composition operators of Faust ($<: \ :\ :>\ ,\ \sim$) are binary operations on signal processors: $\mathbb{P}\times\mathbb{P}\rightarrow\mathbb{P}$. ### Semantic Brackets -In order to distinguish a Faust expression from its _meaning_ as a signal processor, we use the semantic brackets notation $[\![\ ]\!]$. For example $[\![+]\!]$ represents the *meaning* of Faust expression $+$ , a signal processor with the following type and definition: +In order to distinguish a Faust expression from its _meaning_ as a signal processor, we use the semantic brackets notation $[\![\ ]\!]$. For example $[\![+]\!]$ represents the *meaning* of Faust expression $+$ , a signal processor that takes two input signals $x$ and $y$ and produces an output signal by adding together the samples of the input signals: $$ \begin{align} [\![+]\!]&: \mathbb{S}^2\rightarrow\mathbb{S}^1 \\ @@ -44,7 +44,6 @@ $$ \end{align} $$ Numbers are also signal processors. The *meaning* of the Faust expression $1$ is the following: - $$ \begin{align} [\![1]\!]&: \mathbb{S}^0\rightarrow\mathbb{S}^1 \\