Abstract -> semantics

Author: mrcinv

latex/abstract.tex

@@ -1,11 +1,27 @@
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 \begin{abstract}
-   In this work we present a theoretical model for differentiable programming. We present an algebraic language that enables both implementations and analysis of differentiable programs by way of \emph{operational calculus}.
+  In this work we present a theoretical model for differentiable programming. We
+  present an algebraic language that presents formal semantics of
+  differentiable programs by way of \emph{operational calculus}, which enables
+  reasoning about their analytic properties.
 
-To this purpose, we develop an \emph{abstract computational model of automatically differentiable programs} of arbitrary order. In the model, programs are elements of \emph{programming spaces} and are viewed as maps from the \emph{virtual memory space} to itself.
-   Virtual memory space is an algebra of programs, \emph{an algebraic data structure} one can calculate with.
+  To this purpose, we develop an \emph{abstract computational model of
+    differentiable programs} of arbitrary order. In the model,
+  programs are elements of \emph{programming spaces} and are viewed as maps from
+  the \emph{virtual memory space} to itself. Virtual memory space is an algebra
+  of programs, \emph{an algebraic data structure} one can calculate with.
 
-We define the \emph{operator of differentiation} ($\D$) on programming spaces and, using its powers, implement the \emph{general shift operator} and the \emph{operator of program composition}.
-  We provide the formula for the expansion of a differentiable program into an infinite tensor series in terms of the powers of $\D$. We express the operator of program composition in terms of the generalized shift operator and $\D$, which implements a differentiable composition in the language. We prove that our language enables differentiable derivatives of programs by the use of the \emph{order reduction map}. We demonstrate our models algebraic power over analytic properties of differentiable programs by analysing iterators, considering \emph{fractional iterations} and their \emph{iterating velocities}. We than solve the special case of \emph{ReduceSum}.
+  We define the \emph{operator of differentiation} ($\D$) on programming spaces
+  and, using its powers, implement the \emph{general shift operator} and the
+  \emph{operator of program composition}. We provide the formula for the
+  expansion of a differentiable program into an infinite tensor series in terms
+  of the powers of $\D$. We express the operator of program composition in terms
+  of the generalized shift operator and $\D$, which implements a differentiable
+  composition in the language. We prove that our language enables differentiable
+  derivatives of programs by the use of the \emph{order reduction map}. We
+  demonstrate our models algebraic power over analytic properties of
+  differentiable programs by analysing iterators, considering \emph{fractional
+    iterations} and their \emph{iterating velocities}, and explicitly solve the special
+  case of \emph{ReduceSum}.
  \end{abstract}