# Representing and Computing with Types in Dynamically Typed Languages

### From LRDE

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## Abstract

In this report, we present code generation techniques related to run-time type checking of heterogeneous sequences. Traditional regular expressions can be used to recognize well defined sets of character strings called rational languages or sometimes textitregular languages. Newton et al. present an extension whereby a dynamic programming language may recognize a well defined set of heterogeneous sequences, such as lists and vectors. As with the analogous string matching regular expression theorymatching these regular type expressions can also be achieved by using a finite state machine (deterministic finite automata, DFA). Constructing such a DFA can be time consuming. The approach we chose, uses meta-programming to intervene at compile-timegenerating efficient functions specific to each DFA, and allowing the compiler to further optimize the functions if possible. The functions are made available for use at run-time. Without this use of meta-programming, the program might otherwise be forced to construct the DFA at run-time. The excessively high cost of such a construction would likely far outweigh the time needed to match a string against the expression. Our technique involves hooking into the Common Lisp type system via the deftype macro. The first time the compiler encounters a relevant type specifier, the appropriate DFA is created, which may be a ${\displaystyle \Omega (2^{n})}$ operation, from which specific low-level code is generated to match that specific expression. Thereafter, when the type specifier is encountered again, the same pre-generated function can be used. The code generated is ${\displaystyle \Theta (n)}$ complexity at run-time. A complication of this approach, which we explain in this report, is that to build the DFA we must calculate a disjoint type decomposition which is time consuming, and also leads to sub-optimal use of typecase in machine generated code. To handle this complication, we use our own macro optimized-typecase in our machine generated code. Uses of this macro are also implicitly expanded at compile time. Our macro expansion uses BDDs (Binary Decision Diagrams) to optimize the optimized-typecase into low level code, maintaining the typecase semantics but eliminating redundant type checks. In the report we also describe an extension of BDDs to accomodate subtyping in the Common Lisp type system as well as an in-depth analysis of worst-case sizes of BDDs.

## Bibtex (lrde.bib)

@PhDThesis{	  newton.18.phd,
author	= {Jim Newton},
title		= {Representing and Computing with Types in Dynamically Typed
Languages},
school	= {Sorbonne Universit\'e},
year		= 2018,
month		= nov,
abstract	= {In this report, we present code generation techniques
related to run-time type checking of heterogeneous
sequences. Traditional regular expressions can be used to
recognize well defined sets of character strings called
\textit{rational languages} or sometimes \textit{regular
languages}. Newton et al. present an extension whereby a
dynamic programming language may recognize a well defined
set of heterogeneous sequences, such as lists and vectors.

As with the analogous string matching regular expression
theory, matching these \textit{regular type expressions}
can also be achieved by using a finite state machine
(deterministic finite automata, DFA). Constructing such a
DFA can be time consuming. The approach we chose, uses
meta-programming to intervene at compile-time, generating
efficient functions specific to each DFA, and allowing the
compiler to further optimize the functions if possible. The
functions are made available for use at run-time. Without
this use of meta-programming, the program might otherwise
be forced to construct the DFA at run-time. The excessively
high cost of such a construction would likely far outweigh
the time needed to match a string against the expression.

Our technique involves hooking into the Common Lisp type
system via the \texttt{deftype} macro. The first time the
compiler encounters a relevant type specifier, the
appropriate DFA is created, which may be a $\Omega(2^n)$
operation, from which specific low-level code is generated
to match that specific expression. Thereafter, when the
type specifier is encountered again, the same pre-generated
function can be used. The code generated is $\Theta(n)$
complexity at run-time.

A complication of this approach, which we explain in this
report, is that to build the DFA we must calculate a
disjoint type decomposition which is time consuming, and
also leads to sub-optimal use of \texttt{typecase} in
machine generated code. To handle this complication, we use
our own macro \texttt{optimized-typecase} in our machine
generated code. Uses of this macro are also implicitly
expanded at compile time. Our macro expansion uses BDDs
(Binary Decision Diagrams) to optimize the
\texttt{optimized-typecase} into low level code,
maintaining the \texttt{typecase} semantics but eliminating
redundant type checks. In the report we also describe an
extension of BDDs to accomodate subtyping in the Common
Lisp type system as well as an in-depth analysis of
worst-case sizes of BDDs. }
}